FX DERIVATIVES TRADER SCHOOL The Wiley Trading series features books by traders who have survived the market’s ever changing temperament and have prospered—some by reinventing systems, others by getting back to basics. Whether a novice trader, professional, or somewhere in-between, these books will provide the advice and strategies needed to prosper today and well into the future. For more on this series, visit our Web site at www.WileyTrading.com. Founded in 1807, John Wiley & Sons is the oldest independent publishing company in the United States. With offices in North America, Europe, Australia, and Asia, Wiley is globally committed to developing and marketing print and electronic products and services for our customers’ professional and personal knowledge and understanding. FX DERIVATIVES TRADER SCHOOL

Giles Jewitt Copyright c 2015 by Giles Jewitt. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.

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ISBN 9781118967454 (Paperback) ISBN 9781119096610 (ePDF) ISBN 9781119096474 (ePub)

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Printed in the United States of America 10987654321 For my wife and daughters: Laura, Rosie, and Emily.

CONTENTS

Preface xi Acknowledgments xiii

PART I The Basics 1 vii CHAPTER 1 Introduction to Foreign Exchange 3

CHAPTER 2 Introduction to FX Derivatives 11

CHAPTER 3 Introduction to Trading 19 Practical A Building a Trading Simulator in Excel 27

CHAPTER 4 FX Derivatives Market Structure 39

CHAPTER 5 The Black-Scholes Framework 57 Practical B Building a Numerical Integration Option Pricer in Excel 69

CHAPTER 6 Vanilla FX Derivatives Greeks 77 Practical C Building a Black-Scholes Option Pricer in Excel 91

CHAPTER 7 Vanilla FX Derivatives Pricing 103

CHAPTER 8 Vanilla FX Derivatives Structures 121 viii CONTENTS HPE 9F eiaie rcn oes375 355 357 323 Models Pricing Derivatives FX 313 Pricing Derivatives 19 FX CHAPTER Exotic Derivatives FX Exotic 18 CHAPTER 293 IV PART Analysis Market Derivatives FX 261 Correlation and 263 17 Volatility CHAPTER ATM Topics Trading Derivatives 16 FX CHAPTER Vanilla 233 Exposures 241 Trading Derivatives 15 FX CHAPTER Vanilla Trading Derivatives FX Vanilla 14 CHAPTER 193 Option from Function III Density PART Probability a Generating 171 169 Excel in Smile Functions Volatility Density a Probability Constructing G Practical 165 13 CHAPTER F Excel and Practical in Instruments Curve Market ATM Smile an Volatility Constructing 12 CHAPTER Construction Curve ATM Surface E Volatility Practical The 11 CHAPTER Excel in Dates Tenor 159 Generating 137 II PART Topics Miscellaneous Derivatives FX Vanilla D Practical Management Risk Derivatives FX Vanilla 10 CHAPTER 9 CHAPTER HPE 1Erpa iia pin 399 387 Options Digital European Classification Product Derivatives 21 FX CHAPTER Exotic 20 CHAPTER rcsi xe 253 205 Excel in Prices Exposures ix CONTENTS Further ReadingAbout the Companion WebsiteIndex 575 573 577 CHAPTER 29 AsianCHAPTER Options 30 Multi-AssetCHAPTER Options 31 Miscellaneous Options 539 527 555 Practical HCHAPTER 27 Building a VanillaCHAPTER Monte Variations 28 Carlo Option Pricer in Excel Accrual and Target Redemption Options 515 485 497 CHAPTER 24 AmericanCHAPTER Barrier 25 Options ExoticCHAPTER FX 26 Derivatives Trading Topics Window Barrier and Discrete Barrier Options 473 461 439 CHAPTER 22 EuropeanCHAPTER Barrier 23 Options Touch Options 413 423

PREFACE

n 2004 I started on an FX derivatives trading desk as a graduate. I wrote down Ieverything I learned: how markets worked, how FX derivatives contracts were risk-managed, how to quote prices, how Greek exposures evolve over time, how different pricing models work, and so on. This book is a summary of that knowledge, filtered through a decade of trading experience across the full range of FX derivatives products. xi In 2011 I started sending out monthly ‘‘Trader School’’ e-mails to traders on the desk, covering a wide range of topics. The e-mails were particularly popular with new joiners and support functions because they gave an accessible view of derivatives trading that did not exist elsewhere. This book collects together and expands upon those e-mails. Part I covers the basics of FX derivatives trading. This is material I wish I’d had access to when originally applying for jobs on derivatives trading desks. Part II investigates the volatility surface and the instruments that are used to define it. Part III covers vanilla FX derivatives trading and shows how the FX derivatives market can be analyzed. Part IV covers exotic FX derivatives trading, starting with the most basic products and slowly increasing the complexity up to advanced volatility and multi-asset products. This material will mostly be useful to junior traders or traders looking to build or refresh their knowledge in a particular area. Fundamentally, the aim of the book is to explain derivatives trading from first principles in order to develop intuition about derivative risk rather than attempting to be state of the art. Within the text, experienced quant traders will find many statements that are not entirely true, but are true the vast majority of the time. Endlessly caveating each statement would make the text interminable. xii PREFACE uruie.I o rntfmla ihti,teei lnyo aeiloln that online material of plenty practicals is The there detail. do. and this, in functions this with to Applications) covers familiar for thing aren’t Basic right (Visual you VBA If the Excel subroutines. up never set learn is to to ability option the trying require easy you’re When the stuck. taking completely something, are you desk. unless trading website derivatives companion a join you when them running Do ground the hit will you desk: trading structure market the how matter no true changes. hold and book ideas the the However, within market. These explored the clients. on techniques their effects and lasting electronic and banks profound between have increasing distinctions will clear changes less are occur- and transactions market, changes on the visibility in important more ring data, most market electronic The increasing execution, structure. is this market and tives here concern simplified. primary or ignored the are issues is those work analysis if markets cleaner product rate interest Derivative how (i.e., practice). counterparty rates a in interest of and risk owed) the money (i.e., on risk paying. credit defaulting worth notably, price most a analysis, the is within that ignored traders for but this, of result mathematical a Some as throughout. lost the level is is school’’ not rigor mathematics high ‘‘advanced is the accessible this Therefore an mathematics. traders to kept derivative derivatives understanding for fully though, as Importantly same operate. they which in h eybs flc ihyu tde n careers, and studies your with luck of best very The derivatives a on joiner new or student a are you if importantly, most and Finally, deriva- FX the within changes significant causing are technology and Regulations largely are considerations important other some clarity, of interests the in Also framework the of understanding good a have they if successful be only can Traders all ote all them Do . otepracticals the Do norder in a urne hti o opeetepracticals, the complete you if that guarantee can I . ontdwla h pedhesfo the from spreadsheets the download not Do . ie eit odn 2015. London, Jewitt, Giles othem Do . ACKNOWLEDGMENTS

hanks are due to T… those who taught me: Fred Boillereau Jeff Wang Mike Killen Rob Ross xiii Hossein Zaimi … those who supported me in writing the book and helped make it happen: Howard Savery Caroline Prior Vincent Craignou Selene Chong … all colleagues who helped me with content, especially: Chris Potter Charlie Chamberlain Daniela Asikian Allen Li … Marouane Benchekroun and his Quants for the tools I used to produce most of the charts in the book. … those who looked after my girls and I while the book was completed: Frances, Tony, Phil, Gerry, Tim, Jod and Mark. … my family for their support, encouragement and assistance: Mum, Dad, and Anna. xiv ACKNOWLEDGMENTS urn aktrates. market current to solicitation or recommendation, securities. endorsement, financial an any as sell or construed buy be to not are and notice. without only. purposes information only. author the of views the reflects publication This l ae n grsue r o lutaieproe nyadd o reflect not do and only purposes illustrative for are used figures and rates All only purposes educational and illustrative for change strictly to are subject discussed are strategies but Any faith, good for in used given be are should herein and expressed opinions nature Any in educational be to intended is publication This FX DERIVATIVES TRADER SCHOOL

PART I

THE BASICS

art I lays the foundations for understanding FX derivatives trading. Trading Pwithin a financial market, market structure, and the Black-Scholes framework are all covered from first principles. FX derivatives trading risk is then introduced with an initial focus on vanilla options since they are by far the most commonly traded contract.

CHAPTER 1

Introduction to Foreign Exchange

he foreign exchange (FX) market is an international marketplace for trading Tcurrencies. In FX transactions, one currency (sometimes shortened to CCY) is exchanged for another. Currencies are denoted with a three-letter code and 3 currency pairs are written CCY1/CCY2 where the exchange rate for the currency pair is the number of CCY2 it costs to buy one CCY1. Therefore, trading EUR/USD FX involves exchanging amounts of EUR and USD. If the FX rate goes higher, CCY1 is getting relatively stronger against CCY2 since it will cost more CCY2 to buy one CCY1. If the FX rate goes lower, CCY1 is getting relatively weaker against CCY2 because one CCY1 will buy fewer CCY2. If a currency pair has both elements from the list in Exhibit 1.1, it is described as a G10 currency pair. The most commonly quoted FX rate is the spot rate, often just called spot. For example, if the EUR/USD spot rate is 1.3105, EUR 1,000,000 would be exchanged for USD 1,310,500. Within a spot transaction the two cash flows actually hit the bank account (settle) on the spot date, which is usually two business days after the transaction is agreed (called T+2 settlement). However, in some currency pairs, for example, USD/CAD and USD/TRY (Turkish lira), the spot date is only one day after the transaction date (called T+1 settlement). Another set of commonly traded FX contracts are forwards, sometimes called forward outrights. Within a forward transaction the cash flows settle on some future date other than the spot date. When rates are quoted on forwards, the tenor or maturity of the contract must also be specified. For example, if the EUR/USD 1yr (one-year) forward FX rate is 1.3245, by transacting this contract in EUR10m INTRODUCTION TO FOREIGN EXCHANGE 4 Xsasaeqoe nsa on em tedfeec nF ae ewe the between rate) FX in difference terms (the CCY1 terms in point equal swap often in are quoted forward swap are a FX swaps is the FX deal of other the legs the called two buy are and the (e.g., trades on trade two notionals spot The the a maturity. and is specific deal a one to often trade Most sell). a other 100.20). to is 101.20 from moves moves spot spot rate describe spot EUR/USD to USD/JPY used the term if Another higher’’ 1.3145). figure to pips 1.3105 forty from moves jumped rate has ‘‘EUR/USD (e.g., moves 140 at above are the points In swap 0.01. 1yr is EUR/USD pip that a say would (‘‘one-forty’’). places, trader decimal swaps two FX where to USD/JPY, an quoted In example, only 0.0001. FX is usually where pip are a EUR/USD, rates places, krona In decimal FX Swedish pair. four currency to quoted particular usually a are for rates of market, quoted number the usually In rate a 0.0140. FX as are the quoted dollar points Zealand dollar are swap New States 1yr points United EUR/USD swap the 1.3245, is forward 1yr krone points Norwegian forward SEK yen Japanese in currency explained each is in This 5. Chapter forward. balances in the cash detail of the more maturity putting the NZD and until given USD investments spot ‘‘risk-free’’ trading into a to In equivalent Name respective be time). Full the must by year’s linked are one NOK rates in forward rates and USD13.245m rate spot for JPY the pound pair, exchanged British currency Great be will Code EUR10m Euro CCY euros) franc million Swiss (ten dollar Canadian dollar Australian GBP EUR Name Full CHF CAD AUD Code CCY oiini ecie s‘ln e olrcd’ enn S1mhsbe bought been has USD10m meaning dollar-cad,’’ ten ‘‘long as This described CAD9.78m. is selling simultaneously position and USD10m buying means This 0.9780. of given a in rates spot than frequency less far pair. change currency points swap general, In legs. two XII 1.1 EXHIBIT rdrtksu e Xpsto ybyn S1mUDCDso tarate a at spot USD/CAD USD10m buying by position FX new a up takes trader A swap FX FX of magnitude the describe to used also are ‘‘points’’) called (sometimes Pips called are rate forward a and rate spot the between Differences nec urny yan-rirg ruet eieyt h owr maturity forward the to delivery argument, no-arbitrage a By currency. each in enn n ude is(.. ‘S/P a rpe gr’ fthe if figure’’ a dropped has ‘‘USD/JPY (e.g., pips hundred one meaning , EUR10m otat oti w Xdasi poiedrcin oeaby the buy, a (one directions opposite in deals FX two contain contracts 1 Currencies G10 o xml,i U/S pti .15adteEUR/USD the and 1.3105 is spot EUR/USD if example, For . notional U/S ptaantsell against spot EUR/USD ahERwl eecagdfr134 S (i.e., USD 1.3245 for exchanged be will EUR each , pips isaetesals nrmn in increment smallest the are Pips . EUR10m U/S y forward). 1yr EUR/USD legs ftetransaction the of wppoints swap interest or 5 INTRODUCTION TO FOREIGN EXCHANGE ) 0 S if the price of the instrument . Mathematically, the intraday − T S falls ( . 1 120k). CCY makes money CAD is the new spot rate. Notional T S position. The concept of selling something you = 2 CCY CCY1 L & P if the price of the instrument 10m USD/CAD) while the profit and loss (P&L) from the trade is P&L from long USD10m USD/CAD spot at 0.9780 USD position in a financial instrument is the initial spot rate and loses money 0 S long and Exhibit 1.2 shows the P&L from a long spot position. As expected, P&L expressed A USD/CAD spot jumps up to 0.9900 after it was bought at 0.9780: The trader EXHIBIT 1.2 where in CCY2 terms is linear in spot. rises P&L from a long spot position is: USD/CAD spot at 0.9900The results initial bought in USD10m selling and newposition, USD10m sold but the USD10m against initial cancel buying sold out, CAD9.78m leaving CAD9.9m. andprofit. no new This bought net is CAD9.9m USD important: leave CAD120k FXterms transactions (e.g., and positions are usuallynaturally quoted generated in in CCY1 CCY2 terms (e.g., don’t initially own is a strange onefinancial in markets the real where world trading but positions it quicklyposition) can becomes and flip normal short in often (a between net long sold (a position). net bought is a hero! Time to sell USD/CAD spot and lock in the profit. Selling USD10m and an equivalent amount ofsold CAD at has 0.9780 been instead, the sold. position Ifthe is USD10m described long/short USD/CAD as refers had ‘‘short ten to been dollar-cad.’’ Note the that INTRODUCTION TO FOREIGN EXCHANGE 6 togradCY ekr.Ti fetitoue uvtr notePLprofile P&L CCY1 the means 1.4. into Exhibit curvature higher in an introduces (spot shown effect levels, CCY1 as This spot fewer weaker). higher relatively CCY2 worth At and be stronger weaker). will CCY1 CCY2 and of stronger amount CCY2 means lower (spot the Therefore, rate. spot prevailing is: position the spot at a from place P&L takes CCY1 CCY1 and CCY2 between spot. in linear is terms CCY2 also: is position spot falls XII 1.3 EXHIBIT tlwrso ees naon fCY ilb ot eaieymr CCY1 more relatively worth be will CCY2 of amount an levels, spot lower At conversion the terms, CCY1 into back brought is deals in spot these expressed from P&L P&L Again, the If position. spot short a from position. P&L short the a shows denote 1.3 to Exhibit negative be will notional the However, A and short oe money loses oiini nnilinstrument financial a in position & rmsotUD0 S/A pta 0.9780 at spot USD/CAD USD10m short from P&L ftepieo h instrument the of price the if P P & & L L CCY CCY 1 2 = = Notional Notional ae money makes CCY CCY 1 1 rises . . ( ( S S T T h nrdyPLfo short a from P&L intraday The . S − − T S S ftepieo h instrument the of price the if 0 0 ) ) 7 INTRODUCTION TO FOREIGN EXCHANGE : the most ) many times tick cross currency major currency pairs P&L from long USD100m USD/JPY spot at 101.00 . For example, EUR/USD and AUD/USD are majors while EUR/AUD is a Exhibit 1.5 is a mocked-up screen-grab of a market-data tool showing live spot The USD is by far the most frequently traded currencyFX with traders the majority draw of a distinction between rates in major G10 currencya pairs. second. In practice these rates change ( pairs cross. FX rates in cross pairsmajors. are The primarily FX determined market by is theand highly trading AUD/USD efficient activity spot so in is if trading the EUR/USD atat spot 0.8000, 1.5000 is EUR/AUD (1.2/0.8). trading spot will at certainly 1.2000 be trading FX trades featuring USD ascurrency either pair, followed CCY1 by or USD/JPY CCY2. and EUR/USD then GBP/USD. is the mostcommonly traded traded currency pairs, usually against the USD, and The international foreign exchangeworth market of is deals transacted enormous, each with day.is The London, trillions most followed of important by international dollars’ New centerroughly York. for equally In FX important. Asia, Tokyo, Hong Kong, and Singapore are EXHIBIT 1.4 Practical Aspects of the FX Market ■ INTRODUCTION TO FOREIGN EXCHANGE 8 ■ urnyPairs? Different Currency Call Traders FX Do What uulyi S)a auiyrte hntetocs osi eua FX regular (http://www.xe.com/). a XE.com or in (http://finance.yahoo.com/) payment. finance settlement flows the determine cash to used two is country, the appropriate the than in day rather maturity The at settlement. USD) in (usually USD one buy that to Forward in currency functions EM of market number FX the the as how USD/CCY). quoted exactly (i.e., are learn majors to emerging EM an vital in country. is trading When it transactions. currency restricting hours by market or open spot spot selling limited and buying have currencies EM some some and example, For flows. currency their is as GBP since GBP terms against GBP/EUR currency. EUR in notional quotes trade natural corporates market U.K. the some of but majority EUR/GBP often the are example, there rules For convention exceptions. market with Unfortunately, GBP/CAD. as market NOK EUR ordering: this from Open (5 Wellington between day a hours 24 (9 tradable are markets FX G10 The trading. 1.5 EXHIBIT urnypishv ae htaewl salse n ieyue.Standardized used. ‘‘ widely and USD/JPY established well calls are that floor names have trading pairs currency the on Nobody ncrec ar ihrsrcin nso transactions, spot on restrictions with pairs currency In control to place in mechanisms have often countries (EM) market Emerging deduced be can pair currency a quoting for convention market the pairs, G10 In their on restrictions P.M. no with floating freely (mostly) are pairs currency G10 pt-aeF ae a efudo h nentuig o xml,Yahoo example, for using, Internet the on found be can rates FX Up-to-date A.M. > e oktm)o Friday. on time) York New elntn e eln ie nMna hog oNwYr Close York New to through Monday on time) Zealand New Wellington, SEK peg ( NDF hi urnya xdlvlo anani ihnataigbn by band trading a within it maintain or level fixed a at currency their > apeG0so rates spot G10 Sample otat r fe rdd Dsstl noasnl ahpayment cash single a into settle NDFs traded. often are contracts ) fix P.Freape h A gis B Xrt sqoe nthe in quoted is rate FX GBP against CAD the example, For JPY. eeec Xrt ulse tacrantm vr business every time certain a at published rate FX reference a , > GBP > AUD > NZD you-ess-dee-jay-pee-why > USD > Non-Deliverable CAD > ’ Major .’’ CHF > 9 INTRODUCTION TO FOREIGN EXCHANGE over a cable on the Atlantic ocean floor.) Selected EM Currency Pair Names Selected G10 Currency Pair Names EXHIBIT 1.7 EXHIBIT 1.6 communication but it exposes thoseFor who this are not reason, experiencedand market using 1.7 participants. the for common correct G10 and market EM currency terms pair is names. important. See Exhibits 1.6 USD/ZARUSD/BRL ‘‘dollar-rand’’ language ‘‘dollar-brazil’’ is common in financial markets. It enables quick and accurate Currency PairUSD/HKD CommonUSD/CNY Name USD/SGDUSD/MXN ‘‘dollar-honkie’’ USD/TRY ‘‘dollar-china’’ ‘‘dollar-sing’’ ‘‘dollar-mex’’ ‘‘dollar-try’’ or ‘‘dollar-turkey’’ EUR/CHFEUR/NOKEUR/SEK ‘‘euro-swiss’’ or ‘‘the cross’’ ‘‘euro-nock’’ or ‘‘euro-nockie’’ ‘‘euro-stock’’ or ‘‘euro-stockie’’ USD/CADUSD/JPYGBP/USD ‘‘dollar-cad’’ EUR/USD ‘‘dollar-yen’’ AUD/USD ‘‘cable’’ (FX pricesNZD/USD between London and New York used to be transmitted ‘‘euro-dollar’’ ‘‘aussi-dollar’’ ‘‘kiwi-dollar’’ Currency Pair Common Name

CHAPTER 2

Introduction to FX Derivatives

he FX market can be split into three main product areas with increasing Tcomplexity: 1. Spot: guaranteed currency exchange occurring on the spot date. 11 2. Swaps / Forwards: guaranteed currency exchange(s) occurring on a specified date(s) in the future. 3. Derivatives: contracts whose value is derived in some way from a reference FX rate (most often spot). This can be done in many different ways, but the most common FX derivative contracts are vanilla call options and vanilla put options, which are a conditional currency exchange occurring on a specified date in the future.

■ Vanilla Call and Put Options

Vanilla FX call option contracts give the right-to-buy spot on a specific date in the future while vanilla FX put option contracts give the right-to-sell spot on a specific date in the future. The term vanilla is used because calls and puts are the standard contract in FX derivatives. The vast majority (90%+) of derivative transactions executed by an FX derivatives trading desk are vanilla contracts as opposed to exotic contracts. Exotic FX derivatives (covered in Part IV) have additional features (e.g., more complex payoffs, barriers, averages). To understand how call and put options work, forget FX for the moment and think about buying and selling apples (not Apple Inc. stock, but literally the green round things you eat). Apples currently cost 10p each. I know that I will need to buy 12 INTRODUCTION TO FX DERIVATIVES h rc fape saoetesrk tmtrt,tecl pinvlersslinearly rises value option call underlying. the If the maturity, market. of at the value strike in the the cheaper with above bought is be apples can of underlying price the the because value no has option included. premium initial the without but terms diagram the hockey-stick apples, familiar of price the volatile more cost. The will factor. option key call a it the be the long more will crucially How apples plus contract: of level, the price transaction of the the of details covers, the it apples on many depend how will a lasts, option the at call cost: month a the at one of comes premium in uncertainty apples in 100 reduction This buy strike). premium to (the need 10p than I higher will rate circumstances no rate. lower under contract the option known; at call market the the in hence or directly 10p; want apples at don’t 100 them I buy buy 5p), will I at to Instead (e.g., right strike my the use the below to is as need apples known of is (also price apples the seller of if the Alternatively, price from the 10p if will at contract, I 15p) apples the (strike: at of (e.g., transact call maturity strike this to the the buying want above at After apples). I hence, 100 month which (notional: transact one at to option, to level want I want the amount I option), the and which call 10p) apples; at a buy future to purchase want the I (I direction therefore, in the month), date one (maturity: the transaction the contract: complete the within elements a different with option call (1mth) this allow options future put the and for Call controlled. difficult. planning be more makes to company uncertainty uncertainty juice this is apple there and fledgling that cost my is will of point apples The the 999p. or much than 1p, how cheaper 10p, apples cost the will buy they can perhaps I apples, hence the are and buy currently 5p then and be they month will one price wait prevailing simply the I If perhaps time. month’s one in apples 100 h rc fape ttemmn h pincnrc aue;tept ae oget to taken path the matures; contract option irrelevant. the is moment there the at apples of price the and maturity, option the where ti ot oigta h & tmtrt rmtecnrc eed nyon only depends contract the from maturity at P&L the that noting worth is It is: option call this from maturity at P&L the Mathematically, call the (10p), strike the below is apples of price the if maturity, in option presented the maturity, At at option call this from profile P&L the shows 2.1 Exhibit the option, call the buying by Therefore, one-month a is risk the control to purchased be could that contract possible One Notional adufott ucaetecl pin ti o adt mgn htthe that imagine to hard not is It option. call the purchase to upfront paid sepesdi em fnme fapples, of number of terms in expressed is or ehp h rc ilb 5 n ec oeexpensive more hence and 15p be will price the perhaps K P & strike stesrk.Often strike. the is L = Notional f1pada and 10p of exercise . max max ( h aloto ogtadby100 buy and bought I option call the S os-aeprhsn rate purchasing worst-case notional T ( − writer S T K − , K 0 ) , S fteoto contract. option the of ) f10ape.Nt the Note apples. 100 of T 0 stepieo plsat apples of price the is ) swritten is uncertainty ( volatility S expires T − about K ) + or is . . INTRODUCTION TO FX DERIVATIVES 13 is known; under no ) 0 , T S − K ( max . and 1,000 apples will be sold at 10p to the . Imagine you own a forest of apple trees. sell worst-case selling rate Notional = exercised L and 1,000 apples can instead be sold in the market at a & the right to P expire buying P&L per apple at maturity from call option with 10p strike Bringing these concepts into FX world, the underlying changes from the price of Mathematically, the P&L at maturity from this put option is: By buying the put option, the This time, at the option maturity, if the price of apples is below the strike (e.g., Put options are the right-to-sell the underlying. This can be conceptually tricky apples to an FX spot rate. At the option maturity, the prevailing FX spot rate will circumstances will I need toat sell the 1,000 end of apples August. at Theexact a contract cost rate details, of plus, lower buying again, than the this more derivative 10poption volatile will contract (the the cost. will price strike) Exhibit of depend 2.2 apples, on shows the the the more the P&L profile from this put option at maturity. at 5p), the put option willoption be seller. Alternatively, if thethe price put of option apples will is abovehigher the rate. strike (e.g., at 15p), to grasp at first— You know that by theyou end will of then August want youprice to will of sell. harvest apples Again, at is uncertainty least unknown.August arises 1,000 To 31 apples, from could control be which the this bought uncertainty, fact with a that a put notional the of option future 1,000 maturing apples on and a strike of 10p. EXHIBIT 2.1 14 INTRODUCTION TO FX DERIVATIVES mrcnvnlaotosaecvrdi hpe 27. Chapter in covered are a options value. vanilla to of American simpler mathematically mention and manage any risk Henceforth, to easier are they because market exercised be can options vanilla only American exercised be can 2. options vanilla European 1. exercised be will option vanilla a whether determine expired. to or strike the to compared be 2.2 EXHIBIT alo u (call put or Call contract: option pair: FX vanilla Currency a describe to required are details following The options vanilla European option: vanilla of kinds main two actually are There xml,aERUDcl pini culyaERcl n S put. for USD a so, and contract, call EUR the a actually describing is often when option Most call are specified vice-versa. EUR/USD or is options a CCY2 example, direction vanilla on CCY1 put Therefore, a the other. and only CCY1 the right-to-buy on a call right-to-sell a simultaneously a simultaneously is Therefore, pair and sold. currency currency is particular that one a one in and option bought vanilla is a that one currencies: maturity. two at exchange calculated be will option vanilla the of value the which against & e pl tmtrt rmptoto ih1pstrike 10p with option put from maturity at apple per P&L h ptF aei hscrec ari h eeec rate reference the is pair currency this in rate FX spot The = right-to-buy/put vanilla r h tnadpouti h Xderivatives FX the in product standard the are pinmasaErpa-tl contract. European-style a means option taytime any at = right-to-sell): at h pinmaturity. option the eoeteoto maturity. option the before Xtransactions FX INTRODUCTION TO FX DERIVATIVES 15 Londontime. London time, P.M. A.M. ) ) 0 0 or 7 is the strike. The CCY1 , , T K K S A.M. − − T S K ( ( max max . . 1 1 CCY CCY Notional Notional NewYorktime,whichisusually3 = = . Tokyo time, which is 6 2 2 A.M. P.M CCY CCY L L The date on which the owner of the option decides whether & & P P the option, which is explored in Chapter 9. Converting between CCY1 and CCY2 notionals The amount of cash (usually expressed in CCY1 terms) that can be The rate at which the owner of the option has the right to exchange is the spot FX rate at the option maturity and The exact time on the expiry date at which the option matures. The two most common cuts in G10 currency pairs are: T NewYork(NY):10 Tokyo (TOK): 3 depending on the time of year. S ■ ■ CCY1 and CCY2 at maturity. exchanged at maturity. VanillaCCY1 and option CCY2 terms notionals using the can strikeis as be the shown level in converted at Exhibit which 2.3 between since CCY1 the and strike CCY2 are potentially exchanged at maturity. to exercise their optionpartially or exercise let it expire. There is actually a third option to Exhibit 2.6 shows a USD/JPY vanilla contract in an FX derivatives pricing tool. The CCY1 put P&L at maturity is the same as a short FX position below the Likewise, the P&L at maturity from a long (bought) CCY1 put option is: Mathematically, the P&L at maturity from a long (bought) CCY1 call option is: Strike: Notional: Cut: Maturity/expiry: EXHIBIT 2.3 Traders use systems like this to price vanilla option contracts. strike. Exhibit 2.5 shows the(USD put/CAD P&L call). at maturity from a long USD/CAD put option where call P&L at maturitya is forward) the above same the asUSD/CAD a strike. call option long Exhibit (USD FX 2.4 call/CAD position put). shows (to the the P&L maturity at date, maturity i.e. from a long 16 INTRODUCTION TO FX DERIVATIVES XII 2.5 EXHIBIT 2.4 EXHIBIT & tmtrt rmln S1mUDCDptoto ih098 strike 0.9780 with option put USD/CAD USD10m long from maturity at P&L strike 0.9780 with option call USD/CAD USD10m long from maturity at P&L INTRODUCTION TO FX DERIVATIVES 17 is the current date (i.e., today). On the . NY time (since the option is priced to NY A.M (Nov. 22, 2013; the delivery date is calculated . The option owner should let the option expire horizon . In this case the option will be exercised because the option delivery date FX derivatives pricing tool showing a USD/JPY vanilla contract (Nov. 20, 2013) at 10 out-of-the-money (OTM) Note that for a put option with all other contractWithin details the the pricing same, tool, the both ITM and volatility and premium prices are shown for the If the spot rate at maturity is below the strike (80.00), the option is said to If the spot rate at maturity is above the strike (80.00), the option is said to be Within the pricing tool, the because USD/JPY spot can be bought more cheaply in the market. OTM sides flip: the ITM side is below the strike and the OTM side iscontract. above the Some strike. market participants want prices quoted in volatility terms while horizon—see Chapter 10 forowner more will information get on longer USD5m tenor versus calculations),get shorter the the JPY400m opposite while option position. the option writer will be (seller) of the option to inform them if they want toin-the-money exercise (ITM) the option. gives its owner the right tois transact exercised, at on a the better ratefrom than the the spot expiry level. If date the option in the same way that the spot date is calculated from the EXHIBIT 2.6 expiry date cut), the owner (buyer) of this European-style vanilla option will contact the writer 18 INTRODUCTION TO FX DERIVATIVES ■ rcia set fteF eiaie Market Derivatives FX the of Aspects Practical sasnl uulyUD ahflw nG0crec ar ti lopsil ouea use to common. possible less also is is this it paid but pairs is contracts currency that derivative G10 amount settle In to settlement flow. fix a cash determine USD) to (usually used single a is as fix a FX maturity an at (i.e., that meaning flows cash of exchange an exercised, is occurs. option trade) spot the if maturity, are at contracts that long-dated pairs longer. FX even currency or their some years ten in in to out However, volumes traded—sometimes under. trading and higher year one have also to tend forward, spot, transactions markets their markets. of derivatives in swap volumes worth trading dollars’ FX higher U.S. have trading and that of exchange pairs billions foreign Currency of day. total every of hundreds 5% to roughly equating are volumes, volumes trading derivatives FX job volatility. trader’s FX derivative to FX exposure exactly an sell is of and this essence buy but The to terms works. is volatility market in derivatives quoted FX be the would how price equivalent a an that calculate strange to seem used may is formula Black-Scholes the and final premium The option maturity. option the is to rates input forward/interest the market current plus inputs spot, strike, as put), current or takes data: (call type The formula option The maturity, terms. details: premium. contract option and premium vanilla volatility between in link the quoted provides prices want others nsm mrigmre urnypis ail pin are options vanilla pairs, currency market emerging some In usually are pairs currency of G10 in maturities options Vanilla with contracts in occurs trading derivatives FX of majority The volatility nEhbt26 a 2.6, Exhibit In . n h lc-coe oml a hnb sdt aclt the calculate to used be then can formula Black-Scholes the and w-a volatility two-way hsclydelivered physically epandi hpe )i given is 3) Chapter in (explained lc-coe formula Black-Scholes w-a premium two-way ahsettled cash meaning , .It , CHAPTER 3

Introduction to Trading

undamentally, markets are a mechanism to match buyers and sellers in order Fto determine the prices of goods and services. Traders interact directly with financial markets, buying and selling in order to manage their deal inventory and 19 change their trading positions. The process of operating within a financial market is easier to explain using a simple market. Therefore, this chapter uses a market on a single asset, like spot FX or a single equity contract, as the reference. However, the same ideas also apply to more complex markets, including FX derivatives.

■ Bids and Offers

The building blocks of financial markets are two types of order: 1. Bid: a rate at which a price maker is willing to buy 2. Offer (also called Ask): a rate at which a price maker is willing to sell Bids and offers need to have a size associated with them. Saying, ‘‘I will buy apples for 10p each,’’ is interesting to another market participant but not enough information; will you buy ten apples or a million apples? There is an important distinction between price makers (also called market makers) and price takers within financial markets. Price makers leave orders in the market. Price takers come into the market and trade on existing orders. If a price taker wants to buy, the contract must be bought at a price maker’s offer, and if 20 INTRODUCTION TO TRADING re)adi fe lointcerwehra re tapriua ee soeorder one is level together. particular aggregated a orders particular at multiple order any of an collection left whether a has clear or isn’t participant also market often which it market known and this order) isn’t transacting, to it Prior (i.e., order. the each anonymous called with is is associated size This a backgrounds. and market) shaded the with in the right shows the which on book, offers order and left the on bids bid between difference smallest the offer way, best offer). another and and put bid or best spread, The bid–offer contract). tightest the the sell tightest buy the to anyone to form that receive to least pay all to the combine to of willing (i.e., offers willing is highest current market is the all the market of is lowest in the bid the in best is offer anyone The best that contract. The contract). most given the a (i.e., in bids offer current best single and bid best a at transacting in results usually order orders the leaving rate. toward that better move is to to difference market market second the the The requires in level. else this someone Generally, requires order. it your since on all can at trade orders happen leaving not while may instantly and executed longer be take can offers and bids existing on trading you. from buys someone bid). that a hope showing and is offer who an someone Leave to sell (i.e., 2. bid a Give 1. methods: possible two are there again, you. sell, to To sells someone offer). that an hope showing and is bid who a someone Leave from buy 2. (i.e., offer an Pay 1. it: doing the of while ways offer. offer, two maker’s essentially their price Put are a at there bid. at sells maker’s buys and and price bid bid a maker’s price at their a sold at at be sells buys taker must maker price contract price the the sell, way, to another wants taker price a XII 3.1 EXHIBIT xii . hw nphto niaiayitrainlapemre,showing market, apple international imaginary an of snapshot a shows 3.1 Exhibit single the only showing given, often is market current the of view reduced A that is first The approaches. these between differences major two are There buy, to wants trader a if contract, financial particular a for market the Within pl aktodrbook order market Apple two-way depth ntemre ie,alcretbd n offers and bids current all (i.e., market the in rc ntemre ie,tepiewt the with price the (i.e., market the in price INTRODUCTION TO TRADING 21 flow )orders, and the pull transparency market. Markets rarely ) and remove ( allow buyers to define their place inverted limit orders , meaning the bid and offer are at the same level. choice the 11p offer starts being paid, the original 11p offers will be trade with each other for some reason. if . The rate of 11p already has an offer there. By ‘‘joining’’ (i.e., . Five apples can be sold at 9p and five apples can be sold at 8p (so are key considerations. can’t ). fill Note that would lead to potentiallyhigher transacting the offer, at the a lower the better chance rate of transacting (selling and higher), the longer but it the will take. but the transaction will certainly beLeave completed. an offer showing the same offer) theup 11p to offer 40 in apples. 10 apples, the size of that offertransacted will first. The go offer could also be left at a higher level, at 12p or 13p, which Give the bids five of the 8p bid would remain). This averages at a rate of 8.5p to sell the apples Exchanges offer different order types that give additional control around how In the apples example, if the 11p offer were to further increase in size, it is In practice, particularly in faster markets, traders work buy or sell orders Back to our apple order book: As mentioned, selling 10 apples in this market can Trader X wants to buy 100 apples at 8p and trader Y wants to sell 100 apples at For a given contract, offers are (almost) always above bids. However, in some In this example, the best bid is 9p and the best offer is 11p. Most of the time, the market who will push theto price transact an lower. order For in this smaller reason chunks it over is time to sometimes reducea appropriate market bid impact. or offermaximum is purchase processed. For price example, and sellers to define their minimum sale price while depending on how the market is reactinglevel in ( order to get the best possible transaction possible that bids would be pulled,the or large would size be on moved the offer lower, and since conclude traders that will there see are a large number of sellers in the using a combination oflarger trading size, on the orders trader and might placing dynamically orders. leave If the ( order was in 2. of information essentially be done in two ways: 1. bid–offer direction. An invertedparticipants market often occurs when the relevant8p. market If trader X andhappy trader to Y trade each with know each whateach other the other’s at other intentions. 8p. wants For a It to market is do, to they therefore function should vital efficiently, be that both traders know of situations, a market will be Even rarer, the offer canstay be genuinely inverted below for the long, bid—an sincebid the and market offer participants should showing be the happy to inverted trade with each other and hence restore the normal best bid is simply referredbids to and as higher ‘‘the offers bid’’ are and referred the to best as being offer ‘‘behind.’’ is ‘‘the offer.’’ Lower 22 INTRODUCTION TO TRADING ■ therefore is ■ spread Bid–offer things): offer. other or bid (amongst the of position either function the difference on holding a the trades of client risk client, the potential if the a for time maker the over for market called contract the cover is a to offer exists on the spread price and bid two-way the a between makes trader a When Spread Bid–Offer ■ general, In 10p. at trading ■ currently market the for with to, 5p, left at be could apples bid 100 A market. buy current example, the to close left always aren’t offers and Bids word: one Orders into Leaving combined often are factors These rate. liquidity transaction transaction transacting, of and probability size, speed, transaction between relationships the of canceled. is orders fill-or-kill ■ u h oiin ec tpigtels,atog h rgnlpsto a not may position original the closes although then loss, order exist. the the actually and stopping level hence order the position, to the gets out market the if a money called lose is will this that level, market current the below loss sells stop or above buys may order position an When original then the order although the profit, and exist. level taking actually order hence not the position, to the moves market out the closes called if is money this make level, will market that current the below buys or a above sells order an When h hneo h lettaigbtgvsasalrsra opoetfo future from protect to spread smaller a gives but changes. trading price increases client offer) lower the a of and bid chance higher the a (hence spread bid–offer tighter a Showing trade offsetting spread. an bid–offer until the longer wider The the market. found, the be in can found be can trade offsetting) period holding Average volatility Contract rdr loajs hi i–fe ped ae nrs/eadpreference: risk/reward on based spreads bid–offer their adjust also Traders understanding an on based are market certain a within transact to how on Decisions aeprofit take . re.Aan h em‘so os’ipisteei neitn position existing an is there implies loss’’ ‘‘stop term the Again, order. r ihreeue meitl nteretrt,o leteorder the else or entirety, their in immediately executed either are re.Tetr ‘aepot’ipisteei neitn position existing an is there implies profit’’ ‘‘take term The order. h oevltl otat h ie h i–fe spread. bid–offer the wider the contract, a volatile more the : h egho iebfr nofetn o approximately (or offsetting an before time of length the : i–fe spread bid–offer ocpulythis Conceptually . INTRODUCTION TO TRADING 23 Trader A two-way price 200 apples (or fewer) at 11p (i.e., pay trader A’s offer). If trader B buys, 200 apples (or fewer) at 9p (i.e., give trader A’s bid). If trader B sells, trader . Trader B can decide not to buy at 11p or sell at 9p and therefore can walk Sell A buys; hence trader A wouldPass have ‘‘bought at their bid.’’ away from the transaction. This could happen for many reasons; perhaps better Buy trader A sells; hence trader A would have ‘‘sold at their offer.’’ Trader B now has three options: The bid is 9p and the offer is 11p. Therefore, trader A has shown a bid–offer If bids in the market are getting ‘‘given’’ or ‘‘hit,’’ this isThese a terms sign can be the confusing market until they is are used day-to-day, at which point they Showing a wider bid–offer spread (hence athe lower chance bid and of a the higher offer) clientprice decreases trading changes. but gives a larger spread to protect from future EXHIBIT 3.2 2. 3. Hence if trader B transactsspread on from either the midmarket side price. of the price, the trade will contain some 1. directly from each otherTrader and B 1,000 comes apples to havedisclose trader just a A traded buying or requesting in selling the ain preference market Exhibit price so 3.2. at trader in A 10p. 200 makes apples. the two-way Trader price B shown doesspread not of 2p. Trader A is assuming that the midmarket price of apples is still 10p. What follows isimaginary a international apple simplified market. example Within of this market, some traders market-making request prices activity in our moving lower. If offers inthat the the market market is are moving getting higher. ‘‘paid’’ or ‘‘lifted,’’ this isquickly a become sign second nature. Bid and Offer Language When traders pay offers they say ‘‘mine!’’ (i.e.,accompanied they’re buying with it). a This is raised sometimes index(i.e., they’re finger. selling When it), sometimes traders accompanied with give an index bids finger they pointing down. say ‘‘yours!’’ ■ Market Making ■ 24 INTRODUCTION TO TRADING att e n hre.Teeoe rdrAmkstepiesoni xii 3.4, Exhibit in doesn’t shown and price transaction the makes initial A the trader from Therefore, short is shorter. apples is any of get A price to trader the want Plus that market. believes the A in trader occurring, rising transaction initial the to Due 2 Scenario maker market a being called of interests, advantages offsetting the have illustrates counterparties scenario when while This risk) profit. reducing (hence a position in their locking balanced has trader the making, market By size) (contract 200 earned having to with managed counterparties has and to bid) A prices trader two-way directions), (buy/sell showing ‘‘interests’’ By opposite transaction. first the offsetting 3.3. Exhibit in shown price two-way same the quotes A Trader 1 Scenario different three are Here next: A. happens trader what from for apples scenarios 200 possible in price two-way new a requests 2. 1. agreed. is trade the A’s as trader soon later, as until changes occur not may money for apples gone has position apples A’s Trader XII 3.3 EXHIBIT Thistime,traderC and market the enters now C Trader risk. the warehouse to decides A Trader options: two has now broadly A Trader B Trader ls out Close apples). of price the Warehouse useful making. be price can future their it in but information not that passed use does being can B is A trader Trader price bids, transact. the to (called B why why wants trader A know perhaps longer trader to or no to seller) and explain a higher mind to or or of have buyer offers change a (lower a is market had B the has trader in whether traders on other depending by bids shown being are prices elhigh sell h ikb on akit h aktadtaigt fstteexposure. the offset to trading and market the into back going by risk the buys h ikad‘rntepsto’ ie,ke h hne xoueto exposure changed the keep (i.e., position’’ the ‘‘run and risk the rdrAtowypiei cnro1 scenario in price two-way A Trader a h fe) vrl,tae ’ oiini akweei started, it where back is position A’s trader Overall, offer). the (at 0 plsa 1.Teeoe rdrAhssl 0 plsand apples 200 sold has A Trader Therefore, 11p. at apples 200 sells feedback 0 plsa p ec rdrAby 0 pls exactly apples, 200 buys A trader Hence 9p. at apples 200 :I nte rdri,freape hwn better showing example, for is, trader another If ): × shorter p(spread) 2p y20 lhuhteata xhneof exchange actual the Although 200. by = 0pfrteetotransactions. two these for 400p w-a flow two-way exposure otepieo apples of price the to . u low buy a the (at INTRODUCTION TO TRADING 25 trading position the probability of trader C selling but it increased Market two-way price in scenario 3 Trader A two-way price in scenario 3 Trader A two-way price in scenario 2 the amount of spread trader A will capture if trader C does sell. 200 apples at 12p. 200 apples at 10p. . reduced Sell Pass Buy Trader C pays the offer: Trader A showed a higher price but again the offer was Trader C was a buyer, but another trader in the market showed a lower offer, Note how important the flow of information within the market is. Depending If trader C is a seller, trader A is hoping that trader C decides 10p is a good price Again, trader C has three options: EXHIBIT 3.5 EXHIBIT 3.6 of’’ or ‘‘offset’’) the position acquired fromreceived the back previous from two the transactions. market The is rate shown in Exhibit 3.6. Scenario 3 Once again, trader A doeshigher not bid want and a an higher increased offer short as apple shown exposure in so Exhibit 3.5. quotespaid. a Trader A goes to the market to get a price in 400 apples to hedge (‘‘get out on the market structure,or the it previous may trade onlythe at be time 11p between known transaction may by andbecause be the reporting, they are traders known the personally more involved by involved power in ininformation. everyone market more the trades makers and transaction. have hence The have access longer to better so trader C passesprice trader making. This A’s scenario price. illustratesto Trader how influence market A their makers price can use making. their use this information in future 2. 3. at which to sell. Raising thehas bid has with a relatively better (higher)and a bid relatively to worse make (higher) it offer to more make likely it that less likely trader that C trader will C1. sell, will buy. EXHIBIT 3.4 26 INTRODUCTION TO TRADING ■ rc aigadRs aaeetOverview Management Risk and Making Price ead nygetrPLvltlt.Rs iisteeoeke & oaiiyto volatility P&L not Greater keep does line: therefore it in limits but Risk be levels. volatility. reward should acceptable P&L greater limits greater for risk only opportunity and reward, targets the with P&L gives made also limits. risk market: are the decisions risk of trading to reading plus on current reference so their and on sentiment, based liquidity, decisions direction, these make Traders out. it transacts. agement client the when client the to when position opposite activities the position. market-making on a of takes take desk consequence trading to a the decision as their seen generated not As are often analysis. positions is their Rather, it on makers based market money for make however, will and earlier, think market hedge they the that into (e.g., go deals participants They only ‘‘moving straightforward: transact market is or process buy-side management money) For risk losing the money). funds) (when making Traders them (when moves. against’’ them market ‘‘moving the for’’ market and the passes time about as talk money to make attempt that they contracts time financial same the level. best at midmarket the but the show contract, from to earned the spread needs on the trader quoting maximize the traders trade, of all the that the of chances win on is price to the quote it competition, banks increases in many likely turn When contract in more spread. same capturing which the and levels, clients activity, servicing midmarket market successfully current about the has The know market. trader they the from a information information assimilating on more depends making price in Success position. trading the of from control money complete in their lose the being hedge to illustrates not to from part scenario trader arising market This substantial the the position. causes A into apple is which in. short go position higher, locked the traders the moves be As if market will position. and the same 2.5p 12p, positions, the of at has loss apples market average the more an of even 14p, sell at to back want bought not does certainly A e ikmngmn eiin o rdr hrfr involve therefore traders for decisions management risk Key in short) or (long positions take to is management risk successful to key The Trader position. short a with stuck is A trader and rising is apples of price The ri te od,dcdn hnt aeos ikadwe oclose to when and risk warehouse to when deciding words, other in or , risks fbigamre maker market a being of netr man- inventory guarantee greater PRACTICAL A

Building a Trading Simulator in Excel

his practical demonstrates how simple financial markets work and illustrates Tthe differences between price-taking and price-making roles. Task A sets up a ticking (moving) midmarket price. Task B then introduces a two-way price (bid and 27 offer) around the midmarket and price-taker controls whereby the trader can pay or give the market. Finally, Task C adds the ability for the trader to act as both price taker and price maker. This practical links closely to the material discussed in Chapter 3.

■ Task A: Set Up a Ticking Market Price

The trading simulator has one main VBA subroutine that updates the market price. The Application.OnTime command is used to pause between market ticks.

Step 1: Set Up a Ticking Midmarket Spot Setting up the framework mainly requires VBA development. User inputs on the sheet are initial spot, time between ticks, and how much spot increments up or down at each tick. Outputs are the current time step and current spot. Control buttons for Go/Pause and Stop are also required: 28 BUILDING A TRADING SIMULATOR IN EXCEL utni pressed: is button Sub End GoButton1() required: if Sub sheet the initializing and variable MarketOn the flipping (MarketOn ticking (MarketOn currently not is variable or market Boolean True) the C global whether like a defines languages that defines MarketOn line to called second similar The more programming. coding better VBA encourages the makes This statements. Boolean As MarketOn Public Explicit Option VBA). the from ‘‘A5’’ cell (e.g., referencing than flexible more far opment n Sub End StopButton1() Sub h tputnsbotn lasteotusadsostemre fteStop the if market the stops and outputs the clears subroutine StopButton The pressed, is button Go/Pause the when run should subroutine GoButton The Dim using declared be to VBA the within variables all forces line first The this: like start should module VBA devel- The makes cells Naming screenshot. the per as named be should cells input The I aktwspeiul tpe hniiilz it Then initialize "" it. then = start stopped Range("Step") ticking, previously If not was is market market 'If If MarketOn it. Not stop = ticking, MarketOn is market 'If Range("SpotMidMarket").ClearContents Range("Step").ClearContents False = MarketOn tick MarketTick1 market a 'Run If End ag(SoMdakt)=Range("SpotInitial") = Range("SpotMidMarket") 0 = Range("Step") = False). ++ and = BUILDING A TRADING SIMULATOR IN EXCEL 29 0.5) Then > Range("SpotIncrement") Range("SpotIncrement") Range("SpotMidMarket") = Range("SpotMidMarket") + _ Range("SpotMidMarket") = Range("SpotMidMarket") - _ 60 / 60), "MarketTick1" 'Schedule a marketApplication.OnTime tick TimeValue(Now() in + the Range("TickTime") future / 24 / _ 'Move spot upIf or (Rnd() down at random Else End If 'Increment step Range("Step") = Range("Step") + 1 Range("SpotMidMarket") End If If (MarketOn) Then Range("DataOutput").Offset(Range("Step"), 0) = Range("Step") Range("DataOutput").Offset(Range("Step"), 1) = _ 'Store data Within the VBA, the MarketTick code needs to be extended to push the data onto When the Go/Pause button is first pressed, the market should start ticking with The MarketTick subroutine updates the market by moving spot up or down by … … If (MarketOn) Then A time series of spot can nowwith be VBA stored on development. the Columns sheet. can Again, be thisthe set is upper-left primarily up achieved cell to of store the the output step and named: spot ‘‘DataOutput.’’ rate, with the sheet. The .Offset command is used to access the appropriate cell within the sheet: sure everything is wired up properly.works In particular, correctly: check Pressing that the the Go/Pausepressing button again button should restart while the spot ticks. is ticking should pauseStep 2: it; Record then and Chart Spot End Sub the frequency specified in the cell named ‘‘TickTime.’’ Test different inputs to make for the OnTime function, plusthan ‘‘_’’ is one used line: when a code statement goesSub over more MarketTick1() the ‘‘SpotIncrement’’ amount at random on eachtick market is tick. scheduled. Then a Note future how market time is converted from seconds terms into day terms 30 BUILDING A TRADING SIMULATOR IN EXCEL f(urnl otybak osdw,icuigbt tpadso columns: spot and step both including down, rows blank) ticks mostly few (currently a of for simulator the Run charted. be andpauseit.Thenselectcells,startingatthetitleandrunningalargenumber(500ish?) to them enables stored ticks the Having data: stored the clear to extended be to needs code StopButton the And … Wend 0) Range("DataOutput").Offset(Count, While 0 ticks = spot Count the clear and around 'Loop … hntesmltri u,so ik hudb eoddi h table: the in recorded be should ticks spot run, is simulator the When on on 1 + Count = Count 1).ClearContents Range("DataOutput").Offset(Count, 0).ClearContents Range("DataOutput").Offset(Count, <> "" BUILDING A TRADING SIMULATOR IN EXCEL 31 Insert an X-Y Scatter chart with straight lines between points. When the simulator Within the sheet a newmust bid–offer be output: spread input is required, plus bid and offer rates If price takers wantwant to to buy sell in in the theprices market, are market, they set they must up must and pay give the the ability the offer. to bid. give If Within or price this pay takers theStep task, market 1: bid is Set and introduced. Up offer a Two-Way Price is un-paused, the data shouldis plot stored with (up the to chart the automatically number resizing of as rows new originally selected): data Task B: Set Up aFunctionality Two-Way Price and Price-Taking ■ 32 BUILDING A TRADING SIMULATOR IN EXCEL ptmv.Tetae cinte ed ob rcse n h oiinupdated MarketTick4() position Nothing’’): ‘‘Do the Sub to and back processed reset be selection the to (and needs appropriate then if action trader The move. spot VBA: the within referenced be can the and that time cell a output at an selected to be the linked can then choices but is the Excel of selection one in only so These ways grouped menu. be Control different should Form buttons the many from Buttons in transact. Option to uses done here spread implemented be a method crossing could hence choices bid), market these the (at Controlling sell buy nothing, or actions: offer), three market of one the do can (at trader VBA the using tick, updated spot kept each and at addition, outputs In as code. added be can P&L and position their both know simulator to needs trader a manage, risk to order In Functionality Price-Taking Up Set 2: Step cleared. be to StopButton, within need and cells up, output set offer be and to bid need offer the and bid initial the GoButton, Within e B oenest paetePLbsdo h rdrpsto n the and position trader the on based P&L the update to needs code VBA New subroutines. StopButton and GoButton the to added be to needs also code New subroutine: VBA MarketTick the within referenced are cells new These f(aktn Then (MarketOn) If Double As SpotIncrement Dim ag(Ofr)=Rne"ptiMre" ag(BdfeSra" 2 / Range("BidOfferSpread") 2 + / … Range("SpotMidMarket") Range("BidOfferSpread") = - Range("Offer") Range("SpotMidMarket") = offer Range("Bid") and bid 'Calculate … ag(DtOtu".fstRne"tp) )=Range("Position") = 2) Range("DataOutput").Offset(Range("Step"), _ = 1) Range("Step") Range("DataOutput").Offset(Range("Step"), = 0) Range("DataOutput").Offset(Range("Step"), data 'Store Range("SpotMidMarket") position n their and P&L ihnthe Within . BUILDING A TRADING SIMULATOR IN EXCEL 33 0.5) Then > Range("Position") = Range("Position")Range("Pnl") + = 1 Range("Pnl") - Range("BidOfferSpread") / 2 Range("Position") = Range("Position")Range("Pnl") - = 1 Range("Pnl") - Range("BidOfferSpread") / 2 SpotIncrement = Range("SpotIncrement") SpotIncrement = -Range("SpotIncrement") Range("TickTime") / 24 / 60 / 60), "MarketTick4" Range("BidOfferSpread") / 2 Range("BidOfferSpread") / 2 'Schedule a marketApplication.OnTime tick TimeValue(Now() in + the _ future 'Process trader action:If Buy Range("Action") = 2 Then End If 'Process trader action:If Sell Range("Action") = 3 Then End If 'Reset the traderRange("Action") action = 1 End If 'Update P&L Range("Pnl") = Range("Pnl") + Range("Position")'Update * spot SpotIncrement andRange("SpotMidMarket") step = Range("SpotMidMarket")Range("Step") + = SpotIncrement Range("Step") + 1 'Calculate bid andRange("Bid") offer = Range("SpotMidMarket") - _ Range("Offer") = Range("SpotMidMarket") + _ Range("DataOutput").Offset(Range("Step"), 3) = Range("PnL") 'Calculate spot increment If (Rnd() Else End If The P&L and position are now also stored on the sheet and can be displayed in Again, new code must also be added to the GoButton and StopButton subroutines automatically updating charts using the same method as the spot chart: End Sub to set up and clear the data on the sheet as appropriate. 34 BUILDING A TRADING SIMULATOR IN EXCEL rnatmksi ifiutt aemnywti hsfaeok l transactions all framework; this within money make P&L to difficult and it position, makes market, transact between interactions the while with familiar. start become to fine is should moves. position market on trader based the update must and P&L the cross should Also, spread down. there or from up executed increment impact is correctly P&L trade negative a initial time Each an expected. be as work controls buy/sell the euti eaiePLcag oeeytaerdcsepce &.Ti san is This P&L. expected reduces trade every so lesson: trading change real-world P&L important negative a in result tsol eoeovosqieqikyta rsigtebdofrsra to spread bid–offer the crossing that quickly quite obvious become should It ticks between seconds five down; it slow quickly, too happening is everything If and correctly ticks still spot that Check tested. be to ready now is simulator The o’ vrtaewe hr ssra rs involved cross spread is there when over-trade Don’t . BUILDING A TRADING SIMULATOR IN EXCEL 35 The following new inputs should be added to the sheet: This framework seeks to show how a price-making trader must deal with hedge the existing position?pre-position Or for should the risk flows? be immediately offset? Can the trader change when theyparticipants trade. transact they Within do thismade so by at simplified the trader. the framework, market when bid and the offer market ratherunpredictable than flows. at Should a the price trader wait to see if offsetting deals come in to In practice, tradersThis are dynamic sometimes isparticipants’’ price achieved to within takers the the and VBA simulator code. sometimes by These price adding price makers. takers price-taking cause ‘‘market the trader position to Test different combinations of bid–offer spreadtheir and relative spot size increment impacts to trading observe behavior how and performance. Task C: Introduce Price-Making Functionality ■ 36 BUILDING A TRADING SIMULATOR IN EXCEL ■ Extensions h ai rmwr a eetne nnmru ifrn ast aei more ■ it make to ways suggestions: different some numerous are in Here extended realistic. be can framework basic The too are swings P&L the should and deals, trading large active offsetting or too passive for tested. different gets is be wait and selling it preferences and buy-and-sell or when Skewed sit buying only big. should participants position trader market the the other reducing theoretically the equal, of probability roughly the If changes. be must P&L and position the trade, a is there If accordingly. sell. updated or buy participants market ■ ■ xml,prasmre atcpnsaemr ieyt u fso oslower goes spot for if buy so, to behavior likely more participant are market participants market around perhaps a rules example, than interesting rather more approach Introduce volatility-based details. a for H using Practical time see increment, over fixed targets rate P&L spot and the limits Evolve risk misaligned having how performance. push observe then trading line, to impacts in line targets and of limits the out with them Start targets. P&L and limits risk Add hnrnigtesmltrwt hsnwfntoaiy h oeo h trader the of role the functionality, new this with simulator the running When other the whether determine to used is VBA the within number random A … If End Else MarketSignal ElseIf MarketSignal If Rnd() = MarketSignal action market 'Process … ag(MreSlPo" Then Range("MarketSellProb") MarketSignal Range("Message").ClearContents Action No 'Market 2 Sells" / "Market Range("BidOfferSpread") = + Range("Message") Range("Pnl") 1 = + Range("Pnl") Range("Position") = Range("Position") Sells 'Market 2 Buys" / "Market Range("BidOfferSpread") = + Range("Message") Range("Pnl") 1 = - Range("Pnl") Range("Position") = Range("Position") Buys 'Market < < ag(MreByrb)+_ + Range("MarketBuyProb") ag(MreByrb)Then Range("MarketBuyProb") > ag(MreByrb)Ad_ And Range("MarketBuyProb") = BUILDING A TRADING SIMULATOR IN EXCEL 37 the price taker is ashows buyer, a 1.3035 they offer should and tradelow the with probability. price high taker probability. is If a the buyer, trader they should only trade with trading further away from the current midmarket. Most realistic (and mostmaking complicated) prices would with the betrader dealt to price side have and depending thetwo-way the market on trader price current the manually is market 1.3000/1.3030 relationship and between bid–offer. the trader the For shows a example, 1.3015 if offer, if the current and sell if spot goes higherthe or flows becomes vice easier. versa. If the trader knowsIntroduce the different rules, sized notionals. managing In practice, trading in larger size often means ■ ■

CHAPTER 4

FX Derivatives Market Structure

arket structure is a topic that is often skipped over. In practice though, it is Mvitally important because it defines how clients interact with the trading desk and how the trading desk accesses liquidity to hedge their risk. 39 In some financial markets all participants access a centralized market or exchange anonymously on the same terms. The FX derivatives market, however, is an over- the-counter (OTC) market, meaning that there is no centralized exchange and a clear distinction exists between banks and their clients. Note that ‘‘banks’’ here refers to large international banks with FX derivatives trading desks. Fundamentally, bank FX derivatives trading desks transact with clients, aggregate and offset the risk where possible, and close out unwanted residual risk. More specifically:

■ Clients come to bank trading desks for prices, often via a sales desk within the bank. Usually the client simultaneously submits the same price request to multiple banks and deals on the best price as per Exhibit 4.1. Traders usually make two-way prices for clients because they do not know for certain whether a client is a buyer or a seller of a particular contract.

■ Bank trading desks transact with each other either via the interbank broker market or the direct market (a price request directly between a trader at one bank and the corresponding trader at another bank). The majority of bank-to-bank transactions occur in the interbank broker market. This structure is shown in Exhibit 4.2. 40 FX DERIVATIVES MARKET STRUCTURE ■ letTypes Client ■ be generally Corporates: can that reasons speculation. for or derivatives investment, FX hedging, use as classified client of types different Many 4.2 EXHIBIT 4.1 EXHIBIT oslwrtecmaywl erltvl oescesu.Fnaetly the Fundamentally, successful. EUR more relatively the be of EUR/USD will when fewer company Likewise, the worth factories. new lower are less build goes and sales relatively workers from their be pay will received to needed company USD the the rate. higher because exchange goes EUR/USD successful the EUR/USD to when exposed unhedged, is company If This America. Europe in in production sales with but manufacturer car exposures a consider FX example, their For funding. managing and with concerned primarily companies International nebn rkrmre interactions market broker Interbank banks from prices requesting Client FX DERIVATIVES MARKET STRUCTURE 41 dual trading relationship, meaning nonreciprocal (DCDs). Within a DCD the client deposits money in one currency : Professional money managers but typically trade to shorter time : Professional money managers who use foreign exchange as an asset are responsible for keeping desk pricing in line with the market. They : Central banks/NGOs (e.g., IMF/World Bank), interested in FX On the trading desk in each center there are various roles.Traders In practice these roles money either in the depositedeffectively currency sold or a in call afrom option second selling on currency. the the option The deposit gives client the currency, has client and an the enhanced coupon premium on earned the deposit. Individuals who trade simple FX-linkedcurrency deposits investment products, for example, for a fixed term. At the end of the term, the bank has the option to return the Have smaller FX derivatives trading desks or perhaps(i.e., hold no ‘‘back-to-back’’) all risk exposures at all to and larger transfer banks.with Regional international banks banks usually as trade clients inthat a the international bank cannot request prices from the regional bank. Hedge funds horizons than real money. Sovereigns volatility plus potentially manage currency reserves. Real money class. Simply put, they seekmarkets to move. take positions that will generate positive P&L as success of the companyand should sell depend cars on rather their thanforeign ability exchange exchange to exposures rate are design, fluctuations. hedged—potentially manufacture, Therefore, using expected FX future derivatives. and bid–offer spread. often overlap and what is presented in Exhibit 4.3 is enormouslymake simplified. prices for clients and risk manage desk trading positions. In addition they FX derivatives trading desks usually deliverthree follow-the-sun global coverage centers, to normally clients London, from Newwithin York, the and three one centers in are Asia-Pacific. inbest Traders constant possible communication client and service they in aim to terms provide of the pricing consistency between centers, speed, Retail: ■ Regional Banks: ■ ■ ■ Institutional: ■ Bank FX Derivatives Trading Desk Structure ■ 42 FX DERIVATIVES MARKET STRUCTURE neatn ihSlsDesks Sales with Interacting keeping for smoothly. responsible running are management they risk essence desk In accurately. and desk. quickly trading positions the trading by used tools and models analysis and pricing the implement and sales desk. educate trading the also by Structurers offered products. products complex derivative new more and on price solutions to construct and traders design with They work requirements. investment and most hedging York. FX the New Asia- of of out in run out be run sits normally will be block usually normally America) (Latin will block Latam block the while currency AUD Pacific the particular example, for a center; for appropriate book-runner EUR/CAD. possibly and main USD/CAD in be The likely USD/CAD, will risk CAD/CHF, the and of AUD/CAD, majority NZD/CAD, the contain although GBP/CAD, actually EUR/CAD, might block CAD/SEK, CAD CAD/NOK, the example, for so, blocks for currency management, risk or generation. pricing idea their trade and with analysis assist market that example, tasks other various perform 4.3 EXHIBIT ewe h rdn n ae ek r o wl pnbtclaoaini crucial is collaboration but business. interactions the upon of dwelt The success the not relationships. overall are within the client desks managers for sales the relationship and for and trading the responsible desks between sales primarily create the They are manufacturers. is who product it bank sense, but a clients in for are, products desks trading Derivatives FX ideoffice Middle Quants Structurers (EM) market emerging or G10 managing risk for responsible usually are Traders qatttv nlss r sal h-ee ahmtcaswodevelop who mathematicians PhD-level usually are analysts) (quantitative oe na Xdrvtvstaigdesk trading derivatives FX an on Roles okwt ae n eainhpmngr oudrtn clients’ understand to managers relationship and sales with work eae urnypisaepttgte noacrec block, currency a into together put are pairs currency Related . nuetaigpstosaecretadta e el i the hit deals new that and correct are positions trading ensure FX DERIVATIVES MARKET STRUCTURE 43 Other departments within the bank come to the FX derivatives trading desk for The trading desk assists the sales desk by providing them with good information Sales desks build relationships with clients by seeking to understand the clients’ Validate the pricing models (see Chapter 19) used by the trading desk. counterparties. Monitor desk pricing versus independent market sources. Ensure the trading desk ispriced offering and a risk valid managed. range of products that can be properly Deal with recruitment, contracts, and training. Build and maintain the desk technology infrastructure. Monitor trading risk to ensure risk limits areEnsure not broken. the trading desk has booked deals correctly and confirms deals with Monitor bank credit exposures toclasses. different counterparties across different asset Produce official trading desk P&Ls. Ensure the trading desk complies with its international tax obligations. Ensure the trading desk complies with their regulatory requirements. FX spots, forwards, swaps, NDFs,lending interest will all rate be products, traded and regularly cash by borrowing an FX and derivatives tradingadvice, desk. pricing, and execution onfor FX derivatives speculation. transactions. For Sometimes these example, are a trader on the FX spot desk may wish to buy a ■ Trading Internally FX derivatives trading desks generally transact internallythe (i.e., trading within the desks bank) of with other asset classes in order to hedge non-FX derivatives risk. ■ ■ ■ ■ ■ ■ ■ ■ ■ Interacting with Support Functions Trading desks do notdepartments operate within in the isolation; bank. they Forthe require example, following: support there are from separate many teams other that■ do all of given the chance to quote on any FX contracts that theabout client wants the to transact. market,competitive prices coming in order up to help build with the client relevant relationship. trade ideas, and offering quick, business and specifically theirinformation about FX what requirements. is happening They in market provide with clients the aim with that the good trading desk is 44 FX DERIVATIVES MARKET STRUCTURE ■ ■ isfraJno Trader Junior a for Tips Internship Trading a for Tips eciigyu oiin oktae rprytefis ie n eal oexplain times. to all able at be accurately and when position time, precise and first Be P&L trader. the your junior properly a trades for book trait position, possible your worst describing the is This sloppy. be Don’t traders how see to important is it and occur this. to to likely react most is activity market when better. the about floor, understand trading you the more and on The desks), roles credit). trading different equities, and rates, desks (interest (sales classes class asset asset other same the impression. within negative make teams a other and as quants), bad people asked as be to almost will is Speak desk impression the pursue. no on you; people to of of range thought like a they would internship involves what prey the job you of fell trading end career I the the a which At what connections. to learn is to one it there (and whether are folly You and is 2001). project in your internship doing an during just time be whole will the and you there know to get to learn. start you will help they to this likely gives doing more traders By for them. lunches and to coffees exposure getting you thing; same the you. through to went given Everyone ask isn’t do; it an to if what material in told training document be getting to in 40-page is waiting assistance a there material for sit read traders the just junior you’ve don’t of hand, that understanding other off the good On showing material a afternoon. than learning Developing important any race. through more a race far isn’t don’t desk, It trading given: the you’re on start first you When and price will desk have trading may derivatives deal FX finance the trade manage. that risk or ultimately component client transaction FX underlying M&A an structured to an a linked example, are they For often, transaction. More option. FX vanilla short-dated ephl neeo h aktbcueyucnb uete are. always they sure traders, be with can talking you because and market sitting the you’re on eye If something an market. at to half experts reacting the keep become quickly in Traders to occurred learned. weather has be the must or that that sports one about question. is chatting a it from you but flipping asking is skill boss difficult your a and is moving, This is spot shouting, are room the across er ob wr fmlil hnsa ne rkr r huig pttraders spot shouting, are Brokers once: at things multiple of aware be to Learn is This expiries. option major and releases economic over desk the on be Always office, middle (structuring, desk trading the of parts different the with sit and Go Sitting trading. about learning and work project between balance right the Get first. team the put them; do and on get just do, to jobs rubbish given you’re If FX DERIVATIVES MARKET STRUCTURE 45 Always have an opinion about the market and a plan for your trading position. Learn the official desk P&L currency and how the P&L conversion from the Follow as many different financial markets as you can, not just your own. Know Understand what kind of trading desk you are on and how it makes its money. Judging market liquidity is a skill that is acquired over time. Knowing where to Don’t be afraid to admit you’ve made a mistake. Everyone makes mistakes. Fix it, Learn how to round exposures and P&Ls when describing them. Traders usually Never exaggerate or bluff. Experienced traders will pick up on bluffs in Don’t panic under pressure; stay calm and keep your thoughtsDon’t be clear. lazy: If If you something looks lose wrong in the position, investigate and find what Don’t get gripped watching spot moving up and down at the expense of all other natural P&L currency perDoes this currency effect pair create a into trading the exposure which desk needs P&L to beAlways currency managed? know is and handled. be ablemade to in many justify different your ways, current but position. if The you justification can’t can justify be why you have a position, you that P&L between client tradesthe and trading position desk taking? have How in muchare the the of main interbank a client broker presence groups does market (corporate,of and institutional, trades the do etc.) these of direct clients the market? like desk Who to and transact? what kind where ten-year USD rates,best the traders Nikkei, have the a view Vix,different on and parts. the so whole forth market and are see all the trading. connections between The the What were the annual P&Ls of the desk for the last five years and what is the split of three hundred and twenty-onetoo thousand, much five information when hundred ‘‘long and ten bucks’’ seven gets U.S. the required dollars’’ information is over. price a large client traderecycled must by be estimating experienced how to the be learned. market will absorb the risk if it is learn from it, and make suresimilar it mistakes doesn’t happen can again. be Obviously, harmful however, to repeated career progression. only care about their deltas tohow the much nearest risk one million is or being five million, run, depending so, on if asked for a delta exposure, ‘‘long ten million, picoseconds and will delight inme find taking out,’’ is you usually apart acceptable. Lying formoves is it. never in acceptable. Saying, terms Also, ‘‘I describe of don’t market ‘‘it’s know, going what insane’’ let has or actually ‘‘it’s gettingmoves happened 0.2%). completely and destroyed’’ when try implied to volatility avoid hyperbole (e.g., of time. your nerve, you might as well not be there. is wrong; don’t assume problems will fix themselves. work. Learn to be aware oftrading spot without positions staring and at it. pick In practice, approximate traders view spot their levels where they will hedge ahead 46 FX DERIVATIVES MARKET STRUCTURE ■ XDrvtvsItrakDrc Market Direct Interbank Derivatives FX akt hrfr,tedcso ocl ieti sal aebcuei opl the compels it because made usually is direct call to decision the Therefore, market. explained is (this dealt always is call, USD and 7). in assumed a Chapter is quoted case in date the this always of in is occurrence side, next notional the out-of-the-money specified, calls, the not direct is year Within a if expiry. terms; cut CCY1 NY 2014 25, August communication. efficient enables which call, the within systems. made management assumptions risk both between into premium booked option be and then traders must rate) the deal deposit on, the dealt forward, and is themselves (spot, price data volatility implied market an the (hence After short agree price. deal a the to has on seconds) then pass 20 A or and to Trader spread) pair) up terms. currency (approximately volatility that implied time in of trading contract amount currently the are on Trader they price contract. (because a vanilla a makes request on the price up a requests picks and B system request messaging bank recorded to calls a A directly Trader via straightforward: other B is each process The contact contracts. vanilla can simple on banks prices different at traders derivatives FX example, for of, result a positioning. market as or taken flows, client be 17), could Chapter (see positions analysis market Trading it. have shouldn’t 4.4 EXHIBIT A B A B A B A B A hog h rkrmre,o ehp h akti oaieadteei currently is possible there be market. and not broker volatile would the is that in market available way the and liquidity a perhaps size limited in or large once market, in at broker transact banks the to multiple through wants with market. trade trader broker to the the need by because may provided made safety be the might of none decision with This price a make to trader other > > > > > > > > > rdr aeacoc ewe aln ieto sn h nebn broker interbank the using or direct calling between choice a have Traders with Put the Call/JPY Note USD 103.00 B. of and USD75m in A price traders a requested between has call A Trader direct typical a shows 4.4 Exhibit GEDTAK O H ELBIBIBI DEAL THE FOR THANKS AGREED OKAY? SPOT 102.50 SURE 5.5% / 4.9 HIHIHIHI H BIBI THX OKAY? DELTA 40 USD% 0.185 DEPO USD 0.15% SWAP -1 AGREED PLS BUY PLS 75M IN NY 103 AUG 25TH USDJPY PLS USDJPY nebn ietcl rmJl 3 2014 23, July from call direct Interbank crossing rdrB’s trader FX DERIVATIVES MARKET STRUCTURE 47 as shown in Exhibit 4.6. Brokers are usually working has global teams of FX derivatives brokers who between shop contracts make up a large part of the market, too. Specifics are specific , which broker A is interest Market instruments at market tenors are often quoted in the interbank broker The trader at bank A wants to transact a specific vanilla contract and requests Brokers are split by currency block (e.g., G10Communication between majors/G10 broker crosses/Asia and trader EM, has traditionally been done either by In the FX derivatives market there are currently four main interbank brokerage Direct relationships between traders work best when they call each other with There is a well-established etiquette within the direct market: Traders don’t call a price from their brokerA’s at one of theworking multiple interbank interests brokerages. for different This traders isdoes simultaneously. called Note not that trader have the to trader disclosetrader a reveals contract that the size interest at is this in stage; large it size, may but help initially a the standard broker ‘‘market if size’’ the market but nonstandard vanilla contracts. For6mth example: 102.00 July TOK in 23 USD/JPY 1.3250 are NY both specific in contracts. EUR/USD or etc.) and also by option type (e.g., short-date vanillas/long-date vanillas/exotics). voice over recorded fixed telephoneOver lines time, or communication via is moving recordedstandardized away text electronic from messaging messaging. free-form systems. text messaging toward This section explains in detail how the broker market works. firms. Each broker them speak to the FX derivativesshown traders in at Exhibit all 4.5. the banks in the market in a structure The interbank broker market is asince crucial it part is of where the the FX vast derivatives majority market of structure bank-to-bank FX derivative transactions occur. exist but thereprocess. is When always quoting a athis direct tension contract?’’ call, There involved the is often trader thatthe a thinks, interbank arises catch; ‘‘Why broker otherwise, are from market the they and thecalls contract calling are probably could dangerous price on transacted because be they closer making worked can to expose in closely traders midmarket. enough. who Direct aren’t following the market receive a direct call on thefar same from what contract. was Most expected, importantly, the if priceto the making check price trader their should received rate be is to given avoid an opportunity bigger problems later when asimilar mistake frequency is on discovered. similar-size contracts. Mutually beneficial direct relationships each other too often and mosta trading contract desks has rarely been reject in a the price broker request. market Also, and once a price has been made, it is not usual to FX Derivatives Interbank Broker Market ■ 48 FX DERIVATIVES MARKET STRUCTURE XII 4.5 EXHIBIT XII 4.6 EXHIBIT rdrArqetn rc rmterbroker their from price a requesting A Trader structure broker Interbank FX DERIVATIVES MARKET STRUCTURE 49 Brokers going to the market requesting a price on the contract The importance of the trader/broker relationship should be starting to become Note that traders are not compelled to make a price and there are many possible Broker A tells the other brokers at their shop about the interest and the brokers The traders price the contract in their pricing tools and when ready they return Bank B: no price Bank C: 6.75 / 7.25% Bank D: no price Bank E: 6.6 / 7.1% EXHIBIT 4.7 perhaps they are off the desk getting lunch. apparent. If a broker and trader have a good relationship, the trader is both more In trader A’s pricingvolatility. tool, the mid-value of this vanilla contractreasons why is they might 7.0% choose not implied to. Perhapsfor they are a busy pricing client, another contract perhaps they are remarking their curves, or talking to their boss, or ■ ■ ■ ■ go to the relevant tradersin at Exhibit all 4.7. the banks in the market requesting a pricetheir as prices, shown quoted in implied volatility terms: would be assumed. In majorto G10 USD50m currency in pairs, market normalcurrency size pairs, market market is size conditions. roughly is USD30m In smaller. cross G10 currency pairs and EM 50 FX DERIVATIVES MARKET STRUCTURE ftae sasle cle a (called seller a is a A (called trader buyer If a is A trader If Pass. 5. offer. an Show 4. bid. the Give 3. bid. a Show 2. offer. the in Pay shown rate is 1. this process with This offer. A the trader or to bid back the goes is broker bank 4.8. The which Exhibit disclose E. not bank does from but offer and this. managing C for bank vital to are who relationships brokers trader Again, other a rate. bothering a the mean make get will to this must want instances not A some does In broker the rates. rate, maximize for traders composite hence their and tight push market, a the getting from back of possible chance (prices) rates most broker. the the get for prices make to different likely four more are and there from), (remember broker choose particular to a shops from price a request to likely rdrA trader 4.8 EXHIBIT rdrAnwhsfieoptions: five has now A Trader from bid 6.75/7.1%: is made prices two the between rate composite best The To important. also is firm their within broker specific the of power relative The rkr olcigpie rmtemre n eotn h etrt akto back rate best the reporting and market the from prices collecting Brokers eln interest selling uiginterest buying ,otos3 ,ad5aevldchoices. valid are 5 and 4, 3, options ), ,otos1 ,ad5aevldchoices. valid are 5 and 2, 1, options ), FX DERIVATIVES MARKET STRUCTURE 51 The interest negotiating with the trader who made the best price since the order was placed. something has fundamentally changed in the market (i.e., spot has moved sharply) Trader A is a seller of the contract and shows aThe 7.0% broker offer. goes back The to trader process C now only and shows them the 7.0% offer (this would Therefore, in normal markets, transacting on the ‘‘opening rate’’ is rare. It is If trader A pays the offer or gives the bid immediately, the trade is done and EXHIBIT 4.9 be described as a ‘‘seven-oh top’’). Trader C then1. has three Keep main the options: 6.75%2. bid at the Show same a level higher (‘‘stuck3. (‘‘better’’) bid. on the Remove bid’’). the (‘‘pull’’/‘‘ref’’/‘‘refer’’) bid. Orders aren’t usually pulled unless illiquid markets, where the brokers mayhave struggle no to option get but any rates to at trade all, on a an trader opening may rate andbecomes hence a cross negotiation the between full trader spread. the A best and bid) trader via C their only respective (since brokers trader as C per Exhibit showed 4.9. even if they have athe preference interest to is buy a or buyer sell or the a contract; seller they before showing wait their tofar own see more hand. whether likely thata trader process A that will will show result a in ‘‘counter’’ (i.e., transacting a at bid a or better offer) level. to However, start in volatile or the process is complete. However,normally works in slower FX than derivatives that. the Traders usually interbank make broker a market relatively wide initial rate 52 FX DERIVATIVES MARKET STRUCTURE novd nw s‘pitn’ h rd.Ti ssoni xii 4.11. Exhibit counterparties in the shown of is market This any the trade. of in the ‘‘printing’’ names as everyone the known inform mentioning involved, their brokers without half The sell pays transaction, to rate. the therefore managed midmarket about has D their A Trader Trader at EUR50m. position. amount in their full only but fits offer contract 7.0% A’s the trader buying that realizes and recently? market the buyers more in been trader’s contracts there the similar Have Therefore, of vital: interest.’’ is sellers the market or of the ahead of state it ‘‘give current the and about hit judgment to bid good 6.9/7.0%. a is is point 6.9% this at price which two-way rate, the the see tightening market further the of in aim traders the all with letting required means the which in out,’’ it trade ‘‘show sure to make interest the to for transaction offer traded, the or being of bid size to the notional. on potential closer both exists the or gets size discuss transact, price enough to they that the hence start As and will levels. paid broker different is the offer at ‘‘stuck’’ or are given is traders point. bid this prevailing at away the walk a either to show want either can doesn’t similarly A and trader do assuming to stuck, chosen be has or C offer trader better what told then is A Trader XII 4.10 EXHIBIT rdrD h a rvosyntmd ae saetdt hstgtprice tight this to alerted is rate, a made not market. previously the had to out who rate the D, show Trader brokers the how shows 4.10 Exhibit that decide may trader another since A trader for risk a is out rate to the happy Showing are they if traders both ask will broker the stuck, are sides both When until brokers their via C and A traders between forth and back goes process This rkr hwn h aeott h market the to out rate the showing Brokers FX DERIVATIVES MARKET STRUCTURE 53 Brokers printing the trade to the market The brokers continue to match buyers and sellers at the trading level untilThe they final step in the process is to agree contract details. Vanilla deals are The information that the contract is ‘‘sell on’’ is also important. That there are no The level at which this transaction was completed is important information for booking the deal. Therefore, theto traders maturity, and agree deposit midmarket rate (see levels Chaptera for 10) spot spot, to maturity, forward or which forward is thenpremium premium turned (i.e., using into a the premium Black-Scholes paidnot formula. at depend Note on the that the delivery deposit a date rate. forward rather than the spot date) will are sure that everyone ininterested the parties, market the is contract aware dies. of the trade. Once there arequoted no and more traded in implied volatility terms but a cash premium is required when more buyers in the market ismarket. a Likewise, sign if that the traders price havethe of more similar brokers of contracts a report is contract falling the in tosimilar contract the buy, contracts as with is no rising. ‘‘bid If sellers on’’ the and found, would interest report this managed that is to the trade a interest their has sign full been that size, ‘‘taken out.’’ the the broker market for traders in the market. Even iftraders they should have been price paying little uppriced attention the up differently to contract to this in point, the theirsurface trading own needs level, pricing traders tweaking tools. must orbelow If judge trading the the whether midmarket the level contract their contract or volatility is sell represents above. an opportunity to buy EXHIBIT 4.11 54 FX DERIVATIVES MARKET STRUCTURE e-olrtrs enn h ee fcmiso nUS olrtrsfreach for rates brokerage terms the dollar know traders U.S. that important in dollar- is commission It in transacted. of quoted notional of are level USD1m desk) the trading meaning the terms, on ‘‘bro’’ per-dollar called just (often rates Brokerage (including trades. prices reporting and live ‘‘buyer’’), reporting e.g., direction, prices, interest requesting the of away combination is the price Note the trader. (i.e., of off-market traders. ahead to is given protection made or provide recent price paid they on a level), be if based correct might the and contract rate from out, a particular shown whether a traders if for in interest which feel the interested a know be have they also to plus They likely behavior. knowledge, are this market contract. the the all of in have price the brokers of good evolution the However, except exchange, anything foreign or about knowledge contracts, any derivatives or interest, the on price ‘‘correct’’ the about rkr a oka neeti aydfeetwy.Freape rkrcan broker a example, For ways. different many in interest an work can the Brokers liquid more the general, rates. brokerage In the with. lower interest the an pair, partially) currency place least to (at should broker that since which pairs determine currency their for shops broking different in hr sa nrosaon fflxblt ihnti rnato structure. transaction this within flexibility of amount enormous an is There transact. they trades of (size) notional the on commission paid are Brokers seller 9.25/9.45 115.00P 1Y JPY 10:04:01 9.25 paid 103.85P Tue JPY 09:51:52 buyer 8.75/9.25 103.85P Tue JPY 09:51:52 pls 115.00P 12.0/8.3 1Y trades JPY 102.00C 09:33:03 Tue vs 102.25C O/N JPY outright 08:23:25 seller 8.25 pls 102 per yen 50 tues 102.00C 08:43:50 Tue vs 102.25C O/N pls JPY 93.50 08:21:55 Cut TK May 12 CAD/JPY 08:20:10 pls 104P Jun 05 JPY pls 08:07:27 103.25P vs 103P O/N JPY seller 07:56:00 6.25/6.65 116.10C May 16th CHF/JPY buyer 07:55:44 9.05/9.2 0.8875 Sep 04 AUD buyer 07:50:10 9.05/9.3 0.8875 Sep 04 AUD 07:40:29 pls 97.00C May 23 AUD/JPY 07:38:34 9.0 trades aud 1yr 07:33:47 pls 0.8875P Sep 04 AUD 07:24:37 a to activity market reporting chat broker a of example an provides sidebar The opinions have to required brokers were process entire this during point no At xml rkrCa rmJl 3 2014 23, July from Chat Broker Example FX DERIVATIVES MARKET STRUCTURE 55 In general, the more traders put in, the more they get out of the broker market. By give it.’’ ‘‘If I show thisyourself.’’ out, I’m sure it will‘‘The get guy paid is ahead pretty of shaky you. on You his should bid—I pay think it he’s about to ref it. You should and have their owngood levels. pricing up-to-date maximize their chances of transacting at making prices (i.e., being a good liquiditybetter provider), fills brokers will on work the harder to trader’s(and get interests. therefore Traders know the who prevailing follow market the sentiment for broker different market types closely of contract) the broker’s skill sometimesmarket involves in manipulating order to the get trader’s them to impression trade. of■ Sales the 101: Create a sense of urgency: ■ ‘‘build size’’ at ainterest particular to level give (i.e., all get into one USD200m get go) on the or best a they fill particular can possible. work bid However, an for this interest the more flexibility quickly inevitably or means slowly that part of

CHAPTER 5

The Black-Scholes Framework

erivatives products have been traded in one form or another for centuries, but Dthe development of the Black-Scholes model in the 1970s enabled financial derivatives markets to flourish by enabling volatility to be consistently priced. 57 Financial mathematics books generally give the derivation of the Black-Scholes formula and list the reasons why the assumptions underpinning it aren’t correct in practice. Traders don’t need to know how to derive the Black-Scholes formula from scratch. However, it is vital that they understand the features of the Black-Scholes framework since it is the foundation for all derivatives valuation.

■ Black-Scholes Stochastic Differential Equation (SDE)

The Black-Scholes framework assumes that the price of the underlying (i.e., the FX spot rate) follows a geometric Brownian motion. The Black-Scholes stochastic differential equation (SDE) is:

dSt 𝜎 =(rCCY2 − rCCY1)dt + dWt St

where St is the price of the underlying (spot) at time t, dSt is the change in underlying at time t, rCCY1andrCCY2 are continuously compounded (see Chapter 10) CCY1 and CCY2 interest rates respectively, 𝜎 is the volatility of the underlying’s returns, generally just called ‘‘volatility,’’ and Wt is a Brownian motion. Sometimes, rCCY1 58 THE BLACK-SCHOLES FRAMEWORK efcl olw h owr path forward the follows perfectly (forward). F to (spot) S from change variable where that: 1 Chapter from recall Plus by: solved is which forward). the the (i.e., define future spot the of in value maturities different future for expected rates no-arbitrage Forward the gives drift The rate interest the on depends passed: that time component the and deterministic differential predictable, a is Drift Drift 2. an 1. that (note FX for life terms. real absolute in the as in zero, as smaller hit never because get underlying can equity changes model model the zero), the in to spot within Therefore, (closer used smaller are gets changes underlying Relative in ‘‘returns’’). generated naturally called is contracts FX standard on P&L 2, terms. and CCY2 1 Chapters in seen the called is ■ ■ rates: interest constant fCY neetrtsaehge hnCY neetrts h owr path forward the rates, interest CCY1 than higher as are higher spot. moves rates equal will interest path forward CCY2 the If equal, are rates interest CCY2 and CCY1 If ne lc-coe supin,tefradpt sbsdo urn ptand spot current on based is path forward the assumptions, Black-Scholes Under important: is This If parts: two has SDE the of side right-hand The represents SDE the of side left-hand The Uncertainty Drift 𝜎 F = T rmteitrs aedifferential rate interest the from ie,n oaiiy,then: volatility), no (i.e., 0 stefradt time to forward the is foreign rmtevltlt fteunderlying the of volatility the from could T neetrt and rate interest nrae.Ti scalled is This increases. eovltlt osntma htso ssai;i en htspot that means it static; is spot that mean not does volatility Zero ot zero). to go dS S . F t F t T ( T =( rCCY T = = and S rCCY S 0 2 0 rCCY + e − S ( rCCY 0 2 SwapPoints rCCY scretso lsnt h outrageous the note plus spot current is − the 2 2 eaiechanges relative − oiiedrift positive rCCY rCCY 1 ) 1 dt ) domestic . 1 T T ) dt . neetrt eas,as because, rate interest nteudryn (often underlying the in owr path forward . THE BLACK-SCHOLES FRAMEWORK 59 . , that is, options where the that spot takes to get there. path 10%. CCY2 interest rates are higher negative drift = 2 rCCY path-dependent options 0% and = 1 increases. This is called T rCCY Short-term forward path When pricing vanilla options or any product where the payoff depends only on Pushing the maturity out to ten years,This the exponential is nature important of when the function pricing long-dated options. Exhibit 5.3 shows the For example: If CCY1 interest ratesmoves lower are as higher than CCY2 interest rates, the forward path EXHIBIT 5.1 payoff depends not just on spot atMany expiry, but exotic on the options arein path USD/TRY dependent. that Consider will an expire exotic if derivative spot product ever trades above 2.5000. Using constant instruments at different maturities. the spot at maturity,as the the forward forward to path maturityconsideration within when is the pricing correct. model However, isn’t the a forward path concern is so an long important the forward path looks linear as shown in Exhibit 5.1. reveals itself in Exhibit 5.2. USD/TRY forward path generatedBlack-Scholes using versus constant a market rates forward to path the generated 10yr using different tenor interest under rate Within this simplified framework, atinterest a rate given can maturity, be used either to thebe calculate forward used the plus other to interest one calculate rate the orignored two within forward. interest this All rates analysis. issues can regarding credit risk and basisthan risk CCY1 interest are rates and therefore there is positive drift. At shorter time-scales ■ 60 THE BLACK-SCHOLES FRAMEWORK XII 5.2 EXHIBIT XII 5.3 EXHIBIT S/R oe essmre owr path forward market versus model USD/TRY path forward Long-term THE BLACK-SCHOLES FRAMEWORK 61 (also called t W t is normally distributed ) dW 𝜀 𝜎 ). 𝜀 + + √ ) dt now )) ( ,𝜀 t 0 ( ( 𝟏 N to time )∼ rCCY t W now )− − t ( 𝜀 𝟐 + t W ( can be plotted in Excel using code shown in Exhibit 5.4 rCCY (i.e., standard deviation from time W 𝜀 =( W t t S dS are random with this distribution: ). A Wiener process is a continuous stochastic process with stationary Excel setup for generating a realization of a Wiener process W The fact that the Black-Scholes SDE is driven by aA discrete normally realization of distributed pro- Changes in Within the SDE, using full interest rate curves is equivalent to making the interest ‘‘Stochastic’’ means ‘‘it moves.’’ ‘‘Stationary’’ means ‘‘its probability distribution does not change over‘‘Independent time.’’ increments’’ means ‘‘each changeous does changes.’’ not depend on any previ- ‘‘Continuous’’ means ‘‘its path doesn’t jump.’’ EXHIBIT 5.4 cess explains why bell-curveframework. shapes appears repeatedly within the Black-Scholes and a sample realization is plotted in Exhibit 5.5. In words, the change in with mean 0 and variance ■ ■ ■ The uncertainty term inBrownian motion the SDE isindependent driven increments. Translating: by a Wiener process ■ rates functions of time: Uncertainty interest rates under Black-Scholesusing the will full market generate interest rate different curve.in trading Issues practice. like exposures this are than therefore very important 62 THE BLACK-SCHOLES FRAMEWORK ■ ovn h lc-coe SDE Black-Scholes the Solving h lc-coe D ssle sn h ai fIt of magic the using solved is SDE Black-Scholes The more: move to spot causes volatility higher expected, as that, meaning volatility, 5.5 EXHIBIT ihnteBakShlsSE h inrprocess Wiener the SDE, Black-Scholes the Within ln aperaiaino inrprocess Wiener a of realization sample A ( S S T 0 ) = = = dS S ( ( ∫ t t 0 rCCY rCCY =( T ( rCCY rCCY 2 2 − − 2 rCCY rCCY 2 − − rCCY rCCY 1 1 − − 1 𝜎 𝜎 1 2 2 ) 2 2 dt − ) ) + 𝜎 2 T T 2 𝝈 Calculus: o ¯ + + ) dW 𝜎 ∫ dt W 0 W + t T T 𝜎 t ∫ dW smlile ythe by multiplied is 0 T t 𝜎 dW t THE BLACK-SCHOLES FRAMEWORK 63 . 2 2 𝜎 − are normally distributed. This is log returns ). This is important because it shows how the implied volatility. T ) ¯ o correction term: 2 is normally distributed with mean 0 and variance √ 𝜎 T = W 4 𝜎 √ ( Representation of the Black-Scholes framework (i.e., standard deviation Understanding log-normality is important because it impacts distributions and However, if volatility is raised to 30% and maturity is increased to five years, Because we’re now in log-space, spot The previous formula shows that the adjusted forward drift is the central reference Furthermore, this term: T 2 EXHIBIT 5.6 the shape of theapparent distribution in changes Exhibit 5.8. dramatically and the log-normality becomes Greek profiles particularly atplotting higher the terminal volatility spot orthe distribution standard-looking longer for bell-shaped maturity. curve 1mth in For EUR/USD Exhibit at 5.7. example, 8% volatility gives point of the future log-spot distribution,with which a at each wider point and isshown normally wider in distributed distribution Exhibit 5.6. over time due to increasing variance.why This log is returns areChapter always 17. used within the realized spot volatility calculations in 𝜎 volatility and time to expiry aredrift, linked within multiplying the time distribution. Ignoring to the expiry adjusted the by same four way as changes doubling the terminal spot distribution in The key points to note arethe that drift we’ve has moved been from adjusted regular-space by into the log-space It and 64 THE BLACK-SCHOLES FRAMEWORK XII 5.8 EXHIBIT 5.7 EXHIBIT o-omldsrbtoshv ogrti ntetpiei eua ptsaeand space spot regular in topside return zero. the (log below on 2.0 go tail to never longer 1.0 a from have move distributions spot log-normal a to opposite and equal nalgnra ol,aso oefo . o05(o return (log 0.5 to 1.0 from move spot a world, log-normal a In emnlso itiuina ogtnradhg volatility high and tenor long at distribution spot Terminal volatility low and tenor short at distribution spot Terminal =+ .9) Hence 0.693). = 063 is –0.693) THE BLACK-SCHOLES FRAMEWORK 65 : t at time S , one using flat W t dW ) t 𝑡 W ( 𝜎 𝜎 + t + ) 2 2 𝜎 dt ) − 1 1 rCCY − rCCY 2 − rCCY 2 ( e 0 S rCCY = t S =( can be generated using the same t t S S dS Sample Implied Volatility Term Structure Within this basic Black-Scholes framework, there is only a single volatility. Consider the sharply upward-slopingTwo implied realizations volatility of term structure in By taking exponentials, the SDE solution gives this analytic solution for The Black-Scholes formula uses constant volatility. This must be changed to the EXHIBIT 5.9 derivative product where the payoffThe value depends of only the on option can spotagainst be at the obtained the by terminal integrating option the maturity. spotcalculation option multiplies payoff distribution the at as maturity probability of shownpayoff spot at in ending that spot Exhibit up level. at 5.11. This each technique Intuitively, point is by implemented this the in Practical option B. Terminal spot distributions can be used to price vanilla options or any other However, in practice,different vanilla implied options volatilities. with Ingiven different a strike maturities given and expiry currency and date is pair,idea strikes determined the is by have explored the implied volatility in volatility surface Chapter for for 7. that a pair. This Exhibit 5.9. volatility and the other usingExhibit the 5.10. implied Using volatility the term term structure. structureat This of shorter is implied tenors shown volatility and in leads higher to volatility lower at volatility longer tenors. full ATM term structurethis when is pricing equivalent to path-dependent making options. volatility Within a function the of SDE, time: 1mth2mth3mth6mth1yr 5.0% 6.0% 7.0% 10.0% 15.0% Tenor Implied Volatility Calculating Option Values Using Terminal Spot Distributions ■ 66 THE BLACK-SCHOLES FRAMEWORK XII 5.10 EXHIBIT XII 5.11 EXHIBIT elztoso inrpoesuigdfeetipidvltlt emstructures term volatility implied different using process Wiener a of Realizations aun ail pin sn h emnlso distribution spot terminal the using options vanilla Valuing THE BLACK-SCHOLES FRAMEWORK 67 T . (1983) formula ) 1 ) ) d 2 2 𝜎 d ( (− 1 2 T . N N − T ) T . . 2 1 2 1 𝜎 1 2 rCCY rCCY itself, which gives prices for − − rCCY T + Se Ke 1 − √ 2 𝜎 )− )− 2 1 rCCY T d d ( rCCY − √ ( (− N 2 𝜎 T . N + 1 T . Garman and Kohlhagen 2 ) rCCY rCCY S K − ( rCCY ( − Se + ln Ke = ) S K = = call ( T P Black-Scholes formula put ln P √ = 𝜎 = = call 1 − put d 1 d Price Cumulative normal distribution function Price = 2 d These formulas are implemented in Practical C. EXHIBIT 5.12 where European vanilla calls andis puts. the The FX-specific extension toboth the currencies: Black-Scholes formula that uses interest rates in Finally, we arrive at the The Black-Scholes Formula ■ 68 THE BLACK-SCHOLES FRAMEWORK nCatr1 xedBakShlsb eaigdfeetasmtoswti the within assumptions different relaxing by the Black-Scholes framework. of extend discussed after models decades 19 simplicity pricing use the Chapter all the in extendibility; still in its that is is it reason note why Another volatility developed. reasons to was key between it instructive the going of is of one is method It framework a Black-Scholes pricing. as which premium is in and way used main is pricing The itself traders. for formula concern Black-Scholes day-to-day the a isn’t equal that or practice, in than hold less value a have will 1 deviation standard to and 0 distributed normally mean a with that probability variable the gives which is function, formula of distribution the sources normal behind all driver removes main This The cost. volatility. for no except at risk hedged delta continuously be can value option X vntog h supin nepnigteBakShlsfaeokd not do framework Black-Scholes the underpinning assumptions the though Even that assumption the is formula Black-Scholes the of derivation the to key The xii .2sosagaho h uuaienra itiuinfunction. distribution normal cumulative the of graph a shows 5.12 Exhibit . N ( X ) h cumulative the , PRACTICAL B

Building a Numerical Integration Option Pricer in Excel

69

hen an option payoff depends only on the spot rate at the maturity of the Wcontract (e.g., European vanilla options) the price of the option can be calculated using the terminal spot distribution and the option payoff.

■ Task A: Set Up the Terminal Spot Distribution

Step 1: Set Up the Future Spots First, future spot levels must be generated using a log-normal distribution. The inputs to the function are:

■ Spot (S): the current exchange rate in a given currency pair

■ Interest rates (rCCY1 and rCCY2): continuously compounded risk-free interest rates in CCY1 and CCY2 of the currency pair

■ Time to expiry (T): the time between the horizon date and expiry date measured in years

■ Volatility (𝜎): the volatility of the spot log returns 70 BUILDING A NUMERICAL INTEGRATION OPTION PRICER IN EXCEL tnaddvain n aclt h eunlvladcrepnigso ee for level spot to corresponding –5 and value: from level deviation go standard return steps, each the 0.1 calculate with and Starting deviations returns. standard theoretical possible all almost sheet: Excel an in up set be can framework This of return given a For world: log-normal Within aln ihnapriua ag fvle.I Excel, In variable values. random of a range of particular likelihood a relative within the falling gives function density probability The Density Probability the Calculate 2: Step ne omldsrbto,arnefo 5to –5 from range a distribution, normal a Under xetdReturn Expected tnadDeviation Standard eunLevel Return X ptLevel Spot tnaddeviations: standard ( 𝜇 = = )= S 𝜇 = . + 𝜎 ( e eunLevel Return √ X rCCY . T tnadDeviation Standard 2 − rCCY + = 1 tnaddvain covers deviations standard 5 OMDS()gvsthe gives NORMSDIST(X) − 𝜎 2 2 ) . T + 5 BUILDING A NUMERICAL INTEGRATION OPTION PRICER IN EXCEL 71 at or below Note how the data in the rows is lined up; the probability density in aThe given probability row density can be plotted against spot to visualize the terminal spot gives the probability of spotthe ending row up below. between that spot level and the spot leveldistribution: in the probability density) cancumulative probabilities: be calculated by taking the difference between two cumulative normal distribution function,distributed random which variable with is mean thethe 0 and input probability standard level of deviation X. 1 a being Therefore, normally the probability of being between two levels (i.e., 72 BUILDING A NUMERICAL INTEGRATION OPTION PRICER IN EXCEL ■ akB e pteOto aofadCalculate and Price Payoff Option Option the the Up Set B: Task ehdcnb sdt rc n aofta nydpnso pta auiy no maturity, at spot on are: depends examples obvious only most that the payoff but any complicated, price how matter to integration numerical used This be framework. the can into method added be now can payoff option The ■ ■ ■ 5: Chapter ■ in explained As changes. distribution spot terminal the how ■ ■ ■ ■ ihrCY neetrtso oe C2itrs ae hudsitthe shift should rates interest CCY2 lower. lower the moving forward or the shift via rates lower should distribution interest rates CCY1 interest Higher CCY1 higher. lower moving forward or the via rates higher distribution interest distribution. CCY2 wider a Higher to lead should volatility higher distribution. or tighter maturity a Longer to lead should volatility lower or maturity Shorter ail u option: put Vanilla option: call Vanilla forward: Short forward: Long h mlmnaincnb etdb hnigtemre aaadobserving and data market the changing by tested be can implementation The S K T − − max max S K T ( ( K S T − − S K T ,0) ,0) BUILDING A NUMERICAL INTEGRATION OPTION PRICER IN EXCEL 73 Within the numerical integration, multiply the probability of spot falling between Then the option payoff at maturity can be calculated at each spot level: In Excel, add a new ‘‘option payoff’’ column and calculate the payoff at each spot two spot levels at maturity by the average payoff at maturity between two spot levels: level. To price a vanilla call option, the strike must be inputted: Remember that these payoffspips) terms. all return values in CCY2 per CCY1 (i.e., CCY2 74 BUILDING A NUMERICAL INTEGRATION OPTION PRICER IN EXCEL au tmtrt.TeCY isoto au utte epeetvle (see valued present spot: be factor current then discount by dividing must the value using option 10) pips Chapter CCY2 The maturity. at value rbblt-egtdoto ausaete umdt e h vrl option overall the get to summed then are values option Probability-weighted ( e − rCCY 2 . T ) n ovre noCY%by CCY1% into converted and BUILDING A NUMERICAL INTEGRATION OPTION PRICER IN EXCEL 75 0%, and = rCCY2 = rCCY1 100, should give (approximately) zero = K = S at the forward 1.0 should have a value very slightly under 4.00 CCY1%: : A forward payoff struck : A vanilla CCY1 call option with = T value: Test 2 Test 1 Finally, the pricer can be tested: Testing ■

CHAPTER 6

Vanilla FX Derivatives Greeks

t is time to start some derivative analysis. The aim of this chapter is to introduce the Ibasic Greek exposures on European vanilla options. This is stylized Black-Scholes analysis with zero interest rates throughout; hence the forward rate is always equal 77 to the spot rate and discounting considerations can be ignored. The charts within this chapter can be generated in Excel after completing Practical C.

■ Option Value

A vanilla call option gives the right, but not the obligation, at maturity to buy spot (i.e., buy CCY1 versus sell CCY2) at the strike in the agreed notional. Exhibit 6.1 shows the value at maturity of a long (bought) vanilla call option over different spot levels. This value at maturity is often described as the option payoff. For a short (sold) vanilla call option, the value at maturity is reflected in the spot-axis resulting in an increasingly negative value above the strike, as shown in Exhibit 6.2. As expected, a long position plus short position in the same contract results in zero value over all spots (i.e., no position). The initial option premium is sometimes added into these diagrams, as per Exhibit 6.3. Adding the premium into the payoff is appropriate if the plan is to transact the option and then hold it isolated until maturity (see the section on breakevens in Chapter 17). However, in delta hedged trading portfolios, many options are risk managed together. Therefore, it is changes in option value caused by 78 VANILLA FX DERIVATIVES GREEKS XII 6.1 EXHIBIT XII 6.2 EXHIBIT au tmtrt fsotvnlacl pinwt 0.0strike 100.00 with option call vanilla short of maturity at Value strike 100.00 with option call vanilla long of maturity at Value VANILLA FX DERIVATIVES GREEKS 79 plus in spot, optionality . This analysis convex exposures intrinsic value , discussed later in this gamma in Exhibit 6.6. Call option value prior to maturity is Total P&L at maturity (including initial premium) of long vanilla call option with , depending on the spot level at maturity. In the value at maturity : or not Optionality is the ability of the contract to transact spot (e.g., buy spot for a call The value of a vanilla option can be decomposed into So far, so straightforward. Now see how the long vanilla call option value changes A vanilla put option gives the right, but not the obligation, at maturity to sell spot Again, a short vanilla put option has a value at maturity that is reflected in the in the contract. Intrinsic value is the option payoff at maturity. Time value is the value expected to be generated from the remaining option), charts, optionality is represented by the change in angle at the strike. chapter). time value ■ ■ spot-axis. This results in increasinglyExhibit 6.5. negative value below the strike, as shown in prior to maturity implying a positive second derivative (i.e., positive changes in the market data that areis most slightly important, cleaner called if the premium is omitted. (i.e., sell CCY1 versus buyshows CCY2) the at value the at strike maturity in of the a long agreed vanilla notional. put Exhibit option 6.4 over different spots. EXHIBIT 6.3 100.00 strike 80 VANILLA FX DERIVATIVES GREEKS XII 6.5 EXHIBIT 6.4 EXHIBIT au tmtrt fsotvnlaptoto ih100 strike 100.00 with option put vanilla short of maturity at Value strike 100.00 with option put vanilla long of maturity at Value VANILLA FX DERIVATIVES GREEKS 81 in Exhibit 6.8 shows how higher higher volatility Value of long vanilla call option with 100.00 strike and 10% volatility Within the value charts, the strike of the option is fixed and spot is being Looking at call option value at Therefore, the value of a vanilla option can give information about how much If the forward to maturity is far above or below the strike, the optionality on a call option, the premiummoved decreases further because away the from payoff the above forward the to strike maturity. is being there must be minimal intrinsichigh-premium value vanilla and options, minimal time there valueor may both. on be the high contract. intrinsic For value, high time value, changed. If strike were being changedapproximately instead reflected for a in fixed the spot, strike the level. diagram would For be example, if the strike moves higher volatility leads to higher timemore value volatile, as it the will distribution have widens. more Intuitively, chances if to spot go is through theoptionality strike. there is within a particular contract. For low-premium vanilla options, has minimal valuebefore since maturity. there Therefore, is optionthe value little strike converges chance to on ofmaturity intrinsic spot is both value equal away going sides. to from throughExhibit the Maximum 6.7. strike; the time here strike the value optionality is occurs most when valuable, as the shown forward in to EXHIBIT 6.6 82 VANILLA FX DERIVATIVES GREEKS XII 6.8 EXHIBIT 6.7 EXHIBIT au fln ail aloto ih100 tiead2%volatility 20% and strike 100.00 with option call vanilla long of Value volatility 10% and strike 100.00 with option call vanilla long of value Time VANILLA FX DERIVATIVES GREEKS 83 delta for clarity. In symbols: P F 𝜕 𝜕 S P 𝜕 𝜕 )= of the call option value profile chart F Notional spot delta (Δ × % (Δ) = gradient =Δ Delta Cash is the spot rate. Note that traders sometimes say Δ S Forward Delta , sometimes called delta is option price and (i.e., zero delta exposure). is the forward rate to the option maturity. P F At maturity, there is a discontinuity in delta from 0% below the strike to 100% Exhibit 6.9 shows the delta of a long vanilla call option with a 100.00 strike. This For a given option, delta can either be quoted in % terms or cash terms. For When a position is delta neutral, no P&L change results from spot moving higher Delta is the exposure of the option value to the spot rate. The delta amount for It is important to appreciate that forward delta can also be calculated, but this above caused byposition) either or expiring exercising the thenotional option option amount). above Prior below to the the maturity, strikechange the strike (hence occurs delta buying still (hence over spot goes a having from in wider 0% no the to spot full 100% range. but Notice the that the delta at all maturities is or lower. However, this only works(gamma) for is small non-zero. moves in spot if the second derivative delta can be calculatedfrom by Exhibit taking 6.6. the example, if an AUD/USD callbought, has AUD10m 25% AUD/USD spot spot delta mustdelta and be in AUD40m sold the of trading ‘‘on the position the unchanged. contract hedge’’must If in is be the order AUD/USD bought to call on was leave the sold hedge. instead, spot where a given option is thereforethe the opposite equivalent delta direction) spot in notional orderneutral that for must the option be plus transacted the (in spot hedge to be distinction will be overlooked for now: they are, e.g., ‘‘long spot’’practice, to delta mean is ‘‘long quoted exposure either toin as spot’’ the a (i.e., notional % currency long where: of delta). the In notional amount, or as a cash amount important Greeks: where Taking the first derivative of option value with respect to spot gives one of the most Delta ■ 84 VANILLA FX DERIVATIVES GREEKS et.Tedlao h pinvlefo xii .0i hw nEhbt6.11. Exhibit position. spot in with short increases shown a value as is option same 6.10 put the because Exhibit lower, delta spot from negative value have options option put the Intuitively, of delta The delta. up ending of strike. chance 100.00 the does as reduces maturity). delta at option ITM call up ending the in-the-money. of lower, chance goes have 50/50 spot forward) a (the (i.e., As spot 50% to approximately close of strikes delta with a options Call maturity. at (ITM) long money a as same the higher, spot with increases option position. call spot the of value the to since equal delta is maturity) to forward the accurately, more strike. (or the spot when 50% around 6.9 EXHIBIT re olaedlai h rdn oiinucagd fteoto a odinstead, in hedge. sold the hedge was on option the the sold on If be unchanged. bought must position spot be trading the must in delta spot leave AUD/USD to order AUD8m bought, is contract the o xml,i nADUDptoto a 1%so et n U8mof AUD80m and delta spot –10% has option put AUD/USD an if example, For a gives with spot option to respect put with value vanilla option this long of a derivative first of the taking value Again, option the shows 6.10 Exhibit in-the- up ending of chance % the as of thought (approximately) be also can Delta positive a has position option call long a that surprise a be shouldn’t it Intuitively, et fln ail aloto ih100 strike 100.00 with option call vanilla long of Delta VANILLA FX DERIVATIVES GREEKS 85 Delta of long vanilla put option with 100.00 strike Value of long vanilla put option with 100.00 strike and 10% volatility EXHIBIT 6.11 EXHIBIT 6.10 86 VANILLA FX DERIVATIVES GREEKS .20srk nER0’ ahrta ‘’egta135 u nEUR50m’’). in put 1.3250 a got ‘‘I’ve than is rather of option EUR50m’’ terms the in in once strike only puts exposures 1.3250 talk or Greek calls they same of book; the terms trading have in their think strike in usually and don’t maturity traders same reason the this with For puts hedged delta and owning calls as same the where strike—exactly matter the No exercised at strike. be bought the contract. will is at forward put spot bought the maturity, short be at the will up strike, will spot ends spot the and spot and below expired) expired) call is put long spot short the If the (and strike. (and the exercised at be bought will be call long the strike, the ■ a called is ■ This same. the details contract other all with ■ forward the trading ■ by call) notional: a and into changed maturity, put strike, a same (or the put in a into call a from in changed (shown be option put long a strike, gives same the 6.13) with Exhibit 6.14). forward in Exhibit short (shown a spot notional a plus all world, 6.12) and diagram Exhibit by maturity, over hockey-stick in simply delta In (shown option maturity. option put option to a call call forward into long of converted the amount be than notional can option a lower call selling 100% a way, is another delta Put values. option put a the has actually that put’’ delta ‘‘ten a so delta omitted, describing often put when is Plus, the delta. sign 25% lower, off negative a leave the goes has often options, actually call’’ spot put they delta ‘‘twenty-five deltas as a describe and so traders %, ITM), the when practice, up In ending negatively. of increases chance 50/50 (i.e., h aemtrt n tiems lasb auda h aeipidvolatility implied same puts?’’ does the who and eyes. at calls wide explains does with also ‘‘who valued met asking This be and be would forward. desk the trading always trading the via approaching must why possible be strike would and arbitrage otherwise, maturity same the 1%delta. –10% hr call Short call Long put Long call Long nte osqec fptcl aiyi that is parity put–call of consequence Another optionality, no has forward the since that is this of consequence One above is spot if maturity, At forward. synthetic long a for formula the Consider option put a and option call a from constructed be can forward a Additionally, called result powerful a is This except identical are 6.11 and 6.9 Exhibits in profiles delta put and delta call The –50% approximately of delta a have forward the to close strikes with options Put + + + + ogforward long hr put short forward short ogput long = = hr ytei forward synthetic short ogsnhtcforward synthetic long = = ogcall long ogput long u–alparity put–call strikes nwrs ail pincan option vanilla a words, In . alotosadptotoswith options put and options call and ytei forward synthetic notionals eg,‘Iv o a got ‘‘I’ve (e.g., et hedged delta : ; . VANILLA FX DERIVATIVES GREEKS 87 Value at maturity of long put option Value at maturity of short forward Value at maturity of long call option EXHIBIT 6.14 EXHIBIT 6.13 EXHIBIT 6.12 88 VANILLA FX DERIVATIVES GREEKS ■ Gamma ptmvs ec o am.A hre auiis hnteei ihe spot gamma. tighter high a hence is moves, there spot as when as slowly maturities, quickly changes changes shorter delta At delta distribution, gamma. distribution, low spot hence previously, wider moves, a discussed spot is the there As when in differentiated. maturities, seen longer neatly be be can cannot gamma delta in Gamma discontinuity strike. the 6.11. around Exhibit from profile profile delta delta option option put call the the either or the This of 6.9 with 6.15. gradient Exhibit the Exhibit options from taking in put by shown and calculated profile, be call gamma can same parity, gamma the put–call have to maturity and Due strike moves. same spot as is delta stable cash a as or amount, notional the of % a where: currency as notional either the in quoted amount is gamma practice, In with price) option of derivative Greek: second important another the gives (or spot delta to of respect derivative first the Taking XII 6.15 EXHIBIT stm oe oadteoto auiy am nrae n concentrates and increases gamma maturity, option the toward moves time As how of measure a therefore is and spot with changes delta how describes Gamma am fln ail pinwt 0.0strike 100.00 with option vanilla long of Gamma Gamma at curvature Γ auiyi o hw ntegahbcuethe because graph the on shown not is maturity Cash =Γ Γ = (Γ) fteoto au essso hrs At charts. spot versus value option the of % × 𝜕 𝜕 Notional Δ S gamma = 𝜕 𝜕 2 S P 2 nsymbols: In . VANILLA FX DERIVATIVES GREEKS 89 P 𝜕 𝜕𝜎 Notional × % (υ) = =υ Vega Cash υ . In symbols: vega Vega of long vanilla option with 100.00 strike Exhibit 6.16 shows how the vega profile of a long vanilla option position changes Long positions in vanilla options always have long gamma exposure because EXHIBIT 6.16 Note that traders describe their position as,to e.g., ‘‘long implied vol’’ to volatility’’ mean (i.e., ‘‘long long exposure vega). over time. Again, due to put–call parity, since forward contracts have no exposure In practice, vega is usually quotedamount either in as the a notional % currency of where: the notional amount, or as a cash Taking the first derivative ofthird option important value Greek: with respect to implied volatility gives a option, peak gamma occurs at the strikeis because maximized. this is the point at which optionality time value leads to apositions convex in option vanilla value options versus always spot have relationship. short Likewise, gamma short exposure. For a given vanilla Vega ■ 90 VANILLA FX DERIVATIVES GREEKS ■ Summary rmmil ears omil am ik h neto on ewe h two the time between over maturity. point two-month inflection changes around The option occurs vega risk. risk vanilla gamma of lower mainly given types to and a risk gamma on vega mainly risk higher from main relatively the have Therefore, options exposures. vanilla short-dated while 9. Chapter Trading in spot. covered with is moves positions delta gamma how short impacts money and lose gamma long and while rises falls volatility implied volatility if implied position money if trading make will A exposure options. vega vanilla long for a strike with the at maximized therefore are both options and vanilla hedged delta exposures. gamma Selling and exposures. vega shorter gamma in and results vega longer options vanilla in hedged delta results Buying impacted. are exposures gamma and vega the only the hedge from delta vega. generated and gamma, is delta, P&L chapter: this trading in of introduced majority exposures Greek the three derivatives, FX trading When always options vanilla short exposure. Likewise, vega value. short option payoffs have the positive increasing larger brings hence therefore play, and into distribution the widens out- volatility deep higher on (if impact position minimal no has like volatility changing or cases, value. in-the-money) option these deep option of (if the either optionality In forward the of-the-money). a from away like Far either maximized. is is value time and optionality implied in change payoff. a the Intuitively, for impact time. time more to over is volatility reduces there option because maturities vanilla the longer a have at on increases maturity vega vega and peak strike same The the profile. with vega options same put and call volatility, implied to ogdtdvnlaotoshv eaieyhge eaadlwrgmaexposures gamma lower and vega higher relatively have options vanilla Long-dated contract derivative the within optionality the from come both vega and Gamma position trader’s a into booked are options vanilla often, Most because is this Intuitively exposure. vega long have always options vanilla Long where strike the at occurs gamma) (like vega peak option, vanilla given a For onwdascuen e hnet h et xouewti h position, the within exposure delta the to change net no cause deals new so ihteappropriate the with PRACTICAL C

Building a Black-Scholes Option Pricer in Excel

91

uilding a Black-Scholes vanilla option pricing tool is one of the best ways Bto develop an understanding of derivatives pricing. Manipulating inputs and observing the impact on vanilla option prices is far more productive than looking at formulas in a book. This practical links closely to the material developed in Chapter 5. ■ Task A: Set Up a Simple Black-Scholes Options Pricer

Step 1: Set Up Spot/Rates/Time to Expiry The first inputs to the pricer are:

■ Spot (S): the current exchange rate in a given currency pair

■ Interest rates (rCCY1 and rCCY2): continuously compounded risk free interest rates in CCY1 and CCY2 of the currency pair

■ Time to expiry (T): the time between the horizon date and expiry date measured in years 92 BUILDING A BLACK-SCHOLES OPTION PRICER IN EXCEL where ■ terms: pips) CCY2 (i.e., ■ CCY1 per CCY2 in prices option vanilla European ■ ■ maturity at spot using calculated are strike payoffs option vanilla European Pricing Option Vanilla Up Set 2: Step when ■ ■ ■ ■ Price Price Payoff Payoff = = = = sflEclfntosare: functions Excel Useful And FX calculates formula Kohlhagen and Garman the 5, Chapter in described As happens what note and forward the impacts inputs the changing how see to Test maturity to forward the is output first The OMDS()frtecmltv omldsrbto function distribution normal cumulative the for NORMSDIST(X) exponential for EXP(X) root square for SQRT(X) log natural for LN(X) rCCY1 ( put call K 𝜎 put call ) : stevltlt fteso o returns. log spot the of volatility the is = = = = Ke Se = d max 2 max − − rCCY2 = rCCY rCCY ( ( K S ln 1 2 T d . . T − T ( 1 . − N N = K S ( (− S K ) d T 1 , ln , )− d + 0 0 2 ) ( ) )− ( K S Ke rCCY F ) T Se − + rCCY 𝜎 = − 2 rCCY √ ( Se 2 − rCCY . T T 1 ( rCCY . rCCY T N 𝜎 N ( 2 2 d √ − (− 2 − 1 rCCY ) T T − d rCCY : 1 1 ) ) 𝜎 . 2 2 T ) 1 . + T 𝜎 = 2 2 ) d 1 . T − 𝜎 √ T ( N S (X) T ) n the and BUILDING A BLACK-SCHOLES OPTION PRICER IN EXCEL 93 0%, the = rCCY2 = rCCY1 10%, and = 𝜎 1.0, = T 1.0, = K = S : For a 100.00 strike call option, if the strike is moved higher, the Example 1 Once the pricing is correct, flex each parameter and work through a logical Test that if equivalent put option will increase in value: Consider the relative positioning ofimplied forward volatility impact and the strike, terminal spot how distribution, time andfrom discounting to maturity of maturity back the payoff to and the horizon (see Chapter 10). call option price reduces because the forward is further away from the payoff. The argument of how the parameter change impacts call and put vanilla option pricing. option price is very slightly under 0.04 pips (0.0399 pips): 94 BUILDING A BLACK-SCHOLES OPTION PRICER IN EXCEL h ubr elwrdfe.A nPatclB: ■ Practical in As feel. real-world notional, a option numbers CCY1 the a Given naturally terms. are CCY1) prices per the option (CCY2 and pips terms CCY2 CCY1 in in generated quoted Payoff usually CCY1 are to notionals Convert Option and Notional Option in Add 3: to Step due decrease prices option put and call both discounting. but increased unchanged be will forward the play: because into increase payoffs prices larger option bringing put hence and wider, call moves both distribution increases, spot expiry the to time or ocneta pinpiefo C2pp em noCY ahtrs multiply terms, notional. cash CCY1 CCY2 into the terms by pips CCY2 from price option an convert To xml 3 Example 2 Example ahprice cash fCY n C2itrs ae ohmv ihrt h aelevel, same the to higher move both rates interest CCY2 and CCY1 If : o .50srk aloto,i mle oaiiymvshigher moves volatility implied if option, call strike 1.2500 a For : nCY a hrfr ecluae.Ti sueu eas tgives it because useful is This calculated. be therefore can CCY1 in BUILDING A BLACK-SCHOLES OPTION PRICER IN EXCEL 95 short put + K − F = : put pricing Price − call Price To convert an option price fromby CCY2 current cash spot. terms into CCY1 cash terms, divide long forward. However, in terms of Therefore, if the strike is set equal to the forward, the call price and the put price should be equal. This can be checked within the pricing tool: Step 4: Investigate Put–Call Parity In payoff terms, put–call parity is= often stated as, for example, long call ■ 96 BUILDING A BLACK-SCHOLES OPTION PRICER IN EXCEL ■ akB e paVAPiigFunction Pricing VBA a Up Set B: Task o fadtoa eiiiybcmspsil ftepiigcluaini done is calculation pricing the if possible becomes flexibility additional of lot A factor discount CCY2 the ver- The forward using future. the valued the from present in P&L be realized the to is whereas needs difference valued, strike present sus are prices Option ulcFnto pinrc(salA ola,SA obe sDul,_ Double, As K Double, As S Boolean, As the of OptionPrice(isCall cells Function inputs: the following Public the in take constructed should function functions pricing using VBA than The sheet. rather Excel function VBA a within sDul,rC1A obe CY sDul,vA obe sDouble As Double) As v Double, As rCCY2 Double, As rCCY1 Double, As T hntesrk smvdaa rmtefrad uteyrvasitself: reveals subtlety a forward, the from away moved is strike the When ( F − K ) au therefore value ( e − rCCY 2 . T ) : BUILDING A BLACK-SCHOLES OPTION PRICER IN EXCEL 97 ), which will make the formula -10 EXP(X) LN(X) SQRT(X) = = = qr(T) = 0) Then= T 0) = Then 0.0000000001 v = 0.0000000001 > > Application.WorksheetFunction.NormSDist(-d2) - SExp(-rCCY1 * * _ T) * Application.WorksheetFunction.NormSDist(-d1)) Application.WorksheetFunction.NormSDist(d1) - KExp(-rCCY2 * * _ T) * Application.WorksheetFunction.NormSDist(d2)) OptionPrice = (S * Exp(-rCCY1 * T) * _ OptionPrice = (K * Exp(-rCCY2 * T) * _ End If Dim d1 As Double, d2If As (T Double If (v d1 = (Log(Sd2=d1-v*S / K) + (rCCY2If - isCall rCCY1 Then + v ^ 2 / 2) * T)Else / (v * Sqr(T)) Call the function from the cell alongside the existing functions to test that both An explicit check for zero or negative implied volatility or time to maturity The VBA option pricing function should look like this: Helpfully, Excel VBA uses slightly different function names: Sqr(X) is the VBA equivalent of Exp(X) is the VBA equivalent of Application.WorkbookFunction.NormSDist(X) is used to accessnormal the distribution cumulative function. Log(X) is the VBA equivalent of T As Double, rCCY1 As Double, rCCY2 As Double, v As Double) As Double End Function calculations give the same results: 'Garman and KohlhagenPublic Currency Function Option OptionPrice(isCall Pricing As in Boolean, CCY2 S Pips As Double, K As Double, _ should also be included becauseInstead, set they them would to a cause small thereturn positive function value the (e.g., payoff to 10 at throw maturity. an error. ■ ■ ■ ■ 98 BUILDING A BLACK-SCHOLES OPTION PRICER IN EXCEL ■ akC eeaeFrtOdrGreeks First-Order Generate C: Task aulyflxn aaee eg,so rvltlt)asalaon paddown and up amount small a volatility) or spot (e.g., parameter a flexing manually a a using calculated called be is also can This exposures formulas. the generate to tiated = where market in changes are to Greeks price first-order important option most the an 6, of Chapter delta in sensitivity explored As the parameters. are exposures Greek u tcnb ple eeial ocluaeayepsr o n contract. any for exposure any calculate to slower generically is applied approach be difference can finite it The but contracts. exotic pricing when particularly calculate to parameter in change the over price in exposure. change the the of ratio the taking and OMITX ,1 FALSE). 1, 0, NORMDIST(X, o hs rtodrGek,teBakShlsfruacnb ietydifferen- directly be can formula Black-Scholes the Greeks, first-order these For Vega Delta ngnrl h lsdfr prahi atrbti sntawy available, always not is it but faster is approach closed-form the general, In and n ( ( (Δ) X 𝜐 ) stesadr omldniyfnto.I xe hsi cesdusing accessed is this Excel In function. density normal standard the is ) vega stecag noto au o hnei mle volatility: implied in change a for value option in change the is stecag noto au o hnei spot: in change a for value option in change the is . 𝜐 call = Δ Δ 𝜐 put call put = = = 𝜕 𝜕 𝜕 𝜕𝜎 P 𝜕 𝜕 P P call S put call S = = = nt difference finite 𝜕 e e 𝜕𝜎 − − P rCCY rCCY put 1 1 = . . T T [ closed-form Se N N − ( ( d d rCCY 1 1 ) )− 1 . T n 1 prah hc involves which approach, ( ] d 1 ) √ prah h same The approach. T BUILDING A BLACK-SCHOLES OPTION PRICER IN EXCEL 99 6 − parameters respectively. The smaller these parameter vol flex and spot flex v + Vol_Flex) v - Vol_Flex) (Vol_Flex * 2)) / S Dim OptionPriceUp As Double, OptionPriceDwOptionPriceUp As = Double OptionPrice(isCall, S, K,OptionPriceDw T, = rCCY1, OptionPrice(isCall, rCCY2, S, _ K, T, rCCY1,OptionVega rCCY2, = _ 0.01 * ((OptionPriceUp - OptionPriceDw) / _ Dim OptionPriceUp As Double, OptionPriceDwOptionPriceUp As = Double OptionPrice(isCall,OptionPriceDw S = + OptionPrice(isCall, S_Flex, S K, -OptionDelta T, S_Flex, = rCCY1, K, (OptionPriceUp rCCY2, T, - v) rCCY1, OptionPriceDw) rCCY2, / v) (S_Flex * 2) The VBA code for the finite difference exposures should look like this: Once both functions have been implemented in the VBA, check that the values Greek exposures have standard market quotation conventions: Within the pricer both methods can be implemented for comparison. Closed-form Vega is often quoted in CCY1quoted terms as (i.e., a divide % the of functionthe CCY1 result Black-Scholes notional by vega for spot) by a and 100 1% to move get in it implied into volatility standard (i.e., market divide terms). Delta is quoted as a % of the CCY1 notional. Vol_Flex As Double) As Double T As Double, rCCY1 As Double, rCCY2 As Double, v As Double, _ S_Flex As Double) As Double As Double, T As Double, rCCY1 As Double, rCCY2 As Double, v As Double, _ End Function End Function 'Finite Difference OptionPublic Vega Function in OptionVega(isCall CCY1% As Boolean, S As Double, K As Double, _ exactly match and investigate the impactflex of but changing test the what flex happens size. to Start the with outputs a as 10 flex size is increased and'Finite decreased. Difference OptionPublic Delta Function in OptionDelta(isCall CCY1% As Boolean, S As Double, K _ ■ additional flexes, the more accurate the outputs,the unless variables the themselves flex within is the smaller VBA. than the accuracy of ■ exposures can be calculated onis the easier sheet surface in but VBA. the finite New difference VBA approach functions for calculating delta and vega must take 100 BUILDING A BLACK-SCHOLES OPTION PRICER IN EXCEL ■ akD ltExposures Plot D: Task ae,ipidvltlt ees rtm oepr.Teepolsaekyt risk to key are profiles These expiry. options: to interest vanilla of spots, time portfolio different or a to over managing levels, used Greeks be or volatility can prices implied developed option been rates, of have charts that and functions tables VBA generate vega and delta, price, The 0.40%. under shade a is vega and 50%, to close is delta option etta if that Test S = K = 1.0, T = 1.0, 𝜎 = 0,and 10%, rCCY1 = rCCY2 = % the 0%, BUILDING A BLACK-SCHOLES OPTION PRICER IN EXCEL 101 and rCCY1 rCCY1 : Try this for strikes close to spot and strikes further : Look at the formula for vega and confirm the relationship. : The gradient of this chart gives gamma, plus try extreme : Note the location of the vega peak, plus try extreme values. values. rCCY2 Interesting exposures to plot are: away from spot. and Vega versus spot rCCY2 Vega versus time to expiry Option value versus volatility Deltaversusspot ■ ■ ■ ■

CHAPTER 7

Vanilla FX Derivatives Pricing

his chapter introduces two primary responsibilities of a vanilla FX derivatives Ttrader: maintaining volatility surfaces and quoting vanilla price requests. 103 ■ Maintaining Volatility Surfaces

In some financial markets, all relevant market prices can be observed directly in the market. However, in OTC (over-the-counter) derivatives markets, prices are very often requested for contracts that have not been directly observed in the market. This flexibility is a key advantage of the OTC market structure. To quote consistent vanilla option prices for any expiry date and strike, traders keep a volatility surface updated in each currency pair. Exhibit 7.1 shows a three-dimensional representation of the volatility surface in one currency pair. The volatility surface can be split into two components: the ATM (at-the- money) curve and the volatility smile.

ATM Curve ATM contracts are the backbone of the volatility surface; they define the term structure of implied volatility. ATM contracts are vanilla contracts quoted to a specific maturity and they have a strike near (or at) the forward to the same maturity. ATM contracts are the most important price reference points within the FX derivatives market. In the interbank broker market ATM contracts are often quoted in a run of prices at the market tenors, the most liquid expiry dates within the 104 VANILLA FX DERIVATIVES PRICING h T otata ahtnr hs are These tenor. each at ( contract ATM 2mth, the month), (one 1mth 2yr. and 2wk, year), week), (one O/N 1yr (one tenors: 6mth, market 3mth, 1wk at prices tomorrow), ATM of i.e., run a (overnight, gives 7.2 Exhibit example, For market. 7.2 EXHIBIT 7.1 EXHIBIT ar h ‘ade’i iia em(.. ‘U/S o rdsofasxhandle’’ six a off currency trades that vol ‘‘EUR/USD in (e.g., volatility term similar implied a of is ‘‘handle’’ level The approximate pair. the meaning base,’’ ‘‘vol volatility. 6.65% earn 6.9% will cost it will selling contract while The buy 6.65/6.9%: to is volatility price volatility implied two-way cut NY contributed hntaestl,AMipidvltlt sotnsoe bu ntrso its of terms in about spoken often is volatility implied ATM talk, traders When for has currently broker the that offer best and bid best the shows run The yohrbns hrfr,i xii .,teERUD1t ATM 1mth EUR/USD the 7.2, Exhibit in Therefore, banks. other by ) xml T run ATM Example surface volatility Example tradable ae hthv enmade been have that rates VANILLA FX DERIVATIVES PRICING 105 if front- fixed market tenors downward sloping if back-end (e.g., 1yr) ATM volatility to mean 6.1% implied volatility. 1 upward sloping Example ATM curve In general, ATM curves move in an orderly manner: Single tenors rarely move in ATM implied volatilities at different tenorsATM are curves plotted are described in as a curve in Exhibit 7.3. It is important to understand how this structure based around EXHIBIT 7.3 weighted shifts. end ATM volatility is higherend than back-end ATM ATM volatility volatility. tends Inmarket quiet to markets, conditions, be back- higher ATM than curvesvolatility. front-end can ATM become volatility. ‘‘inverted,’’ In with stressed higher front-end isolation and changes in the ATM curve are often characterized as either parallel or strike of the ATM contract changestracts too. are not This standardized, is although note the that nature strikesfive are of often or an rounded ten to OTC pips the market: so nearest Con- the ATM strike does not exactly change pip for pip as spot moves. is higher than front-end (e.g., 1mth) ATM volatility and as works in practice. As the horizon datechanges changes, the accordingly expiry date (the for each methodology marketin tenor for Chapter calculating 10 tenor andquoted implemented expiry in in dates the Practical market is today D). given haveor Therefore, different those expiry the quoted tomorrow. dates contracts Plus from as liquidly those spot quoted (and yesterday hence the forward outright) moves, the means the implied volatility indown, EUR/USD traders is often 6-point-something-%). use, When for written example, 6 106 VANILLA FX DERIVATIVES PRICING ■ ■ smile: volatility the the describe through in to used shown Slicing instruments as the tenor. smile given volatility a a produces at maturity particular 7.6. volatility Exhibit a implied at surface ATM volatility the to relative The Smile Volatility 7.5. Exhibit in demonstrated as maturities far at volatility ATM than 7.4. Exhibit in demonstrated as down or up amount 7.4 EXHIBIT hnhge tie tagvnmtrt eas qiistn orlysol and slowly rally to tend equities volatility because implied maturity higher given have a to at tend strikes strikes higher lower than derivatives, equity In curve. itdtevltlt ml is). smile volatility the the describes tilted contract reversal risk The are). smile volatility the of sides the the describes contract butterfly The utryadrs eeslcnrcsaeqoe tmre eoslk h ATM the like tenors market at quoted are contracts reversal risk and Butterfly the market, broker interbank the In a Within a Within oaiiysmile volatility parallel weighted aallAMcreshift curve ATM Parallel T hf,teAMvltlt talmtrte oe h same the moves maturities all at volatility ATM the shift, ATM enshwsrksaa rmteAMsrk r priced are strike ATM the from away strikes how defines T hf,AMvltlt tna auiismvsmore moves maturities near at volatility ATM shift, ATM wings butterfly skew ftevltlt ml ie,hwsteep how (i.e., smile volatility the of ftevltlt ml ie,how (i.e., smile volatility the of and ikreversal risk otat are contracts VANILLA FX DERIVATIVES PRICING 107 Example volatility smile Weighted ATM curve shift EXHIBIT 7.6 EXHIBIT 7.5 108 VANILLA FX DERIVATIVES PRICING tie otmr nipidvltlt em hndwsd tie se .. the e.g., (see, strikes downside than topside terms so volatility tilted implied usually are in smiles more volatility cost this pair For strikes higher. currency jump EM to the spot versus USD/CCY, causes USD that as devaluation reason, quoted currency EM pairs sharp a currency volatile). is market more risk main be emerging will major generally or in jump example, to likely For more is spot 7.10. which Exhibit in direction in the shown as crisis) financial 2008 the of height the at AUD/JPY 7.9. Exhibit in shown as 2014) July in USD/HKD as 2014) July in 7.8. Rand] Exhibit African in [South shown USD/ZAR 6mth (e.g., strikes downside than higher shown have as 2014) strikes July in downside EUR/USD that 7.7. 1yr Exhibit (e.g., such in volatility strikes tilted topside shaped than differently are volatility have smiles implied pairs volatility currency Some different FX smiles. in but quickly, drop 7.7 EXHIBIT S/A xml nEhbt7.8). Exhibit in example USD/ZAR oaiiysie aehge mle oaiiyo h ‘ekr’sd fso (i.e., spot of side ‘‘weaker’’ the on 1yr volatility (e.g., implied higher volatility have smiles high Volatility have and/or 1mth skewed highly (e.g., are volatility smiles low volatility have Some and/or symmetric are smiles volatility volatility Some higher have strikes topside that such tilted are smiles volatility Some xml onadsoigvltlt smile volatility downward-sloping Example VANILLA FX DERIVATIVES PRICING 109 Example symmetric volatility smile Example upward-sloping volatility smile EXHIBIT 7.9 EXHIBIT 7.8 110 VANILLA FX DERIVATIVES PRICING rcn osd tie thge mle oaiiyta eoe h oaiiysmile volatility The before. than volatility implied higher as at lower strikes moved topside be pricing is must tenors curve shorter ATM at the volatility of 7.11. implied at Exhibit end that marked in near shown signal is the a ATM therefore as 1mth and this current reducing takes the trader surface The volatility 5.0%. trader’s the In 4.8/5.0%. the trading a hits represents volatility price/trade the midmarket that the conclude may that opportunity. trader such the or inputs price trader surface market the either desk volatility observed, the are the by differences If updates generated contract. volatility same the midmarket for the surface with volatility interbank level the volatility that is compares information this for source primary market. which The broker for pairs responsible. currency the are in information they market volatility new for watch Traders Surface Volatility the Updating 7.10 EXHIBIT hr r ifrn asi hc hscnb civd o example: For achieved. be ■ can this which in ways different are 7.12. there Exhibit in shown is trade the as tenor same the at h T a emvdhge,hnemvn h nievltlt ml higher smile volatility entire the 7.13. moving Exhibit in hence shown higher, as moved be can ATM The h oaiiysiea hstnrms eudtdt i hstaiglvlbut level trading this hit to updated be must tenor this at smile volatility The 2 Example 1 Example new a When ail rd cusi h aktwihsget httemre is market the that suggests which market the in occurs trade vanilla A : h S/A mhAMcnrc w-a rc sqoe at quoted is price two-way contract ATM 1mth USD/CAD The : xml xrm oaiiysmile volatility extreme Example price per ra or appears trade cusi h rkrmre,tetrader the market, broker the in occurs VANILLA FX DERIVATIVES PRICING 111 Existing volatility smile plus new volatility market information Changing the ATM curve to match new volatility market information The trader chooses between these three possibilities (or maybe a combination of volatility smile is tilted suchstrikes that below strikes the above ATM are the priced ATMExhibit lower). 7.15. are Note priced that the higher ATM while stays unchanged in The wings of the volatility smileATM can are be priced moved relatively higher higher (i.e., strikes than away the from ATM)The the as skew shown of in the Exhibit volatility 7.14. smile can be moved more ‘‘for the topside’’ (i.e., the them) when updating the volatility smile. As mentioned, single tenors within the EXHIBIT 7.12 ■ ■ EXHIBIT 7.11 112 VANILLA FX DERIVATIVES PRICING information 7.14 EXHIBIT 7.13 EXHIBIT information 7.15 EXHIBIT hnigtewnso h oaiiysiet ac e oaiiymarket volatility new match to smile volatility the of wings the Changing information market volatility new match to ATM the Changing hnigtese ftevltlt ml omthnwvltlt market volatility new match to smile volatility the of skew the Changing VANILLA FX DERIVATIVES PRICING 113 vega . The bid–offer spread for a P 𝜕 𝜕𝜎 implied volatility. A bid–offer spread is two-way price (υ) = Vega midmarket Standard ATM Bid–Offer Volatility Spreads Exhibit 7.17 shows that looking at ATM volatility spread alone is misleading Recall from Chapter 6 that vega is the first derivative of option value with respect Therefore the ATM premium spread at a given tenor is calculated by multiplying ATM volatility spreads are wide in short-dates, reduce to a minimum from 1mth In practice, standard market bid–offer spreads are observed in the interbank EXHIBIT 7.16 2yr3yr4yr5yr 0.35% 0.4% 0.45% 0.5% TenorO/N1wk2wk1mth to 1yr ATM Volatility Spread 3.0% 1.0% 0.3% 0.6% is unchanged as implied volatility changes. when comparing bid–offer spreadshave at lower different vega. tenors In because most shorter currency pairs maturities the 1mth ATM is the most liquid ATM to implied volatility: the ATM volatility spread by the vega. Note that this calculation assumes that vega spread and ATM bid-offer spread becausewhen the delta spot hedging bid-offer the spread ATM must contract be to crossed maturity. to 1yr, then go wider againis at used longer to tenors, but go why? from To ATM get volatility a spread better to intuition, ATM premium spread in Exhibit 7.17. the strike to the ATM. market. Bid–offer spreadspairs for under ATM ‘‘normal’’ contractsSimple market interpolation at conditions can market bebetween often tenors tenors. used In look in general to there generate liquid similar is also bid–offer G10 a to strong spreads relationship Exhibit between for spot expiry bid-offer 7.16. dates Vanilla Bid–Offer Spreads The volatility surface generates a then applied around the mid-rateparticular to give vanilla a contract changes depending on its maturity and the proximity of volatility surface rarely moveover in multiple isolation tenors so at the once.the volatility trader By smile determines observing is how many best usually to prices updated update and the trades entire in volatility surface. the market, 114 VANILLA FX DERIVATIVES PRICING xml,i h y T a .5 oaiiysra n .0 rmu spread, premium 0.10% and spread volatility For 0.25% spread. has premium ATM constant as 1yr much the as if not in example, but shown terms, volatility as in widens vega spread lower to due spread premium bid–offer 7.20. Exhibit tighter a have would 7.19. Exhibit in shown as the vega had lower tenor 0.44% to 1yr due the spread at strikes all If 0.36% spread. premium same bid–offer 0.28% 0.10% a to 7.18. equates Strikes Exhibit in strikes. 0.20% shown ATM as on vega 0.12% occurs lower have exposure ATM vega the maximum from away the maturity, 0.08% given a At 0.06% spread volatility 0.05% Constant 0.87% 2. spread premium Constant 0.03% 0.79% 0.05% 1. the 0.06% 0.69% spread: of bid–offer shape the relative generating of 0.06% the methods 0.56% but pairs. tighter, 0.40% currency across be consistent will fairly is 7.17, pairs Bid–offer curve Exhibit spread in currency spread. 0.28% shown bid–offer liquid those premium than most wider bid–offer be the 0.20% will tightest 0.5% and pairs currency the liquid has less 0.16% in 0.45% therefore spreads it and 0.4% 0.10% contract 0.08% 0.06% 0.35% 0.3% 0.02% 0.3% 5yr 0.3% 4yr 0.3% 3yr 0.3% 2yr 0.6% 1yr 1% 6mth 3% 3mth 2mth 1mth 2wk 1wk Volatility ATM O/N Tenor y 0dlavnlaoto ih ae03%vltlt pedadhne0.06% hence and spread volatility 0.35% have might option spread. premium vanilla delta 10 1yr a XII 7.17 EXHIBIT npatc,frsrksaa rmteAMi iudcrec ar,tebid–offer the pairs, currency liquid in ATM the from away strikes for practice, In same the had tenor 1yr the at strikes all If spread volatility bid–offer ATM 0.25% a so vega 0.40% has option ATM 1yr A possible two consider maturity, given a at ATM the from away strikes For rmu i–fe spread bid–offer premium Spread tnadAMBdOfrVltlt ped n rmu Spreads Premium and Spreads Volatility Bid–Offer ATM Standard igsrkswudhv ie i–fe volatility bid–offer wider a have would strikes wing , CY%2d.p.) 2 (CCY1% Vega oaiiybdofrspread bid–offer volatility ped(C1 d.p.) 2 (CCY1% Spread Premium ATM ,wingstrikes VANILLA FX DERIVATIVES PRICING 115 Volatility spread versus put strike delta Vega versus put strike delta EXHIBIT 7.19 EXHIBIT 7.18 116 VANILLA FX DERIVATIVES PRICING ■ ail rc Making Price Vanilla h eal i ntecnrc s64%adtesse w-a rc is price two-way system the and 6.45% is improved. contract a been takes called the is tool This on default the 6.25/6.65%. mid a in default applies shown and The price surface volatility The views desk trader tool. the derivatives pricing spread. from FX bid–offer desk the volatility received, midmarket their is the in request contract price vanilla the new a When Overview on Making Price based live. prices or hedged their delta adjust transacted be traders either can derivatives, contracts plus FX sentiment, vanilla market making price When 7.20 EXHIBIT xml,b eln ea uiggma rbyn pin ihtpiestrikes), topside with making. options price within buying reflected or are gamma, preferences (for these buying position vega, their change selling the to aims by is trader a example, adjustment If position. the trading current within the to factors reference main the of One preference quote. to happy are h ako h rdri hnt ofo h eal ytmrt oart they rate a to rate system default the from go to then is trader the of task The GBP20m. in call GBP 1.6800 2mth GBP/USD a requests client a example, For ftetae.A ecie nCatr3 rc aigi efre with performed is making price 3, Chapter in described As trader. the of rmu pedvru u tiedelta strike put versus spread Premium eta rate neutral eas ete h i o h fe has offer the nor bid the neither because position VANILLA FX DERIVATIVES PRICING 117 . market sentiment (i.e., the bid shown is 12.6/12.65%—before at the time. If there is then … to 1mth ATM (i.e., a contract (i.e., the offer shown is equal to bid on midmarket bid similar 12.6/12.7% … involves market-wide preferences to, for is the current market preference for various midmarket offer appropriate quoted rate might be 8.1/8.25%. These effects usually apply over and abovesurface the default and modeling therefore of need the to volatility be adjusted for within price making. types of contract—that is,prefers is to buy), the or market offered (the currentlyshort-dates? market back-dates? neutral, prefers skew? to bid wings? sell) For (the for example, specific7.5/7.7% suppose market event but the dates? it 1mth last ATM traded is ata 7.6% new and client was price requestwith in a something similar expiry datebe and a similar reasonable hedge), strike it forIf is which appropriate the the to 1mth default show ATM a system would better bid two-way on rate the contract. on the new contract is 8.0/8.25%, an example, buy very low-premiumsmile options or buy on options the near high the side ATM of on the the volatility low side of the volatility smile. Market sentiment should be reflected within price making because it makes The trading level and the speed of transaction are linked and they give information Market sentiment is most easily judged by observing the interbank broker market. Structural market sentiment Temporary market sentiment The other important factor within vanilla option price making is If traders need to transact quickly,from they midmarket. will usually need to cross a larger spread If traders can wait to transactwork and slowly want toward to a deal at midmarket the trading best level. possible level, they will risk easier to recycleOccasionally and prices will enables be better quoted (tighter) withequal either to prices a the to system midmarket) be orthe quoted a system for midmarket). clients. ■ At the extreme, if thesignal,andiftheopeningofferispaidstraightaway,thatconstitutesastrongbuysignal. opening bid is hit straightaway, that constitutes a strong sell around 30 minutes but it can take anything from a fewabout minutes the to current a market few sentiment: hours. ■ Traders learn how theworks. normal For transaction example, processrelatively the for wide—12.45/12.75%—and implied different over volatility vanilla timeand rate contracts it on offers will tighten an are upeventually ATM as shown—12.55/12.7% trading contract better somewhere bids will within start the original rate. This process usually takes There are two types of market sentiment: structural and temporary: 118 VANILLA FX DERIVATIVES PRICING ti motn httaesko h aktcnetosi h urnypisthey pairs currency the in conventions market the know traders that important is It Conventions Market (i.e., has live 2 transact Leg to hedge. going the is forward if hedge). counterparty a delta appropriate the a with data; if without hedged market appropriate vanilla delta data; choiced same market transact has the spread 1 to shows Leg going 7.21 tool: is Exhibit pricing counterparty spread. a bid–offer in premium forward twice total the option the from of come amount can substantial spread a since the important in particularly crossed be must hedged. spread delta that is since trade rates, the volatility when forward implied market and two-way spot a only two-way not also using the but calculated is when premium delta two-way the the case hedges then and moves machine spot a as (increasingly) trades. price or client maturity salesperson the until a updating isolation Either in with trade manner. deals future a this hold hedge in to to transact plan want who typically who clients for institutional clients neutral or Corporate delta flows package position. FX the forward trader’s make or to derivatives spot order FX appropriate in market the the the live, in trades transacted be client must a deal When hedge). delta a was without deal the which at manner. rate volatility this FX same trade in the who transact clients at typically institutional delta, and transacted option market is broker and interbank hedge The premium forward priced. option or calculate spot to a used then is formula Black-Scholes conditions. The market normal in the minutes stable few stay reduces a can option this to contract up given the so perhaps a changes, longer, rounded for for quoted forward) usually volatility and the implied is maturity the therefore premium However, fixed impact). (and the a (although rate with changes spot contract premium vanilla the a time makes For this each because easier. strike, terms process volatility making in price prices the quote typically in traders quoted derivatives either FX are requests price derivative FX Vanilla Live or Hedged Delta Transacting rd.Freape ifrn T otat r rddi ifrn urnypairs: currency different ■ in traded are contracts ATM different example, For trade. ie,acl n u ihtesm auiyadsrk;AMsrdlsare straddles ATM strike; and maturity same the 8). Chapter with in put detail more straddle a a in as covered and traded is call contract ATM a the pairs, (i.e., currency EM) some (and G10 all In o ogdtdotos sn h orc w-a owr ihnpiigis pricing within forward two-way correct the using options, long-dated For hedged be must delta the live, trades client the If traded are terms premium in quoted are that requests price Vanilla traded are terms volatility in quoted are that requests price Vanilla as h eli rnatd nthis In transacted. is deal the volatility or et hedged. delta premium live terms. (i.e., VANILLA FX DERIVATIVES PRICING 119 ). Therefore, if a vanilla option has a strike lower absolute delta Traded vanilla option contracts Vanilla option pricing with and without delta hedge This is important because, although the Greeks on a delta hedged call and a delta When trading vanilla options with strikes away from the ATM, the market option with the strike equal to current spot. In other EM pairs, theforward) ATM vanilla contract option with is the traded strike equal as to a the single forward,The plus ATMF a final (at-the-money- forward hedge. ATM contract is ATMS (at-the-money-spot), which is a single vanilla EXHIBIT 7.22 EXHIBIT 7.21 the ATM, it will bein traded Exhibit as 7.22. a CCY1 put/CCY2 call. Traded contracts are shown hedged put are the same (due toa put–call smaller parity), premium the and out-of-the-money a direction smaller expected has payoff at maturity and hence has less credit convention is to always tradewhichever the out-of-the-money has side the (i.e., tradeabove the the call or ATM, put, it will be traded as a CCY1 call/CCY2 put. If the strike is below ■ ■ 120 VANILLA FX DERIVATIVES PRICING ocp ndrvtvspiig ti laseult udet foepretof percent one of hundredth a to equal always is notional. it the pricing; derivatives in frequently. nearest a concept less the (0.005%) to updated half rounded be or usually (0.0025%) are can prices quarter terms clean terms, premium point volatility making CCY1% inflection price in in quoting volatility quoted the When keeps prices with rounding that tenors, This ensures longer tenor. and in 2mth 0.025% the around and usually tenors shorter in 0.05% for important used particularly rates is interest this the options. and in-the-money that risk premium this vital credit high within is counterparty covered it the isn’t contracts reflect it derivatives correctly badly Although FX earn crisis. trading to financial when mechanism 2008 book, traded? a the be as during options side cash deep-in-the-money conventional needed sell the to wouldn’t tried Why banks clients: Some in-the-money by when requested careful be are should options Traders direction. in-the-money the than risk hnqoigi oaiiytrs rcsaeuulyruddt h nearest the to rounded usually are prices terms, volatility in quoting When ai point basis 00%.Absspiti key a is point basis A (0.01%). CHAPTER 8

Vanilla FX Derivatives Structures

121 anilla options can be combined to create different payoffs. Some of these Vcombinations are so common that the resultant structures have standardized names that are requested by clients or quoted in the interbank broker market. For an FX derivatives trader, it is most important to understand how these combinations of long and short vanilla options impact the exposures in the trading position. Within this chapter, vega is the primary focus. If structures are traded to shorter maturities, the most important exposure will be gamma. As observed in Chapter 6; for vanilla options, gamma profiles and vega profiles have similar shapes but they evolve differently over time.

■ Straddle

A straddle contains two vanilla options with identical contract details (same currency pair, buy/sell direction, notional, expiry, strike, and cut) except that one is a call and the other is a put. Exhibit 8.1 shows the value at maturity of a long USD/JPY 100.00 straddle in notional N per leg:

■ Leg 1: Buy USD call/JPY put with strike 100.00 in notional N.

■ Leg 2: Buy USD put/JPY call with strike 100.00 in notional N. 122 VANILLA FX DERIVATIVES STRUCTURES u hyaenteatytesm.Rcl rmCatr5ta h owr sderived is forward the that 5 Chapter from Recall same. the maturity exactly same not the are for forward they the but to close positioned is strike straddle zero-delta The Placement Strike Straddle means Zero-Delta actually zero-delta straddle. EUR/USD a a as forms in which traded contract put, is and ATM contract call 1mth a ATM transacting a the buying pairs, currency Therefore, EM straddle. some that and such set G10 are strike all contracts its straddle with traded straddle commonly a most the far By Straddles if ATM legs. Zero-Delta and, both made on is premiums the with quote as determine volatility notional to same two-way combined used single the is the A volatility is in that cut. option straddle dealt, and vanilla a strike, single expiry, for a same price for the volatility price a volatility a making making parity, put–call to Thanks Making Price Straddle 8.1 EXHIBIT rmso n neetrates: interest and spot from au tmtrt fln 0.0straddle 100.00 long of maturity at Value F T = Se Δ ( rCCY call 2 − =−Δ rCCY 1 ) . T put n therefore and eodlastraddles zero-delta Δ straddle = .In 0. : VANILLA FX DERIVATIVES STRUCTURES 123 in T 1and rCCY T . ) 2 𝜎 1 2 + is volatility. 1 𝜎 T . ) rCCY − S ] K ) 2 2 1 ] ( 1 𝜎 is the spot, and rCCY 1 2 ln S ( )− S . Therefore: K 1 )− 0 Se than the forward due to the + ) 1 d ] 1 put ( d 1 1 = =− = = d ( ( N [ T N K T . T N [ . T . . ) )− T T 2 . 1 rCCY higher . 1 T ) 1 ¯ 𝜎 1 o correction within the Black-Scholes ) =−Δ d 2 1 2 2 − ( √ rCCY 𝜎 + 𝜎 rCCY rCCY 1 N 2 − call 𝜎 1 2 1 2 − e − e e Δ 1 2 rCCY + + ), − = = rCCY 1 2 1 K ( )=− )=−[ )= (measured in years), S call put S 1 1 1 rCCY P P + 𝜕 d d d 𝜕 ( T rCCY rCCY ( ( ( 𝜕 𝜕 e T ) N N N − − S √ T K = = . 2 1 2 𝜎 ( put call 0. So ln rCCY Δ Δ − rCCY rCCY e = = ( ( 1 1 , which is linked to the It d d + T 2 ) S 𝜎 K 1 2 ( , then is the cumulative normal distribution function and + 1 2 ln ) is the forward to time x ( T N F )= 1 2 are continuously compounded interest rates (see Chapter 10) to time d ( Unfortunately, this is not quite the end of the story. The above is standard Black- Recalling the shape of cumulative normal distribution function from Chapter 5, At the zero-delta straddle strike ( Dipping briefly into Black-Scholes mathematics: N framework (see Chapter 5). Scholes mathematics, which assumes the premium is paid in CCY2. In, for example, Therefore, the zero-deltaadjustment straddle strike is if where where rCCY CCY1 and CCY2 respectively. 124 VANILLA FX DERIVATIVES STRUCTURES h eapa cusa h strike. a the of expected, at profile As occurs vega maturity. peak and the vega strike, to notional, the identical (combined) is same It the straddle. with a vanilla of single profile vega the shows 8.2 Exhibit Exposures Trading Straddle larger. get expiry to time and volatility as increases strike straddle adjustment: the is to strike straddle due zero-delta forward the the CCY1, than in paid is premium the when Therefore, as: through works which delta): CCY1. (USD in premium USD/JPY pay example, to for is in, convention but market case; the the premium), is this premium), (USD EUR/USD XII 8.2 EXHIBIT nbt ae,tedfeec ewe h owr n h T zero-delta ATM the and forward the between difference the cases, both In adjusted premium on details more for 14 Chapter (see pairs premium CCY1 In eapol fln 0.0straddle 100.00 long of profile Vega Δ Δ call put = = 𝜕 𝜕 𝜕 𝜕 K P P S put call S = = = Se ( e e − rCCY − rCCY rCCY − 2 − 1 1 . rCCY 2 1 . T T 𝜎 [ N N 2 1 ( T − ( d d . 1 1 2 1 )− 𝜎 )− 2 ) . T P 1 call S ]− P S put lower VANILLA FX DERIVATIVES STRUCTURES 125 .Both )25%. − ( = . . N different strikes N 2 𝜐 2 2 𝜎 𝜐 25% and put delta + + two-way volatility is quoted. If dealt, 1 1 = 𝜐 𝜐 has (approximate) implied volatility: 1 2 𝜎 single K = and 1 strangle K 𝜎 Value at maturity of long 90.00/110.00 strangle per leg: N A strangle containing strikes Strangles are often quoted for a given delta. For example, a 25 delta strangle is Exhibit 8.3 shows the value at maturity of a long USD/JPY 90/110 strangle in Leg 2: Buy USD put/JPY call with strike 90.00 in notional Leg 1: Buy USD call/JPY put with strike 110.00 in notional EXHIBIT 8.3 Strangle Price Making When making a price onthat volatility a is strangle, used to a determinequoted the on premiums a on strangle both in legs. volatility The termsthe bid–offer same will maturity spread usually because be strikes wider away than from the the ATM ATM have spread less to vega (see Chapter 7). ■ constructed with strikes such that call delta strikes are placed out-of-the-moneythan and the put therefore strike. the call strike is alwaysnotional higher ■ A strangle is like a straddle except that the call and put have Strangle ■ 126 VANILLA FX DERIVATIVES STRUCTURES ■ utry(Fly) Butterfly ■ strangle: a and straddle ■ a of combination a is contract butterfly The in shown as strikes two the vega from the peaks deltas distinct lower two At wider the ATM. has is 8.4. with the profile Exhibit strangle straddle from vega the a away strangle of out of the spread profile profile that are vega except strikes the leg the to per because similar notional is and strangle maturity a same of profile vega The Exposures Trading Strangle the to close be volatilities. will strike volatility individual strangle two the the vega, equal roughly with strikes volatility strike where XII 8.4 EXHIBIT hr butterfly Short butterfly Long 𝜎 2 n vega and K eapol fln 00/1.0strangle 90.00/110.00 long of profile Vega 1 a mle volatility implied has = = ogstrangle long hr strangle short υ 2 hrfr,i h tagecnan qa et aladput and call delta equal contains strangle the if Therefore, . + + hr straddle short ogstraddle long 𝜎 1 n vega and υ 1 hl strike while K 2 a implied has average of VANILLA FX DERIVATIVES STRUCTURES 127 . . . . N N N N the strikes for a given delta are contract is the most commonly but broker fly Value at maturity of long 90.00/100.00/110.00 equal notional butterfly butterfly: In the interbank broker market, the Leg 2: Sell USD put/JPY call with strikeLeg 100.00 3: in Sell notional USD call/JPY put with strikeLeg 100.00 4: in Buy notional USD call/JPY put with strike 110.00 in notional Leg 1: Buy USD put/JPY call with strike 90.00 in notional generated using a differentmore calculation. detail The in broker Chapter 12. fly product is examined in far squint. traded butterfly contract. Anotional broker on fly the has ATM equal straddleBroker notional fly is contracts set strangle are such strikes quoted that and for the a the package given delta is initially vega-neutral. ■ ■ ■ Legs 1 and 4 form a longis strangle and called legs a 2 and butterfly 3 because form a its short value straddle. The at contract maturity looks like a butterfly if you really Buying the butterfly means buying the strangle, or putExhibit another 8.5 way, buying shows the wings. the valueequal at notional maturity of a long USD/JPY 90.00/100.00/110.00 ■ EXHIBIT 8.5 128 VANILLA FX DERIVATIVES STRUCTURES ■ ikRvra (RR) Reversal Risk te sapt lsoei ogtwieteohri od h otati aldarisk a called is strikes. contract two The the the sold. and between is call other optionality a the transfers is while it bought one because is reversal strikes, one different notional, plus have put, pair, a legs currency is two same other the the However, with cut. options and vanilla expiry, two contains reversal risk A 8.6. Exhibit the in in shown vega as long strikes and strangle construction by long vega the ATM from flat wings often is position butterfly long A Exposures Trading shown. Butterfly be often will spread bid–offer bid–offer A strangle isolation. the in half component than strangle less the spread than tighter significantly quoted is is contract the Plus volatility strikes. the ATM as the the that quoted and such are quoted strikes contracts strangle fly the broker between on differential prices market, interbank the In Making Price Butterfly XII 8.6 EXHIBIT h rkrfl a eovg n iia am ycntuto.Teeoe it Therefore, construction. by gamma minimal and vega zero has fly broker The eapol fln 00/0.0100 eanurlbutterfly vega-neutral 90.00/100.00/110.00 long of profile Vega utrynotional butterfly sthe is tagenotional strangle . VANILLA FX DERIVATIVES STRUCTURES 129 price ,which choice . . N N zero-premium collar of the single leg notional of spot must be sold. % 50 = per leg: 2 N contract. It has two legs: one bought, one sold. There are %× spread Value at maturity of 90.00/110.00 risk reversal (buying topside) Corporates often call risk reversals ‘‘collars’’ when they are used as instruments Risk reversals are usually quoted to a specific delta. They are constructed using Exhibit 8.7 shows the value at maturity of a USD/JPY 90/110 risk reversal Leg 1: Buy USD call/JPY put with strikeLeg 110.00 2: in Sell notional USD put/JPY call with strike 90.00 in notional A risk reversal is a offsetting gamma and vega exposures betweenquoted tighter the than legs the and equivalent contract therefore where thelegs either both product are legs is sold. are When bought or quoting both aprices price on on both spread, legs, rather the than market quoting convention tighter is two-way to quote one leg with a to hedge future cash flows. Acaps popular and structure floors is the a P&L from an FX exposure as demonstrated inRisk Exhibit Reversal 8.8. Price Making a long call and athe short put two (or legs short therefore call compoundnegative). and To (i.e., long calculate put) the the delta and delta hedge the on on thesummed. delta risk For both exposures reversal, example, the legs from to delta is of delta the hedge either twoand a legs a positive 25d is short or 25d risk put, reversal, 25 containing a long 25d call (buying topside) in notional ■ ■ EXHIBIT 8.7 130 VANILLA FX DERIVATIVES STRUCTURES seaie nfrmr eali hpe 12. Chapter in detail product more reversal far risk volatility The in the calculated. examined determines be is can that premiums and both agreed hence is strike; other strikes the the of of is one there on hence volatility and the terms, delta same the vega have net strikes This minimal reversals example). an risk USD/BRL the above as when the quoted appropriate in is often 3.35/3.85% (e.g., are strikes delta two the given between a for reversals Risk ■ ■ ■ strangle. equivalent quoted the usually of are reversals spread Risk bid–offer legs. overall the the of half one and around onto CH) with put denoted is often spread sell, bid–offer or the buy all to a possible with is leg it other which the at volatility single a (i.e., 8.8 EXHIBIT ttesotsrk.Ti ssoni xii 8.9. Exhibit peak (negative) in a shown with is the exposure This at vega strike. peak short short a a the with generates at exposure leg vega sold long the a while generates strike reversal long risk the in leg bought The Exposures Trading Reversal Risk y 5 ikrvra mle oaiiy 43 48%v.1.%CH 11.0% vs. 14.85% / 14.35 volatility: implied reversal risk 25d 11.25% 1yr / 10.75 volatility: implied put 25d 1yr 14.85% Outright / 14.35 volatility: implied call 25d 1yr Outright o xml,UDBL1r2drs reversal: risk 25d 1yr USD/BRL example, For egn nF xouewt olr(ikreversal) (risk collar a with exposure FX an Hedging ntesrcue netetaei gedi oaiiydifferential volatility in agreed is trade the Once structure. the on spread rc ie,atowyvltlt rc) nohrwords, other In price). volatility two-way a (i.e., price mle oaiiydifferential volatility implied VANILLA FX DERIVATIVES STRUCTURES 131 . . N N Vega profile of a 90.00/110.00 risk reversal This product is popular with clients hedging FX flows because the underlying can Leveraged forwards can be decomposed into a forward in the matched notional A leveraged forward is an extension of a synthetic forward in which the two Leg 2: Sell USD put/JPY call with strike 100.00 in notional 2 Leg 1: Buy USD call/JPY put with strike 100.00 in notional volatility price will also beunmatched notional. equal to the volatility price of the vanillabe option transacted in at a the better ratefunded than by the the increased forward notional for on zero the premium. sell The leg. better rate is ■ plus a vanilla invega the trading unmatched risk notional. is Since equal forwards to have a no optionality, vanilla the in the unmatched notional and the implied notional amounts are no longer equal. A 200%is leveraged forward bought in which at USD/JPY 100.00 issold leg: constructed using two legs with double■ the notional on the Recall from Chapter 6call that and a short synthetic vanilla forward put with is all constructed other using contract details a the long same. vanilla EXHIBIT 8.9 Leveraged Forward ■ 132 VANILLA FX DERIVATIVES STRUCTURES ■ ■ alPtSpreads Call/Put Spread Calendar ATM oiinwl aemnyi h mhAMvltlt ie oeta h 6mth the than more rises volatility ATM This 3mth spread. the calendar if volatility. ATM bucketed ATM money vega-neutral the they make 3mth/6mth shows since will 8.10 long curve position Exhibit the ATM tenors. from the two profile of in vega shape positions the vega to offsetting exposures contain give spreads calendar ATM Exposures Trading Spread Calendar ATM same with the leg, in far the contracts on choice. ATM shown quoted be two leg will near than spread the bid–offer tighter vega standard quoted the since offsetting Perhaps therefore substantial spread direction: are have is choice. and usually straddle shown risk spreads ATM is Calendar vega straddle far maturities. ATM longer the at other spreads increases the calendar on and shown ATM vega is for price more Therefore, (two-way) with spread straddle A ATM apply: the rules pricing spread vanilla Standard straddle. ATM Making long Price a Spread form Calendar 4 and ATM 3 legs and straddle ATM short a form 2 and 1 Legs ■ ■ ■ straddle: ■ a as traded is ATM ATM the vega-neutral assuming a legs, becomes it structure, the on vega net spread. zero calendar is there that and such date’’) ‘‘far language, trading old-school and In bought maturities. one called different are contracts, to spreads ATM spread), calendar a two (hence of sold combinations one are spreads calendar ATM h aemtrt,ete ohclso ohpt.Oelgi u n h te sa is other the and buy a is leg One to puts. and notional both same or the calls in both legs either two maturity, have same spreads the put vanilla and spreads call Vanilla e :By6t T S u/P ali notional in call put/JPY USD ATM 6mth Buy notional 4: in Leg put call/JPY USD ATM 6mth Buy notional 3: in Leg call put/JPY USD ATM 3mth Sell notional 2: in Leg put call/JPY USD ATM 3mth Sell 1: Leg og3t/mheulntoa T aedrsra scntutduigfour using constructed is spread calendar ATM notional equal 3mth/6mth long A Buying h aedrsra means spread calendar the selling h hre ae(h ‘erdt’) ftentoasaeset are notionals the If date’’). ‘‘near (the date shorter the oiotlspreads horizontal buying . h ogrdt te‘bc ae’or date’’ ‘‘back (the date longer the N N N N . . . . VANILLA FX DERIVATIVES STRUCTURES 133 . . N N per leg: N Value at maturity of long 100.00/110.00 call spread Bucketed vega profile from long 3mth/6mth vega-neutral ATM calendar spread Buying call/put spreads is popular with institutional clients because they enable Exhibit 8.11 shows the value at maturity of a long USD/JPY 100.00/110.00 Leg 1: Buy USD call/JPY put with strikeLeg 100.00 2: in Sell notional USD call/JPY put with strike 110.00 in notional ■ ■ a directional view with no possible downside other than the initial premium at sell; hence the structure isleg a is spread. always more When expensive, buying and the hence call/put further spread, in-the-money the than the bought USD sold call leg. spread in notional EXHIBIT 8.11 EXHIBIT 8.10 134 VANILLA FX DERIVATIVES STRUCTURES ■ Seagull ti ototnapoutsl o eoiiilpeimt netr h atto want who investors to premium initial zero for exposure. FX sold underlying product an spread a hedge call/put often long most a is is It contract seagull long A same the generates with strike reversal short the risk and exposure. a exposure vega vega short note from vega long a the to exposures a generates portfolio, interesting vega strike trading the long is hedged the as It strikes; delta same leg. a the other in are the placed exposures are choice spreads too: and spreads call/put vega put once more and that spreads with call leg to the apply rules Spread quoting spread vanilla standard The Making Price Spread Call/Put ■ ■ call/put language, trading old-school In vanilla. outright called are the spreads than cost cheaper a XII 8.12 EXHIBIT ogCY u pedpy u fso sblwtehge tiewt h payoff the with strike higher the strike. below lower is the spot at if out capped pays spread put payoff CCY1 the long with A strike lower the above strike. is higher spot the if at out capped pays spread call CCY1 long A flf negdutlmaturity: until unhedged left If au tmtrt fln 00/0.0100 seagull 90.00/100.00/110.00 long of maturity at Value etclspreads vertical . plus nadtoa hr put/call. short additional an VANILLA FX DERIVATIVES STRUCTURES 135 . . . N N N per leg: N Exhibit 8.12 shows the value at maturity of a long USD/JPY 90.00/100.00 Leg 1: Sell USD put/JPY call with strikeLeg 90.00 2: in Buy notional USD call/JPY put with strikeLeg 100.00 3: in Sell notional USD call/JPY put with strike 110.00 in notional Seagull Price Making On vanilla spreads with morequote than prices two but legs generally there the are price feware is easier concrete quoted to rules choice. understand on Plus if how as the to price many legs legs quoted with as in the possible volatility most terms vega will be should the be tightest. spread because their ■ ■ ■ Legs 2 and 3 form a long call spread. /110.00 seagull in notional

CHAPTER 9

Vanilla FX Derivatives Risk Management

137 X derivatives trading portfolios contain different types of deal: vanilla options, Fexotic options, spots, forwards, and so on. To risk manage derivatives positions traders use Greeks; the exposures of the position to market changes. Greeks are calculated on each deal in the portfolio and then aggregated together. For a given market move, some deals in the portfolio will make money or, for example, get longer vega, and others will lose money or get shorter vega: Traders only care about the net impact from all deals. For this reason traders primarily describe their positions in terms of long or short positions in aggregated Greek exposures. For example, an FX derivatives trader may describe their position in a given currency pair as ‘‘flat delta, short topside gamma, and long vega.’’ As markets move, positive or negative P&L is generated from different aggregated exposures within the position. The most important exposures are to spot (delta exposure) and ATM volatility (vega exposure). There are also exposures to CCY1 and CCY2 interest rates (rho exposures), exposures to curve moves in these instruments, plus exposures to the shape of the volatility surface. Finally, exposures are not static: Recall the gamma and vega profiles for vanilla options from Chapter 6; exposures change as the market moves or time passes. 138 VANILLA FX DERIVATIVES RISK MANAGEMENT ■ rdn Gamma Trading hw &,dla n am vrarneo ptvle.Ti lostaesto traders allows This values. spot of range a over gamma and and delta delta, P&L, their shows managing by change changes to P&L P&L these hence control exposures. and gamma Traders options delta These moves. cause spot. spot current that to as exposures close fairly gamma strikes significant with so generate or month into next strike the the within the at concentrates how and recalls the increases 9.1 Therefore, option maturity. Exhibit vanilla a exposures. from gamma exposure vanilla short gamma short give while exposures always gamma positions long option give always positions option vanilla Long ■ and spot to ■ exposures from comes The usually folly. P&L volatility: obvious trading implied is derivative exposures focusing FX first-order though, of neglecting Fundamentally, majority while risk. risks of higher-order types different on these all investigate must ■ ■ ■ ■ together: ■ Greeks of types similar consider often traders analysis, simplify pcns ls fapge rmngdcrec arhssgicn uprisk, jump significant has pair currency managed spot or appropriate the pegged wider a the volatility, if spot the Plus, higher spacings. The particular, pair. currency In the exposures. moves. in delta spot their hedge as to when change determine will to ladders position spot and use traders P&L their how anticipate uv.Teeepsrsmil oefo ogdtdvnlaoptions. vanilla long-dated from come mainly exposures These to curve. P&L causes then options. The vanilla exposure short-dated from delta come resultant mainly exposures The These change. moves. spot as significantly The changes) volatility risk Cross-exposure risk rate Interest risk Smile risk ATM risk Short-date h asbtenso eesi h adrsol eaindwt h ptvolatility spot the with aligned be should ladder the in levels spot between gaps The a is position short-date the viewing of method common One trader a position, trading derivatives vanilla a in risk trading the understand To To straightforward. not is position derivatives FX an trading reasons, these For T position ATM hr-aeposition short-date eawihe eabcee vega vega/bucketed vega/weighted : xoue otesaeo h oaiiysurface volatility the of shape the to exposures : delta/gamma/theta : h/wppoints rho/swap : ossso ail pin htgnrt xoue oteATM the to exposures generate that options vanilla of consists eg,tecag nitrs aepsto hnAMimplied ATM when position rate interest in change the (e.g., hr-aeposition short-date ossso ail pin htcuedlat change to delta cause that options vanilla of consists anycnit fvnlaotosexpiring options vanilla of consists mainly ptladder spot that VANILLA FX DERIVATIVES RISK MANAGEMENT 139 EUR20m 1wk ATM position, long Initial EUR/USD spot ladder with long gamma exposure Gamma profile of long vanilla option with 100.00 strike EXHIBIT 9.2 are generated assuming marketadditional data trades (apart are from executed spot) as remains spot unchanged moves and from no its current level. This is a long appropriate. Trading Long Gamma Exhibit 9.2 shows a EUR/USD spot ladderwith from a current spot highlighted in the middle of the ladder. Values in this spot ladder EXHIBIT 9.1 two ladders—one with tight spacings and one with wide spacings—might be 140 VANILLA FX DERIVATIVES RISK MANAGEMENT 2. 1. ■ ■ for by so, increases spot, delta 1.3130), in to move long 1.3000 1% (from (i.e., higher per derivative move quoted spot approximately second 1% is the a gamma on long positions, example, being trading of In characteristic gamma). is This direction. initial so construction by spot neutral delta current zero. is at is contract is delta The gamma contract. ATM Peak an gamma. is long it therefore because is and position option vanilla XII 9.3 EXHIBIT oe,ago hn)a eni h ptladder. spot the in seen as thing) good a delta money, long and higher Spot rises. spot gamma long and higher Spot h rdrnwhsadcso omake: to decision a has now trader The 9.3: Exhibit in shown as 1.3065, to 1.3000 from rises spot EUR/USD Assume either in moves spot further the faster occurs change P&L moves, spot As et oiint cuuaetetae a nicesdepsr otefuture the to exposure allowing increased By an spot. balanced. has not of trader is direction the position accumulate the to hedged, position not delta is a delta positive If the lower, lost. back be retraces will a spot P&L if but for money, more equal even make approximately will delta position ‘‘balanced.’’ now the is is hedge position the not change spot; Do in P&L move the down or shown that up is same-sized Note position the 9.4. 1.3065, start Exhibit at can process spot the in hedging EUR/USD (offset) delta EUR2.8m the selling hedge and After will flat again. approximately spot to EUR/USD back position EUR2.8m delta Selling profit.’’ ‘‘taking as spot Sell ntemre ordc h ogER.mdlaepsr,known exposure, delta EUR2.8m long the reduce to market the in U/S ptlde ihso higher spot with ladder spot EUR/USD + U56 cretgamma). (current EUR5.6m ie,lttepsto u) fso otne ihr the higher, continues spot If run). position the let (i.e., ( ( 𝜕 𝜕 𝜕 𝜕 price spot delta spot ) ) produces ed oa nraigln et oiinas position delta long increasing an to leads oiiePLchange P&L positive ie,making (i.e., VANILLA FX DERIVATIVES RISK MANAGEMENT 141 again, as seen in positive P&L change leads to an increasing short delta position as produces ) ) spot delta 𝜕 spot 𝜕 price 𝜕 𝜕 ( ( EUR/USD spot ladder with spot lower EUR/USD spot ladder after rebalancing delta in the market to reduce the delta exposure. Again, this is taking profit but the trader does not haveafter to buying completely back hedge just the EUR2m delta EUR/USD exposure. spot For at example, 1.2935, the position is shown Buy spot Once again, spot has moved, a delta position results from the long gamma, and Now, from the same starting point, assume EUR/USD spot falls from 1.3000 to the spot ladder. Spot lower and long gamma spot falls. Spot lower and short delta EXHIBIT 9.5 the trader has two choices: 1. 1.2935, as shown in Exhibit 9.5: ■ ■ EXHIBIT 9.4 142 VANILLA FX DERIVATIVES RISK MANAGEMENT hngmaepsr.Wt pta .01(bv h tie,dlaepsr is exposure delta strike), the (above 1.3001 at spot With exposure. gamma than viewed a is as horizon viewed the is on expire that options position trading strike. the the shows around 9.7 Exhibit balanced Then, contract, delta hedging. ATM the delta long with the frequent from of gamma more date the to expiry pass, lead the days will on as which strike increases, the option to vanilla close the stays spot if example, done, option if single 1.3000, at EUR5m buy done, if 1.3130). 1.3130, position, at EUR5m at gamma sell EUR5m long sell a (e.g., as trade used known to often process market a the in in day goes orders the it of when higher. course spot goes the it buying over when to spot leads selling naturally and exposure lower gamma long a from resulting up either moves 2. 9.6 EXHIBIT otaiggma xetthat except gamma, trading to the change, delta EUR20m a + U1m ihso t129 blwtesrk) et xouei –EUR10m: is exposure delta strike), the (below 1.2999 at spot With EUR10m. rdn iki iwda faseichrzndt uulytdy n h ikon risk the and today) (usually date horizon specific a of as viewed is risk Trading this In frequently. more hedged delta are exposures gamma larger with Positions spot if change P&L positive a causes gamma long balanced, initially is delta the If oiinwl aeee oemny u fso ercsbc ihr the higher, back retraces spot if but lost. money, be will more P&L not even positive is make position will the delta position amount, the was. delta hedge it full than not balanced the Do more back is it buying but balanced, not completely By 9.6. Exhibit in U/S ptlde fe atal eaacn delta rebalancing partially after ladder spot EUR/USD strike or ;an on uts oga tmvssmwee egn h delta the Hedging somewhere. moves it as long so just down, ntnaeu et jump delta instantaneous ie,lttepsto u) fso otne oe,the lower, continues spot If run). position the let (i.e., notional all h et hneocr ttestrike. the at occurs change delta the fteoto.Taigti oiini similar is position this Trading option. the of uiglow Buying tepr time expiry at qa oteoto oinlrather notional option the to equal rdn h gamma the trading and opdtake-profit looped eln high selling hrfr,teoption the Therefore, . hnleaving When . ok nP&L in locks resare orders VANILLA FX DERIVATIVES RISK MANAGEMENT 143 time of holding the cost , also known as ) time price 𝜕 𝜕 ( theta . For a long gamma position, theta is the position: decay EUR/USD spot ladder at maturity long strike or just If realized volatility is smaller than implied volatility there will be fewer If realized volatility is larger than implied volatility there will be more opportu- Trading a long gamma position looks easy—buy vanilla options to get long In general, P&L volatility from a single vanilla option increases as time to expiry For a If spot goes above thelong strike, delta the exposure. trader can sell spotIf in spot the goes market below to the hedgeshort strike, the delta the exposure. trader can buy spot in the market to hedge the opportunities to delta hedge,long which gamma usually exposure results than is in paid smaller in P&L theta, hence from generating trading negative a P&L overall. decay trading position from one trading day to the next. nities to delta hedge, which usuallyexposure results than in is larger P&L paid from in trading theta, a hence long generating gamma positive P&L overall. from trading the gamma prior to expiry. gamma, and if spot moves higher ormeans lower, spot you make can money. Long be gamma naturally boughtcourse, low there and is sold a high, cost hence for this generating and positive that P&L. cost Of is Once the delta position is fullyuntil hedged spot back goes to back zero, through it the is strike. not possible to hedge again gets shorter if spot remainsP&L volatility near from the trading the strike. strike Plus, at expiry for tends a to be given larger ATM than P&L vanilla volatility option, ■ ■ EXHIBIT 9.7 144 VANILLA FX DERIVATIVES RISK MANAGEMENT ■ ■ ■ ■ to how and factors: ■ when different on on based decisions are though, exposure practice, delta In the trade Reading). Further (see ‘‘ this book a on Sinclair Euan of The size. frequency spread hedge bid–offer delta option optimal Therefore, the volatility. FX day. the to throughout are 5 frequently derivatives hedged exposures Chapter FX delta An and are from derivation). positions skill, (recall formula focus, Black-Scholes volatility the main realized of trader’s and essence the volatility is implied this that of would P&L function practice), in a possible Alternatively, (not be etc.). continuously checking, performed were limit hedging generation, delta is P&L if between data for difference market used the (end-of-day reference of spot official function end-of-day the a today’s P&L be and no will spot assuming P&L end-of-day (and daily yesterday’s day the trading source), the other throughout any performed from is hedging delta no If Practice in Hedging Delta odtosi sdbtbehwmc au rdrcnad u hnunexpected market when but stable add, In can trader higher. a 8% value much over how central spot debatable Swiss is sent the it before and conditions shortly intervened at spot is friend unexpectedly EUR/CHF spot A ‘‘trading selling bank face) this. and straight doing Mozart’’ a by (with playing value proclaiming like add boss they their that reported bank believe another they and day the throughout na tep ognrt agrPL,bttwr h pinepr hy‘‘scalp’’ they expiry aggressively. option more the delta) toward but the P&Ls, (trade larger generate to attempt an in derivative. second positive day their of of Time advantage traders take released, to is further data run economic spot important let when often jumps spot if But moves. spot jumps and missed. volatility not spot are Expected market hedge delta spot to the zone, opportunities time in that book-runner’s bid–offer orders ensures main spot orders leave the leaving If outside to Plus, so. sense directly. do more transacting they makes than rather time often each it spread wide, are a spreads cross must they because hedge spreads bid–offer Spot oeagesvl,weesi pti rnigsrnl,te a e h et run delta the let may hedging. they strongly, before trending further is spot if whereas aggressively, more moving hedging. before is accumulate deltas spot larger How let or may positioned traders be and will further barriers run to where levels) spot levels spot spot high rounded and (i.e., low market recent spot the in levels Key aydrvtvstaeslv rdn ptt aaeterdlaexposures delta their manage to spot trading love traders derivatives Many determining in factor primary the walk, random mathematical a followed spot If ttesato h rdn a,taestn oltdla u further run deltas let to tend traders day, trading the of start the At . fso swipn paddw,taeswl et hedge delta will traders down, and up whipping is spot If . h ie h ped,tels fe rdr iet delta to like traders often less the spreads, the wider The . fso rastruhkylvl,ti fe causes often this levels, key through breaks spot If . uigqitrtms rdr a eg smaller hedge may traders times, quieter During . oggamma long oaiiyTrading Volatility oiinwudb h spot the be would position ’hsago section good a has ’’ VANILLA FX DERIVATIVES RISK MANAGEMENT 145 . . EUR20m 1wk ATM position. short negative P&L change negative P&L change theta but loses money throughout the trading day earns the delta (i.e., buy high/sell low) or let it run and potentially Initial EUR/USD spot ladder with short gamma stop-loss A short gamma position If spot moves in either direction, the delta change from a short gamma position P&L change, delta, and gamma are all equal and opposite (negative) to the long Spot lower and short gamma leads to an increasing longSpot delta lower position and as spot long falls. delta again produces Spot higher andspot short rises. gamma leads toSpot higher an and increasing short delta short produces delta position as EXHIBIT 9.8 as spot moves. Again,generated if is primarily the a position function of is implied delta volatility and hedged realized frequently, volatility, but the this overall P&L lose even more at an increasingtrading rate short due gamma to the is shortearned often second will made derivative. be In easier lost practice, through byavoid stopping accepting large out. negative that Trading P&Ls. at with close least stop-loss half orders the helps theta ■ ■ produces a negative P&Lwhether change. to The decision for the trader therefore becomes ■ ■ while: Trading Short Gamma Exhibit 9.8 shows a EUR/USD spot ladder from a ATM position: events occur, the abilitycontext of can a be human an advantage. trader to rapidly process new information in 146 VANILLA FX DERIVATIVES RISK MANAGEMENT ■ rdn h hr-aePosition Short-Date the Trading aigi airt iulz h oiinn fsrks hsi hw nEhbt9.12. Exhibit in shown is This strikes. of positioning the visualize to easier it making 9.11. Exhibit in shown as portfolio higher. topography spot strike with longer jumps it delta Moreover, the it. since through strike jump long delta spot a a vanilla be as causing a must position (i.e., ‘‘strike’’ longer the a profiles in jumps be today) Gamma must delta expiring 9.10. there option Exhibit Therefore, the smooth. in yet are highlighted options as topside, vanilla 0.9205 from the to 0.9177 to from gamma higher goes short is position The ■ ■ ■ ■ is position the vanillas. topside maturities short shorter and at vanillas that downside implies long profile net gamma This topside. the trading to derivatives FX AUD/USD an trading from main ladder the spot identify a position. to shows order 9.9 in Exhibit position risks. short-date their investigate Traders below is volatility realized if P&L positive higher volatility. generate implied will position the time XII 9.9 EXHIBIT hr am n pthigher spot and gamma Short lower spot and gamma Short lower spot and gamma Long higher spot and gamma Long tietpgahe ipa rdo xiydtsadsrksi n urnypair, currency one in strikes and dates expiry of grid a display topographies Strike a using checked be can position the in options specific of Details up: tie positions gamma and delta the that is check to thing first The gamma short but downside, the to and spot current at gamma long is position The U/S ptladder spot AUD/USD rd ur sue ortr eal feeyoto nthe in option every of details return to used is query trade A . → → → → et esshorter gets delta et eslonger gets delta et eslonger gets delta et esshorter gets delta rd query trade or VANILLA FX DERIVATIVES RISK MANAGEMENT 147 AUD/USD strike topography AUD/USD spot ladder with delta jump highlighted AUD/USD trade query These views confirm that the position is long AUD10m of 0.9200 strike expiring There may be just one option or multiple long/short options at a particular expiry at NY cut today.requires This particular is attention important from information theto because trader. the the Strikes delta delta are jump closely jumpsstrike at risk they requires. 0.9200 managed This generate. due long The strike larger will the generate notional, negative (paying) the theta, more but attention it a gives date and strike ininformation the plus strike other topography; missing detailsby only drilling (e.g., the down cut into net or a notional given counterparty) expiry is can and displayed. be strike level. This obtained EXHIBIT 9.12 EXHIBIT 9.11 EXHIBIT 9.10 148 VANILLA FX DERIVATIVES RISK MANAGEMENT motn ocnetaeo otoln h & ntoeaesrte hno pure on than rather areas balance. those P&L in down P&L up/spot more are spot the is there controlling it if reached, on be and concentrate P&L conceivably to range, might within important which spot somewhere considered levels wider spot often a at volatility P&Ls over are negative look implied 1% extreme traders to with standard though, 0.75% practice, pairs one-and-a-half around In moves balance. approximately currency spot of In 10%, to move down. Traders close spot same. and the one-day up roughly is a deviations moves at spot down look and usually up equal-sized on change P&L likely most position. are the jumps in P&L risk exotic unexpected by investigated; caused be should discrepancies Any ■ ■ ■ ■ no has trader the assumes course, of This, changes strong moves. P&L spot similar down and generates up position similar-sized the for whether assessing involves balance P&L Balance P&L consider. to areas common cost. reasonable isn’t at liquidity like the do because they like position particularly a don’t get to they However, available position position. a with the up change put traders to executed be should the trades like, doesn’t trader is, position—that the of course the over delta trading expires). by it it (before from day back money make to opportunity the ptsol ebuh.Byn ptcagstedlaepsr n ec adjusts hence 9.14. and Exhibit exposure in shown delta as the spot over changes profile spot P&L Buying the bought. be 8,000/(0.9233 should lower), spot move spot 9.13. equivalent Exhibit (the in 0.9068 highlighted as better it balance to bought be could spot additional hr et n pthigher spot and delta Short lower spot and delta Short lower spot and delta Long higher spot and delta Long oices h & t093 yUDkadhnerdc h & yUDkat USD8k by P&L the reduce hence of and bit USD8k by a 0.9233 at but P&L balanced, the increase roughly To is P&L the position, AUD/USD example the In the that checked be can it direction, spot on opinion no has trader the Assuming up: tie should positions delta and P&L The several are there but differently position short-date a assess will traders Individual they whether is trader a for question key the general, In cost opinion faheigapeeal oiinms etknit con.Vr often Very account. into taken be must position preferable a achieving of nftr ptmvs & aac a ecekdi h ptladder. spot the in checked be can balance P&L moves. spot future on ili eeaeaprofit? a generate it will → → → → & decreases P&L & increases P&L & increases P&L & decreases P&L fteeaeapcso h oiinta the that position the of aspects are there If − 0.9150) like h short-date the = AUD960k VANILLA FX DERIVATIVES RISK MANAGEMENT 149 from the short-date position if the delta is initially from the short-date position if the delta is initially lost made AUD/USD spot ladder with better P&L balance highlighted AUD/USD spot ladder with P&L balance highlighted If a position is mainly short strikes and short gamma, the short option values If a trading position is mainly long strikes and long gamma, the long option In practice, the delta and hence the P&L profile over different spot levels can the maximum that canbalanced be since delta hedging the long gamma position will make moneyreduce back. over timemaximum and that theta can will be be positive. The positive theta is roughly the Theta In trading positions, thetatrading is day to quoted the as next for the all cumulative deals in change the position. invalues value reduce from over time one and theta will be negative. The negative theta is roughly be adjusted by buyingproblem. or The selling spot gamma but profilevolatility full and must the P&L be volatility balance smile. considered is as a well multi-dimensional as impacts from ATM EXHIBIT 9.14 EXHIBIT 9.13 150 VANILLA FX DERIVATIVES RISK MANAGEMENT oreo h a,tesrksbigdatwt e lsradcoe ocretspot the current traders. Over to to closer level. office and middle spot closer from passes current get responsibility with the point dealt some from being at strikes far and the very day, are the of that course strikes exercise to other NY is pairs York. G10 called New in each simply and is time London of if cut cut in them NY time desks common before tell expiry period most to 20-minute the The writer the option. the at and cut contacts the fix, option exercise a each to than of want owner rather they the cut position, a the in against option settle that options For than larger hedging spot Strikes delta the from at loss looking a cause gamma, spot short back will is theta? make earned, the position to theta the enough the move If and spot hedging? ladder will delta paid, from theta the theta and the ladder spot the at looking ■ levels. spot different at ■ changes exposure gamma the that: how Recall shows ladder spot The Gamma so and 14. balances, Chapter cash in funding explored curve, are ATM factors the These down forth. roll decay, smile example, for position. vanilla gamma short-dated existing large, the too reduce is to If volatility transacted moves. P&L be spot hence with should how and options to way, theta exposed another that more decides is Put trader trader volatility. a the gamma P&L negative higher or positive hence more and exposure gamma higher like, you position theta!’’ gamma any trader any pay A have don’t can money. you ‘‘You as cost told, long will once so was position bank gamma another short at the friend hedging delta since balanced xrieo xietecnrc smd ae ntepealn ptlevel. spot to prevailing decision the the on arrives, based time made cut is the contract the as expire Then or option strike. exercise the the with on ‘‘hold’’ beforehand, shortly to established asking be owner will writer option the and owner h estebetter. the less the gamma Short better. the more the gamma Long fso svr ls oapriua tie omncto ewe h option the between communication strike, particular a to close very is spot If each contact desks trading bank day, trading the of start the from practice, In gamma, long is position the If make: to decision a has trader the Fundamentally, theta, generate also can gamma than other factors that here noting worth is It implies theta Higher proportional: are theta and gamma world, Black-Scholes In → → atso omv oeta mle oaiiysget twill, it suggests volatility implied than more move to spot want atso omv esta mle oaiiysget twill, it suggests volatility implied than less move to spot want expiries nms trading most on VANILLA FX DERIVATIVES RISK MANAGEMENT 151 . For example, if an partially exercised So-called ‘‘bad dates’’ are expected to be quiet and therefore spot is expected So-called ‘‘good dates’’ have events (data releases) on them and spot is therefore For options that settle against a fix rather than a cut, if the option is in-the-money It is also possible for an option to be After expiries, if there were close strikes, the delta position may need to be This process is usually smooth but it can get dangerous when a trading position currency pairs for the next month at least. to move less than usual,New Year’s. for example, days over Easter or between Christmas and expiries on (what traders call)want ‘‘good’’ or to ‘‘bad’’ dates unwind or in offset the these future, positions. a trader might expected to move moreGDP than data usual, is released. for Such example,new events information. days can Traders on cause must which spot therefore to employment know jump or the as big the events market coming adjusts up to in their Gamma/Strike Profile A strike topography can betime. used Strikes to disappear judge from the howto position the produce as gamma more they profile gamma expire will as and they changecan options move judge over ‘‘behind’’ start closer how to their gamma the position horizon. will Based evolve. on Also, this, if a there trader are significant option at expiry, rathergenerates than a generating single a cashfrom spot the payment. FX option disappears In transaction, andtransacted practice, the must off be this cash-settled the replaced same means option with fix an as that FX the option the trade settled, which delta hence is minimizing usually position risk. rebalanced. For example, if astrike, trading position delta is may long USD100m bethe USD/JPY strike. positioned 105.00 After long expiries, when USD50m theremain option with above has no and ‘‘rolled protection off,’’ from the short theto delta strike USD50m flat exposure and by will below hence trading the spot. delta must be hedged back option notional is AUD50m,AUD30m. the When option this owner occurs, couldmust the partially be new exercise established delta as it exposure quickly inis within as requested only possible. the on a Remaining trading large calm position strike when is a a partial rare exercise skill. keep an unchanged delta positioncontract the to trader exercise must or wait expire for beforebank. they the Often can bank it pass that is the is obvious same whatthe long decision strike, action the onto this should the can be other be performed, a buthiding difficult if under situation spot the to desk is manage. and exactly There refusing on are to stories come about out traders until expiries are finished. has multiple offsetting strikes at the samea level. For contract example, to a trader one might be bank short and long the same contract to a second bank. In order to 152 VANILLA FX DERIVATIVES RISK MANAGEMENT ■ rdn h T Position ATM the Trading otesur oto ie(oeg,te3t T eai afte1rATM 1yr 0.02% the roughly cost half will volatility is = head: implied vega 10% their costing in ATM option ATM points 3mth overnight reference the vega e.g., ATM proportional approximate (so O/N is these vega time have that of remember Traders to since root vega). important expected square is as It the is vega. EUR80k to 0.40% roughly has of ATM wider ladder, vega 1yr spot Total a the of appropriate. focus are the now spacings is spot vega Since 9.16. Exhibit normal in a shown 9.15. position like Exhibit looks per as option strike vanilla the a at on vega weighted profile peak with and vega bell-curve, the vega distribution that to uses 6 exposure Chapter position position’s from ATM a Recall ATM the is the Vega assessing within exposures. of volatilities vega way implied the standard to The exposures curve. contains position ATM The may positions. existing Traders out it. close take to ‘‘axes’’) can (called clients they to earned), prices spread improved market even show broker interbank also (or the cross in or spread clients minimal with contract with a out close to opportunity an ■ position: cleaner much ■ a prefer others while book), the in strikes answers that 17. Analysis Chapter value? in good introduced New Non-Farm represent is and these that a Christmas like does buy between questions but Monday can volatility, holiday with Anyone 3% a considered for sell position. or Year’s be the volatility must achieve 25% and at this to more day paid However, Payroll moving price less. is the moving spot to when is reference gamma spot long when be gamma will short position the because days bad xii 9.17. Exhibit only. options .0 npeimtrs lhuhnt htti prxmto ok o ATM for works approximation this that note although terms, premium in 0.20% oe & oaiiybtmr pedcospi wyt bantecenposition. clean the obtain to away paid cross spread more but volatility P&L lower position in strikes Few → position in strikes Many f1rAMipidvltlt hnrsst 05,tepsto hne sper as changes position the 10.5%, to rises then volatility implied ATM 1yr If trading EUR/USD the gives volatility implied 10% at ATM 1yr EUR20m Buying is there if so position, their in have they options vanilla big the know must Traders open of lots have (i.e., risk strike of lots run to happy are traders some Also, short and days good long is that position short-date a manage risk to easier is It ihrPLvltlt u essra rs adaa laigteposition. the cleaning away paid cross spread less but volatility P&L higher = .2,1mth 0.02%, = → .0,3mth 0.10%, → oiinese omng ic et hne less changes delta since manage to easier position oiinhre omng ic et hne more changes delta since manage to harder position = .0,1yr 0.20%, aallshifts parallel = .0.Teeoe an Therefore, 0.40%. nteAMcurve. ATM the in × → 10 VANILLA FX DERIVATIVES RISK MANAGEMENT 153 EUR/USD trading position with higher ATM implied volatility EUR/USD long vega trading position Vega profile of long vanilla option with 100.00 strike EXHIBIT 9.17 EXHIBIT 9.16 EXHIBIT 9.15 154 VANILLA FX DERIVATIVES RISK MANAGEMENT egtdvg,ipyn httemi hr eaepsrsaea ogrmaturities. longer at are exposures vega short main the that implying buckets). vega, 6mth weighted position. and AUD/USD the 3mth from the exposures in vega bucketed exposures the bucketed with shows as option 9.19 vanilla appear Exhibit a from will vega expiry the (e.g., 5mth them a between split be maturity will vega at the tenors, bucketed is depends price option the Therefore, options, maturity. vanilla vanilla option the For to position. volatility ATM implied the the on of only view fuller a get to vega. used weighted on detail more long-dated for than 14 more Chapter move See volatilities. volatilities curve implied implied ATM the ATM short-dated if shift, but weighted parallel, a in moves curve ATM the if hedged is position The 3. options, ATM 1yr than exposure gamma higher a have options 3mth ATM the 3mth from Since distribution vega 2. short tighter the lower, or higher moves spot If 1. ways: three least at in 9.18. flat Exhibit not in shown is This volatility. implied 10% at again ATM, 3mth USD) (vega) (in change P&L example, where effects: second-order no Assuming XII 9.18 EXHIBIT h oiini hr ohvg n egtdvg,bti ssotrvg than vega shorter is it but vega, weighted and vega both short is position The is position the importantly but flat roughly is of spot EUR40m current sell at to exposure be vega would The exposure vega ATM long the hedging of way One higher. is volatility implied and vega long is position the because risen has P&L uktdvg exposures vega Bucketed vega Weighted oe nannaallmne,aPLcag ilb generated. be will change P&L a overall manner, nonparallel is a in position moves the transacted, been has 3mth of gamma. short notional the double and overall. position vega long a in results ATM × P & .00(ovrinfo U oUSD) to EUR from (conversion 1.3000 L Δ sPLchange, P&L is U/S rdn oiinwt mhvg hedge vega 3mth with position trading EUR/USD steepsr to exposure the is 𝜎 Δ P ie,tevg xouea ahmre eo)are tenor) market each at exposure vega the (i.e., sipidvltlt hne and change, volatility implied is = & fteoto xiyi ewe w market two between is expiry option the If . L . ipidvltlt change) volatility (implied 0.5 Δ weighted = 𝜎 Δ ×υ hne nteAMcre Within curve. ATM the in changes = USD51,863. l eafraparticular a for vega all υ × svg.I the In vega. is EUR79,789 VANILLA FX DERIVATIVES RISK MANAGEMENT 155 = in change EUR10m × 100,000/0.40%) = 1.3450) − If there are aspects of the position will it generate a profit? P&L is calculated. This involves periodically recalculating the AUD/USD bucketed vega profile the ATM position— like For example, EUR/USD spot mid is 1.3450. A trader crosses a two-pip spread to Throughout the day, P&Ls are updated as trading positions are refreshed with live In general, it is not possible to trade implied volatility with the same frequency –USD2k: the spread crossed to transact the deal. The trading position now has a within the risk management system. sell EUR10m EUR/USD spot at 1.3448. When thisempty trade is trading entered into position, a previously the P&L changes byshort (1.3448 EUR10m delta. If the mid EUR/USD spot then falls to 1.3445 and the trader total value from somesince reference yesterday’s end-of-day point snapshot), to month-to-date,monitored, now. with and traders year-to-date Typically primarily tracking P&Ls daily daily are P&L (the within their P&L risk management. change market data. If new dealsis are calculated entered as into the the difference position, between the the P&L traded from price the and new the prevailing deal mid-price If trades are transacted and monitoredbe calculated individually, and the tracked total over time. P&L However, oncontain since each FX many derivatives trade trading (potentially can positions thousandsmark-to-market of) trades that aretotal all risk mid-value managed of together, all contracts in the position. The P&L is then the they that the trader doesn’t like, trades should be executed to change the position. 1yr tenor since it is1yr the ATM could largest be single bought short back bucket. to AUD25m hedge the ( vega exposure. as spot; thereposition. is Trading less ATM and liquidity smilethe positions and short-date is it position. a However, longer-term is endeavor the harder than key trading to question for take a profit trader remains and whether stop-loss the EXHIBIT 9.19 If a trader wanted to flatten this position, they should buy back an ATM vanilla in the FX Derivatives Trading P&L ■ 156 VANILLA FX DERIVATIVES RISK MANAGEMENT ■ XDrvtvsMre Language Market Derivatives FX haeWa tMeans It What Phrase wisdom: and slang market derivatives FX common some is Here bids better if possible be However, market. only USD125k. would the profit in of that appear profit updated realizing and a is position shows data the simulation out market closing the this position, Once trading mid). 16.0/16.5% the from (16.5% within rises 16.25/16.75% market the to in mid) price less ATM in (16.25% in USD/SGD vega USD500k 5yr particularly long The occurs is tenor. position 5yr This trading the realized. USD/SGD a be example, mid ‘‘paper’’ For actually a markets. could between liquid differences that be P&L can and there because valuation done is This account. into position trading the in data. contracts market all of of set where value relevant the the P&L using summing EOD by total calculated today’s the is and taking P&L by P&L total calculated EOD then total is book yesterday’s trading between a difference for P&L daily official The the data. (EOD) of value the than more (10.2%). would volatility be midmarket that would premium the volatility) the instead, at because contract implied 10.3% vega 10.3% contract paid for the at premium of bought (calculated tenth the been a paid of as had change value P&L midmarket contract negative mid a the current generate same If the the contract. at has the position transacted for option was long deal The as the ‘‘marked’’volatility: described because is sometimes causes 10.2%, ATM system at deal 1yr management ATM This that 1yr sheets.’’ say AUD100m ‘‘at buys Traders trading trader surface. A volatility 10.2%. implied desk at 10.2% is the mid in ATM 1yr volatility AUD/USD example, For considered. the additionally on depends moves spot each deals. as all change positions, from P&L trading exposure net derivative delta the aggregated FX exposures; by in delta rise deals own will many their deal with are this from there P&L practice, the In position, USD5k. trading the within data market updates ‘lt’Small. 7.55%). is double’’ ‘‘seven (i.e., 0.55% volatility/2-month implied ATM 1-month ‘‘Flot’’ ‘‘Double’’ ‘‘Ones’’/‘‘Twos,’’ etc. ial,i sipratt oeta fca &sadtoal aebdofrspread bid–offer take additionally P&Ls official that note to important is it Finally, end-of-day the taken: is data market of snapshot full a day, each of end the At the with contracts, derivative to applied is methodology same The T mle oaiiy etc. volatility, implied ATM oPLchange P&L no si setrdit h risk the into entered is it as premium VANILLA FX DERIVATIVES RISK MANAGEMENT 157 exposures in order to make backP&L. a negative market moving lower. market moving higher. higher. Specifically used to describe FXpoints swap moving less negative or more positive. uncertainty (around a market event orpolitical a situation) and sold when the uncertainty is removed. that in the FX derivatives market,that expiry occur cuts earlier in the tradinghave day a always lower price. buying something very cheap. There isto a this, truth involving the optimal balancepremium between and payoff. trend. instrument for which the price is going lower. instrument for which the price is going higher. brokers. cable’’). bucks’’). lower, specifically used to describe FXpoints swap moving more negative or less positive. ‘‘More buyers than sellers.’’ A true but unhelpful explanation for the ‘‘Short-and-caught.’’‘‘When in trouble, double!’’ Holding a Dreadful short trading‘‘More sellers position advice than in about buyers.’’ a doubling financial A true but unhelpful explanation for the ‘‘The trend is your friend.’’‘‘Long-and-wrong.’’ Standard market banter about following a Holding a long position in a financial ‘‘It’s cheap because it’s rubbish.’’ Used as a rebuke to a trading idea that involves ‘‘Buy the rumor and sell the fact.’’ Optionality should be bought when there is ‘‘The first cut is the cheapest.’’ A reworking of a song lyric to reflect the fact ‘‘Market moving left’’‘‘Market moving right’’ Generally used to describe a price moving Generally used to describe a price moving ‘‘Yard’’‘‘Quid’’‘‘Buck’’ One billion. One million GBP (i.e., ‘‘I’m long ten quid One million USD (i.e., ‘‘The notional is fifty Phrase‘‘Touch’’ What It The Means tightest price on a contract shown by the

CHAPTER 10

Vanilla FX Derivatives Miscellaneous Topics

159

everal FX derivatives pricing technicalities have thus far been brushed under the carpet. In this chapter, present and future valuing, tenor expiry date calculations, Sand premium conversions are examined.

■ Present Valuing and Future Valuing

It is well understood that $10 received today is worth more than $10 received in the future due to the time value of money. The core of the argument is that money received now has interest-earning potential if it is placed on deposit (of course, assuming interest rates are positive).

■ Present valuing involves bringing a cash value in the future back to its equivalent value at the present day. In derivatives, option values are often calculated at maturity and must then be present valued. 160 VANILLA FX DERIVATIVES MISCELLANEOUS TOPICS where case: this rate interest where using: calculated is factor deposits where where called are rates interest is the interest maturity, If deposit balance. the at cash rates interest payment deposited one the in on paid compounds all interest how on depends Discount denominated. in and: value 1 cash than the less which generally in are currency factors the in maturity given a ■ o neetrt rdr u hyaentdyt-a ocrsfrms FX most for concerns day-to-day not are they but important bootstrapped are traders be methods traders. derivatives must These rate techniques. they interest quantitative and of used for variety are a curves using rate together interest Different this. than tt oeftr date. future some to it valuing Future npatc,itrs aecrebidn n acltosaefrmr complicated more far are calculations and building curve rate interest practice, In As of frequency compounding regular a given general, In an on compounded is interest If calculation exact the but rates interest from calculated are factors Discount a uses valuing future and present for calculation The m T rCCY r r m A eslre,i h ii h aebcmsa becomes rate the limit the in larger, gets stetm odpstmtrt esrdi er and years in measured maturity deposit to time the is sa neetrt ntegvncmonigrate. compounding given the in rate interest an is sa nulycmone neetrt.Temre ntuetcalled instrument market The rate. interest compounded annually an is as ald‘dps’ olwti convention. this follow ‘‘depos’’) called (also sacniuul opudditrs rate. interest compounded continuously a is h aeue ihnteBakShlsmteaia rmwr.In framework. mathematical Black-Scholes the within used rate the : n h icutfco scluae using: calculated is factor discount the and steopst prto:tkn ahvletdyadpushing and today value cash a taking operation: opposite the is Value df df present nulbasis annual df df = = = = ( = ( ( 1 1 e 1 df − + + + rCCY 1 1 1 . r m h icutfco scluae using: calculated is factor discount the , r r m Value 0 . A T ) ) T T ) Tm future otnoscompounded continuous m icutfactors discount ie er h discount the year, a times r 0 sazr neetrate. interest zero a is ( df zero ) to VANILLA FX DERIVATIVES MISCELLANEOUS TOPICS 161 1 must be a business day and also not a + (see Chapter 27 for examples of different late delivery 1 settlement (e.g., USD/CAD), the spot date is one 2 settlement, the spot date is two days after the horizon. + + 2 must be done by considering each currency within the + Timeline of the four key dates within market tenor calculations : the date on which the final transfer of funds generated from the late delivery : the date on which the contract expires and any final transfer of funds : the date on which the initial transfer of funds (the premium) often takes : the date on which the trade originates (i.e., today) the future and test again. The calculation of T pair separately. For USDhorizon there and the must spot beclear date working one days and clear between for the working all horizon day and non-USD the between currencies spot date. the there must be two day after the horizon.U.S. In holiday. this If case, an T unacceptable day is encountered, move one further day into The term ‘‘business day’’ is used to describe a day that is not on a weekend and These four dates are summarized on the timeline shown in Exhibit 10.1. This If the final transfer of funds takes place after the natural delivery date, the option Expiry date is known. Delivery date contract usually takes place and the date on which forward hedges usually settle. Horizon Spot date place and the date on which any spot hedge settles. EXHIBIT 10.1 2. If a currency pair has T The spot date is calculated from the horizon1. (T). There are If two a possible currency cases: pair has T the spot date. is also not a holidayversion in of either these currency tenor within calculations the is relevant implemented currency in pair. Practical A D. stylized Calculating Spot Dates is described as vanilla options). All these datesnot can open over only the ever weekend. be weekdays since the FXtimeline market may is be different for overnight options since the expiry date can be before ■ ■ tenors within the FX derivatives market. Four dates■ are defined: ■ A well-defined logic exists for calculating expiry and delivery dates for market Market Tenor Calculations ■ 162 VANILLA FX DERIVATIVES MISCELLANEOUS TOPICS h eieydt sannbsns a raUS oia,mv owr ni an until forward move holiday, U.S. a or day non-business a is date delivery forward the moving then and date, a with trade a For Years for date date). expiry spot the its find be the would (i.e., from date operation calculated delivery date’’ is the spot which date ‘‘inverse expiry an until the using forward Finally, move date found. holiday, delivery U.S. is a date or delivery day acceptable non-business a an is date delivery the forward If moving date. then and date, a spot with trade a For horizon. Months the the in from case date calculated which expiry is in the date from 1, spot calculated January the then as or is way weekend date same delivery a The is invalid). date is tenor tenor expiry the the this case (unless which an horizon in with the 1, trade after January a or for weekend and a invalid) is is date expiry this (unless horizon a with trade a For Weeks and Days date The spot the horizon. as way the same horizon. after the the in weekday from date calculated next expiry the is the from calculated is then date is date expiry delivery the trades, overnight For Overnight 1. January except currencies, depending the dates of delivery and tenor. both expiry the or calculating on one for any in conventions be holiday differing can date are a expiry There is the it general, In if years. terms even or in months, weekday or ‘‘overnight’’ weeks, as days, either of number quoted a are of contracts option FX for tenors a Market not is Dates Delivery USD and if Expiry even Calculating holiday holidays, U.S. U.S. a on on (settle) pair. occur currency clear cannot the can within date currency money spot no the that currencies, meaning most for addition, In eieydt sn n‘ivreso ae’oeain(.. n h xiydt for date the date). expiry spot from the its calculated find be would (i.e., is date operation delivery date date’’ the spot expiry which ‘‘inverse the an Finally, using date found. delivery is date delivery acceptable zyears days v months y eo,teepr aei on yfis acltn h spot the calculating first by found is date expiry the tenor, eo,teepr aei h day the is date expiry the tenor, eo,teepr aei on yfis acltn the calculating first by found is date expiry the tenor, z weeks x er rmteso aet h eieydt.If date. delivery the to date spot the from years y otsfo h ptdt otedelivery the to date spot the from months eo,teepr aei 7 is date expiry the tenor, v aedrdy fe the after days calendar x aedrdays calendar VANILLA FX DERIVATIVES MISCELLANEOUS TOPICS 163 , for time target x months 1 month months forward from the spot date month if the tenor is x to lie assuming all days are business days:to If expiry the implies spot a date delivery is datethe January of expiry 30, February date a 30; becomes however, February this 28 doesn’t (in exist a and non-leap year, obviously). target month. For example, assumingis all days April are 30, business a days:This If one-month is the time described spot as to date trading expiry ‘‘end-end.’’ will make the deliverydate date is May beyond 31. theby end convention of to the be target the month, last then business the day delivery of date is the defined target month. For example, then the delivery date is defined by convention to be the last business day of the Finally, a quick word for anyone wondering why this section looks similar to the Also, expiry date and delivery date calculations sometimes adjust in different time Pips prices have different notionalUSD10m, and premium 52 premium JPY currencies. pips Example: implies Notional a cash premium of JPY520m. % prices have theUSD10m, premium same 0.40 USD% notional implies a and cash premium premium of currency. USD40k. Example: Notional ■ FX option premiums can benumber quoted of in CCY2 four for ways: one CCY1%,number CCY1, of CCY2 as CCY1 pips spot for (meaning one is a CCY2). quoted), CCY2%, and■ CCY1 pips (the on the trading desk manyand years put it ago. up Some on kind Wikipedia. soul obviously took the document date may be adjusted totrading USD/JPY avoid for JPY NY holidays, cut, butfor the market once expiry tenors date London can will comes change change. in not Therefore, only and expiry from starts dates day to dayWikipedia but entry within on the this trading subject: day. Documenting this process was one of my first jobs zones. For example, when trading USD/JPY for Tokyo cut in Asia time, the expiry 2. If the spot date falls before the end of the month but the resultant delivery month example, if the spot date monthtarget is month February is and May. the tenor is 3M (three months),1. the If the spot date falls on the last business day of the month in the currency pair, Special Cases There are two special casesmonth involving and trades have that a take tenor place around defined the in end month of or the year multiples. Defining the Option Premium Conversions ■ 164 VANILLA FX DERIVATIVES MISCELLANEOUS TOPICS ti motn ont htteecnesosaeol osbei h pincontract option the if possible only are strike. conversions a these has that note to important is It is beeps.’’ contract ‘‘twenty-five a as of described price verbally the be if might example, it For EUR%, 0.01%). 0.25 (i.e., percent pairs. a (RHS) of side one-hundredth right-hand called are CCY2 in while pairs, paid (LHS) is side pips). premium left-hand CCY2 the called (i.e., are where terms CCY1 pairs USD be in in paid usually will is quoted will premium be premium the notional will where premium the the Pairs the EUR/USD and but in EUR EUR while in quoted CCY1%) in be (i.e., quoted in terms example, be EUR For in usually pair. quoted will currency the notional in the convention EUR/JPY market the on depending terms, 10.2 EXHIBIT xii 02soshwt ovr pin rmusqoe ndfeetterms. different in quoted premiums options convert to how shows 10.2 Exhibit term the %, in premiums quoting When pips CCY2 or CCY1% in quoted usually are derivatives FX vanilla on Prices omlsfrcnetn pin premiums options converting for Formulas ai point basis sotnue omean to used often is PRACTICAL D

Generating Tenor Dates in Excel

o build a volatility surface or quote prices based on market tenors, the expiry dates corresponding to each tenor must be calculated. In Excel, dates are Tinternally stored as integers with 0 = Jan 1, 1900, 1 = Jan 2, 1900, and so on. 165 Current dates are therefore over 40,000 (e.g., June 11, 2014 is 41,801). Within VBA code, dates can be represented using variables with type Long. First, VBA functions are required to:

■ Increment a date to the next business day.

■ Decrement a date to the previous business day. Note that these functions don’t take holidays into account. The built-in VBA function Weekday is used to check the input day of the week:

Function nextBusinessDay(InputDate As Long) As Long

If Weekday(InputDate) = 7 Then 'Input Date = Saturday nextBusinessDay = InputDate + 2 ElseIf Weekday(InputDate) = 6 Then 'Input Date = Friday nextBusinessDay = InputDate + 3 Else nextBusinessDay = InputDate + 1 End If

End Function 166 GENERATING TENOR DATES IN EXCEL ucinbsnsDyermn(nuDt sLn,_ Long, As businessDayDecrement(InputDate Function Function End _ Long, As T businessDayIncrement(InputDate always is Function date spot the horizon): that the assumed after is days business it two code (i.e., this In days. business of number ■ ■ Function End Long As Long) As previousBusinessDay(InputDate Function ucingtptaermoio(nuDt sLn)A Long As Long) As getSpotDateFromHorizon(InputDate Function Function End ermn sLn)A Long As Long) As Decrement Long As Long) As Increment aclt h oio aefo ptdate. spot a from date horizon the Calculate date. horizon a from date spot the Calculate hscnb civduigVAfntosta nrmn n ermn given a decrement and increment that functions VBA using achieved be can This to: required also are Functions etCount Increment Next To 1 = Count For InputDate = businessDayIncrement Long As Count Dim If End Else Then 2 = Weekday(InputDate) ElseIf Then 1 = Weekday(InputDate) If eSoDtFoHrzn=bsnsDynrmn(nuDt,2) businessDayIncrement(InputDate, = getSpotDateFromHorizon Count Decrement Next To 1 = Count For InputDate = businessDayDecrement Long As Count Dim uiesaDceet=previousBusinessDay(businessDayDecrement) = businessDayDecrement nextBusinessDay(businessDayIncrement) = businessDayIncrement 1 - InputDate = previousBusinessDay 3 - InputDate = Monday previousBusinessDay = Date 'Input 2 - InputDate = Sunday previousBusinessDay = Date 'Input + 2 GENERATING TENOR DATES IN EXCEL 167 Today() can be used. It is nice to format date cells = DeliveryDate = DateAdd("yyyy",getExpiryFromTenor Count, = SpotDate) getHorizonFromSpotDate(DeliveryDate) MsgBox "Invalid Tenor" getExpiryFromTenor = -1 getExpiryFromTenor = nextBusinessDay(Horizon) Count = Left(Tenor,getExpiryFromTenor Len(Tenor) = - Horizon 1) + CountCount * = 7 Left(Tenor,SpotDate Len(Tenor) = - getSpotDateFromHorizon(Horizon) DeliveryDate 1) = DateAdd("M",getExpiryFromTenor Count, = SpotDate) getHorizonFromSpotDate(DeliveryDate) Count = Left(Tenor,SpotDate Len(Tenor) = - getSpotDateFromHorizon(Horizon) 1) Else End If If UCase(Tenor) = "ON" Then ElseIf Right(UCase(Tenor), 1) = "W" Then ElseIf Right(UCase(Tenor), 1) = "M" Then ElseIf Right(UCase(Tenor), 1) = "Y" Then Dim Count AsDim Long SpotDate As Long, DeliveryDate As Long getHorizonFromSpotDate = businessDayDecrement(InputDate, 2) The horizon must be input and column headers for the tenors and expiry dates The expiry dates for market tenors can now be set up in an Excel sheet. It is Market tenors can be specified in terms of a number of weeks (e.g., ‘‘2W’’), neater to use a subroutinefunctions that in places the expiry cells. dates onto the sheet rather thanmust using be named TenorRefuser and input or ExpiryDateRef the respectively. Excel function The horizon can be a End Function Therefore, the getExpiryFromTenor function must containdifferent different cases logic using for these theDateAdd rules is outlined used in to Chaptertrading 10. go ‘‘end-end,’’ and The from so forth, built-in spot are VBA date all ignored function to in delivery thisFunction code: date, getExpiryFromTenor(Horizon and As special Long, cases around Tenor As String) As Long End Function months (e.g., ‘‘6M’’) or years (e.g., ‘‘5Y’’), or the overnight tenor (e.g., ‘‘ON’’). End Function Function getHorizonFromSpotDate(InputDate As Long) As Long 168 GENERATING TENOR DATES IN EXCEL n Sub End populateExpiryDates() Sub a with cells formatting by achieved is This week. dd-mmm-yy’’: the ‘‘ddd of e.g.: day format custom the show also they so h olwn uruiecnb sdt ouaeepr ae ntesheet: the on dates expiry populate to used be can subroutine following The Wend 0) Range("TenorRef").Offset(Count, While 1 = Count Long As Count Dim on on 1 + Count = Count _ = 0) Range("ExpiryDateRef").Offset(Count, ag(Tnre".fstCut 0)) Range("TenorRef").Offset(Count, _ getExpiryFromTenor(Range("Horizon"), <> "" PART II

THE VOLATILITY SURFACE

X derivatives trading desks maintain volatility surfaces in all tradable currency Fpairs in order to determine the implied volatility for vanilla options with any expiry date and strike. It is therefore important that traders understand details about how volatility surfaces are constructed since it is a vital part of all FX derivatives valuation. Fundamentally, a volatility surface is constructed along two axes: maturity and strike. The ATM curve forms the backbone of the volatility surface along different expiry dates and the volatility smile defines the implied volatility for strikes away from the ATM strike. Volatility surface construction is usually split into these two separate considerations.

CHAPTER 11

ATM Curve Construction

TM curves can be constructed in two steps. First, a core ATM curve is Aestablished. Then additional parameters are introduced so the correct ATM implied volatility is generated for all possible expiry dates. Note that within this 171 chapter, some calculations are approximate.

■ Variance

The key measure for building ATM curves is

variance = 𝜎2T

where 𝜎 is the ATM implied volatility to time T (measured in years). For example, the variance of a 3mth ATM option with 12.0% implied volatility is 0.122 × 0.25 = 0.0036. Variance can be thought of as a measure of cumulative spot movement. It has two powerful properties: 1. Variance over any time period must be nonnegative. 2. Variance is additive (i.e., variance over two days = variance on first day + variance on second day). Variance can be used to calculate the forward ATM implied volatility (usually called forward implied volatility or just ‘‘forward vol’’ by traders) between 172 ATM CURVE CONSTRUCTION ■ oeAMCreConstruction Curve ATM Core nEhbt1..Tevrac rfielosraoal,rsn vrtm sexpected. as time over rising reasonable, looks profile variance The 11.3. Exhibit in can methodology interpolation tenors. market linear between The seen 11.2. clearly Exhibit be in at shown defined as interpolation curve ATM upward-sloping A—an curve tenors. tenors. market ATM market between shows interpolation using 11.1 constructed Exhibit be can curves ATM O/N Core Interpolation are: Using Curve years ATM two Core a to Constructing up 2yr. tenors and market 1yr, 6mth, standard 3mth, 2mth, the 1mth, 2wk, that 1wk, I (overnight), Part from Recall a Use 2. curves: ATM core generate to 1. used be can that approaches main two are There the 12.8 then is 11.7%, 1yr to is 6mth volatility from volatility implied implied ATM forward 1yr the and 10.5% is volatility implied between volatility between variance Therefore, ■ ■ volatility implied ATM Given volatility future. the in dates two tec xiydt ie h rfiesoni xii 11.5. Exhibit in shown profile 15% the is gives date point expiry data each next at the then and 1yr, tenor. to 2yr the up at tenors volatility market implied at 20% is volatility ainefo oio to horizon from Variance to horizon from Variance nepltn T uv sn iervltlt n hncluaigtevariance the calculating then and volatility linear using B curve ATM Interpolating implied ATM tenors. market at defined B curve ATM new a shows 11.4 Exhibit calculated is date expiry each at variance further, interpolation this investigate consider To first tenors, market between dates expiry For akttenors. market tenors. market the between dates expiry for Input % . h T uv ttemre eosand tenors market the at curve ATM the 𝜎 model 2 otime to ognrt h T uv and curve ATM the generate to T 1 T 2 and ,and T 2 T T T = 1 2 1 < T = = √ 1 T 𝜎 𝜎 and 𝜎 2 2 1 : 2 2 2 2 T T T T 2 1 T 2 2 2 − − = T 𝜎 1 1 𝜎 √ 2 2 T 2 1 T 11.7 o xml,i h mhATM 6mth the if example, For . output 2 interpolate − % 𝜎 𝜎 2 1 1 2 h T oaiiya the at volatility ATM the × otime to T 1 1.0 1.0 n owr implied forward and ogtAMvolatility ATM get to − − iervolatility linear T 10.5 0.5 1 T implied ATM , % 2 × 0.5 = ATM CURVE CONSTRUCTION 173 Variance 0.046 = 1.5 × 0.04 0.045 = = (interpolated) 2 1.0 2.0 × × 2 2 17.5% = 20% 15% = = (i.e., the change in variance for each expiry date) gives . Therefore, linear volatility interpolation has failed to build a ATM curve A generated using linear volatility interpolation ATM curve A defined at market tenors daily variance Calculating Up to 1yr, implied volatility is constant so variance rises linearly with maturity, Variance between horizon and 1yr Variance between horizon and 18mth Variance between horizon and 2yr valid ATM curve from valid inputs (variance to■ 2yr is larger than variance to 1yr): ■ ■ bad news in Exhibit 11.6. but from 1yr to 2yr variance rises andmust then falls. be This sets nonnegative alarm bells ringing: EXHIBIT 11.2 EXHIBIT 11.1 174 ATM CURVE CONSTRUCTION rfiersligfo iervrac neplto fAMcreBi hw in shown is B curve ATM of interpolation variance linear a from methodology: resulting interpolation profile new a suggests This 11.4 EXHIBIT 11.3 EXHIBIT iervrac methodology. variance linear a too. good, looks 11.8 Exhibit in shown curve 11.7. Exhibit on akt T uv ,tepol hw nEhbt1. sgnrtdusing generated is 11.9 Exhibit in shown profile the A, curve ATM to back Going volatility implied ATM the and variance daily negative no has profile variance This ainepol o T uv eeae sn iervltlt interpolation volatility linear using generated A curve ATM for profile Variance T uv enda akttenors market at defined B curve ATM iervariance linear h variance The . ATM CURVE CONSTRUCTION 175 Daily variance profile for ATM curve B generated using linear volatility Variance profile for ATM curve B generated using linear volatility interpolation EXHIBIT 11.6 interpolation EXHIBIT 11.5 176 ATM CURVE CONSTRUCTION XII 11.7 EXHIBIT XII 11.8 EXHIBIT ainepol o T uv sn iervrac interpolation variance linear using B curve ATM for profile Variance T uv eeae sn iervrac interpolation variance linear using generated B curve ATM ATM CURVE CONSTRUCTION 177 ATM curve A generated using linear variance interpolation does not ensure positive forward variance. variance (given validATM inputs) curves. but does not always create intuitively correct These daily variance patterns are not realistic. Intuitively it does not make sense Comparing these two interpolation methodologies: The ATM implied volatility between market tenors in Exhibit 11.9 looks odd. In practice, trading desks use a combinationcurves of with these approaches no to negative produce intuitive in forward variance variance. ATM terms curves butvariance are more evolves generally over sophisticated time. constructed schemes are used to control how daily 1. Linear volatility interpolation often produces intuitively correct ATM curves but 2. Linear variance interpolation produces ATM curves that ensure positive forward that daily variance should jump immediatelyany past special each market factors, tenor why date. would Excluding be daily significantly variance different to one daily day variance priorthe one to core day the after daily the variance 3mth 3mth function tenor tenor should date date? be Ideally, smooth. The linear variance methodologyrises generates sharply an initially ATM and impliedATM volatility then curve. profile flattens Why that off is betweenExhibit this tenors 11.10. happening? for Consider this the upward-sloping daily variance profile shown in EXHIBIT 11.9 178 ATM CURVE CONSTRUCTION a efdwt the with fed be can and speed, is where curves): ATM arbitragable generate could it because model. most a form use functional to the long-term is fundamentally curve but a possible ATM involves often are core models a different constructing Many of method possible Model Another a Using Curve ATM Core a Constructing interpolation 11.10 EXHIBIT prahrqie h oe aaeest eclbae omre T implied ATM market to calibrated be to parameters model the requires approach 11.12. Exhibit per as volatilities h T uv snwan now is curve ATM The Higher The practice in used be never would (that approach simple possible one is Here 𝜎 ( 1 short atr n a and factor, − 𝜆 and e asstefnto omv rm0t oeqiky hsfunction This quickly. more 1 to 0 from move to function the causes T − 𝜆 stm oepr esrdi years. in measured expiry to time is T 𝜎 short-term ) al ainepol o T uv eeae sn iervariance linear using generated A curve ATM for profile variance Daily long ucinmvsbten0ad1a hw nEhbt11.11. Exhibit in shown as 1 and 0 between moves function T r hr-emadln-emAMvltlte respectively, volatilities ATM long-term and short-term are sfo akttnrepr ae ocluaeAMimplied ATM calculate to dates expiry tenor market from ’s speed 𝜎 t = atr(ol evltlt,vrac,o al aine,a variance), daily or variance, volatility, be (could factor fmvn rmsott long. to short from moving of 𝜎 output short +( rmtemdlrte hnbiga nu.This input. an being than rather model the from 𝜎 long − 𝜎 short ) . ( 1 − e − 𝜆 T ) 𝜆 ATM CURVE CONSTRUCTION 179 at market tenors are also required to ensure the overrides ATM curve output at market tenors Function used within a simple ATM curve model Within this approach, model is set up and all tenorsis closely 0.1% match the lower market in except for thea 2mth ATM, market –0.1% which than override the at model thethat suggests. 2mth the 2mth The tenor. ATM This trader is is therefore relatively useful inputs cheaper information than because other it tenors. suggests volatilities. This can bemodel parameters time change consuming as the initially market but ATM curve traders moves. soonsystem learn ATM implied how volatility the hits market mid values. For example, the ATM curve EXHIBIT 11.12 EXHIBIT 11.11 180 ATM CURVE CONSTRUCTION ■ T uv osrcin Short-Dates Construction: Curve ATM pcfi tiems iea ogrmtrte ic ainems iea longer at rise must variance since a arbitrageable. maturities is for curve longer premium ATM the at option Otherwise, vanilla rise maturities. the must framework, strike the specific from removed are discounting ATM 8-day the 12.0%, is volatility be ATM must 7-day volatility the If volatility: implied ATM 8-day the is What variance). zero (i.e., day 8th volatility? the implied ATM during 8-day static completely be will spot each volatility? on implied volatile ATM equally 8-day the is is spot What because Assume variance). move daily variance). not equal zero does (i.e., (i.e., weekday spot closed where is days market weekend the two and weekdays five contains 11.13. Exhibit shorter in at diagram applied the be per can as framework scales, variance time same the how demonstrate examples be must information This this within curve. and curve. times the ATM volatility different the spot over even into expected incorporated (or control different dates have sufficient expiry dates) traders different expiry because give required weights to is or order parameters control additional in constructed, introduced been has are curve ATM core the Once XII 11.13 EXHIBIT u otepoete fvrac,ti fetvl om oe on nthe on Variance bound lower a forms effectively this variance, of properties the to Due 2 Example 1 Example following The dates. expiry shorter at important mainly is control additional This w 7dy T mle oaiiyi 20.Asm ti nw that known is it Assume 12.0%. is volatility implied ATM (7-day) 1wk : w 7dy T mle oaiiyi 20.A1koto always option 1wk A 12.0%. is volatility implied ATM (7-day) 1wk : and hr-aevrac xmlsframework examples variance Short-date tleast at pinpremium option 𝜎 𝜎 8 − 8 a ATM day − 11.25%. a ATM day = = √ √ √ √ √ √ √ √ √ √ variance variance ( r lsl ikd ffradditand drift forward If linked. closely are ( 365 365 8 8 1 wk ) 1 ) wk . = 5 6 = 11.25 12.3 % % ATM CURVE CONSTRUCTION 181 K K Impact of forward drift on option prices A better strategy would be to sell the second option as the first option expires. If both options are left unhedged until expiry, profitability depends on how spot In practice, however, with forward drift reintroduced, the situation becomes Short 7-day call option with strike Long 8-day call option with strike EXHIBIT 11.14 This would offer a near-certainextreme nonnegative forward P&L, drift but prevents Exhibit a 11.14expiry guaranteed shows (shown profit. how in The an vanilla leg call 1)overnight option has expiry) value a due at higher to the value large than negative the forward second drift. vanilla call option (now moves between the two expirymore dates. in-the-money An (ITM) overall at profit theloss will second will be expiry be than generated generated the if if first. spot spotis However, is no is an more guaranteed profit ITM overall locked at in. the first expiry than the second. There how can a guaranteed profit be generated from these■ trades? ■ more complicated. Consider vanilla optionsthe same on strike. two If consecutive the following expiry trading dates position with can be achieved for zero premium, 182 ATM CURVE CONSTRUCTION eiaie market. derivatives ( pricing, expiry for formula to Black-Scholes the time using when market, derivatives FX the Within Pricing Market Derivatives FX the as maturities longer within at observed dampens commonly stabilizes. effect ratio the is market-open-to-total-days although 11.15, market, Exhibit derivatives FX in ratio. the shown the market-open-to-total-days than saw-toothing, horizon, higher volatility a fixed ATM implied has a This Friday lower For the a because week. have it always working following almost the Friday will over expiry dates Monday expiry future future a for rise to tends market. days. derivatives weekend FX two the 1wk of contains feature the date observed than expiry commonly higher a 1wk is is the This volatility because implied volatility ATM implied overnight ATM (one-day) the words, In weekend: the over variance no and variance) daily day equal market-open (i.e., each volatile Assume overnight has equally volatility. (one-day) ATM is date Consider 1wk the expiry volatility. to specific implied compared volatility ATM a ATM the to on days impact market-closed important and an market-open of number The Week the over mean Patterns Volatility really Implied P&L. negative they with year when the end get arbitrages occasionally can often bank as desks on books’’ Traders trading trading prices derivatives ‘‘arbitrage premium. the result, or zero a As love for opportunities.’’ trades trading ‘‘fantastic would spread describing traders 8-day back, away versus step carried 7-day a this taking However, transact account. to into taken are spreads n vrih YctAMipidvltlt s1.% h vrih expires overnight The 15.0%. is volatility implied ATM so tomorrow, cut NY overnight and osdrastainweetecrettm s9 is time current the where situation a Consider volatility implied ATM why explains also ratio market-open-to-total-days The to hard is it practice In T T variance = sseie in specified is ) 365 ATM 1 and: O 1 ∕ guarantee wk N variance 2 = . ATM ( ATM 0.15 365 7 iceediysteps daily discrete 1 1 1 wk rfi nti rd,priual nebid–offer once particularly trade, this on profit a 2 wk wk ) × < = = = ( ATM ATM ATM variance 365 1 O O O ∕ ∕ ∕ ) N N N O 2 = ∕ × . N hsi e etr fteFX the of feature key a is This . ( 6.1644 √ × 365 A.M. 1 5 7 5 ) odntm nMonday on time London × × 5 10 − 5 ATM CURVE CONSTRUCTION 183 A.M. 6 − 10 % 6 × − 10 14.75 × = 2.0548 6 − = 5 10 ) − 5.9589 × London time, so this overnight option 1 10 = 365 × ( P.M. 30 hourly 5.9589 √ √ √ √ 6.1644 variance = = × hourly NATM 29 ∕ O 𝜎 variance Monday to Friday ATM saw-toothing is unchanged within this calculation due to being specified in discrete T Note that values within the Black-Scholes formula for pricing, implied volatility reduces forward one trading day, thevariance. ATM implied Then volatility jumps over highergradually due the moves to course lower increased due of to reducinga the fixed variance. expiry Variance trading date (and reduces day, hence as premium)T time the to passes but ATM because the implied market uses volatility constant daily daily steps. At the start of each trading day, when the overnight option expiry moves After one hour passes, the remaining variance on the option is: which implies a new overnight ATM volatility of: actually expires in 30 hours. Assumingcut spot tomorrow: is equally volatile between now and NY EXHIBIT 11.15 This variance can now be splitNew into York even smaller time, time which intervals. is NY cut (usually) is 3 at 10 184 ATM CURVE CONSTRUCTION ahrta n.Teeoe namre lc-coe rcn world, pricing Black-Scholes market a later in days Therefore, three one. Monday; than a rather day. on following expires the option ‘‘overnight’’ expires the option Friday, overnight on the However, Friday, from apart weekday any On Friday a on ATM (O/N) Traders Overnight stable. risk trading management. the risk their keeps within it cuts effect but at different this inconsistent, at for date is expiring adjust horizon this options date, the has expiry position on the trading expiring the on options If times. all expiry show own positions their trading that date reason horizon the the on expiring options that reason the generate is This in steps. value time daily any discrete kill will transactions options; two vanilla the short-dated in in trade. involved flow the cross two-way spread good liquid same the in is the applicable otherwise only selling there is then technique where This and pairs day. day trading trading currency the of the end ‘‘ of the start be at back the can contract at gamma ATM day, overnight the gamma the buying of long course reduce the to over market deltas. the their into trade to come struggling volatility quickly are traders they more implied where as falls market positions rate’’ often the ‘‘natural volatility implied and its static, positions than is their spot hedge if Alternatively, to an positions increases. to market gamma short due the spot with higher into or Traders moves move volatility. come generally spot spot volatility large future implied a increased is range, of there expectation recent If its important. of also out is behavior breaks spot but day, the of benefit. minimal only into for accuracy market the more of introducing complexity Therefore, static. completely be However, is not day. the volatility throughout spot stable more since be would volatility implied short-dated short-dated in visible only is impact the overnight. but the tenors particularly all options, at occurs effect This instead. hrfr inrn discounting): (ignoring Therefore Vega assuming performed usually also is positions derivatives FX of management Risk should it than slower falling is volatility implied short-dated believe traders If course the over lower drift to tend does volatility implied short-dated practice, In accurate more a used market the if that consider to interesting is It ( et jumps delta 𝜐 ) safnto of function a is hog hi tielvl(sse nCatr n ) ti also is It 9). and 7 Chapters in seen (as level strike their through 𝜐 call o constant not = √ 𝜐 1 𝜐 − T put a ATM day : = 𝜕𝜎 hogottedy mle oaiiywould volatility implied day, the throughout 𝜕 . P √ = 3 Se = − rCCY 𝜐 rented 3 − 1 a ATM day . T n example, for involves, This .’’ ( d 1 ) √ T T T T = o pricing, for dst the to adds 365 3 . ATM CURVE CONSTRUCTION 185 213% of the = 3 71% of the correct × = each day and therefore: ATM premium 𝜐.𝜎 position. The jump from Friday 3 = GMT) day ATM √ − GMT) 1 P.M. 𝜎 ATM put A.M. = Price weekend decay = day ATM − 3 𝜎 3. Furthermore, the bid–offer spread shown on a ATM call √ on ATM options, so for a given tenor, roughly: ) Price 𝜕𝜐 𝜕𝜎 ( Tokyo time (often 6 New York time (often 3 P.M. A.M. Amazingly, in a sophisticated financial market in the twenty-first century, this In practice, the market pricing of the overnight ATM contract on a Friday In practice this means that the market overnight ATM implied volatility quoted Assuming there is no variance over the weekend and each weekday is equally There is no volga TOK cut: 3 NY cut: 10 Make money on Tuesday through totheta is Friday paid as per only day. (5 / 7) Lose all additional profitcorrect the theta following is paid Monday from as Friday (5 into Monday. / 7) That is, NY cut options contain an extra nine hours of optionality. Tokyo (TOK). The New Yorkbe cut analyzed versus using Tokyo the cut same ATM variance volatility framework: differential can ■ ■ effect still produces tradingcheap opportunities on Friday as as short-dated some options banks oversell can to become reduce theirNew too weekend York theta. Cut versus Tokyo Cut Pricing In G10 currency pairs the two most common expiry cuts are New York (NY) and ■ ■ is closely related toend-of-day to the Monday market’s morningfor covers within three risk management days. systems, If theta from thislarge. Friday In is to a Monday not simplified will correctly world be with artificially adjusted the no same adjustment amount for of this gamma effect, each a day position will, that on is average: long three-day overnight should be tighterpremium in spread. volatility terms Again, in the orderthe level to market-open-to-total-days of ratio, show or ATM the put implied same another(time way, volatility the adjusted is ratio to of being consider economic market impacted time activity by only) to calendar time (see Practical E). on a Friday cannot be directlyquoted compared on with the other overnight ATM days.it implied volatility To must get be the multiplied Friday by overnight ATM into the same terms volatile, the O/N ATM contract will have the same 186 ATM CURVE CONSTRUCTION rcda h aeipidvltlt asmn oeet,ec,o h xiydate). expiry the approximately: on tenor, three-month etc., the events, at no example, (assuming For volatility implied same the at priced ■ ■ ■ ■ day. trading given a of course the over and framework, pricing Black-Scholes where premium. lower a priced and are variance less both has because volatility daily implied discrete same ATM the cut using York New the than discount’’) / O cut TOK O/N cut NY O/N cut TOK O/N / Ycut NY O/N o auiisps he ots e okadTkoct ilgnrlybe generally will cuts Tokyo and York New months, three past maturities For 5 At increases differential volatility 9 At cut Tokyo versus cut York New the Therefore, volatile: equally always is spot Assuming a at (‘‘trades lower always is volatility implied ATM cut Tokyo the Therefore, T NY T P.M. A.M. stera iet xiyt h e okcut. York New the to expiry to time real the is market GMT: GMT: stetm oepr esrdi er sdwti h market the within used years in measured expiry to time the is = = = = 3 3 𝜎 6 6 P.M. P.M. TOK A.M A.M. 𝜎 𝜎 = M h etday next the GMT M h etday next the GMT TOK TOK M h etday next the GMT . M h etday next the GMT 𝜎 𝜎 T variance TOK NY = = u h oy u cusfis ntedyadtherefore and day the in first occurs cut Tokyo the but T T 2 . TOK 𝜎 𝜎 TOK . √ T NY NY T 𝜎 market TOK TOK TOK 2160 2151 . . √ √ stera iet xiyt h oy cut Tokyo the to expiry to time real the is = = = 22 13 30 21 hours hours 𝜎 𝜎 variance NY = = NY = = T = = T 2 2hours 22 hours 30 . 𝜎 𝜎 NY . √ = NY T NY NY 1hours 21 3hours 13 market NY 𝜎 T × × T NY TOK NY 0.77 0.84 × 0.998 ATM CURVE CONSTRUCTION 187 Stylized intraday hourly realized variance The intraday variance patterns are different in emerging market currency pairs Decreases after GMT 15:00afternoon (NY cut) to the end of the day in the New York Peaks around GMT 08:00 as Europe/London come in Dips during Europe/London lunch around GMT 11:00 Picks up again in theGMT afternoon 14:00 with New York in and reaches day highs around Starts low and builds up during Asia trading time pricing framework for maximum accuracycut when times. pricing options expire at different ■ where trading is concentrated inrestricted. one Such region variance or patterns the should spot be market opening taken hours into are account within the option ■ ■ ■ Intraday Variance Patterns The simplifying assumption that spotis is obviously equally not volatile correct throughout infollows the practice. a trading In fairly day liquid well-established G10 pattern currency shown in pairs, Exhibit realized■ 11.16 variance in which it: EXHIBIT 11.16 188 ATM CURVE CONSTRUCTION ifrnl vrtecus ftedy ro oteeet hr-ae T implied ATM short-dated event, the to Prior day. the of course implied the ATM over differently cut TOK versus cut the date, NY but larger expiry event far usual. the this a than from to On differential leads variance volatility 11.18. This spot Exhibit not. expected does then in additional cut and shown the TOK sharply as contains data cut day increases the NY normal variance until the day’’ the spot ‘‘normal to realized a back release, on reverts data variance spot the the Over than release. lower slightly or exact similar The curve ATM it. market following the therefore dates and expiry beforehand information. known also this is incorporates and events of released) time occurs been and cut the date has (if date event expiry that the on increases expiring after turn options in for This volatility released. implied is ATM data the the after spot immediately period which the in in specifically 2013, May public. employment from made is U.S. month) data the of economic the on gauge after Friday important immediately first jumps (an the release on released data usually Payroll reaction Non-Farm spot USD/JPY the the shows to 11.17 Exhibit information. new to adjusts market Events Holidays and Events XII 11.17 EXHIBIT h rsneo neetas asssotdtdipidvltlt odecay to volatility implied short-dated causes also event an of presence The usually is variance spot realized releases, data important containing days On curve, ATM the within variance higher assigned therefore are days Event eooi aarlae,eeto eut,ec)cueso omv sthe as move to spot cause etc.) results, election releases, data (economic S/P ptoe o-amPyoldt ees rmMy2013 May from release data Payroll Non-Farm over spot USD/JPY ATM CURVE CONSTRUCTION 189 also impact realized volatility and variance. There is often Stylized intraday hourly realized variance on Non-Farm Payroll day Like the NY cut versus TOK cut implied volatility differential, the day of the Public holidays Events usually occur in a particular currency. For example, European employment week, as shown in Exhibitbe 11.19. more This effect data occurs releases partiallyremoved, later because Mondays in there are the tend often to less week. volatile However, than even other weekdays. with theweek effect of of events a particular expiry date matters more at shorter tenors than at longer pairs. Therefore, public holidaywithin days the in ATM a curve. particular currency have lower variance Weekday Variance Patterns The FX spot market often exhibits increased realized variance later in the working perfectly synchronized manner, EUR/AUD realized volatility also increases. significantly less spot activity in pairs containingthere the are holiday fewer currency simply market because participantspublic operating that holidays day. are In important addition, U.K. enough and to U.S. reduce spot activity across all currency volatility can drop sharply as expected future spot variance reduces. data primarily impacts spot inmost currency important pairs events, that crosses includemove EUR. can in However, an also for asynchronous exhibit manner. the but increased For if example; volatility EUR/USD Non-Farm and if Payrolls AUD/USD impacts the are USD, both majors more volatile but they do not move in a EXHIBIT 11.18 volatility will move lower only slightly but after the event has occurred ATM implied 190 ATM CURVE CONSTRUCTION a ocluaetepeimo aedyoto st tr iha overnight an with start volatility. to implied the is adjust option to same-day framework variance a the of use and premium option the calculate to way terms. formula: volatility pricing option in Black-Scholes quoted hence the be zero; in that cannot is 5 options days Chapter same-day market of from standard Recall therefore number the within and The impossible zero framework. is today pricing later Black-Scholes expire that options Pricing Options Same-Day Pricing ATM 6mth the and example, concern. expiries for major Friday of, a weekday few not the next is but the contract expiries buy Monday to few next preference the a sell has often market The tenors. 11.19 EXHIBIT hsi hw nEhbt11.20. Exhibit in shown is This ln ( h / YAMipidvltlt s1% Therefore: 12%. is volatility implied ATM NY O/N The Example So-called K S ) + ⎝ ⎜ ⎜ ⎛ rCCY :At9 𝜎 2 √ aedyoptions same-day − T rCCY variance vrg al ptvrac o 1 ar n2012 in pairs G10 for variance spot daily Average A.M. 1 + 𝜎 2 O 2 M letrqet rc nasm-a Yctoption. cut NY same-day a in price a requests client a GMT ∕ ⎠ ⎟ ⎟ ⎞ NNYCut . T ol ra eas h eoiao szero. is denominator the because break would = utteeoeb utdin quoted be therefore must 0.12 2 × ( 365 1 ) = 3.945 × premium 10 − 5 em.One terms. d T 1 = is ATM CURVE CONSTRUCTION 191 6 − 10 × 7.89 % = 5 5.3 − = 10 6 × − ) 10 1 × 3.945 365 ( × 7.89 ) 6 30 √ √ √ √ ( = = day ATM 1 𝜎 day NY Cut − Same Pricing a same-day vanilla option variance In general, same-day options are nonstandard and a wider bid–offer spread should This implied volatility can then be used to price an overnight option which gives standard option expiry time? the same-day option premium. Note thatthe interest same-day pricing rates since should forward be drift set and to discounting zero will within have nobe impact. charged. Plus it is vitalaccount. to take Be expected suspicious: intraday Why variance wouldn’t profiles and the events counterparty into be happy to wait until the Assuming each hour has equal variance: Therefore, the equivalent one day ATM volatility is: EXHIBIT 11.20

PRACTICAL E

Constructing an ATM Curve in Excel

ithin this practical, three methods of constructing an ATM curve are Wdeveloped. First, an ATM curve is constructed using interpolation between market tenors. Then, an ATM curve is constructed using a parameterized model. 193 Finally, weights are added to a simple ATM curve to demonstrate how ATM curves are maintained by traders in practice. These steps mirror the material developed in Chapter 11.

■ Task A: Constructing an ATM Curve Using Interpolation

When constructing an ATM curve based on market tenors, the expiry date for each market tenor must first be calculated using functions developed in Practical D. The ATM implied volatility is then manually inputted at each tenor. For the purposes of testing, a simple upward-sloping ATM curve can be used initially: 194 CONSTRUCTING AN ATM CURVE IN EXCEL ucingtTVlQeyaeA og sDouble As Long) As getATMVol(QueryDate Function volatility ATM generate tenors: market to between at used dates volatilities interpolation expiry ATM linear for and with cells, dates named expiry using the tenors references function This date. any sn hs nus B ucincnitroaet ieteAMvltlt for volatility ATM the give to interpolate can function VBA a inputs, these Using Wend _ 0) dates Range("ExpiryDateRef").Offset(Count, Expiry While the 1 (requires = row Count date Expiry relevant Double the As 'Find VolHigh Double Double, As As TimeHigh VolLow Double, Dim As TimeLow Long Dim As Count Dim Else Then QueryDate = 0) Range("ExpiryDateRef").Offset(Count, ElseIf 0) Range("ExpiryDateRef").Offset(Count, And 1 = Count ElseIf Then "" = 0) Range("ExpiryDateRef").Offset(Count, If urDt Then QueryDate 0) Range("ExpiryDateRef").Offset(Count, And ordered) be to ieo Rne"xiyaee".fstCut-1 )-_ - 0) 1, - Volatility (Range("ExpiryDateRef").Offset(Count Implied = ATM TimeLow get to 'Interpolate 0) Range("ATMVolRef").Offset(Count, = Found getATMVol Date Expiry 'Exact Date -1 Expiry = Minimum getATMVol before Date 'Query Date -1 Expiry = Maximum getATMVol beyond Date 'Query 1 + Count = Count ag(Hrzn) 365 / Range("Horizon")) <> < urDt _ QueryDate "" > _ CONSTRUCTING AN ATM CURVE IN EXCEL 195 "" <> getATMVol(Range("ChartExpiryDateRef").Offset(Count, 0)) Range("Horizon")) / 365 VolLow, VolHigh, (QueryDate - Range("Horizon")) / 365) Range("ChartATMVolsRef").Offset(Count, 0) = _ Count = Count + 1 TimeHigh = (Range("ExpiryDateRef").Offset(Count, 0) -VolLow _ = Range("ATMVolRef").Offset(CountVolHigh - = 1, Range("ATMVolRef").Offset(Count,getATMVol 0) 0) = LinearVolatilityInterpolation(TimeLow, TimeHigh, _ (QueryTime – TimeLow) / (TimeHigh – TimeLow) Dim Count As Long Count = 1 While Range("ChartExpiryDateRef").Offset(Count, 0) Wend LinearVarianceInterpolation = VolLow + (VolHigh – VolLow) * _ End If The getATMVol function can be tested by querying for implied volatility in four TimeHigh As Double,QueryTime VolLow As As Double) Double, As VolHigh Double As Double, _ End Sub Sub populateATMImpliedVolatilities() The ATM volatility for dailyfor expiry dates two (starting years) at can the nowpopulates overnight be the tenor calculated. ATM and This implied going subroutine volatilities: (run by pressing the button) 2. An expiry date after3. the maximum An tenor expiry expiry date date at4. a tenor An expiry expiry date date between two tenor expiry dates different cases: 1. An expiry date before the minimum tenor expiry date End Function End Function Function LinearVolatilityInterpolation (TimeLow As Double, _ 196 CONSTRUCTING AN ATM CURVE IN EXCEL u populateVariance() Sub subroutine: this using sheet the onto chart: a in plotted be can data This ainefrec xiydt a lob acltd(e hpe 1 n pushed and 11) Chapter (see calculated be also can date expiry each for Variance hl ag(Catxiyaee".fstCut 0) Range("ChartExpiryDateRef").Offset(Count, While 1 Double = As Count vol Double, As T Long Dim As Count Dim Rne"hrEprDtRf)Ofe(on,0 _ - 0) (Range("ChartExpiryDateRef").Offset(Count, = T ag(Hrzn) 365 / Range("Horizon")) <> "" CONSTRUCTING AN ATM CURVE IN EXCEL 197 ) T 𝜆 − e − 1 ( . ) short 𝜎 − long 𝜎 +( short 𝜎 = T 𝜎 vol = Range("ChartATMVolsRef").Offset(Count,Range("ChartVarianceRef").Offset(Count, 0) 0) =Count T = * Count vol + ^ 1 2 QueryVariance As Double (QueryTime - TimeLow) / (TimeHigh - TimeLow) Dim VarianceLow As Double, VarianceHighVarianceLow As = Double, TimeLowVarianceHigh _ * = VolLow TimeHigh ^ * 2 VolHighQueryVariance ^ = 2 VarianceLow + (VarianceHighLinearVarianceInterpolation - = VarianceLow) Sqr(QueryVariance * / _ QueryTime) Wend Finally, linear variance interpolation can be used instead if required: There are many possible ATMterizations curve introduced models. in One Chapter of 11 the is: simplest possible parame- End Function Double Function LinearVarianceInterpolation(TimeLow AsDouble, Double, VolLow TimeHigh As As Double, _ VolHigh As Double, QueryTime As Double) As _ End Sub Task B: Constructing an ATMModel Curve Using a ■ 198 CONSTRUCTING AN ATM CURVE IN EXCEL h uptcnb lte nachart: a in plotted be can output The stylized this within intervals 1/12 (use intervals monthly framework): in displayed time and with respectively, volatilities ATM from long-term reversion and short- the are where 𝜎 T steAMipidvltlt ttime at volatility implied ATM the is 𝜎 short to 𝜎 long hsmdlcnb mlmne na xe sheet, Excel an in implemented be can model This . T maue nyears), in (measured 𝜆 stesedof speed the is 𝜎 short and 𝜎 long CONSTRUCTING AN ATM CURVE IN EXCEL 199 ’s) to T calendar time Within this model, a single flat volatility is used. By introducing a separate row can therefore be calculated: the expected varianceexample, assigned assign to low variance individual toevent weekends/holiday dates. days days. This and One high control variance commondiscrete to is daily major way chunks used this based on to, can the for weight be assigned achieved to each is day. for by each date splitting starting one variance day into after the horizon (for at least a year), In practice, traders keep their ATM curves aligned with the market by controlling The function can thencalculate ATM be implied attached volatility: to the market expiry dates (and their Task C: Adding Weights to an ATM Curve ■ 200 CONSTRUCTING AN ATM CURVE IN EXCEL ifrn at ftedy o o,tog,stwihso nec a ocalendar so day each on identical: 1 are of weights time over set economic though, models, and now, time sophisticated For more day. in the of or, parts to days different variance different allows over time distributed economic unevenly Controlling be models. curve ATM real within used calculate Sub End populateDayWeights() reference Sub offset the generate to cell: used correct cunningly the is to function VBA Weekday the how h a egt pt ie eo a hnb umdaddvddb 6 to 365 by divided and summed be then can tenor given a to up weights day The Note dates. expiry the onto weights day the push to used be can subroutine VBA A table: a in defined are These added. be now can weights Day Wend 0) Range("DateRef").Offset(CountExpiryDates, While 1 = CountExpiryDates Long As CountExpiryDates Dim onEprDts=Cutxiyae 1 + CountExpiryDates = CountExpiryDates _ = 2) Range("DateRef").Offset(CountExpiryDates, cnmctime economic fstCutxiyae,0) 1) 0)), Offset(CountExpiryDates, _ Range("DayWeightRef").Offset(Weekday(Range("DateRef"). hsclna ievru cnmctm ehiu is technique time economic versus time calendar This . <> "" CONSTRUCTING AN ATM CURVE IN EXCEL 201 changes from: t dt i 𝜔 T 1 2 n = 𝜎 i ∑ 2 = 𝜎 T = var T var . 1 365 = dt When plotted with constant day weights of 1, the ATM volatility is flat as ATM volatility is then calculated from total variance using calendar time: When weights are added, total variance to (calendar) time expected: into where 202 CONSTRUCTING AN ATM CURVE IN EXCEL eovrac oteweedbcueteei ml hneta nxetdnews unexpected that chance small a is there non- because but weekend small a the assign to usually variance desks trading zero practice, In Sunday. or Saturday on open sheet: the derivatives in FX data real the the in at observed Look saw-toothing market. ATM the contains now output The the check and subroutine, populateDayWeights the again: graph using weights day date expiry hsefc sajse o hnteeaetre oaiiylvl htms ehit. be must that levels volatility target are there practice, In when 1. for than less adjusted being is time effect calendar to this time economic of ratio the it. to due preceding input Friday the than lower be to morning. expiry Monday Monday implied on the ATM for thing the volatility causes first ratio sharply economic-time-to-calendar-time move the in to reduction spot The cause will weekend the over nti ae T oaiiytnstwr au hti oe hntefltvolatility flat the than lower is that value a toward tends volatility ATM case, this In isn’t market FX the because weekend the over stops time economic model, the In the repopulate zero, to weights day weekend the Set magic: the comes Now CONSTRUCTING AN ATM CURVE IN EXCEL 203 rises: and subsequent dates Daily variance can now be calculated by taking the difference in variance between The ATM volatility for the Non-Farm Payroll date itself moves higher plus the Finally, consider how the model is adjusted when there is an event. On Thursday, subsequent expiry dates, which in turn canThe be daily used to ATM calculate volatility daily ATM is volatility. effectively the implied volatility for a strip of forward increased variance causes subsequent days toobserved move when higher, building too. ATM curves: This If is expected athe variance real ATM for feature volatility a given for date that increases, date on that date is higher andThis the ATM is volatility achieved for that within date is the correspondingly higher. model by moving the weight for that date higher: July 3, 2014, Non-Farm Payrollsoccurs (a on big the USD first economic Friday indicator, of which the normally month) is released. Therefore, expected variance 204 CONSTRUCTING AN ATM CURVE IN EXCEL iia otoedvlpdi akAo akBi hspatclwl euulybe variance usually forward be nonnegative will that way practical a this such in in B approach guaranteed. top is Task An on or volatility. added A weights flat desks Task with just Trading in taken, not specified, day. developed curve, often those same ATM to the are core similar within events sophisticated cuts of a different require times of also Exact pricing model. correct the the enabling within included usually is variance. are are expected and schedules reduced too, the advance, release reflect in to far when known weights are lower weights currency assigned particular higher a in assigned days future Holiday are plus known. market, dates broker interbank release the in economic observed prices market match to model events. over underpriced or volatilities overpriced ATM is curve overnight ATM forward the these whether determine use to Traders contracts. ATM overnight npatc,taigdssuefaeok iia oti u oegranularity more but this to similar frameworks use desks trading practice, In curve ATM their within weights update actively traders derivative FX Vanilla CHAPTER 12

Volatility Smile Market Instruments and Exposures

205 n the interbank broker market, at each market tenor, three market instruments Idefine the volatility smile: 1. At-the-money (ATM) contracts define the implied volatility for a specific strike close to (or exactly at, depending on the market conventions for a given currency pair) the forward for the given tenor. 2. Butterfly (Fly) contracts define the implied volatility differential between the wings of the volatility smile and the ATM—a measure of the height of the wings of the volatility smile. 3. Risk reversal (RR) contracts define the implied volatility differential between strikes above and below the ATM—a measure of how skewed or tilted the volatility smile is. Butterfly and risk reversal contracts are most often quoted at 25 delta (25d) and 10 delta (10d) strikes. An example run of market instruments at market tenors is shown in Exhibit 12.1. Exhibit 12.2 shows the relative positioning of different deltas within a stylized volatility smile. Recall that it is the market convention to trade the out-of-the- money side. 206 VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES a rteohr smnindi hpe ,ptdla r fe utdwithout quoted often are deltas one put smile 7, the Chapter the tilts in reversal lifts mentioned risk As butterfly the other. the and the point, or sides, reference way both on central higher the symmetrically represents wings ATM the words, In 1997: in Malz M. smile. Allan volatility the into fit instruments market these how shows 12.3 Exhibit ■ ■ Therefore: ■ ■ tenor: the given for a volatilities implied the with instruments 12.2 EXHIBIT 12.1 EXHIBIT a ssoni xii 24 n n tiea httnrwl easge h same the assigned be will tenor that at strike any volatility. and implied 12.4, midmarket Exhibit in shown the as in flat used are 100% and 0% between values delta formula. Malz put Positive sign. negative the 𝜎 𝜎 𝜎 𝜎 fbtefl n ikrvra otat taldla r eo h oaiiysieis smile volatility the zero, are deltas all at contracts reversal risk and butterfly If by delta any for formula single a into generalized were approximations These following The Fly RR Put Call 25 25 25 25 d d d d 𝜎 = = = = XDeltaPut 𝜎 𝜎 ( 𝜎 𝜎 Call ATM Call ATM etsqoe ihntevltlt smile volatility the within quoted Deltas tenors market at instruments market EUR/USD Example 25 25 d + approximations d + + 2 = − 𝜎 𝜎 𝜎 Put Fly 𝜎 𝜎 Fly 25 ATM Put 25 d 25 ) d 25 d − − + d + 𝜎 2 2 1 ATM 2 1 𝜎 𝜎 𝜎 RR ikteAM 5 utry n 5 ikreversal risk 25d and butterfly, 25d ATM, the link RR RR 25 25 25 d d d . ( X − 50 )+ %) outright 16 𝜎 5 aladptotosat options put and call 25d Fly 25 d . ( X − 50 %) 2 VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES 207 Volatility smile with zero risk reversal and zero butterfly 25 delta market instruments within the volatility smile With a positive risk reversal, strikes above the ATM have a higher implied volatility With a negative risk reversal, strikes below the ATM have a higher implied If the butterfly increases, the wings of the volatility smile rise symmetrically as than the equivalent delta strikes below the ATM. This is shownvolatility in Exhibit than 12.6. theExhibit 12.7. equivalent delta strikes above the ATM. This is shown in EXHIBIT 12.4 shown in Exhibit 12.5. EXHIBIT 12.3 208 VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES XII 12.6 EXHIBIT 12.5 EXHIBIT oaiiysiewt eors eesladpstv butterfly positive and reversal risk zero with smile Volatility oaiiysiewt oiiers reversal risk positive with smile Volatility VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES 209 since ) delta 𝜕𝜎 𝜕 ( . 𝜕𝜎 ∕ ) P spot 𝜕 𝜕 ( : sensitivity of vega to changes in spot. Vanna can also be thought = : sensitivity of vega to changes in implied volatility. Volga is the Volatility smile with negative risk reversal ) ) : sensitivity of price to changes in implied volatility. spot spot 𝜎 vega ) vega d 𝜕 𝜕 𝜕 P 𝜕 𝜕 𝜕𝜎 ( ∕ ( ( ) P 𝜕 𝜕𝜎 Volga second derivative of price with respect tovolga changes in is implied volatility. to Therefore, implied volatilityvolatility as rises, the gamma expected is P&L to from spot a and long volga as trading the position volatility increases. of implied Vega Vanna of as the( sensitivity of delta to changes in implied volatility, i.e., ATM Exposures The vega profile forspot. a Exhibit 12.8 long shows how, ATM at vanillaspot higher distribution volatility, option the is has vega wider, profile but a is vega wider single is since unchanged peak the at around the initial current spot. ■ ■ ■ The reason for describing the volatilityinstruments smile with becomes ATM, clearer butterfly, and whenmarket risk instruments reversal the are examined. implied The volatility key implied exposures volatility exposures of are: the three EXHIBIT 12.7 Market Instrument Vega Exposures ■ 210 VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES hrfr,ti pinhsaln an exposure. ending vanna rises. strike long delta the a hence has and of option widens chance this distribution the Therefore, the rises, as volatility increases implied maturity at If volatility in-the-money delta. implied 25% current has At spot). strike above the strike (i.e., in option call shown topside is the-money option ATM long into a rises from vega profile since vanna positive 12.9. The is Exhibit strike peak. the topside below (now) spot the with Vanna peak. downside ■ ■ ■ 12.8 EXHIBIT itiuinwdn n ec et al.Teeoe hsoto a hr vanna short the a as has decreases option maturity this Therefore, at volatility falls. in-the-money implied exposure. delta ending If hence strike delta. and 75% widens the distribution has of strike chance the the volatility rises, implied current At spot). og tedfeec ewe h eapolsfrdfeetipidvolatility implied exposure different no for level): spot profiles initial vega the at the levels between difference (the exposure Volga no chart): vega/spot the of gradient (the Vanna exposure positive Vega: ieie osdradwsd i-h-oe)cl tie(.. tiebelow strike (i.e., strike call (in-the-money) downside a consider Likewise, as vanna of interpretation dual (now) the Recalling the into rises vega since negative is strike the above spot with Vanna exposures: following the has inception at contract ATM long a Therefore, eapol rmln T tdfeetipidvltlt levels volatility implied different at ATM long from profile Vega 𝜕 𝜕 vega spot or 𝜕 𝜕𝜎 delta osdra out-of- an consider , VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES 211 risk reversal position can give either a long or long the risk reversal means buying the topside strike versus buying Vanna profile from long ATM Notice that these vega profiles aren’t perfectly rotationally symmetric since vega In this instance, Volga with spot above or below the ATM strike is positive since long wing vanilla if downside strikes have adownside.’’ higher In implied, a traders currency say pairreversal the where position risk the initially reversal gives risk is, a reversal yes, short is vanna ‘‘for for position as downside, shown a inpersists long Exhibit risk further 12.12. to the topside. This occurs because the Black-Scholes formula is selling the downside strike butother in way different currency around. pairs Therefore, or a short tenors vanna this exposure, may depending be on the implied whether volatility topside strikes than are the athave equivalent higher a delta higher or implied downside lower volatility, traders strikes. say the If risk the reversal is topside ‘‘for topside,’’ whereas strikes For a risk reversal contract,wider again, but higher implied at volatility initialin moves spot the Exhibit vega the 12.11. profile vega Itgenerated exposure with is is fixed important unchanged strikes, toafter equivalent at trading understand to the zero. that contract. calculating This these the is exposures exposure shown immediately profiles are options generate positive volga. The volgain profile Exhibit from 12.10. a long ATM option is shown Risk Reversal Exposures EXHIBIT 12.9 212 VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES XII 12.10 EXHIBIT oaiiylevels volatility 12.11 EXHIBIT og rfiefo ogATM long from profile Volga eapol rmrs eesl(uigtpie tdfeetimplied different at topside) (buying reversal risk from profile Vega VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES 213 rotationally is Vega versus spot profile from risk reversal (buying downside) The vanna exposure on a risk reversal does not persist over allTherefore, spot a levels. long Rather risk reversal contract at inception has the following exposures: Volga: no exposure Vega: no exposure Vanna: positive or negative exposure‘‘for topside’’ depending or on ‘‘for whether downside’’ (i.e., thepriced whether risk at topside higher reversal or implied is downside volatility within strikes the are volatility smile) short straddle (ATM) with thevega-neutral ATM and the notional call and set put such legs inExhibit that the the strangle 12.15 have structure the shows is same notional the initially andstrikes delta. vega are profile fixed fromdifferent and a implied hence volatility long levels the butterfly. after trading. Again, chart the shows butterfly how the vega exposure changes at ■ Butterfly Exposures A long butterfly contract is constructed using a long strangle (long wings) and a ■ ■ stated in log-return terms, which causeszero. distances A in stylized spot space vega to versussymmetric compress spot as toward log-return shown in graph Exhibit for 12.13. a risk reversal it is maximized at the initial spot as shown in Exhibit 12.14. EXHIBIT 12.12 214 VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES XII 12.13 EXHIBIT XII 12.14 EXHIBIT eavru o ptpol rmrs eesl(uigdownside) (buying reversal risk from profile spot log versus Vega an rfiefo ikrvra byn topside) (buying reversal risk from profile Vanna VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES 215 . ) spot vega 𝜕 𝜕 ( of implied volatility because their main level . ) P 𝜕 𝜕𝜎 ( Vega profile from long butterfly at different implied volatility levels In practice this means that: The volga exposure on a butterfly does not persist overA all long spot butterfly contract levels. at Rather inception it has the following exposures: exposure at inception is vega Risk reversal contracts arerelationship because used their main to exposure at trade inception is the vanna spot versus implied volatility ATM contracts are used to trade the Volga: positive exposure Vega: no exposure (by construction) Vanna: no exposure ■ adaption (explained in Chapter 14) andin broker this fly chapter), strike the placement three (explainedexposures later different at market inception shown instruments in give Exhibit the 12.17. three unique vega ■ ■ Summary Within this stylized analysis using a flat volatility smile and ignoring issues like is maximized at the initial spot as shown in Exhibit 12.16. ■ ■ EXHIBIT 12.15 216 VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES ■ 12.17 EXHIBIT 12.16 EXHIBIT h aesaea h eapol fters eeslwietevlao h ATM the of volga the butterfly. while the reversal of risk vega the the as of shape profile same vega the takes the as shape same the utrycnrcsaeue otaetevltlt fipidvltlt because volatility implied volga is of inception volatility at exposure the main trade their to used are contracts Butterfly ial,i smll neetn oosreta h an rfieo h T takes ATM the of profile vanna the that observe to interesting mildly is it Finally, og rfiefo ogbutterfly long from profile Volga eaepsrsfo aktinstruments market from exposures Vega ( 𝜕 𝜕𝜎 vega ) . VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES 217 the selling the risk reversal in the volatility smile buying skew strikes—the same strikes as if same-tenor and same-delta the call or put strike with the higher volatility, and also quoted. For example, in USD/ZAR, if the 1yr 25d call outright buying direction Additionally, if a currency pair has a large forward drift, at longer maturities the If a currency pair had a completely flat volatility smile, the risk reversal strikes When trading a risk reversal, particularly if it is long-dated, it is important to pay The delta used to calculate the risk reversal strikes is sometimes spot delta and In the interbank broker market, risk reversals are quoted in positive terms, The FX derivatives market expresses the amount of ATM strike will be farforward from drift current is spot large and positive, it the 35d is put possible strike that, is for positioned example, close if to the current spot. would be positioned approximatelyspace. symmetrically Therefore, around the the topside ATMdownside strike strike strike will in in be log- regular further spotlonger away space. maturities At from the short impact the can maturities ATM be this than significant. effect the is small but at delta convention is used totransaction generate if the dealt. strikes will also be used to delta hedgeattention the to exactly which strikesreversal are are being the transacted. Strikescall traded or within put a vanillas risk are traded in isolation. other leg. sometimes forward delta, depending on market convention.G10 Most often, risk short-dated reversals areemerging quoted market using risk reversals spot are delta quoted strikes using while forward delta long-dated strikes. G10 Whichever and over,’’ meaning that the USDIn some call currency volatility pairs, is it higherin is than market CCY2 convention the terms. to USD USD/JPY quotecalls put risk the volatility. risk over’’ reversals reversal if are direction quoted thethe as, implied implied for volatility example, volatility for ‘‘1.4%always for JPY the means the downside strike topside is strike. 1.4% As higher noted, than between the call strike implied volatilitysame and tenor the and put delta. strike implied volatility for the with the is priced atimplied 15.5% volatility, implied the 1yr volatility 25d and risk reversal the would 1yr be quoted 25d as put ‘‘4.25% USD is calls priced at 11.25% In practice, implied volatility changes depending (amongstmoves. other Plus, things) in on how the spot market theredownside is optionality, often which differing supply leads and to demand an for asymmetric topside volatility or smile. via the risk reversal contract. Specifically, the risk reversal gives the differential The Black-Scholes formula assumes that the volatility of the underlying is constant. Risk Reversal Contract ■ 218 VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES ■ ■ reversal reversal. risk risk downside the large a and with 8) pair Chapter (see forward the also. lower than straddle, relatively zero-delta lower are a strikes is is ATM strike the and ATM CCY1 the is currency premium the If placement. ■ placement: strike ■ reversal risk impacts also smile volatility the contract, reversal XII 12.18 EXHIBIT h U u tiei oae nasepysoigpr ftesie othe so smile, the premium) of CCY1 part a sloping by (caused steeply higher. strike a significantly lower is relatively on the located from is volatility implied strike not put is premium) AUD CCY1 The a by implied the (caused so strike smile, different. the lower too of relatively part flat the relatively from a on volatility located is strike call AUD The moves in-the-money strike up the ending smile, of the chance volatility. decreasing on lower the lower at about is think delta ATM; the given up to a ending closer for of volatility implied chance the increasing If moves the strike about the smile, volatility. think higher the ATM; at on in-the-money higher the is from delta away given a further for volatility implied the If xii 21 hw yia oaiiysiei U/P— C1premium CCY1 AUD/JPY—a in smile volatility typical a shows strike 12.18 Exhibit reversal risk in role important an play conventions market Finally, risk a within traded always are strikes out-of-the-money that Remembering U/P oaiiysmile volatility AUD/JPY VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES 219 relative strength spot goes lower, if spot goes lower, spot will be if ; see Chapter 17). Usually, the larger the interest rate differential carry Historically, interest rate differentials and risk reversals were highly correlated This idea of the relative strength of currencies also links to the interest rate differ- The risk reversal contract can be thought of as a measure of the At longer tenors the risk reversal is largely driven by expectations of spot moves At shorter tenors the risk reversal is largely driven by expectations of spot moves This effect causes AUD/JPY risk reversal contracts to be valued at higher implied since low interestcountry rate with currencies lowerBRL (e.g., growth or JPY potential TRY) implied or whilesocial, a high CHF) or country economic interest with implied instability. higher rate However, a growth since currencies potential more the but (e.g., 2008 stable more financial political, crisis most G10 in a given currencymore pair, important at the longer largerthe tenors. low-yielding the Buying currency risk to the benefit reversal. higher-yielding fromfor the currency This carry and protection and relationship then selling buying from becomes thecurrency risk a reversal pairs. blowup is a classic trading strategy in emerging market positive because there is amarket far currency higher (i.e., chance spot jumps of higher) avolatility than sharp invariably the rises devaluation USD. When and of spot therefore the jumps, aexposure emerging implied long to risk the reversal topside position will with make a money. long vega ential (i.e., implied volatility will rise more. Oris it an may increased imply chance that that the spot market will expects move that there lower. of the twocurrency currencies pair in (e.g., the USD/TRY or currency USD/BRL), pair. the risk In reversal a will invariably USD be versus emerging market downside, that may imply themore market volatile. expects Or that itchance may that imply spot that will move the lower. market expects that thereand is implied an volatility increased changes. Forfor example, if downside, longer that tenor risk may reversals imply go more that the market expects that versus downside optionality. This preference is aChapter function 17) of but market it positioning also (see dependsrealized on the spot market’s volatility, perception and of implied expected volatility spot changes. moves, and realized spot volatility. For example, if shorter tenor risk reversals go more for reversals or comparing risk reversalsconventions. between currency pairs with different market What Drives the Risk Reversal in theThe Market? risk reversal contract expresses the prevailing market preference for topside volatility levels and themeans impact that gets care larger must be for taken long-dated when options. assessing the In term-structure practice of this long-dated risk 220 VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES o ofl noteta fbleigta h ttsqowl rvi indefinitely. prevail will quo status the that believing of trap the into fall to not reversal tenors. risk longer a at often particularly is increases, there premium ranges risk market trading the extreme recent makes as likely of overvaluation which most out the position, breaks from spot reversal protection When offers risk moves. a reversal long is risk from there a the back that since holding made sense implies with be This can associated relationship). than premium volatility position risk implied reversal versus risk spot (see the the volatility hold trading and implied buy than to lower more often costs is details), volatility for realized 17 that Chapter way same the In Reversal Risk the the Trading of 70 in 2014. reversals 1, October risk the of delta shows as 25 pairs 12.19 1yr currency Exhibit and liquid most factor. rates reversals interest important risk 1yr an and between remains carry relationship between it link although the weakened, and rates has interest low have pairs currency 12.19 EXHIBIT oaiiisms eare.Freape w ak ol ge otasc an transact to agree could banks two example, For implied agreed. actual the be agreed, is must transaction a volatilities After strikes. two the between years differential ten over reversals risk points. delta various 25 at 1yr comments USD/JPY trader of with chart a shows 12.20 Exhibit nteitrakboe akt h ikrvra stae ntrso h volatility the of terms in traded is reversal risk the market, broker interbank the In careful be must traders instruments, financial other all for as reversals, risk For y neetrt ifrnilvru y 5 ikrvra cte plot scatter reversal risk 25d 1yr versus differential rate interest 1yr elzdskew realized sotnls than less often is mle skew implied ie,it (i.e., VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES 221 bank, the higher the agreed implied buying . The call and put strikes are backed out of an inverted Black- (explained in Chapter 14). Buying a risk reversal results in a USD/JPY 1yr 25d risk reversals from May 2002 to November 2012 short adapted vega exposure.the Therefore, highest the implied riskthe volatilities reversal highest possible buying possible bank so wants level.lowest they implied Likewise, volatilities get possible the short so they riskpossible adapted get long level. reversal vega adapted vega selling from from the bank lowest wants the positioned. The higher volatilitylower side of volatility the side. volatilitythe smile Therefore, ATM, is the by steeper risk reversal than pushinghigher buying the (i.e., bank the better) gets on a strikes the long volatilityAdapted strike further smile. which vega is away marked from even Strike placement Scholes formula. For the risk reversal volatilities, the further away from the ATM both risk reversal strikes are Traders must check the proposed market data and only agree to trade at correct implied volatility levels. There will be occasionsmore where transacting important the than risk reversal these is second-orderthe effects but P&L traders should difference always fromadditional calculate ‘‘spread’’ the mid transaction is implied costing. volatility levels so they know how much 2. the RR selling banktwo wants reasons: 11.4% on the AUD put. This1. disagreement occurs for EXHIBIT 12.20 AUD/USD 1yr 25d RR atbank 2.6% wants AUD 11.6% puts over, on but the then AUD the put risk (and reversal buying therefore 9.0% on the AUD call) while 222 VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES lotalcrec ar,a oe et h ikrvra ut nrae,a shown as increases, quote reversal risk the 12.22. delta Exhibit lower in at pairs, called currency are all deltas almost relationships different The at market. reversals broker risk interbank 25d the between only in pairs, quoted many regularly In are smile. reversals volatility risk the understanding for apart. useful further is are relationships strikes the when less exposures offsets vega vega peak the the since and ATM larger the be from to away further positioned be to peaks vega reversals. risk 10d and 25d from profiles vega shows 12.21 Exhibit Contracts Reversal Risk 10d the versus effect, 25d gamma. In long spot. being to higher equivalent the be changes at will delta sold creates delta vanna position be higher, reversal if can risk 1% CCY1, delta long approximately additional is is and USD volatility longer Assuming implied USD20m delta. and longer USD20m be long will is position their will that not volatility implied knows vanna has that long trader the clear the to is Due it if higher. market even be the higher, in jumps prices trader volatility spot a implied and if any topside, reversal seen for risk reversal risk short-dated a long with pair is currency a in example, For positions. XII 12.21 EXHIBIT h ikrvra utsa ifrn etsaelne.Ivsiaigthese Investigating linked. are deltas different at quotes reversal risk the The causes reversal risk delta 10 the within strikes the of positioning wider The delta their manage to exposures reversal risk short-dated their use traders Finally, 5 ikrvra eapol ess1drs eeslvg profile vega reversal risk 10d versus profile vega reversal risk 25d ( 𝜕 𝜕𝜎 delta ) xouefo h ikrvra h trader the reversal risk the from exposure ikrvra multipliers reversal risk .In VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES 223 × 0.775 AUD% = d 25 Fly 𝜎 2.56 0.31% vega + × d d 25 25 RR Fly 𝜎 𝜎 d 0.8 25 2.56 + RR + 𝜎 d ATM 25 𝜎 1.6 RR = 𝜎 = d 10 0.8 Put Delta Put − 𝜎 % − 90 ATM d 𝜎 𝜎 25d and 10d risk reversals on the volatility smile 10 = = Call 𝜎 ) 0.975 of the 25d RR in premium terms. If the 10d RR costs (0.775% = = ) 0.755% in premium terms, that equates to a 10d risk reversal quote of Delta Put Delta Call d = 10 % % 10 10 RR The 1yr AUD/USD 25d RR has –4.2% vanna while the 1yr AUD/USD 10d RR The 1yr AUD/USD 25d risk reversal is –2.5%. Therefore, this risk reversal This 25d/10d multiplier of 1.6 is a touch lower than values typicallyAnother observed method for calculating risk reversal multipliers is to assume that the cost Within the Malz volatility smile formula, substituting 10% put delta and 10% call 𝜎 𝜎 𝜎 more to buy in premium terms than if the risk reversalhas was –4.1% 0%. vanna. If the(4.1/4.2 premium cost of vanna is0.975 constant, the 10d RR should cost deltas are linked. Exhibit 12.23 showsvanna vega for for AUD/USD AUD/USD 1yr 1yr long outrighthence risk strikes short reversals and vanna). over a range of deltas (AUD puts over contract ‘‘costs’’ approximately 2.5% volatility in the market for liquidreversal currency multipliers pairs are where usually fairly the stable value in is liquid usually currency around pairs. 1.8.of Risk vanna remains constant.circular, This but method it is gives back-of-the-envelope, some old-school, intuition and as to how risk reversal contracts at different ■ ■ Therefore: ■ EXHIBIT 12.22 delta into the formula gives: 224 VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES nohr tfist rdc ml ls otemre.Amr fetv variation effective more A market. the but to well close works smile this a cases produce some to a In fails assuming it spots. smile others pair volatility in cross major a the builds between and correlation tenor static given the by a generated at functions) smiles density volatility probability major on information more for 13 volatility. Chapter implied low the with pair of risk currency EUR/USD majority managed off a vast clearly generated is the be is USD/HKD can contribute since there reversals reversals will if risk that EUR/HKD suitable example, pair is For the methodology skew. This within pairs. currency major dominant the a of one in reversal reversal risk 25d the calculated? can how be USD/JPY, EUR/JPY and in EUR/USD in reversals risk 25d Given Reversals Risk market. Cross the in observed often values to close are multipliers These 12.24. Exhibit in reversal. risk vega (0.755%/0.17% 12.24 EXHIBIT 12.23 EXHIBIT hnueti ognrt rs ikrvras(aigtelvlo h T noaccount into and ATM the currency, of level each the (taking for reversals risk parameter cross ‘‘strength’’ generate to relative this use a then imply other, each against level. absolute the generate to copula the uses nte osbeapoc st oka ytmo ikrvrasi aycurrencies many in reversals risk of system a at look to is approach possible Another A risk the to offset fixed a as calculated be can reversals risk cross cases, some In shown multipliers reversal risk gives methodology above the AUD/USD, 1yr For copula prah nisms ipefr,tkspoaiiydniis(see densities probability takes form, simple most its in approach, U/S y ikrvra multipliers reversal risk 1yr AUD/USD vanna reversal risk long 1yr and vega strike outright 1yr AUD/USD = 44% hc s(–4.45%/–2.5% is which –4.45%, ) changes ntecosrs eeslrte hngenerating than rather reversal risk cross the in = 1.775 ) × h 25d the VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES 225 taken into not but this instrument is the outright 25d call and not strike fly broker fly volatility is equal to the + 0.40%: + 1.3810. The 25d broker fly call strike is further away 1.3690 (10.80% volatility) 1.1695 (13.15% volatility) = = = broker fly volatility is used to generate the call and put strikes + . The strikes within the 25d broker fly are : In EUR/USD, spot is 1.2600, the 1yr forward is 1.2660, the 1yr ATM : The ATM The broker fly is the impliedput volatility generated at and which priced the premium using ofpremium the the of ATM call the and same the strikes on the full volatility smile. If the 25d broker fly volatility is Example This statement can be broken downPart into 1 two parts: The broker fly is a messy concept, but put as simply as possible: Strike placement is very important within the butterfly contract. The butterfly The FX derivatives market expresses the height of the wings of the volatility smile Cross risk reversals are a tricky area and this section barely scratches the surface. Outright 25d put strike Broker fly 25d call strike from the ATM strike than the outright 25d call strike since it is generated using Outright 25d call strike ■ ■ account within broker fly strike placement. strike is 1.2740, and the 1yr ATM implied■ volatility is 11.5%: within the broker fly. Crucially, this means the risk reversal/skew is reversal and broker fly25d are call different. and The 25d butterfly putrarely constructed strikes traded is using in sometimes practice. the called outright a via the butterfly contract, quoted asvolatility the and average put of strike the implied same-delta call volatility strike less implied the ATM volatility atcontract a given that tenor. is quotedbroker fly and traded in25d put the strikes. interbank Therefore, broker the market strikes is within the called same-tenor the and same-delta risk The Black-Scholes formula assumes constantimplied volatility. and In realized) practice, itself volatility ispriced (both at volatile. higher This implied causes volatility wing thanof the vanilla implied ATM volatility) options due they to to contain. the be volga often (second derivative Different banks take different approaches but flexibilitymethod is that important; works finding for a all single crosses all the time is very difficult. each time). Alternatively, the realized historicbe spot used versus to volatility imply relationship a can cross risk reversal using a regression-style calculation. Butterfly Contract ■ 226 VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES ■ ■ ■ 19.25%: is volatility ■ implied ATM 5yr the and 58.20, is strike large. be can strikes fly broker and strikes outright between broker showing diagram be a will gives placement. volatility 12.25 strike strike Exhibit fly fly volatility. broker strike from since smile away lower strike the further outright than the be higher delta fly will on same fly broker While the broker since the than volatility. within ATM strike strike strike the outright the smile smile, delta the volatility same the than of the lower side than be ATM will the volatility to strike closer be will fly ■ XII 12.25 EXHIBIT rkrfl 5 alstrike call 25d fly Broker strike put 25d fly Broker strike call 25d Outright strike put 25d Outright 11.9% using generated is it smile. since the on strike volatility put 13.15% than 25d rather outright volatility the than strike ATM strike put 25d fly Broker smile. the on volatility 10.8% volatility ATM (11.5% volatility 11.9% Example difference the pairs currency skew high in or maturities long-dated at Particularly broker the within strike the smile, volatility the of side higher the on general, In nADJY pti 02,te5rfradi 38,te5rATM 5yr the 63.85, is forward 5yr the 80.25, is spot AUD/JPY, In : rkrfl tieplacement strike fly Broker = = = = = 60 2.5 volatility) (22.55% 46.05 91 1.5 volatility) (14.15% 79.10 96 3%fraddlao h smile) the on delta forward (30% 49.60 the to closer is strike put fly broker 25d The 1.1775. 28 2%fraddlao h smile) the on delta forward (20% 82.80 + .%boe yvltlt)rte than rather volatility) fly broker 0.4% VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES 227 broker fly + broker fly contract contains vanna 2.84%). + 5.65% = Pricing tool showing broker fly premiums broker fly volatility is equal to their combined premium priced using the : The combined premium of the call and put options priced using the when valued on the smile caused by the strike positioning. + In a CCY1 premium pair a long broker fly contains long vanna exposure because The broker fly strikes are generated in leg 1 and inputted in legs 2 and 3. Look at Some long-dated AUD/JPY volatility surface instruments are shown in Part 2 EXHIBIT 12.26 the CCY1 premium pulls all strikes lower. This makes the long topside strike Exhibit 12.27. The ATM and RRflies are are both going rising more at negative. longer This tenorsand is but larger a the skew counterintuitive 25d result is broker because intuitively a linkedfact, higher with the ATM higher broker wings within fly the goesexposure volatility more smile. negative In because the the premiums: The cost ofvolatility the is strangle the (the same call as the plusthe cost the full of put) smile the at (8.49% call ATM using the full smile plus the cost of put on ATM full volatility smile. Exhibit 12.26 shows this in a pricing tool. 228 VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES rdn h Butterfly the Trading reversal risk long position. a gamma holding long long of a a holding cost/benefit holding or the of position with cost/benefit compared The is volatility. implied position and butterfly spot in moves sharp from vega. wing for preference plus market be, prevailing will volatility the is implied reflects butterfly volatile also the how it tenors of expectations longer market at by while driven spot) largely current of from away perception gamma market’s (i.e., gamma the market on of depends function changes. also a volatility it implied is and but realized preference expected 17) This Chapter optionality. (see option- ATM positioning wing with for preference compared market prevailing ality the expresses contract butterfly The Market? be the in will Butterfly the quote above Drives the What fly in broker as exposure), reversal the reversal risk risk downside, Fly (short (long 25d vanna for vanna long case. long is AUD/JPY the the to reversal the due to topside, lower risk due pulled for higher the is pulled reversal If be risk exposure). will the If quote pairs. fly currency broker premium CCY1 in long contract a downside, for is exposure. reversal RR vanna 25d risk long the contains fly if broker whereas 19.25% exposure, vanna short contains 18.3% in resulting away, 17.0% further relatively strike 16.6% downside vanna. long long the and 15.3% closer relatively ATM 5yr 4yr 3yr 2yr 1yr Tenor e asa ie oee,taesudt h igprmtr ihntheir within parameters wing the update traders for risk However, trade as time. not frequently might a pair at as currency days traded given few not a a in are contracts contracts Butterfly butterfly contracts. reversal market, interbank the In XII 12.27 EXHIBIT neegn aktcrec ar,taesbytebtefl otata protection as contract butterfly the buy traders pairs, currency market emerging In wing for preference market prevailing the reflects butterfly the tenors shorter At fly broker the of price volatility fly the broker impact long significantly a exposures vanna topside, These for is reversal risk the if pair, premium CCY2 a In U/P oaiiySraeInstruments Surface Volatility AUD/JPY − − − − − 8.4% 8.1% 7.8% 7.2% 5.7% + − − − − 0.1% 2.0% 1.5% 1.0% 0.5% VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES 229 , plus a wider distribution. The 25d equal wing notional 25d butterfly vega versus 10d butterfly vega profiles Finally, it is worth noting that the broker fly strike placement prevents the Exhibit 12.29 shows how the long volga at current spot goes flat and then short As shown in Exhibit 12.28, both 10d and 25d butterfly contracts give sharp changes EXHIBIT 12.28 existence of stable 25d/10d butterfly multiples in most currency pairs. as spot moves awaywider from (better) the volga current level distributionmaximum in volga than exposure the 25d occurs wings at flies. current andis spot. Within how to Therefore, a if get 10d the longer butterfly flies aim volga of25d give contract, the at butterfly) a trader the is higher not or necessarily lower the spot best contract levels, to the trade. butterfly (particularly the butterflies can therefore be tradedtrading in position. large Within 25d size broker withoutcan butterflies significantly be in impacting CCY1 positioned the premium so pairsexposures. closely the Believe together strikes me, that I found they this can out the generate hard large way. localized vanna 25d versus 10d Butterfly Contracts Exhibits 12.28 and 12.29 give the vega and volga profiles for 25d and 10din butterflies. vega away from current spot.wings The versus 10d the flies 25d have flies over for double the peak vega in the volatility surface more frequentlytrading levels in on specific order contracts. to match implied volatility prices or 230 VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES ■ oaiiySieRs Management Risk Smile Volatility .Rega 2. Vanna also 1. and therefore is changes, position exposures: volatility the of smile sets The implied how two themselves. and using understand instruments monitored spot smile to the as important to impacted exposures is the be it will position exposure derivatives vega FX an trading When 12.29 EXHIBIT ihr(.. rm8 o9) et ilgtlne yUSD1m. 1% by moves longer volatility get will implied delta ATM 9%), 1% if to moves Or, 8% spot USD10k. from implied if (i.e., by ATM vanna, higher longer in USD1m get change long will 1% is vega a position higher, for trading a delta if in Therefore, change volatility. the vanna, of interpretation dual Vanna htrg ersnstePLgnrtdfo eaun l otat nthe in contracts. contracts reversal all risk delta revaluing 25 changed from with surface generated meaningful. volatility is a P&L rega using delta the position 25 represents only smile, rega volatility delta the That risk delta build whichever 25 to only to if used quoted example, are For are reversals surface. volatility sega the build and to used rega are Both contracts change. prices butterfly and changes. volatility implied and ( ( suulyqoe scag nvg o hnei pt r ealn the recalling or, spot, in change a for vega in change as quoted usually is 𝜕 𝜕 𝜕 RR 𝜕 P vega spot ) ) 5 utryvlavru 0 utryvlaprofiles volga butterfly 10d versus volga butterfly 25d n sega and n volga and ( ( 𝜕 𝜕 Fly P 𝜕 𝜕𝜎 vega ) ) xli o & hne sters reversal risk the as changes P&L how explain xli o h eapsto hne sspot as changes position vega the how explain VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES 231 1.0%, a P&L change + , etc.). However, when rega 𝜕𝜎 𝜕 or 0.8% to of its vega since, for example, + spot vanna 𝜕 half 𝜕 is usually quoted as a change in vega for a 1% move in ATM implied is usually quoted as the sensitivity to a 1% change in the volatility price is usually quoted as the sensitivity to a 1% change in the volatility price USD30k will be generated. 0.1% change in the risk reversal will roughly move the implied volatility for Risk reversal contracts have vanna and rega exposures while butterfly contracts The quotation conventions used for these smile exposures will differ from trading For reference, the rega on a risk reversal is approximately the average of the Sega Like other exposures, within a trading position; vanna, volga, rega, and sega Rega Volga + + However, when trading atrading portfolio exotic contracts of the links vanilla between options vanna andcan in rega and break high-skew between down. volga pairs and For sega or example,Therefore, a when traders trading must position monitor might all be these flat exposures volga within but their risk long management. sega. thought of as thewithin sensitivities the to volatility surface the construction. parameters that control themainly skew have and volga wings and sega exposures.positions It that is have therefore natural vanna totrading exposures assume positions that also trading that have have corresponding long rega volga exposures exposures and also have long sega exposures. the 25d call strikedown up by by 0.05%. 0.05% and the implied volatility fordesk to the trading desk. 25d Plus, although put rega andto sega strike here the are specifically market the sensitivities risk reversal and butterfly instruments, they can more generally be two absolute strike vegas whilethe the two sega on wing a strike butterflywill vega is have exposures. approximately the In a sum addition, 25d of a a rega 25d approximately topside equal strike to in isolation will not be static; theyhave will their change own as higher-order spot derivatives or (e.g., analyzing ATM derivatives implied trading volatility positions moves, it so isexposures often all within better a to view, spot for ladderhigher-order example, or sensitivities vanna implied at current volatility spot ladder only. rather than considering of of the butterfly instrument. Therefore,delta if sega, a trading if position the is 25dwill long butterfly be USD100k generated. moves of lower 25 by 0.1%, a P&L change of –USD10k derivative like gamma. A long volgawhen exposure implied therefore volatility means rises that and vega can bought be when sold implied volatility falls. of the risk reversal instrument. Therefore,25 if delta a rega, if trading the position 25d is risk long reversal USD150k moves of from volatility. Therefore, if a trading position is short USD250krises volga, if by implied volatility 0.1%, vega will get shorter by USD25k. As mentioned, volga is a second 232 VOLATILITY SMILE MARKET INSTRUMENTS AND EXPOSURES ■ oaiiySieCntuto Methods Construction Smile Volatility hthsdvlpdadbcm h tnadoe ie u te parameterizations other convention but valid. market time, equally over a be standard is would the approach It become smile. butterfly and volatility and developed the has reversal, representing that risk of way ATM, the from possible about the only nothing distances makes equity is the there that percentage in that market understand different while to derivatives important FX at standard is It quoted market price. stock the is current the become volatility has implied model derivatives, SABR the derivatives, strike terms, delta e.g., terms. in parameter interpolating model this, or for terms, methods own their market. develop the desks match that tenors such market model at model the the of by parameters output the instruments market updating the traders with model, SABR ward’s information. more for Reading Further controlled in the be book on Gatheral’s either calls (SVI)-see Volatility must Stochastic Inspired delta like strikes model 10 a using delta volatilities and automatically generated 10 or downside surface, extrapolation the using the volatility beyond on the volatilities are puts Implied build delta instruments topside. 10 to market between delta used only 10 are defined and are instruments placement 25 market strike only if and since conventions liquid, approach, market for this adjust Within must issues. that process a in directly or input an either be more can far smile is volatility process the output. the curve, volatility an practice ATM a in the construct with but to As instruments possible complicated. market is the it from how directly shows smile formula smile volatility Malz The te se lse xrs oaiiysrae ndfeetwy.I neetrate interest In ways. different in surfaces volatility express classes asset Other Trading tenors. between interpolated be must smile volatility the addition, In Wood- and Lesniewski, Kumar, Hagan, as such models use desks trading Other instruments market the using smile volatility the express desks trading Some PRACTICAL F

Constructing a Volatility Smile in Excel

233 onstructing a volatility smile using the Malz smile model builds understanding of the volatility surface market instruments. The Black-Scholes framework Ccan then be used to calculate strikes for different deltas to show how the market instruments impact strike placement within the volatility smile.

■ Task A: Set Up the Malz Smile Model

Recall the Malz formula for implied volatility at a given (positive) delta put from Chapter 12:

𝜎 𝜎 𝜎 . 𝜎 . 2 XDeltaPut= ATM + 2 RR25d (X − 50%) + 16 Fly25d (X − 50%)

This formula can be coded up in Excel: 234 CONSTRUCTING A VOLATILITY SMILE IN EXCEL ■ n netgt Parameters Delta Investigate versus and Volatility Implied Plot B: Task n utrypie hne hsbcmsese ftevltlt ml hr splaced is chart smile volatility the if easier becomes This change. prices butterfly and to 0% from smile volatility full a generate to delta: extended 100% be can output function The ■ approximations: ■ standard the with up matches volatility implied delta) 𝜎 𝜎 hc httevltlt ml pae sepce hnteAM ikreversal, risk ATM, the when expected as updates smile volatility the that Check plotted: be then can smile volatility The that Check Put Call 25 25 d d = = 𝜎 𝜎 ATM 𝜎 ATM 50 % + + et Put Delta 𝜎 𝜎 Fly Fly 25 25 d d = − + 𝜎 2 1 2 1 𝜎 ATM 𝜎 RR RR 25 n h 5 u et n 5 aldla(5 put (75% delta call 25% and delta put 25% the and 25 d d CONSTRUCTING A VOLATILITY SMILE IN EXCEL 235 ) X . ( T . N ) 2 2 𝜎 + 1 rCCY T − 2 √ 𝜎 . A strike input is rCCY ) ( X + ( T ) . 1 S K ) − ] ( 2 2 𝜎 is implied volatility, N 1 ln + 𝜎 1 = )− 1 ) 1 rCCY 1 d d − d ( 2 ( N N [ rCCY T T . . ( 1 1 − S T 2 are continuously compounded interest rCCY rCCY √ − − .𝜎 e ) e 1 rCCY + = = NORMSINV(X) for put Δ = T call S put S . 1 P P 𝜕 𝜕 1and 𝜕 𝜕 rCCY e and ( 1 = = ) is time to expiry (in years), − rCCY X N T e ( put call can be inverted to get the strike from the put delta: N Δ Δ = put K Δ is strike, K is the inverse cumulative normal distribution function. ) X ( 1 − is spot, N S Note that the put delta used within Black-Scholes formulas is its true (negative) These functions can be implemented first on the Excel sheet using The formula for NORMSDIST(X) for value rather than the positive quoted put delta. If the strike input and outputformulas are are equal correctly as implemented: other inputs change, this confirms that the where = used to generate the put delta, which is itself then used to generate a strike output. rates in CCY1 and CCY2, is the cumulative normal distribution function, and where The Black-Scholes framework candelta. be Recall used that: to get the equivalent strike for a given next to the inputsimplied volatility on axis the in the same chart Excel are fixed sheet, rather and than automatically the rescaling. low and high values of the Task C: Use Black-Scholes to Get Strike from Delta ■ 236 CONSTRUCTING A VOLATILITY SMILE IN EXCEL ■ akD wtht B ucin n ltImplied Plot Strike and versus Functions Volatility VBA to Switch D: Task ucinSrkFoPtet( sDul,Ptet sDul,rC1A _ As rCCY1 Double, As PutDelta Double, As StrikeFromPutDelta(S Function it make to parts debug: three and into follow split to is easier calculation The function. worksheet distribution Function End _ As rCCY1 Double, As K Double, As PutDeltaFromStrike(S Function function: worksheet Function End _ As Fly25d Double, As RR25d Double, VBA As as neater MalzSmileVol(ATM much Function are they simple: is but function sheet volatility the smile in Malz The messy functions. and long are functions These n Function End obe CY sDul,TA obe sDul)A Double As Double) As v Double, As T Double, As rCCY2 Double, Double As Double) As v Double, As T Double, As rCCY2 Double, Double As Double) As PutDelta Double, h tiefo u et B ucinacse h nes uuaienormal cumulative inverse the accesses function VBA delta put from strike The distribution normal cumulative the uses function VBA strike from delta put The 1=(o( )+(CY CY )*T v*Sqr(T)) * (v / T) * 2) / 2 ^ _ v * + T) rCCY1 * - Exp(-rCCY1 (rCCY2 = + PutDeltaFromStrike K) / (Log(S = d1 Double As d1 Dim _ * 16 + 0.5) - (PutDelta * RR25d * 2 + ATM = MalzSmileVol tiermuDla=S/EpApiainWrsetucin_ Exp(Application.WorksheetFunction / S = StrikeFromPutDelta 1 + Sqr(T) PutDelta * * part3=(rCCY2-rCCY1+0.5*v^2)*T v Double T) = As * part2 part3 Exp(rCCY1 Double, = As part1 part2 Double, As part1 Dim NrSn(at)*pr2-part3) - part2 * .NormSInv(part1) 1) - (Application.WorksheetFunction.NormSDist(d1) 2 ^ 0.5) - (PutDelta * Fly25d CONSTRUCTING A VOLATILITY SMILE IN EXCEL 237 Once matches are confirmed, the VBA functions can be combined to plot implied It is not possible to find strikes for 0 or 100 delta options, so replace 0% delta Again, the VBA functions can be tested by placing them alongside the existing volatility versus delta and implied volatility versus strike. with, for example, 0.01% and replace 100% delta with, for example, 99.99%: Excel functions: 238 CONSTRUCTING A VOLATILITY SMILE IN EXCEL ■ akE netgt oaiiySieStrike Placement Smile Volatility Investigate E: Task etsaeruhyeulysae,wt eaieysihl agrdfeecsfor differences distribution: larger spot terminal slightly the of relatively log-normality these the with for to strikes spaced, due the strikes equally topside smile, volatility roughly the no use are With to strike strike. deltas remembering developed, the investigated, calculate functions be to the delta can Using negative smile delta 90d. volatility to 25 75d, equivalent the and are 50d, within delta strikes 25d, placement these 10 10d, framework are Malz deltas: tenor the put In given these ATM. a the plus at calls, strikes and important puts most the practice, In CONSTRUCTING A VOLATILITY SMILE IN EXCEL 239 Changing the risk reversal moves the strikes for a given delta further away from Increasing the butterfly causes the strikes to move further from the ATM, with a Increasing implied volatility or time to maturity causes the terminal distribution Reducing implied volatility or time to maturity causes the terminal distribution the ATM on the highside side of of the the volatility volatility smile: smile and closer to the ATM on the low larger impact at lower delta strikes due to the higher implied volatility: to widen and hence the strikes are positioned further from the ATM: to tighten and hence the strikes are positioned closer to the ATM: 240 CONSTRUCTING A VOLATILITY SMILE IN EXCEL effect. lower: moves smile volatility whole the hence and lower move to forward oigCY neetrtshge rCY neetrtslwrhsteopposite the has lower rates interest CCY1 or higher rates interest CCY2 Moving the causes lower rates interest CCY2 or higher rates interest CCY1 Moving CHAPTER 13

Probability Density Functions

volatility smile at a given maturity can be converted into an equivalent Aprobability density function (pdf). The probability density function contains useful information because integrating an area under the curve gives the likelihood 241 of spot being within the given range at maturity. Starting with the simplest case, Exhibit 13.1 shows a 1yr volatility smile with 10% volatility for all strikes (i.e., pure Black-Scholes world). This volatility smile generates the standard log-normal bell-shaped pdf shown in Exhibit 13.2. The method of generating pdfs from option prices is explored in Practical G. Exhibit 13.3 shows implied volatility rising to 15% for all strikes. Increased volatility widens the distribution and the pdf extends out on both sides as shown in Exhibit 13.4. The area under the pdf represents a probability space. Therefore the total area under the pdf is always equal to 1 and the pdf function can never go negative. If the pdf does go negative, that indicates a potentially arbitrageable volatility surface. This can manifest itself in many different ways within pricing or risk management systems, most visibly via incorrect implied volatility or unstable gamma exposures. Exhibit 13.5 shows how the volatility smile changes when positive wings are added. As shown in Exhibit 13.6, higher wings in the volatility smile causes the pdf to rise in the wings (often called ‘‘fat tails’’) and rise around the ATM but fall in between, 242 PROBABILITY DENSITY FUNCTIONS lhuhtettlae ne h d utrmi nhne.Dsrbtoswith Distributions unchanged. remain must pdf the under area total the although 13.2 EXHIBIT 13.1 EXHIBIT oaiiysd,tepfsrthsfrhra xetdsnetehge oaiiycauses higher volatility the higher On the tilt. since to expected pdf as the further causes stretches also pdf skew the side, The volatility 13.7. Exhibit in shown as other, as markets. known are shape this digpstv rngtv kwcue h oaiiysiet itoewyo the or way one tilt to smile volatility the causes skew negative or positive Adding oaiiysiewt a 0 mle volatility implied 10% flat with smile Volatility rbblt est ucinfo a 0 oaiiysmile volatility 10% flat from function density Probability leptokurtotic n hyaeotnosre nfinancial in observed often are they and PROBABILITY DENSITY FUNCTIONS 243 Probability density functions from flat 10% and flat 15% volatility smiles Volatility smiles with flat 10% and flat 15% implied volatility the distribution to widen but theoccurs peak because of the the no-arbitrage pdf moves conditionfuture the ensures value opposite that of direction. spot. the This Therefore, forward if is probabilityof mass the the moves smile expected into due the to wings higher onother volatility, one way. the side center This of is the shown probability in mass Exhibit must 13.8. shift the EXHIBIT 13.4 EXHIBIT 13.3 244 PROBABILITY DENSITY FUNCTIONS ■ a-aldDistributions Fat-Tailed XII 13.6 EXHIBIT 13.5 EXHIBIT h elzddsrbto ftnyaso al o-eun nUDJYso versus spot USD/JPY volatility. in same log-returns the daily with distribution of normal normal years theoretical ten the the shows of 13.9 versus Exhibit distribution phenomenon. kurtosis realized well-known the excess a is (i.e., markets financial distributions in distribution) fat-tailed of existence The oaiiysie ihflt1%adpstv igipidvolatility implied wing positive and 10% flat with smiles Volatility rbblt est ucin rmflt1%adpstv igvltlt smiles volatility wing positive and 10% flat from functions density Probability PROBABILITY DENSITY FUNCTIONS 245 volatility of (in emerging market spot jumps Probability density functions from flat 10% and negative skew volatility smiles Volatility smiles with flat 10% and negative skew implied volatility (in freely floating currency pairs) or Fat tails in the spot distribution can be most easily explained by The difference around the peak is easy to see, but to see the wings more clearly a change to asample, log-scale the largest for down frequency moves occurred is roughlythan ten required a million normal times as more distribution in frequently would suggest. Exhibit 13.10. Within this volatility EXHIBIT 13.8 EXHIBIT 13.7 246 PROBABILITY DENSITY FUNCTIONS 03lgscale) 2013/log 13.10 EXHIBIT 13.9 EXHIBIT itiuin osdrasml oe hr h ptdfuinhsa qa hneof chance equal an has diffusion spot the where model financial simple a within consider distribution, isolation in applied be to these tend 19). of Chapter they combination (see simplicity a models for usually is but reality effects The two pairs). currency pegged/managed or oudrtn o oaiiyo oaiiygnrtsfte al ihntespot the within tails fatter generates volatility of volatility how understand To S/P elzdvru hoeia o al hnedsrbto 20 o2013) to (2003 distribution change daily log theoretical versus realized USD/JPY S/P elzdvru hoeia o al hnedsrbto 20 to (2003 distribution change daily log theoretical versus realized USD/JPY PROBABILITY DENSITY FUNCTIONS 247 Three-state volatility model average pdf Three-state volatility model pdfs The probability density function of the three-state model is an equal combination having 5%, 10%, or 15% volatility.are The shown three separate in probability Exhibit density 13.11. functions of the three states.a Therefore, fat-tailed Exhibit pdf 13.12 compared shows to how a averaging static 10% the volatility pdfs distribution. gives The low-volatility EXHIBIT 13.12 EXHIBIT 13.11 248 PROBABILITY DENSITY FUNCTIONS ■ ofiec Intervals Confidence ugssta,freape hr sa5%cac fso en ihnte50% the surface. downside volatility within the GBP/USD being the to in spot further skew stretch of downside bounds the chance to the 50% due surface how a Note volatility bounds. is The interval there GBP/USD. confidence example, in for the chart that, 90%), interval suggests or 13.13 confidence Exhibit 50% example levels. (e.g., spot spot an appropriate intervals suggests the shows probability for smile queried given volatility is function For the density future. probability how the visualizing in of move way may charts, interesting interval an confidence generate are to used which be also can functions density Probability 19 models. Chapter See pricing tails. derivatives fatter FX gets on pdf details the the more and occur. frequency, higher for not or move does magnitude smile jump volatility in the a increase of if jumps wings spot or increases current volatility around of it volatility keeping As and occurs jump the a contributes if wings state high-volatility the while peak tails. higher fatter the contributes state XII 13.13 EXHIBIT h rsneo ptjmshsasmlripc,psigpoaiiyms nothe into mass probability pushing impact, similar a has jumps spot of presence The B/S ofiec intervals confidence GBP/USD PROBABILITY DENSITY FUNCTIONS 249 2.65%6.05% 0.25% 0.40% 0.70% 1.85% − − 1.75% 3.45% − − EUR/CHF 1mth and 1yr Market Instruments at June 1, 2011 EUR/CHF spot in 2011 and 2012 However, there are instances where these parameterizationsIn cause mid-2011, EUR/CHF problems. was freely floating and theThe EUR/CHF market corresponding instru- volatility smile and probability density function for 1yr EXHIBIT 13.15 1mth1yr 11.15% 12.10% EXHIBIT 13.14 Tenor ATM 25d RR 10d RR 25d Fly 10d Fly Exhibit 13.14 shows EUR/CHF spot from 2011 and 2012. ments at June 1, 2011 were as per Exhibit 13.15. EUR/CHF are shown in Exhibit 13.16. As expected, the higher implied volatility number of parameters. On some trading25d desks the RR, five 10d market instruments—ATM, RR,parameter, 25d Fly, one 10d Fly—are wingcurrency used; parameter, pairs on these reduced-form and others, parameterizations perhaps work oneof well just information and skew one they to allow be ATM parameter a expressed lot in are an efficient used. manner. In liquid Within FX derivatives trading desks, volatility smiles are expressed using a limited Limitations of Volatility Smile Parameterization ■ 250 PROBABILITY DENSITY FUNCTIONS nevninlvlwsrmvd hr ssgicn iko ptjmiglwr as the lower, if jumping because spot tail, of longer risk a significant practice. the has is in above, happened there pdf but removed, the to, was of close level is downside intervention year the one Plus, in position level. spot intervention likely most The 13.17. a Exhibit have could spot point which at policy, its change to decides above bank freely central floats The spot level; 2. 1.2000 the defends successfully bank could central that The scenarios two 1. are there world out: simplified played vastly have a potentially In it like? prevent look to function order in market 2015. the early in until actively 1.2000) spot they below EUR/CHF which going 1.2000, bought at market they spot (i.e., the in defended floor a establish density bank central probability Swiss the causes downside. reversal) the to risk more stretch negative to (i.e., function strikes downside for 13.16 EXHIBIT ye fvltlt ml a egnrtd hsmasta nycrantpsof types certain only that means This generated. be can smile volatility of types 13.19. Exhibit in intuition the onto overlaid is quotes these hncntutn oaiiysrae sn iie aaee e,ol certain only set, parameter limited a using surfaces volatility constructing When from derived pdf The later. year one instruments market the shows 13.18 Exhibit like look to expected be intuitively would pdf the scenarios, these Given density probability the should what defended, actively was floor the While the saw onward 2011 late from announcements and interventions of series A i dutetlower. adjustment big lower. any go cannot but level this U/H y mle oaiiysieadpfa ue1 2011 1, June at pdf and smile volatility implied 1yr EUR/CHF PROBABILITY DENSITY FUNCTIONS 251 EUR/CHF 1mth and 1yr Market Instruments at June 1, 2012 EUR/CHF 1yr actual and intuitive pdfs at June 1, 2012 EUR/CHF 1yr intuitive pdf EXHIBIT 13.18 EXHIBIT 13.19 Tenor1mth1yr ATM 2.00% 25d 6.70% RR –0.70% –4.60% 10d RR –1.40% –9.65% 25d Fly 1.15% 1.80% 10d Fly 3.75% 6.10% EXHIBIT 13.17 252 PROBABILITY DENSITY FUNCTIONS osntmthtemre,o ieversa. levels vice spot or key market, around the strikes match individual not for does volatility implied the but values way market no is there but parameterizations. level standard intervention within an information significantly this beyond is include just level to ending intervention spot an inside than take just likely ending should more spot construction that surface pdf fact volatility complex the more the account a Ideally into similar, reality. or fit level, better intervention often floor, would peg, currency In a dynamics. is spot there complex where more pairs which with pairs generated, currency be in can problem function a presents density probability peaked) single (i.e., unimodal h yia ann ino hsisei htmre ntuet i h correct the hit instruments market that is issue this of sign warning typical The PRACTICAL G

Generating a Probability Density Function from Option Prices in

Excel 253

his practical introduces a method of generating a probability density function from a volatility smile by numerically differentiating vanilla option prices twice Twith respect to the strike. The code reuses volatility smile functions developed in Practical F and vanilla options pricing functions developed in Practical C. Probability density functions are explored in detail within Chapter 13. First, volatility smile inputs and market data must be defined within the Excel sheet. Then a range of delta values is established, from 0.1% to 99.9% in tight steps of 0.1%: 254 GENERATING A PROBABILITY DENSITY FUNCTION n Sub End a populateSmileStrikesAndVols() takes Sub StrikeFromPutDelta that input: Note as StrikeFromPutDelta used. value and delta being MalzSmileVol put can negative The F surface. Practical values sheet from the calculate the functions to subroutine on VBA a them use place to better and is it sheet the on data of amount h mle oaiiyadsrk utb acltdfrec et au.Det the to Due value. delta each for calculated be must strike and volatility implied The hl ag(VltltSiee".fstDlaon,0) Range("VolatilitySmileRef").Offset(DeltaCount, While 1 = DeltaCount Long As DeltaCount Double Dim As ImpliedVol Double Dim As InputPutDelta Dim Wend etCut=Dlaon 1 + DeltaCount = DeltaCount _ = 2) ImpliedVol Range("VolatilitySmileRef").Offset(DeltaCount, = 1) Range("VolatilitySmileRef").Offset(DeltaCount, _ Range("RR25d"), MalzSmileVol(Range("ATM"), = 0) ImpliedVol Range("VolatilitySmileRef").Offset(DeltaCount, = InputPutDelta ag(rC1) ag(rC2) ag(T) ImpliedVol) Range("T"), Range("rCCY2"), Range("rCCY1"), _ -InputPutDelta, StrikeFromPutDelta(Range("Spot"), InputPutDelta) Range("Fly25d"), <> "" GENERATING A PROBABILITY DENSITY FUNCTION 255 Dim PDFStrikeCount As Long, SmileDeltaCount As Long A VBA function is used to calculate the implied volatility and equivalent option Next, in a new column, define equally spaced strikes for calculating the probability volatility since delta has small increments.Cisreused: The OptionPrice function from Practical Sub populatePDFImpliedVolsAndPrices() price at each strike level. It is okay to use linear interpolation to generate the implied density function (pdf): 256 GENERATING A PROBABILITY DENSITY FUNCTION n Sub End hl ag(SrkRf)Ofe(DSrkCut 0) Range("StrikeRef").Offset(PDFStrikeCount, While 1 Double = As PDFStrikeCount ImpliedVol Double, As Double InputStrike As Dim HighVol Double Double, As As HighStrike LowVol Double, Dim As LowStrike Dim Wend DSrkCut=PFtieon 1 + PDFStrikeCount = PDFStrikeCount _ = 2) ImpliedVol Range("StrikeRef").Offset(PDFStrikeCount, = 1) Range("StrikeRef").Offset(PDFStrikeCount, Wend _ 0) 1, + Range("VolatilitySmileRef").Offset(SmileDeltaCount While 1 = SmileDeltaCount -1 = 0) ImpliedVol Range("StrikeRef").Offset(PDFStrikeCount, = InputStrike ag(rC1) ag(rC2) ImpliedVol) _ Range("rCCY2"), Range("T"), Range("rCCY1"), InputStrike, Range("Spot"), OptionPrice(False, <> mlDlaon mlDlaon 1 + SmileDeltaCount = SmileDeltaCount If End Vol Implied (InputStrike get If to Interpolation 'Linear _ Range("VolatilitySmileRef") = HighStrike _ Range("VolatilitySmileRef") = LowStrike _ Range("VolatilitySmileRef").Offset(SmileDeltaCount = 1) HighVol Range("VolatilitySmileRef").Offset(SmileDeltaCount, = LowVol "" < 2) 1, + .Offset(SmileDeltaCount 2) .Offset(SmileDeltaCount, +1,1) ihtie Then HighStrike) mleVl=Lwo HgVl-Lwo)*(nuSrk _ - (InputStrike * LowVol) - (HighVol + LowVol = ImpliedVol oSrk)/(ihtie-LowStrike) - (HighStrike / LowStrike) > oSrk)Ad(nuSrk _ (InputStrike And LowStrike) <> "" GENERATING A PROBABILITY DENSITY FUNCTION 257 Note that the probability density (second derivative of option value with respect Probability density functions can then be compared by copying the output values. The probability density can now be calculated by finding the second derivative of When making changes torerun the both volatility VBA smile subroutines or in order market to data correctly inputs, set up remember the to calculation. to strike) takes a similarwith shape respect to to that spot). of gamma (second derivative of option value price with respect to strike on the sheet: 258 GENERATING A PROBABILITY DENSITY FUNCTION ltvltlt ml essln osd ikrvra oaiiysmile: volatility reversal risk topside long versus smile volatility Flat smile: volatility wings long versus smile volatility Flat GENERATING A PROBABILITY DENSITY FUNCTION 259 Finally, as discussed in Chapter 13, the area under the probability density function level and summingstrike the spacings are total. tight This enough, theand indicates total 1.01. probability how value should accurate be between the 0.99 output is. If the should equal one sincemultiplying it the average represents pdf a between probability strikes mass. by This the change can in be strike checked at by each strike

P A R T III

VANILLA FX DERIVATIVES TRADING

he material up to this point has been developed within a stylized framework. This is important because the increased clarity makes learning easier. However, Tit is now time to confront some of the real-world issues faced by vanilla FX derivatives traders.

CHAPTER 14

Vanilla FX Derivatives Trading Exposures

263 n Part I, Greek exposures were examined within a stylized framework. In practice, Ieven the simplest Greek exposures have additional layers of complexity that must be understood by FX derivatives traders within their risk management.

■ Delta

Delta (Δ) is one of the most important exposures to a derivatives trader—the sensitivity of price to a change in the underlying. It is a simple concept but there are many possible variations in exactly what the delta exposure represents.

Spot Delta versus Forward Delta Apologies if this is obvious, but the spot delta on a spot deal is 100% of the notional. Therefore, buying GBP50m GBP/USD spot results in longer GBP50m GBP/USD spot delta exposure within the trading position. For this reason, FX spot traders talk only in terms of net long or short positions, rather than their exposures. The forward delta on a forward outright contract is 100% of the notional. Selling USD100m of 1yr USD/CAD forward outright results in a shorter USD100m USD/CAD 1yr forward delta exposure within the trading position. 264 VANILLA FX DERIVATIVES TRADING EXPOSURES sused: is terms. CCY2% and CCY1% between where used: is is option GBP/USD a GBP/USD. on delta in the given if as example, quoted also For mistakes. is cause example, can delta For terms clear. CCY2 option immediately the are has direction when option and so an deals amount forward hedge terms, the and CCY1 terms Spot in CCY1 pair. CCY1 traded currency with that generally terms cash for are CCY1 convention or market CCY1% by in delta determined option quote often most Traders Conventions Quoting Delta points swap and rates G10 interest in to particularly exposures monitored. delta, case, additionally hedge the be must is to this used When often pairs. most currency is exposure spot delta because spot management residual a swap, AUD100m FX versus notional spot equal an of in remains. AUD100m traded when is example, around forward for is of factor that, discount means AUD delta This 5yr spot and the 0.84. a 1 when has AUD42m below outright approximately forward is of 5yr factor exposure AUD/USD discount of AUD50m the buying are), example, usually For they (as positive are and delta, spot the where trade: specific a on hl C2i od rvc es ihna Xtransaction. FX an within versa vice or sold, is CCY2 while where h eaiesgsaepeetwti hs omlsbcueCY sbought is CCY1 because formulas these within present are signs negative The formula following the delta spot CCY2% to delta spot CCY1% convert To risk derivatives FX within used often most exposure delta the is delta Spot Therefore, flows. cash like valued future or valued present be can exposures Delta ocnetCY%fraddlat C2 owr et h olwn formula following the delta forward CCY2% to delta forward CCY1% convert To F df S steforward. the is sso and spot is stedson atri h et urnyt h owr maturity, forward the to currency delta the in factor discount the is short long S1m h et eg tcretso ol eto be would spot current at hedge delta the USD16m, Δ C1dlaexposure: delta CCY1 K F stefraddla hnitrs ae ntedlacurrency delta the in rates interest When delta. forward the is stesrk.Nt htwhen that Note strike. the is Δ Δ CCY F CCY Δ=Δ 1 1 % % =−Δ =−Δ Sell F . CCY F CCY df pto h eg.Qoigdlain delta Quoting hedge. the on spot 2 % 2 % . K S . K F S = K h et sunchanged is delta the , | sell Δ | GBP10m < | Δ Δ F | is . VANILLA FX DERIVATIVES TRADING EXPOSURES 265 50% around + Premium − Premium 2 CCY exposure is generated since USD are paid out =Δ Premium 1 CCY Δ 100% (with spot far above the strike), hitting short USD delta + Long vanilla call delta (premium paid in CCY2) with 1.0000 strike The intuition that the inclusion of premium moves delta shorter is as follows: In general: This analysis assumes that the option premium is paid in CCY2In (the USD/JPY, domestic consider a long USD call option with JPY as the natural P&L At higher spot, the USD premium will(domestic) be terms. relatively more expensive to pay in JPY At lower spot,(domestic) the terms. USD premium will be relatively cheaper to pay in JPY EXHIBIT 14.1 ■ where delta and premium are quoted in the same terms. ■ currency but premiumoption, paid an additional in USDin by the premium. market convention. When buying this below the strike) to the ATM, as shown in Exhibit 14.1. currency). If the option premiumexposure is changes. paid in CCY1 (the foreign currency), the delta CCY1 versus CCY2 Premium Delta The standard delta profile for a long vanilla call option goes from 0% (with spot far 266 VANILLA FX DERIVATIVES TRADING EXPOSURES et Bleed Delta currency. G10 the EUR ordered: be approximately will be currency > premium can the currencies spot then G10 USD, Other with contains USD. pair impact currency larger the If a convention. has 14.4. time Exhibit premium in this CCY1 shown as a which downside Again, the delta, strike). to the short in-the-money above additional far spot an (with causes 0% 50%. stylized to the strike) from the away below far tenors be longer to At options ATM 14.3. of Exhibit in-the-money delta in the deep shown cause is is can This effect option premium. this the large when a has large therefore particularly and be will adjustment delta 14.2 EXHIBIT rdn as pcfial,taesrl h oio aefradna h n fthe of end the near forward date over horizon constant the roll few stay next traders the Specifically, not over days. (bleed) do change trading exposures positions how investigate trading therefore Traders derivatives time. within exposures Greek NZD nG0vru mrigmre urnypis h rmu suulypi in paid usually is premium the pairs, currency market emerging versus G10 In market by determined is pair currency particular a in currency premium The far spot (with –100% from goes profile delta stylized the option, put long a For The 14.2. Exhibit in shown is profile premium option CCY1% call long The > CAD ogvnlacl pinpeimwt .00strike 1.0000 with premium option call vanilla Long > CHF > NOK > SEK > JPY. > GBP > AUD VANILLA FX DERIVATIVES TRADING EXPOSURES 267 Long vanilla put option delta (premium paid in CCY1 or CCY2) with Long vanilla call option delta (premium paid in CCY1 or CCY2) with EXHIBIT 14.4 1.0000 strike EXHIBIT 14.3 1.0000 strike 268 VANILLA FX DERIVATIVES TRADING EXPOSURES ntemnya auiy oe iet auiygvsso estm omove to time less spot gives maturity to time Lower maturity. at in-the-money ■ ■ ■ ■ and spot ■ option: of call positioning vanilla relative long the a for of cases strike possible the three are there date expiry to the them allows This day. trading next the particularly on bleed, look position will assess position their how see to day 14.5 EXHIBIT ■ Therefore: strike. the through o na-h-oe AM pt(.. ptcoet h tie,dlaremains delta strike), the to close spot time. over (i.e., unchanged spot bleeds roughly (ATM) delta at-the-money strike), an below For spot (i.e., spot (OTM) shorter. out-of-the-money an For o nOMoto,a iet xiysotn,tepoaiiyo nigu ITM up reduces. ending delta) of the probability hence the (and shortens, expiry maturity to at time as option, OTM an For bleed) delta –2% (i.e., delta 29% put: call/USD EUR 1.3750 6-day delta 31% put: call/USD EUR 1.3750 7-day longer. bleeds delta strike), above spot (i.e., spot (ITM) in-the-money an For nutvl hsmkssnewe et stogto stecac fedn up ending of chance the as of thought is delta when sense makes this Intuitively Example to Prior time. over changes option call vanilla a on delta how shows 14.5 Exhibit U/S pt 1.3650. spot: EUR/USD : ail aloto et ih100 tieoe time over strike 100.00 with delta option call Vanilla et bleed delta . VANILLA FX DERIVATIVES TRADING EXPOSURES 269 50% of the + traders with their risk management: assists 50%. On the expiry date itself and with spot unchanged delta 50%. On the expiry date itself and with spot unchanged delta + + 100% above the strike. Therefore, delta bleed will be + : EUR/USD spot: 1.3650. In each of these scenarios, the delta bleed matches a natural risk management Generally, delta bleed The delta bleed for a particular strike increases as the expiryConsider a date long approaches call option: Example For a long put option the ITM versus OTM side is switched (e.g., spot below the delta will bleed longer. If the trading position isdelta will mainly bleed long shorter. topside or short downside vanilla options, approximately will be 0% below thenotional. strike. Therefore, delta bleed will be –50% of the option If the trading position is mainly long downside or short topside vanilla options, On the day beforeapproximately maturity, if spotwill is be slightly aboveoption the notional. strike, delta will be On the day before maturity, if spot is slightly below the strike, delta will be 7-day 1.3750 EUR put/USD call: –69% delta 6-day 1.3750 EUR put/USD call: –71% delta (i.e., –2% delta bleed) For an ITM option, asat time maturity to (and expiry hence shortens, the the delta) probability increases. of ending up ITM For an ATM option,ITM as at time maturity to (andmately expiry hence 50%. the shortens, delta) the remains probability roughly of unchanged ending at up approxi- preference to run deltatopside into gamma) short and to gamma rundelta areas delta (i.e., if away running long from long long downsidebalanced gamma delta P&L gamma). profile. areas if Delta (i.e., short running bleed long therefore often helps to produce a ■ ■ ■ ■ with the most dramatic deltaoptions bleed with occurring large into notionals, the traders expiry pay date close itself. attention On to vanilla these delta changes. the delta bleed on call options anddue put to options put–call ends parity. up being equivalent, as expected ■ ■ ■ strike is OTM for a call but ITM for a put) but additionally, put deltas are negative so ■ 270 VANILLA FX DERIVATIVES TRADING EXPOSURES ■ am n Theta and Gamma fgma ihrvolatility higher gamma, of strike the into rising profile; gamma corresponding the maturity. at to shape identical more an the takes paid moves, be must spot theta gamma. as more long make way, changes more another to of delta Put privilege ability hedging. the more delta for the the from gives made and be gamma moves can money long spot e.g., of out because, money sense intuitive makes This theta. (negative) paying causes gamma (positive) long where per (as change value of source volatility additional implied an day, is 11). the Chapter maturities options throughout shorter This Second, vanilla at theta). date. out-of-the-money lower (i.e., horizon moving slightly jump new for P&L the large a on particularly is expiring there be forward, will rolled jump is P&L horizon the when First, rate interest and forward? volatility rolls implied date the horizon the within the used to as clear are happens curves seems assumptions what what This example, know day. (positive traders For trading that benefit calculation. full important net one is or for it position but theta) enough trading (negative the cost a holding net from from the P&L theta) represents the therefore as It quoted date. usually is it spot. tools in move 1% a for quoted usually is it tools management time. over exposures, theta gamma exposures, short earns gamma produces which options long vanilla Selling produces time. options over theta vanilla costs buying which 9, Chapter in discussed As i et hneoe ie ptcag,weesi oaiiyi ih hr will there high, is volatility a if be whereas will change, there spot low, is given volatility a If over profile: change delta delta the big consider why, understand To have they therefore and volatility away. implied decay to higher value at more premium higher have options diinly o h aeaon ftea ihrvltlt asslwrgamma. lower causes volatility higher theta, of amount same the for Additionally, amount same the for that is formula gamma-theta the from note to thing next The 14.6 Exhibit in shown profile theta the proportional, are theta and gamma Since proportional are theta (negative) and gamma formula, the at Looking formula: this by linked are theta and gamma Black-Scholes, Under ways. different two in time with changes value position cumulative practice, In Theta Gamma 𝜎 ( svltlt and volatility is 𝜃 ) (Γ) stert twihpiecagsoe ieadi ikmanagement risk in and time over changes price which at rate the is stert twihdlacagsa ptmvsadi risk in and moves spot as changes delta which at rate the is S sso.Tengtv ineit eas,frexample, for because, exists sign negative The spot. is ( 𝜎 ) assteat iengtvl.Itiiey vanilla Intuitively, negatively. rise to theta causes 𝜃 =− 2 1 Γ 𝜎 2 n-a shift one-day S 2 owr ntehorizon the in forward Γ∝− ∝ (Γ 𝜃 ) . VANILLA FX DERIVATIVES TRADING EXPOSURES 271 Long vanilla option with 100.00 strike theta over time In practice, this effect occurs within the entire volatility surface as all strikes roll Traders actively track the gamma/theta ratio in their trading positions. Within Forward Roll Theta With everything else fixed constant, butforward the outright horizon shifted to forward a by fixed one expiry day, date the drifts toward spot. For a given option this is positions) as the ATM impliedcloser volatility to the drops horizon. over If time thethan as ATM 1mth curve the is ATM), expiry downward date the slopingcurve, moves (e.g., opposite the 1yr larger effect ATM the impact. lower occurs. Exhibit 14.7 The demonstrates more the ATM steeply curve roll sloped process. to the closer ATM maturities. ATM Curve and Volatility Smile Roll Theta Over time, options roll downexample, the 365 ATM days curve to as 364 days, an1yr and option so ATM maturity on. higher goes If than from, the ATM 1mth for long curve ATM), vanilla is option this positions upward (and effect sloping additional (i.e., causes theta to additional be theta earned on to short vanilla be option paid on financial crisis, buying 1mthwhile buying AUD/JPY 1mth ATM USD/HKD ATM at today at 50% 0.25% volatility volatility gives gave 490%the 3% gamma. Black-Scholes gamma framework, thetacomes simply from pays different aspects for of gamma, the trading but position. in practice theta EXHIBIT 14.6 be a much smaller delta change over the same fixed spot change. During the 2008 272 VANILLA FX DERIVATIVES TRADING EXPOSURES pin auduigteAMcre(called curve ATM the time. using over valued options hold position. to smile expensive long gamma-theta are a holding positions the of smile costs long (recall the pairs, of currency volatility one high-skew as implied In of thought higher be can the This to formula). due ratio gamma/theta ratio the gamma/theta on impact significant a the with is position trading a of aspect One Gamma Smile are they how know understand traders and that position important their is in It day-to-day. balances future. managed cash against the these usually into monitor traded, balances to be cash how can push swaps to FX USD, Alternatively, hence the cash. balances, cash the short on deposit, offset over to on interest borrowing balances paying position and cash cash long trading the putting on a by interest earning managed in hence be accumulate can balances payments deals, cash forward cash These or time. spot other settled or from and balances premiums, at AUD10m Cash option settle date. long spot position into the trading settles on spot USD9.5m derivatives short AUD/USD a 0.9500 in AUD10m deals Long spot maturity. when happens what Consider Theta Balance Cash differential rate interest high in but important. theta, from be option impact can total P&L it the The pairs of points. currency part swap small one-day a the only to is equal this move spot free a to similar 14.7 EXHIBIT ■ the is exposurecalculatedfromoptionsvaluedusingthefullvolatilitysurface.Thedifference oaiiyado hr ail pin rcda oe mle oaiiyo the implied on higher volatility negative. at be implied will lower priced effect gamma at options smile priced net vanilla the options smile, long vanilla mainly short is and/or volatility position trading the If ail rdr oeie oka hi am xouecluae rmvanilla from calculated exposure gamma their at look sometimes traders Vanilla ml am effect gamma smile ml position smile T uv roll curve ATM wigotoshg ntevltlt ml eut nalower a in results smile volatility the on high options Owning . h muto am oigfo h smile: the from coming gamma of amount the ; T gamma ATM n opr tt h gamma the to it compare and ) VANILLA FX DERIVATIVES TRADING EXPOSURES 273 from a short gamma trading position is made Stylized P&L/theta distribution from long gamma position from a long gamma trading position (ignoring P&L from vega, rho, new lost Similarly, the most P&L that can be A common issue for risk managers is a low gamma/theta ratio within their trading Ifthetradingpositionismainlylongvanillaoptionspricedatlowerimpliedvolatilityand/or short vanilla options priced at higher impliedsmile volatility on gamma the effect smile, will the be net positive. EXHIBIT 14.8 contains many small lossesdistribution is and zero. few large gains, although thetheta expectation while of (much) the more can be lost if spot is highly volatile. Exhibit 14.9 shows P&L Distributions from Long Gamma or ShortOn Gamma a given trading daycan be in a stylized Black-Scholesdeals, etc.) world, is the theta while maximum more P&L (potentiallyvolatile. much that Exhibit more) can 14.8 be shows made if the spot is P&L highly distribution from a long gamma position; it higher premiums priced usingoptions the have smile almost volatility no (e.g., smilebecause 0.10% gamma they to effect are 0.20%). (they low contain These negative delta), little theta. gamma yet of long any positions kind in them can contribute significant position. To fix thisnegative issue, smile traders gamma search effect.smile for It is options high is compared in worth to the their noting volatilityhave base, position that very this with approach low in can premiums large currency overlook priced options pairs that using where the the ATM volatility (e.g., up to 0.02%) but ■ 274 VANILLA FX DERIVATIVES TRADING EXPOSURES ■ eaadWihe Vega Weighted and Vega feta ogrmtrte.I steeoentsf oasm htvg nATM on vega that shows 14.10 assume Exhibit to example, vega. safe ATM For AUD/JPY maturities. large not long-dated longer a therefore at have is increases can always It discounting changes contracts practice maturities. generally in longer vega time, at stylized of while effect root so square expressed, the is to vega it proportionally delta, currency Like vega. the is in contracts derivatives discounts most on exposure important most The whichever though, with time. the interview, trading at happy job risk/reward equally best a are the In you gives position positions that positions. gamma obviously gamma gamma is long short answer trade correct trade to the to prefer prefer traders others Some while books. trading their zero. position is distribution the and of gains expectation small the many again, contains although, It losses, position: large gamma few short a from distribution P&L the 14.9 EXHIBIT fteAMcremvsi egtdmne,i steeoeepce ht for that, expected therefore is it manner, weighted a in moves curve ATM the If vega weighted vega a EUR/USD, in preferred. example, be For may currency. USD P&L in the quoted is exposure that if CCY2 in viewed be tnadvg aclto sue h T uv oe nprle.A parallel. in moves curve ATM the assumes calculation vega standard A will vega sometimes but terms, CCY1 in exposures vega view often most Traders managers risk how impacts distribution P&L gamma short or long a for Preference tlzdPLteadsrbto rmsotgmaposition gamma short from distribution P&L/theta Stylized aclto sue h T uv oe rprinlto proportional moves curve ATM the assumes calculation √ 1 t . VANILLA FX DERIVATIVES TRADING EXPOSURES 275 a in neither into weighted vega t will most accurately predict at time ) 𝜐 vega ( .𝜐 . t weighted t will most accurately predict P&L changes weighted t √ = weighted weighted vega 𝜐 AUD/JPY ATM vega . It is therefore important to know which reference pillar the weighted at the weighted reference time ) Exhibit 14.11 gives vega multipliers for converting bucketed vegaIf at the ATM market curve is moving in a parallel manner, The following formula is used to convert vega The weighted vega calculation folds all exposures into a single reference pillar weighted 𝜐 caused by changes to the ATMparallel curve. nor Most weighted often, manner the ATM butview curve some both moves combination in measures of within the their two. riskcurrently Traders management moving and therefore when assess deciding which how exposure the to ATM pay curve closest is attention to. tenors into 1mthcalculate weighted weighted vega vega. on each The option individually. same methodologyP&L can changes also caused by bemoving changes in used a to weighted manner, the to ATM curve, whereas if the ATM curve is ( (usually 1mth or 3mthcurrency pair). depending For on example, whichfull a vega tenor 3mth exposure weighted is in vega most thethe of 3mth position liquid USD400k tenor is in implies equivalent that a tovega the calculation a particular uses. vega exposure of USD400k EXHIBIT 14.10 example, the 3mth ATM movesexposures twice where as 3mth much ATM vega as is the half 1yr that ATM. of This the links 1yr to ATM. vega 276 VANILLA FX DERIVATIVES TRADING EXPOSURES ■ dpe Greeks Adapted osdrafiiedfeec ptdlacluaini hc pti ee pand up flexed is spot which in calculation delta spot difference finite a Consider Delta Adapted long a on arguments. delta logical the these of following think when to easiest option is call it vanilla and volatility smile the using given Black-Scholes exposures correct the using manage risk behavior. can market they prevailing so exposures adapted 0.129 and 0.204 methodology. so-called delta These sticky 0.289 a moves. assuming forward the as constant 0.333 0.408 within used is framework. volatility 0.577 calculated constant Black-Scholes one are the only Greeks since methodology Black-Scholes strike 0.707 Standard sticky a moves. assuming forward the 1.000 as constant 2.236068 stays 1.472 a 1.414214 using 2.082 calculated be 1.000000 can delta options 5.515 sticky vanilla for exposures 0.866025 Greek 5.00000 0.707107 2.00000 0.500000 1.00000 0.408248 0.288675 0.75000 0.196116 0.50000 0.138675 5yr 0.25000 0.052342 2yr 0.16667 1yr 0.08333 0.03846 9mth 0.01923 6mth 3mth 0.00274 2mth 1mth 2wk 1wk O/N Tenor h mle oaiiyfrtefie tiecagsa ptflxsu n on hsis This down: calculation and up delta flexes adapted spot an as changes in strike but the fixed constant the for remains volatility valued implied is the vanilla volatility implied the to the which flex calculation, spot delta at the Black-Scholes by standard divided the is In change delta. premium calculate resultant the and amount small a down XII 14.11 EXHIBIT hogotti analysis, this Throughout exposures standard between differences the understand traders that important is It and date expiry given a for volatility implied the delta, sticky Under and date expiry given a for volatility implied the strike, sticky Under adaption methodology. . t egtdVg utpir 1t reference) (1mth Multipliers Vega Weighted (years) lc-coe delta Black-Scholes √ t eest h et acltdunder calculated delta the to refers dpe Greeks adapted 1t reference) (1mth Multipliers Vega Weighted tcystrike sticky r calculated are delta strike stays ra or VANILLA FX DERIVATIVES TRADING EXPOSURES 277 for downside 𝜎 dS d 𝜐. + Scholes − quantity, consider a finite Black 𝜎 dS d =Δ ) S S ( 𝜎 𝜕 dS P d multiplied by the change in implied . 𝜕 P ) 𝜐 𝜕 implied volatility. This makes the vanilla = 𝜕𝜎 ( implied volatility. This makes the vanilla + S P Scholes lower − 𝜕 𝜕 higher = Black Δ )) S ( , the whole volatility smile moves lower with the flex , the whole volatility smile moves higher with the flex dS ,𝜎 up S down ( dP = Adapted delta finite difference calculation shown on volatility smile is positive. 𝜎 Adapted dS d Δ Within the volatility smile: In symbols: Adapted delta is such a well-accepted concept within the FX derivatives market and the fixed strike rolls downoption to premium a relatively lower. When spot is flexed and the fixed strike rollsoption up premium to relatively a higher. When spot is flexed EXHIBIT 14.12 Therefore, ■ ■ The formula for adapted deltais comes equal from to the chain Black-Scholesvolatility rule. delta In for plus words, a vega adapted changedifference delta delta in calculation within spot. a volatility To smileshown with in understand the Exhibit risk this 14.12. reversal that in someinterbank emerging broker market market with an currency adapted delta pairs, forward hedge. risk reversals are traded in the 278 VANILLA FX DERIVATIVES TRADING EXPOSURES on ftevltlt ml,tcnclycle ‘ttebto ftebucket.’’ the of bottom the ‘‘at called technically smile, volatility the of point skew levels zero with spot smile volatility different symmetric wings. a over positive shows and 14.13 sharply Exhibit change ATM. the differences around delta Black-Scholes versus ■ ■ wings. the in zero to reduce differences delta adapted ■ ■ adaption: the Black-Scholes under versus spot delta the adapted from comes the difference of delta direction the Therefore, positive. always XII 14.13 EXHIBIT fters eesli for is delta. Scholes reversal risk the If for is delta. reversal Scholes risk the If fixed a for volatility and implied rises the spot as strikes, fall higher will at strike increases fixed volatility a implied for If volatility and implied rises the spot strikes, as rise lower will at strike increases volatility implied If hna T otati rdd h mle oaiiywl ea h lowest the at be will volatility implied the traded, is contract ATM an When delta adapted skew, lower and wings higher relatively with pair currency a In wings: lower and skew higher relatively versus with delta pair currency Black-Scholes a so in Therefore, strike the from away decreases vega addition, In vega formula, delta adapted the at back Looking ymti oaiiysmile volatility Symmetric downside topside d dS 𝜎 d dS d dS 𝜎 𝜎 uniy h hl oaiiysiemvswith moves smile volatility whole The quantity. dpe et susually is delta adapted , ilb negative. be will ilb oiie(si h rvosexample). previous the in (as positive be will dpe et susually is delta adapted , ( 𝜐 ) o ogvnlaoto is option vanilla long a for shorter longer hnBlack- than hnBlack- than VANILLA FX DERIVATIVES TRADING EXPOSURES 279 Scholes − Scholes − Black Δ Black > Δ —the difference < Adapted Δ Adapted Δ → → gamma adaption effect and the Scholes Scholes − − Black Black positive Δ Δ gamma adaption effect > < is positive and: Adapted Adapted 𝜎 dS d Δ Δ is negative and: 𝜎 dS d fixed strike becomes downside fixed strike becomes topside → → Adapted gamma finite difference calculation for a symmetric volatility smile Consider a symmetric volatility smile withAs no described skew in the but previous positive section on wings. adapted delta, The for a strike that is initially If spot moves lower, the volatility smile at the fixed strike becomes locally sloping If spot moves higher, the volatility smile at the fixed strike becomes locally sloped Spot flexed higher Spot flexed lower EXHIBIT 14.14 ■ ■ Therefore, under adaption, deltaHence moves more around for the both ATM up and point down there spot is flexes. between adapted gamma and Black-Scholes gamma. adapted gamma argument is visualized in Exhibit 14.14. ATM: Adapted Gamma Adapted gamma takes intosmile account in exactly how the a samea fixed way risk as strike management adapted rolls perspective delta around does. is the The the important volatility measure from up to the topside, hence higher to the downside, hence 280 VANILLA FX DERIVATIVES TRADING EXPOSURES XII 14.16 EXHIBIT 14.15 EXHIBIT uvtr ntevltlt ml ie,maximum (i.e., smile volatility the in curvature symmetric a within shows strike 14.15 ATM Exhibit an smile. for the smile. gamma volatility of adapted wings and the gamma pronounced Black-Scholes more the larger is effect h ags oiiegmaaato fetocr ttepito maximum of point the at occurs effect adaption gamma positive largest The dpe am essBakShlsgmai ymti oaiiysmile volatility symmetric a in gamma Black-Scholes versus gamma Adapted aiu uvtr onsfrsmercadnnsmercvltlt smiles volatility non-symmetric and symmetric for points curvature Maximum d dS 2 𝜎 2 .We kwcomponent skew a When ). VANILLA FX DERIVATIVES TRADING EXPOSURES 281 gamma gamma adapted short positive negative it is possible for a trading position to be theta—a difficult situation to risk manage. Adapted gamma versus Black-Scholes gamma profiles for a non-symmetric In extremis paying ) is the exposure to ATM volatility changes. Consider a finite difference vega 𝜐 This results in the gamma profiles shown in Exhibit 14.17,In which general, in the turn gamma results adaption effect follows these rules of thumb: Strikes on the highadaption effect. side of the volatilityStrikes smile on usually the haveadaption low a effect. side of the volatility smile usually have a Vega ( calculation in which ATM impliedand volatility the is resultant flexed premium up changethe and is standard down divided Black-Scholes a by vega small the calculation amount ATM the flex implied to volatility calculate at vega. which In a vanilla Therefore, buying vanillas high onabove the the volatility ATM) smile results (i.e., inBlack-Scholes. at a an implied worse volatility gamma/theta ratiogamma yet under adaption than under Adapted Vega ■ ■ is added into thelower volatility volatility smile, side of this the pointdownside volatility of and smile. the maximum In point Exhibit curvature of 14.16, maximum moves the curvature to risk has reversal moved the is to the for in topside. the gamma adaption effect profile shown in Exhibit 14.18. EXHIBIT 14.17 volatility smile 282 VANILLA FX DERIVATIVES TRADING EXPOSURES eesl nUDJYters eesli o onies eln h ikreversal risk the selling so downside risk for 25d is 1yr reversal of risk USD100m/leg sell the are and USD/JPY reversals market In risk the USD/JPY reversal. into 1yr go that they concludes Therefore and high. analysis too some done has trader A Greeks Adapted with Managing vega Risk adapted long a gives contract reversal risk sold short a exposure. a and gives contract exposure reversal risk vega bought adapted have a when smile frequently volatility most the effect of this observe side low is the effect on This strikes implied smile. and the the of causes side high and the on wider 14.19. less smile Exhibit rise in to volatility shown strike whole specific a the for pushes volatility This ATM. the rebuilds curve calculation volatility vega adaption. whole adapted the an the under in because but differently flex changes ATM volatility the implied as the much as exactly changes priced is 14.18 EXHIBIT ie tiesasucagda ptmvs(.. tcystrike). sticky (i.e., moves spot as for unchanged are volatility stays 14.20 assumes strike Exhibit that given in methodology risk a exposures Black-Scholes The The results standard 14.20. the This strike. Exhibit using forward. in calculated 1yr topside shown USD/JPY position the of trading USD50m the buying selling in by versus hedged strike delta is downside reversal the selling involves hrfr,srkso h ihsd ftevltlt ml have smile volatility the of side high the on strikes Therefore, from away move delta specific a for strikes the higher, flexed is ATM the When am dpinefc rfiefrannsmercvltlt smile volatility non-symmetric a for profile effect adaption Gamma 𝜐 Adapted >𝜐 𝜐 Adapted Black − Scholes <𝜐 Traders . Black − Scholes VANILLA FX DERIVATIVES TRADING EXPOSURES 283 Spot ladder from selling USD/JPY 1yr risk reversal (Black-Scholes exposures) Volatility smile adjustment at higher ATM volatility Put another way, if the method used to calculate P&L within the spot ladder and However, the P&Ls within Exhibit 14.20 are generated by changing spot to each delta is roughly flat atsignificant 101.38 negative and P&L yet at there 104.42. is significant positive P&L at 98.33the and method used to calculateexposures the and exposures P&L within changes the will ladder not are be not aligned. aligned, delta level in the ladder,reference rebuilding point, the and revaluing volatility all the smileto options using within a the the portfolio. sticky new This delta is spotdo equivalent methodology level not and as tie therefore the in the delta between exposures different and spot P&L levels change within the spot ladder. For example, EXHIBIT 14.20 EXHIBIT 14.19 284 VANILLA FX DERIVATIVES TRADING EXPOSURES lc-coe xoue raatdepsrs o ikmngmn ti etto trading: best currently is is market it the how management judge risk and exposures For both exposures. view adapted or exposures Black-Scholes ■ reversal: ■ risk downside money with cost pair probably currency will positively. a changes decays in volatility value since implied smile ATM short moves, the spot as as static However, stays everything if time over money ■ is reversal risk the ■ in risk strikes be moves: two spot will the as for therefore change volatility to trade expected implied reversal how risk on This based considered. managed additionally be to exposures. delta need the with aligned are changes P&L and shorter is spot 101.38 at delta 14.21 EXHIBIT ■ ■ fteipidvltlt ufc nusaecagn sso oe uhthat balanced. such looks the position moves constant, delta spot the (roughly) and remains as used strikes be changing should transacted exposures are the Black-Scholes for inputs volatility surface implied volatility the implied the If position delta the balance to short back too bought be looks adapted better. could position moves, forward or trading spot spot the as some adaption, and changing delta Under not used. are be inputs should surface exposures volatility implied the If h oaiiysiea ptmvs lc-coe xoue hudb used. be should exposures by Black-Scholes suggested moves, as spot changes as ATM smile the volatility particularly the and changes surface volatility should the exposures If adapted moves, spot as changing used. not be is surface volatility the If higher Spot lower Spot rdr fe aeacoc hte ors aaevnlapstosusing positions vanilla manage risk to whether choice a have often Traders earn to book trading the cause will position reversal risk short the Holding changes surface volatility 14.21, Exhibit per as constructed is 14.21, ladder Exhibit spot in a as When instead ladder spot the in displayed are exposures adapted If → → ptlde rmsligUDJY1rrs eesl(dpe exposures) (adapted reversal risk 1yr USD/JPY selling from ladder Spot T mle oaiiyhge hl h oiingt hre vega. shorter gets position the while higher volatility implied ATM T mle oaiiylwrwietepsto eslne vega. longer gets position the while lower volatility implied ATM VANILLA FX DERIVATIVES TRADING EXPOSURES 285 positive zeta negative zeta AUD/JPY 1yr volatility smile → → ATM ATM >𝜎 <𝜎 Smile Smile In AUD/JPY, the risk reversal isThe for equivalent downside zeta as shown curve in shown Exhibit 14.22. in Exhibit 14.23 reflects this. Of course, the The zeta for a vanilla option is approximately (ignoring second-order effects) vega In practice, for small spot moves the volatility surface often remains unchanged 𝜎 𝜎 EXHIBIT 14.22 zeta for an ATM strike is zero. volatility smile. multiplied by the smile volatility less ATM volatility■ difference. Therefore: ■ The zeta ofthe a option vanilla priced using optionvolatility. the Zeta is smile therefore the volatility quantifies premium less the difference its value value between in priced the the using option value the which of ATM is due to the use Black-Scholes exposures). The speedvolatility of spot reacts moves more also matters, ifsurface since therefore spot implied contains moves information about moreor expected lower. quickly. speed In of spot some moves sense, higher the volatility (i.e., use adapted exposures), butreacts for larger such spot that moves the the volatility implied surface volatility often for specific strikes remains constant (i.e., Zeta ■ 286 VANILLA FX DERIVATIVES TRADING EXPOSURES pin n u et ail pin,bt ntehg ieo h ml.Teidea The smile. the ■ of side high follows: the as on is both strategy options, this behind vanilla delta 2 buy and options the around options vanilla smile. buy volatility and the around smile on options volatility point vanilla the cheapest sell on to point try expensive often most participants the market because terms delta and ■ ■ 14.23 EXHIBIT ■ h o on ftezt uv sotnlctdaon h 5 on ntelow the on point 35d the around located often smile. the is of curve side zeta the the of more on point smile. decay point low the to The 15d to options due causes the value curve around additional zeta their the located of on often because high quickly Being is smile. curve the zeta of side the high of point high The ik pi au a oeta h 5 pina h T n ikreversals risk and ATM the as relatively option option 15d 2d the the price. than smile, in more sharply volatility increase the far contracts of value side in high up the picks to premium. up lower far blows a spot has If than option quickly 2d more the (positively) since decays option option 2d long 15d the short the move, doesn’t spot If nte oua taeyi ihse urnypisi osl 5dlavanilla delta 15 sell to is pairs currency high-skew in strategy popular Another strike in are points these where approximately know traders that important is It pairs: currency high-skew In U/P y eaprofile zeta 1yr AUD/JPY VANILLA FX DERIVATIVES TRADING EXPOSURES 287 are sensi- ) 2 𝜌 ( and rho2 ) 1 𝜌 ( Volatility smile with positive wings and no skew The following analysis examines rho exposures for vanilla contracts using stylized Finally, in a currency pair with no risk reversal,This the results in volatility the zeta profile smile shown in is Exhibit 14.25, sym- which is roughly symmetric Black-Scholes analysis wherecurrency are premiums ignored. and hence subtleties around the P&L tivities to changesinterest in the rates CCY1 respectively.to and In consider: CCY2 FX continuously one derivativesquoted compounded for in there risk-free per each are basisinterest currency always rates. point in (pbp) two terms—the the rho P&L values currency change pair. for a Rho 0.01% is change most in often and always positive. Within standard Black-Scholes mathematics, rho1 on very low delta optionstheir on theoretical the midmarket high (see side Chapter 15). of the risk reversal aremetric often well and above theExhibit 14.24. implied volatility never goes below the ATM as shown in EXHIBIT 14.24 Unfortunately, the market does not make implementing this strategy easy: Offers Interest Rate Risk (Rho) ■ 288 VANILLA FX DERIVATIVES TRADING EXPOSURES ogVnlaCl Option Call rho. Vanilla USD Long pbp at USD13.1k long forward and rho 1yr EUR EUR/USD pbp EUR10k EUR100m short buying generates 1.3100 example, For separately. currency amount cash fixed a of delivery taking future. versus when the position in because cash confusing the be holding can USD from This example, comes rho. for USD that, is longer cash indicates get USD to formula forward the sold in be sign must negative The date. delivery where terms: point basis per In tenor. short-dated and notional than proportional to risk linearly exposure rho rate negative generate interest payments cash more Future contracts. have contracts long-dated generally, Very Forwards and Cash Future 14.25 EXHIBIT h e ai on h xoueo ogvnlacl pinpirt auiyis maturity to prior option call ■ vanilla long a 14.26: on Exhibit exposure in shown rho point basis per The ftefradt xiyi a eo h tie h ogCY alvnlahsno has vanilla call CCY1 long exposures. the rho strike, no the and below risk, far no payoff, is expiry to forward the If h aefruacnb ple ofradcnrcsb osdrn each considering by contracts forward to applied be can formula same The Notional held sftr ahntoa and notional cash future is ihrrtr sgnrtdwt ihrUDrts h difference The rates. USD higher with generated is return higher , eapol o oaiiysiewt oiiewnsadn skew no and wings positive with smile volatility for profile Zeta 𝜌 Cash ≈− 0.01 % T . stetm i er)t h uuecash future the to years) (in time the is Notional . T VANILLA FX DERIVATIVES TRADING EXPOSURES 289 T T . . Notional . Notional . 1 2 . F Δ % . % 0.01 0.01 ≈− ≈− Call 1 Call 1 CCY : Per basis point rho is 0.01%. ATM CCY 𝜌 𝜌 : Per basis point rho is 0%. : Per basis point rho is 0.005%. Long 1yr call option with 1.3100 strike rho is the forward delta expressed in notional currency %. F Δ As noted, longer tenor options have higher rho exposure. Therefore, over time The rho exposure on a vanilla option contract is linked to its forward delta At 0% forward delta Around the ATM At 100% forward delta behaves like a long forward andCCY2 hence rho generates exposures. negative CCY1 rho and positive If the forward to expiry is far above the strike, the long CCY1 call vanilla payoff the rho exposures on a vanilla reduce as shown in Exhibit 14.28. and where ■ ■ ■ Therefore: exposure. Exhibit 14.27 shows how adelta long that 1yr produces CCY1 short call CCY1 vanilla rho has and a long long CCY2 forward rho: EXHIBIT 14.26 ■ 290 VANILLA FX DERIVATIVES TRADING EXPOSURES ogVnlaPtOption Put Vanilla Long 14.28 EXHIBIT 14.27 EXHIBIT n ors.I h owr oepr sfrblwtesrk,teln C1put CCY1 14.29. rho Exhibit long CCY1 in the long shown strike, are generates profiles the hence rho and below These payoff rho. position far no CCY2 forward is short has short and expiry vanilla a put to essentially CCY1 forward the is the strike, vanilla maturity, If the to above risk. Prior far option. no put is and vanilla expiry long to a forward for the rho if the with surprises no are There og1rcl pinwt .10srk owr delta forward strike 1.3100 with option call 1yr Long og1rcl pinwt .10srk h vrtime over rho strike 1.3100 with option call 1yr Long VANILLA FX DERIVATIVES TRADING EXPOSURES 291 Long 1yr put option with 1.3100 strike rho In general, if there is a negative correlation between spot and CCY1 interest rates Exhibit 14.31 shows how the ATM straddle rho exposures change with implied The crossing point occurs around the ATM as shown in Exhibit 14.30. It is At higher implied volatility,widens. the rho exposures stretchAt out as lower the implied distribution tightens. volatility, the rho exposures compress as the distribution shorter, and CCY2 rho gets longer. With a lower forwardlonger, to and maturity, CCY2 rho forward gets delta shorter. gets shorter,With CCY1 a rho higher gets forward to maturity, forward delta gets longer, CCY1 rho gets ■ ■ or a positive correlationstraddle between results in spot a and desirable trading CCY2 position. interest rates, buyingvolatility: an ATM instructive to note that thethis same straddle, rho a exposures forward-hedged as CCY1vanilla option. Exhibit call 14.30 option are or generated a by forward-hedged CCY1 put Combining long CCY1 call andmaturity gives CCY1 a put long vanillas ATM straddle with position: the same■ ATM strike and ■ EXHIBIT 14.29 Long ATM Straddle 292 VANILLA FX DERIVATIVES TRADING EXPOSURES ed hog oteroepsrso nAMstraddle: ATM an on exposures rho the to through feeds levels 14.31 EXHIBIT 14.30 EXHIBIT fetol eoe infiata ogrtenors. longer at significant becomes contains only position effect straddle ATM long a Hence ■ ■ C2rtshigher rates CCY2 higher rates CCY1 ial,cagn neetrtswt xdso mat h owr n this and forward the impacts spot fixed a with rates interest changing Finally, og1rAMsrdl ih130 tierho strike 1.3100 with straddle ATM 1yr Long og1rAMsrdl ih130 tieroa ifrn mle volatility implied different at rho strike 1.3100 with straddle ATM 1yr Long → → owr higher forward lower forward → → ogrCY rho CCY1 longer ogrCY rho CCY2 longer ogitrs aegamma rate interest long lhuhthis although , CHAPTER 15

Vanilla FX Derivatives Trading Topics

293 he following topics highlight common situations that FX derivatives traders Tcome across when risk managing their positions.

■ Understanding the FX Derivatives Market

Traders must learn the characteristics of the markets in which they are operating. Every market is different and this guides how traders interact with the market and how positions are risk managed. Within foreign exchange there are enormous differences between the spot, forward, and derivatives markets in different G10 and emerging market currency pairs.

Speed G10 spot markets ‘‘tick’’ (change) many times a second. They are very fast, simple (in the sense that the market consists of a single contract per currency pair) markets in which high-frequency algorithmic trading plays a large (and increasing) role. 294 VANILLA FX DERIVATIVES TRADING TOPICS i–fe pedaealcoeylne.Temr iudacnrc,telre the larger the contract, a liquid and more size, The speed, linked. closely Liquidity, all consideration: are important spread enormously bid–offer an is Liquidity Liquidity an within trader individual the and bank shouldn’t important. the This is of banks. market other reputation OTC of makes The eyes it does: the that it in small small-fry but so like matter, is look contract trader that the and because bank transacting decide the than may position rather trader their risk a clean the flipside, to run USD/JPY the interbank to ATM On the 1mth away. USD15m in sell traders initially to other size wants scare who full may their that reveal EUR/CHF because not with EUR750m market buy may interacts broker to they trader have size), a they (large way knows ATM trader the O/N a impacts if This example, is, For market. contract the a small or large relatively enormous. be would USD/VND tenor example, any for in, in but transaction traders; USD30m other a move to Dong), not interest (Vietnamese will much (the EUR30m of ATM be buying or 1mth instrument), market and in EUR/USD the tenor, trade in pair, example, pairs currency liquid For currency most sizes. different transaction in standard instruments different different derivatives, FX Within Size ■ ■ ■ reasons: several rdn oiin tmnmlcs.Frti ob h aeteems ego sized good be must there level. case midmarket the the be exit around to tight and this offers the enter and For tighter smoothly bids cost. to the minimal traders and at enables market positions liquidity the trading simply, moving Put without spread. transacted be bid–offer usually can that size rdn ae tmr prpit o agrsz ob rnatda given a at transacted levels. be different to at tickets size smaller multiple larger than for rather level generate appropriate volatility to implied more after details used it contract is agreeing makes and data generating trading market of process prevailing This then details. contract and full terms volatility implied in rounding This level. terms. structure given Quoting a volatility at stickier implied be rounded to prices in volatility causes quoted are prices clients, possible different of volatility implied range Quoting wide very traded. a be therefore can that is contracts There strike. and maturity range Contract ti motn httaesko tnadmre ie n hrfr how therefore and sizes market standard know traders that important is It for market spot FX the than slowly more moves market derivatives FX OTC The ail pincnrcshv w anadtoa dimensions: additional main two have contracts option Vanilla . o h aoiyo rnatos ail pintae r agreed are trades option vanilla transactions, of majority the For . nteitrakboe aktadfrms institutional most for and market broker interbank the In . VANILLA FX DERIVATIVES TRADING TOPICS 295 Another important issue is that liquidity is often not symmetric around the In FX derivatives, one of the most striking liquidity reductions occurred in Liquidity can also be impacted by the amount of pain the market isJudging collectively liquidity within a market is a vital trading skill. In general, it is dangerous The manner of market moves is also important: Large jumps in the price level, Liquidity does not stay constant within a given market; it changes as market This leads to decreased trading size and wider bid–offer spreads. become comfortable with thesubsequent trading market levels moves. andbid–offer spreads. This require less leads spread to toHigher increased market cover volatility often trading leads to size decreasedspread liquidity to as and cover traders subsequent require market tighter more moves and remain within their risk tolerance. Lower market volatility often leads to increased liquidity over time as traders too much. midmarket level. Most FX derivatives tradersexpiring will on be important happier event to dates buymay (i.e., vanilla occur) an options than expiry sell date them, on so the which size a that large can spot be move transacted at the bid will be larger transactions within the market costspot huge amounts and of implied bid–offer volatility spread. overmoments USD/KRW that this the period best is traders already shownwing have vega), in the or Exhibit right they position 15.1. (long are Itoccurring gamma able and is and to aggressively long at quickly seek these to appreciate obtain that the an right extreme position market before event liquidity is reduces to the trading levelsbecause the and underlying was many so static—1mth marketUSD30m. ATM Then participants was quoted the were (e.g., FX 2.95/3.05% happy in marketexploded jumped to to sharply around sell higher 100% volatility andspreads in ATM-implied a volatility widened matter enormously—1mth ofUSD10m. weeks. ATM The market Liquidity liquidity was situation vanished had now changed and completely and bid–offer quoted, performing any e.g., 60/120% in to assume that theexperienced current correct market positioning liquidityexiting will being the always position undone prevail. at by the Every optimal a trader moment. has lack ofUSD/KRW liquidity during preventing 2008. Priorvolatility to was creeping 2008, lower USD/KRW toward 3%. FX The was market had stable become and very accustomed implied back into the middle of a well-established range may lead toexperiencing. increased If liquidity. traders are losing money,each they other. are This less can likely be to a provide nasty liquidity self-reinforcing to cycle for the market as a whole. and hence increased uncertainty, normally lead tolevel a larger of reduction the in market liquidity. also The hashighs an or lows impact. will For often lead example, to spot reduced moving liquidity while and spot breaking moving new the same distance ■ conditions change. In general: ■ 296 VANILLA FX DERIVATIVES TRADING TOPICS ■ rdn h Overnight the Trading tie tms oefrenough far move must it initial strike; the (including diagram P&L 15.2. hockey-stick Exhibit in standard shown the premium) by given sound is doesn’t maturity answer This higher. strike. initial moves spot topside An if a out options? with pay How option vanilla options day. call call trading next since overnight unreasonable the by an of buy expressed to course best be the might be over higher opinion rise this slowly can will spot thinks trader A their within this for size. adjust larger in Traders requests offer. price the quoting at when transacted making be price can that size the than 15.1 EXHIBIT h rdrsie a htso ol ie ahrta hpigu n on If down. and up whipping than rather but rise, traded would be can spot gamma that the was Intraday topside. idea the trader’s to the is peak gamma the that notice in shown is position trading hedged delta initial 15.3. the Exhibit position, the in are trades other U/S pti .10adAD0 fON090 a enpurchased; been has 0.9200 O/N of AUD50m and 0.9150 is no spot Assuming AUD/USD live? traded being than rather hedged the delta to is higher option move to the spot if for What enough not at is it P&L trade, total this from the money hedge), make To delta (without live traded is option call long the If S/R ptadipidvltlt rm20 o2009 to 2006 from volatility implied and spot USD/KRW through h tiet locvrteiiilpremium. initial the cover also to strike the VANILLA FX DERIVATIVES TRADING TOPICS 297 Inception trading position from buying O/N topside call option P&L from buying topside call option It turns out that if the trader’s view is that spot will rise slowly—sometimes The horizon is now equal to the expiry date of the option and the option is for spot to move throughcan the happen strike. is If that no spot spotspot is rises ending bought up up back, at to a the the strike worst strike, at thing but expiry that is not called through beingcalled it. ‘‘pinned.’’ In ‘‘grinding higher’’—buying market-speak, a wing vanilla option in that direction is a bad zero within the risk management system.is The the loss theta in paid option (negative value USD18.5kdue from P&L) to the previous the day change to today.trader from The still long delta thinks has delta that bled spot on shorter willbuying the rise, spot. previous the Then short (again) day delta the to should only be zero chance hedged to delta back make today. to any flat If money by out the of this position is spot remains static until the endas of per the Exhibit day, 15.4. the next morning the trading position is out-of-the-money at current spot. Therefore, both option value and option delta are EXHIBIT 15.3 EXHIBIT 15.2 298 VANILLA FX DERIVATIVES TRADING TOPICS andadmr rfi smd fso rnshge,eatyfitn h rdrsview. trader’s the fitting exactly higher, grinds spot if made is profit more and earned 15.5. Exhibit in shown position initial the to in is results trade better far A trade. 15.5 EXHIBIT 15.4 EXHIBIT oeta hr eln nPLi ptge bv h hr tie eln strike a Selling strike. short the above goes spot if P&L in decline sharp potential and xii 57sostePLpolsfo uigso nyvru uigspot buying versus only spot buying from profiles P&L is the Theta shows 15.6. 15.7 Exhibit in Exhibit shown is forward rolled horizon the with position The eln h osd ail pin eln h pinersteabtcue a causes but theta earns option the Selling option. vanilla topside the selling rdn oiinfo ogtpiecl pino xiydate expiry on option call topside long from position Trading neto rdn oiinfo eln osd aloto n uigspot buying and option call topside selling from position trading Inception sell h vrih osd pinand option topside overnight the buy pt which spot, VANILLA FX DERIVATIVES TRADING TOPICS 299 P&L profiles from different trading strategies Trading position from short topside call option and long delta on expiry date Selling strikes close to currentvolatility spot and increases introduces the larger median negative P&L P&L scenarios. but it increases P&L EXHIBIT 15.7 closer to current spotthe would topside result would in more happenrisk/reward balance theta at based earned on closer a but trader’s spot expectations the of levels. P&L where/how spot Strike decline will move. positioning to is therefore a EXHIBIT 15.6 300 VANILLA FX DERIVATIVES TRADING TOPICS ■ ata’ ue fTrading of Rules Gartman’s ■ ■ ■ ■ ■ ■ ■ most I the sense. opinion my common In trading ■ Internet. of the on deal list great full are: a the important for contain trading searching of rules advise list These strongly updated year. an publishes every fame) rules Letter’’ ‘‘Gartman (of Gartman Dennis ■ epyu ehia ytm ipe opiae ytm re confusion; breed systems Complicated simple. elegance. breeds systems simplicity technical your it. accept Keep trading; of trades nature well-researched times,’’ the most ‘‘good is the This In even awry. times,’’ poorly. go ‘‘bad trading in profitable; when are when modestly errors aggressively and even and small large Trade trade bad. well; higher. most sell trading good; ‘‘high.’’ to is some price and cycles: what in high know runs buy we Trading can to Nor ‘‘low.’’ but is high, price sell what and know never low can buy job We difficult to a not such is is objective mental trading The why of reasons key sums are immeasurable moves market costs costs from positions result it losing but Holding is capital, two. mental capital. actual the the of capital, of of sums expensive types measurable and two important the more Of Capital. the be Real position; trumps vega short Capital or Mental vega long hand. a positions upper example, flip the for actively to, gained to attached willing has too side getting willing not one be about and when here side readily winning the sides on change fight must to We soldier. mercenary a like Trade ruin. losing to to Adding lead position—ever! will losing position a to add circumstances, any under Never, l hs ue r en ob rknbtol eyifeunl.Genius infrequently. very rules. only these break broken—but to when be and to how knowing meant in comes are news rules bad these that is All markets most of only lesson loss bad. The the follows cockroach! avoid usually one to just trying never trades; right is the There doing by chin easy the on is finds agony it the there crowd it take If prolongs exposures. the take, if gamma to which or and loss vega that a reduce don’t; and to is Payrolls) sell, do Non-Farm (e.g., to to days hard easy event important is is that it trade If the objectionable. trade: Do right don’t. the buy, to is trade hard The h mtosta eutfo u fbglse n h mtoa wnsthat swings emotional the and losses big of run a from result that emotions The nF eiaie,a xml fti ssligsrkswt xiison expiries with strikes selling is this of example an derivatives, FX In . . . ei talking is He VANILLA FX DERIVATIVES TRADING TOPICS 301 this occurs as but Trading position from buying 1yr ATM and selling a gamma-neutral amount of 15%. The trader makes a profit from the long wing gamma as spot moves + ATM implied volatility is unlikely to jump higher if spot is static or range-bound. Money has been made from the vega position but not as much as could have The trader is proved correct; implied volatilityOver the does course jump of a few days, USD/TRY spot jumps to 2.3000, the 1yr ATM 5% to EXHIBIT 15.8 3mth ATM hedging, DCDs) means thatoften if drifts lower spot over time. stays within a range, ATM implied volatility volatility smile. This is a keyfirst reason place; why the options skew that of may thehigher volatility pick levels smile up of exists implied more in volatility. sharply the in value on a spotIn jump addition, trade the at supply of optionality into the market (from, e.g., corporate FX but the implied volatility for14% due the to purchased the increased 2.2800 topside strikemarked skew. at only The strike a moves has lower from become implied downside 9% volatility and than to is the now ATM, as shownbeen in made Exhibit from 15.9. owning anmoves, option with the a biggest higher strike. P&L During is most major generated market from wing options on the high side of the shown in Exhibit 15.8. USD/TRY spot moves sharply higher due to a sudden TRY devaluation. implied volatility jumps from 9%+ to 18%, and the 1yr 25d risk reversal goes from is at 2.2900.generates A positive USD/TRY P&L FX if1yr derivatives implied ATM trader volatility (2.2800 wants goeslong strike) a significantly vega. is trading higher. However, bought position Therefore, gamma-neutral this at that notional causes of 9.0% significant 3mth implied ATM theta volatility is (paying) sold. in This so order results to in to mitigate the get trading this position a In USD/TRY the risk reversal is for topside, spot is at 2.1200, and the 1yr forward Vega Positioning ■ 302 VANILLA FX DERIVATIVES TRADING TOPICS ■ hr-aeTaig ogAMvru Short Wings versus ATM Long Trading: Short-Date nlqi 1 akt n oua taeyi h hr-ae st e pteposition the up set to is short-dates the in strategy popular one markets G10 liquid In out close completely to rarely effort are concerted position. of deals trading years for derivatives even offsets FX or exact an months since take and can warehoused it found are deals derivatives FX of trading; spot FX implied or trading and exchange It rates, zero. interest to spot, as exposures change Greek the exposures current will how the move. position examine volatility How all trading also flattening change? or derivatives to simply vital skew FX lower of is large the matter moves a a will spot never flattening How is generally, if change? More example, ATM change? the For wings will together. how moves higher, volatility implied and 15.9 EXHIBIT ih ag,tegmacnb rdd ls esteai adoealfrtegamma the fairly for a overall paid within is down theta and less Plus, up traded. moves be spot can gamma if the delta, range, starting tight balanced in a shown From is lower. position trading a such of example An 15.10. wings. Exhibit short versus ATM long h oiinis position The to relative trading derivatives FX OTC of aspects toughest the of one is This spot consider traders portfolio, trading derivatives FX an positioning When S/R oaiiysiepe n otso jump post–spot and pre– smile volatility USD/TRY oggamma long rudteAMand ATM the around oiin antb aiycoe down closed easily be cannot positions hr gamma short ihso ihror higher spot with h majority The . VANILLA FX DERIVATIVES TRADING TOPICS 303 Trading position from long ATM and short wings (i.e., transact if the premium of the structure reaches a certain (i.e., transact if spot reaches a certain level). This involves the FX Premium firm level). This involves the FXprevailing derivatives market trading rate desk when transactingthe the a client hedge structure trading at the premium the volatility option hits and spot structure the cannot live. be target known, Since level this future requires and the levels option of structure premium the implied Spot firm derivatives trading desk transacting a deltaclient hedge trading at the a option specified structure spot levelat live. and Since the the the order implied level volatility cannotstructure when cannot be spot be is known guaranteed. with certainty, the premium of the option This strategy ties in with the fat-tailed distributions observed in the spot market. One trader I know has implemented this strategy repeatedly in liquid G10 pairs 2. 1. Rather than transacting directly, clientstransactions sometimes provisional leave on option certain orders,main market which types are of conditions option being order: met. There are three simplest ideas are often the most effective when well executed. Traders like selling short-dated wingsof in time liquid can currency pass between pairs spot because moves long large enough periods to cause significant negative P&L. long ATM vanillas decay (negatively). for almost 20 years. He trades theif gamma spot as spot moves moves into in a theto long short get gamma gamma area, the area, and position he back aggressively to long makes ATM prices versus in short the wings brokers again at minimal cost. The EXHIBIT 15.10 because short wing vanillas usually decay (positively) relatively more quickly than Client Option Orders ■ 304 VANILLA FX DERIVATIVES TRADING TOPICS ■ utn ail Spreads Vanilla Quoting ntne w-a rcswudb utdo ohlg ihn tightening. no with legs both on against is quoted notionals vega) be equal would 0.02% in prices and vega) two-way gamma 0.20% instance, and 100% gamma (e.g., (15% ATM have ATM O/N must 3mth request example, the For spread, risk. a vega as quoted be vega to long but always is vanilla long a since sold are gamma. legs long some and and bought a are called legs is some request price a 8, offsetting Chapter in seen As contracts the all on traded. are rates be quotes can others trader one the only the that but but transacted, means prices multiple which requests is OCO, client prices the sub-orders requests when used the is terminology of same The one canceled. When sub-orders. spot. off-market multiple an using level. is exchange client delta the off-market that delta the valid at is the hedge it from check delta P&L Plus, the the versus for spot adjusted current be at must hedge premium structure the then contract, the than off-market lower or level? higher reference the the be to at likely moves Looking volatility spot reference? ATM if pricing spot level of the current new level is the the assumptions at will smile, surface What volatility volatility care. the with about done making be tool must spot) (off-market level risk. some taking desk trading the without firm 3. w-e pedrqetwl eqoe ihtelgwt otvg hnethe level). (hence same the vega at most offer and with bid leg (i.e., CH the Usually, denoted with sometimes transact. quoted to wants be risk) client will most the request around spread way two-leg which a known isn’t it sell—but h otcmo pedvrathstovnlalg—n u,teohra other the buy, a legs—one vanilla two has variant spread common most The compound, risks where requests price equivalent than tighter quoted are Spreads leave sometimes clients Finally, an at exchange delta a with structures option deal to want clients Occasionally, current the from away reference spot a off premium orders and option firm client spot firm both spot be Pricing to order option an for possible not is it that Note pinsrcuedlahde ihteF eiaie rdn desk. trading derivatives FX vanilla the simple with a hedged trading delta client the structure involves option This level). certain a reaches structure the against firm checked Volatility and recalculated continuously) level. target (ideally periodically be to am rvg ikbtentelg.Teeoe ihnavnlaspread vanilla a within Therefore, legs. the between risk vega or gamma ptlvl nti iuto,tecretso hudb sdt rc the price to used be should spot current the situation, this In level. spot spread ie,tasc noto tutr fteipidvltlt o the for volatility implied the if structure option an transact (i.e., ie,bdadofra ifrn ees n h te leg other the and levels) different at offer and bid (i.e., OCO oecnesohr)odr incorporating orders others) cancels (one spread significant fi a utpelg with legs multiple has it if fstiggmaor gamma offsetting not ped nthis In spread. a choice , VANILLA FX DERIVATIVES TRADING TOPICS 305 implied volatilities volatility differ- exact terms. A broker has a price on 1mth ATM versus 2mth ATM: Once a contract has been traded in differential terms, the If a trader cannot trade a spread as a package, they may try to ‘‘leg into’’ the spread. If the broker is trading a 1mth versus 2mth ATM spread at 7.6% versus 8.0%, In the interbank broker market, vanilla spreads are quoted in It is worth briefly noting that there are occasions where spreading the leg with For example, a vanilla spread request might be 1mth 1.3500 versus 2mth 1.3500 as ‘‘0.05/0.15% around favoring 2mth.’’ In‘‘around’’ means this case one the side price of is‘‘favored’’ the side –0.05/0.15%; has price the is higher negative mid and volatility. the other is positive. The If the price made is 6.95%as CH ‘‘0.05/0.25%, 2mth versus over.’’ 7.00%/7.20%, the price would beIf quoted the price made is 6.95%as CH ‘‘flat/0.2%, 2mth versus over.’’ 6.95%/7.15%, the price would beIf quoted the price made is 6.95% CH versus 6.90%/7.10%, the price would be quoted In most major currencyfive pairs years there in is the only interbank good broker liquidity market. for However, maturities there out are to currency pairs, This means trading one leg ofand the then spread the at 2mth a ATM timeapproach later (i.e., is with trading a the that 1mth different additional ATM counterparty).spread first bid–offer The has downside been spread to fully must transacted this the usually trading be position has paid additional plus risk. before the have to be agreed betweendeal, the the two trader who traders. gets Ifvolatility long there on vega both is from legs a while the net theto spread vega trader set will exposure who a try gets relatively on to high short the set volatility vega a on from both the relatively legs. spread low will try they might say they are ‘‘buying twos, selling ones, paying 0.4%.’’ ■ ■ ■ three-day spread, the price10/13% of and the it O/Nchoiced would three-day could be mid implied be more volatility. well natural known to in quote theential 10/13% market (or as similar) versus a better bid or better offer) ratethen of pays 6.95% or CH versus gives 6.95%/7.15%. thedirection. The 2mth price (the taker spread leg) with the 1mth tradedthe in smaller the vega opposite is more appropriate. For example, when quoting an O/N versus NY Cut. The midmarketthe implied volatility 2mth for 1.3500 the is 1mth 1.3500 7.05%. is A 6.95% and trader for might therefore quote a neutral (i.e., no Trading Long-Dated FX Derivatives ■ 306 VANILLA FX DERIVATIVES TRADING TOPICS ■ rdn egdCrec Pairs Currency Pegged Trading lohls oiin r tesdwt ifrn aktdt cnro nodrto order in scenarios data market often scenarios. different danger approach main with the This flat. stressed identify P&L fall. are the will positions keep to helps; volatility attempting also implied than example, practice and for in rise view; better high-level works will a rates is around interest position trade the managers USD against in risk market smile elements Often the these vital. against as between therefore offset vega elements relationships different against the the Understanding delta from P&Ls moves. running the possible: that such as exposures rates well as position the ■ ■ ■ FX long-dated ■ a even or trading years when 30 suggestions to position. high-level out derivatives some traded be are can Following options longer. which in JPY, involving particularly neegn aktcrece n hr r ayvrain ntemtosused methods happens the often on most variations This many bank. are there central and country’s currencies pegged the market is by emerging pair way in currency some a in when managed presented or are challenges management risk Unique aemnmlroepsrsadteso suulycoet h forward. the to close vital. is usually liquidity and is pricing interest contracts their different spot and understanding these instruments markets the rate derivatives because FX and long-dated derivatives exposures in FX However, rho vanilla minimal short-dated have trading mainly if the of changes. understanding convention deep the forward A when at-the-money jump strike to relative an curves impacts ATM This being causes maturity. to which ten-year placement, the straddle past zero-delta hedge forward a plus strike as traded being the from liquidity, the Learn of limits reduces liquidity the how markets. and understand stretched positioning, market in to the by order caused skews price in standard prices make actively rates interest USD ten-year Understand if position vega sense market a the 0.25%. get to rise must happen traders will example, on what For risks of the essential. market-making of absolutely leaves understanding is which detailed products a products, standard Therefore, of positions. range similar limited with a banks trade markets. to derivatives FX tend long-dated the Clients in important particularly is it but generally Understand rdn ogdtdF eiaie oiin sotnamte ftyn obalance to trying of matter a often is positions derivatives FX long-dated Trading aktconventions market aktliquidity market aktpositioning market neetrt markets rate interest o xml,teAMcnrc nG0pisgoes pairs G10 in contract ATM the example, For . naewt h nebn rkrmre and market broker interbank the with Engage . hsi motn nls iudmarkets liquid less in important is This . sntasltl necessary absolutely not is cnroanalysis Scenario VANILLA FX DERIVATIVES TRADING TOPICS 307 (i.e., in front of the reduction in liquidity that will likelyoften occur once involves spot jumps. simulating Risk management 2%,the 5%, P&Ls 10%, over these 20%Holding scenarios. spot a Controlling revaluations position these for and which P&Lscertainly checking the cost is P&L money inevitably over is time. a positive for balance: all of these spot jumps will pairs. In terms of pricing, market prices forprices exotic generated contracts can (see be far Part from model extreme IV). gamma Within exposures from risk barriers management, close to the expiry. A biggest key danger consideration is withinpreparation pegged/managed for currency a pair large risk spot management revaluation. This is is especially important considering construct valid volatility surfaces. Bank tradingof desks expend quant enormous amounts brainpowerpegged on currency pairs. generating Traders learn validis the close warning arbitrage-free to signs breaking, that volatility often the via volatility surfaces unstable surface GreekCare in exposures. must be taken when trading exotic contracts in pegged or managed currency This causes large gamma exposurescertain and benchmark it options often (i.e., makes a senselevel) 1yr in to option premium track with terms the the rather cost than strike of implied at volatilityFor the terms. intervention currency pairs withthe very standard low volatility volatility but surface relatively building high methodologies skew or sometimes wings, struggle to NDF can move freely outsideinterest the rates band become or driven range. by Thefuture the consequence value forward of of rather money this than in is being the that often two a far currencies. reflection higher Also, of than the the the volatility volatility of of the spot. forwardPegged is currency pairs typically have very low implied volatility (e.g., below 1%). When trading amechanisms used pegged to currency controlplace it, pair originally, the and context it the of historyis why of is important how these because the good measures these mechanisms decisions were evolved to arebe put over often monitored linked time. in understand for to This potential political changes the factors in that approach. must exact From a risk management perspective often spot is curtailed but the forward or One popular risk management techniquethe is trading to position. place In positive vanilla essence spreads this within involves placing long vanillas ■ ■ ■ ■ ■ ■ to control the currency. Followingderivatives are in pegged some currency high-level pairs. suggestions for trading FX Positive Vanilla Spreads ■ 308 VANILLA FX DERIVATIVES TRADING TOPICS ■ rtn-f ail Risk Vanilla Writing-Off ptcniustruhtesotstrike. the short of the through front strike. continues in spot closer strike a long of of profile. P&L AUD50m amount the buying smaller smoothing of hence a However, strike, effect date. short buy the expiry to same shows the is 15.12 for Exhibit approach strike same preferable the potentially back buy a to be would risk this strike. wing downside as negative then and positive strikes. goes two P&L the through so moves vanillas spot short than) spot current to closer 15.12 EXHIBIT 15.11 EXHIBIT ailsotosaermvdfo h antaigbo tzr rmu ie(i.e., live premium zero at book trading main the from removed are options vanillas Writing-off rfi smd hog h ogsrk,wihte rvdsPLpoeto if protection P&L provides then which strike, long the through made is Profit hedging of method One strike. the through sharply drops spot of that AUD100m is short danger is The that position trading a shows 15.11 Exhibit example, For sapwru ikmngmn ehiu.Teie sta eylow-value very that is idea The technique. management risk powerful a is rdn oiincnann hr oniewn tieadaln tiei front in strike long a and strike wing downside short containing position Trading strike wing downside short containing position Trading VANILLA FX DERIVATIVES TRADING TOPICS 309 write-off ). From there the options are usually left until maturity as lottery tickets that Writing-off changes the P&L distribution of the trading book by paying small It is also possible to write-off long vanilla strike spreads with the same expiry The writing-off technique works best when traders are engaged with their For example, a trading portfolio is long a low premium and low delta topside When implied volatilityinvestigated. falls, both downside and topside strikes should be When spot moves significantlyand higher, very downside low premium strikes options should can be be writtenWhen investigated off. spot moves significantly lower,very topside low strikes premium should options be can be investigated written and off. main trading book thatresulting can in be positive used P&L. to risk manageIf the spot moves position through (close the strike itrealized close out?), at to the the option option expiry. maturity, the full P&L can be If spot does not go uppremium to has the been strike lost. prior to the option maturity,If spot the moves (low) up option to theback strike prior into to the the main option maturity, book, the again option can at be zero sold premium. This generates a profit in the ■ amounts of P&L (the premiums) for the (low) chance of making big returns. It ■ ■ date. This can be usedcost to money clean in up the strike form riskor of would in spread cause the cross increased portfolio to P&L close that volatility. out would in otherwise the interbank brokerposition: market ■ from the write-off book must not be incorporated into■ the main book risk. ■ up, but crucially the longexpects gamma it and will negative be thetawith will difficult a also to higher increase. spot. make The Therefore,dedicated back trader the write-off the option book theta is at from soldbook zero equal from trading premium. to the the the This main prevailing causes long value tradingbook a of gamma is book the loss risk to vanilla in managed option. a as From the here normal main the without trading main the trading removed vanilla. Importantly, the risk lottery tickets pay out over theNote course that of because long the option year, positions theycan are can never moved generate have live sizable at negative profits. no P&L. cost the write-off book vanilla option in large notional. If spot rises sharply, the value of the vanilla will pick no delta hedge) and placed inbook a separate trading book (sometimesmay called pay the out if spot moves through the strike prior to maturity. If only a few of these 310 VANILLA FX DERIVATIVES TRADING TOPICS ■ ■ o et ail Options Vanilla Delta Low Risk Pin Vanilla pin ic hr svr itepeimere oprdt h oeta negative potential the to compared earned premium little very is a there since within options moves spot The as change directly: not price will premium premium is rounded the range. the this reasonable quote that terms so small also volatility zero) so could implied is below delta In trader go EUR%. the cannot 0.005/0.03 but bid show the 3.75%/5.4%, to (since want symmetrically they it instead spread 15.13. cannot Exhibit in they shown but as 4% around is Delta point) 4.1%. basis of (one volatility EUR% 0.01 implied approximately an is off 1.3530 at spot with option vanilla put often in it them strikes delta on low prices very Outside quote terms for topside. volatility to and the sense on vega calls little more strikes delta have for makes 10 options market and the vanilla downside in levels the liquid those on reasonably puts and delta defined 10 well between is smile volatility The takes move spot large a then profits. it; significant for generates theta and much strike paid the have through to spot spot from away far too spot large hence a and P&L. on size positive pinned large large being reasons, generate in opposite can market and strike the equal short in the may be For This it. to strike). toward known topside gravitate is the bleed may long strike delta if the the delta if unwind long applicable run to (i.e., traded. be is the strike be situation the from never this into can delta away from run delta something bleeds and that salvage and such delta to spot strike way The the current only date. toward to The dribbles expiry close then the fairly spot and into strike strike its theta traders. with sizable derivatives option FX paying vanilla an hence for long nightmare a being is involves strike This long large a on pinned Being trend play. or into jump strikes to tendency written-off a brings has which spot strongly, where pairs currency in well particularly works ag muta rna otesse imre.Hwvr obyte,a them, buy More paid. to be However, to need midmarket. usually lopsided system sell will to midmarket in the possible system to results usually the above is near This far It or relatively strike. contracts. price at these the for amount through levels large offer jumps a and spot bid the if at result liquidity may that P&L aktpriiat fe ontlk osl eylwpeimwn vanilla wing premium low very sell to like not do often participants Market contract the on spread bid–offer premium 0.025% a show to wants trader The Example be to strike the for is book the in strikes long with trader a for situation best The h ytmmdpeimfraERUD1k136 U call/USD EUR 1.3665 1wk EUR/USD a for premium mid system the : atclrya hre tenors. shorter at particularly , rmu terms premium ahrthan rather implied VANILLA FX DERIVATIVES TRADING TOPICS 311 . leverage Pricing tool showing low delta 1wk EUR/USD vanilla option Another important factor is how the volatility base determines the range of vanilla Within the interbank broker market, oncethe a transaction remaining is inputs agreed incalculate to volatility the terms option the premium. Black-Scholes formula must be agreed in order to it is hard to transact suitablespot options level to because adequately even hedge vega vanillabecome risk deep options at in-the-money that the after were post-jump the quite spot jump. low delta to start with can larger than the potential upside (the premium), sometimes called options that can bein traded emerging in market the currencyoptions pairs. interbank that When can market. be implied This liquidly volatilityfairly traded is close is in particularly to the low, current relevant interbank spot. the market If range is the constrained of main to risk strikes is that spot suddenly jumps a long way, EXHIBIT 15.13 generally, most traders dislike doing trades where the potential downside is far Agreeing Broker Market Data ■ 312 VANILLA FX DERIVATIVES TRADING TOPICS osatyloigfrwy nwiht ana de omte o small. how are matter and no edge, smart an extremely gain to are which data participants in market ways fair market for that looking Most assume constantly not suggested: Do be important exists. is always issue it this will but that use, to understand data to market traders which for about debate little be can there because will level. they market deal, current the hedge the from at volga agreed exactly short the gets reference trader volatility from a the If jumps want possible. level. effectively market as current volatility level the midmarket implied to level current deal, from the far booking as On be to reference volatility implied exactly reference spot the want spot. will market they deal, current spot. the the market from at current gamma the short to the gets level booking trader hedge On a agreed possible. If the as from level jumps midmarket effectively current spot the deal, from far as be to reference remdaktdt suulyes ooti,s hsiserrl assproblems causes rarely issue this so obtain, to easy usually is data midmarket True the want naturally they transaction, the from volga long gets trader a if Likewise, spot the want naturally they transaction, the from gamma long gets trader a If CHAPTER 16

ATM Volatility and Correlation

orrelation is an important measure within FX derivatives. The ATM volatility Cand correlation framework is often used to calculate ATM volatility in cross- currency pairs. Dephased vega exposures are also calculated within the same 313 framework.

■ ATM Volatility Triangles

In trigonometry, the cosine rule relates the lengths of sides of a triangle to the cosine of one of its angles as shown in Exhibit 16.1. If the length between point A and point B is denoted AB, the cosine rule states that:

BC2 = AB2 + AC2 −(2 . AB . AC . cos x)

The cosine rule can also be applied to ATM implied volatility in three currency pairs at the same tenor as shown in Exhibit 16.2. The distance between EUR and USD represents EUR/USD ATM volatility; the longer the length, the higher the ATM volatility. The angle 𝜃 is the inverse cosine (cos−1) of the correlation (𝜌) between spot log returns in EUR/USD and GBP/USD. Going forward, 𝜌 is described as the correlation between the major currency pairs (EUR/USD and GBP/USD in this example) and the output pair (EUR/GBP) is called the cross-currency pair.Bewareof the overlap in symbols between correlation and rho exposure (see Chapter 14); the correct meaning should be obvious from the context. 314 ATM VOLATILITY AND CORRELATION rmwr a lob sdt aclt nipidcreainfo he ATM three from correlation implied an calculate to The used market. volatilities. broker be pairs interbank also the cross-currency in can in traded volatility are framework contracts ATM of the number calculate limited a to where used be can formulas These Rearranging: 2010). Edition, 1st Sons, & Wiley book, (John Clarke’s J. Iain in found be 16.2 EXHIBIT 16.1 EXHIBIT ■ ■ ■ 𝜌 𝜎 𝜎 Example example, for therefore, is volatilities ATM for rule cosine The can valid is volatility ATM to trigonometry from transformation this that proof A UUDUSDCAD AUDUSD USDCAD AUDUSD acltn mhADCDATM: AUD/CAD 1mth Calculating : 1 1 t ATM mth t ATM mth 𝜎 EURGBP T mle oaiiytriangle volatility implied ATM triangle rule Cosine 1 mth = = 2 =− 4.25 6.30 = 𝜌 𝜎 50 = % EURUSD % % 𝜎 EURUSD 2 + oeg xhneOto rcn:APattoesGuide Practitioners A Pricing: Option Exchange Foreign 2 𝜎 2 𝜎 GBPUSD + EURUSD 𝜎 GBPUSD 2 .𝜎 −( 2 GBPUSD 2 − 𝜎 𝜎 EURUSD EURGBP .𝜎 2 GBPUSD .𝜌 ) ATM VOLATILITY AND CORRELATION 315 %) 50 %× 4.25 %× 6.30 ). This is shown in Exhibit 16.4. o × 90 2 = −( 0 2 1 − % 4.25 is used in the formula to take the currency pair + 2 % % 6.30 5.55 √ = = negative correlation Realized AUD/USD versus USD/CAD historical spot correlation mth ATM 1 AUDCAD 𝜎 If correlation is more unstable, this implies that a widerThe bid–offer triangle representation spread of should implied ATM volatilities explains the link between When correlation is an input to the formula, it is often not straightforward to Note that the EXHIBIT 16.3 angle within the triangle is 90 degreesNote (cos that the cosine rule reduces to Pythagoras’s Theorem in this case. historical spot correlation using different sampleadjusted windows, for and any then strong take an recent average trend in the data as perbe Exhibit applied 16.3. in the cross-currency pair. cross volatility and correlation. If the correlation between major pairs is 0%, the is used. determine what value to use. Theconsistent implied so correlation between interbank tenors broker isbe often prices used fairly in to the calculate cross-currency it. at If other market tenors prices could are not available, traders often assess the quotation order into account.and The CAD/USD formula takes (the correlation commonthe between currency AUD/USD market as quotes CCY2 in AUD/USD both and cases). USD/CAD, However, the since negative of the correlation Therefore: 316 ATM VOLATILITY AND CORRELATION urnyAMvltlt ERGP eue ssoni xii 16.6. Exhibit in shown as reduces (EUR/GBP) volatility ATM currency 16.5. Exhibit per as tightens triangle volatility 16.5 EXHIBIT 16.4 EXHIBIT nefc,AMipidvltlt eescet onso h correlation. the on bounds value. create negative levels a volatility square-rooting implied to ATM effect, due In breaks formula cross-volatility the 100%, is static. be will GBP/USD EUR/GBP and pair EUR/USD cross-currency when the way, together another perfectly Put move 16.7. spots Exhibit in shown as other fERUDadGPUDAMvltlte r o qa u hi correlation their but equal not are volatilities ATM GBP/USD and EUR/USD If each of top on be will cross- lines the and the degrees 0 constant, is angle the remaining 100%, is lengths correlation the major If the with tightens, angle the implied the As within angle the higher, goes pairs major between correlation the As T mle oaiiywt eocorrelation zero with volatility implied ATM orlto n nes oiefunction cosine inverse and Correlation ATM VOLATILITY AND CORRELATION 317 : wider angle in ATM volatility triangle tighter angle in ATM volatility triangle → → Correlation and cross volatility move in opposite directions ATM implied volatility triangle with 100% correlation ATM implied volatility triangle with correlation above zero lower cross volatility higher cross volatility In practice, this framework is often used to generate ATM curves in cross-currency If EUR/USD ATM volatility is 10% and GBP/USD ATMThis volatility is is a 10%, key the result: If the correlation between major pairs is –100%, the angle is 180 degrees and As the correlation between major pairs goes lower, the angle widens and the → Lower correlation between major pairs → Higher correlation between major pairs ■ pairs using the ATM curves in the two major currency pairs and a term structure of directions as shown in Exhibit 16.9. relationship between correlation and cross volatility is shown in Exhibit 16.10. ■ major lengths stay constant;Exhibit hence 16.8. the cross-ATM volatility increases as shown in the cross ATM volatility iscorrelation the sum implies of the EUR/USD two and major ATM GBP/USD volatilities since spots –100% move in perfectly opposite EXHIBIT 16.7 EXHIBIT 16.6 318 ATM VOLATILITY AND CORRELATION XII 16.9 EXHIBIT 16.8 EXHIBIT XII 16.10 EXHIBIT T mle oaiiytinl ih–0%correlation –100% with triangle volatility implied ATM zero below correlation with triangle volatility implied ATM rs-urnyAMvltlt esscreainprofile correlation versus volatility ATM Cross-currency ATM VOLATILITY AND CORRELATION 319 . The . The 3 2 3 3 ) only, 3 CCY CCY CCY CCY 2 1 2 1 CCY calculated; 2 3 CCY CCY is calculated; CCY CCY CCY 3 CCY 𝜎 𝜕𝜎 𝜕𝜎 𝜕𝜎 𝜕𝜎 2 . CCY . 2 3 3 CCY 𝜎 CCY CCY CCY 2 2 𝜎 CCY CCY vega vega = = 2 3 ) are held constant and hence the is also calculated. The major ATM CCY CCY 1 3 1 𝜌 CCY CCY CCY 1 CCY 𝜎 therefore depends on the correlation between therefore depends on the correlation between 3 2 and CCY CCY 1 1 2 dephased_vega dephased_vega CCY CCY CCY 1 is a term from physics that roughly means ‘‘shifting an CCY 𝜎 Dephasing dephased_vega dephased_vega only is flexed a small amount and the impact on only is flexed a small amount and the impact on 3 2 CCY CCY 1 1 CCY CCY Within the dephasing calculation, the system of ATM volatilities and correlations For example, the vega on a EUR/AUD contract can be split into equivalent The sensitivity to changesimplied in volatilities correlation ( dephased correlation causes a change in thewhich cross-ATM changes volatility contract ( values. volatility is measured using magnitude of pairs CCY1/CCY3 andCCY2/CCY3 CCY2/CCY3. cross The vega will higher dephase into the CCY1/CCY3 vega. correlation, the more magnitude of pairs CCY1/CCY2 andCCY2/CCY3 CCY2/CCY3. cross The vega will higher dephase into the CCY1/CCY2 vega. correlation,𝜎 the more that is, the other major ATM volatility is flexed and its impact on the cross-ATM 𝜎 that is, one major ATMvolatility is volatility measured using: is flexed and its impact on the cross-ATM ■ ■ rency pairs, CCY1/CCY2 andcurrency pair: CCY1/CCY3, CCY2/CCY3, as and shown in a Exhibit vega 16.11: ■ exposure in the cross simultaneously. Usually the cross pair is split intothe two case; USD pairs for but example, this USD/PLN is would notEUR/PLN. always most naturally be split into EUR/USD and is kept constant except for one element, which is flexed. Given two major cur- vega exposures. exposure into its core constituents.’’ EUR/USD and AUD/USD vegaversus exposures AUD/USD plus correlation. anliquid This cross-currency exposure transformation pairs or to is when the risk particularly in EUR/USD many useful currency in pairs is less being considered should disregard the effect of the USD event. The same ATM volatility triangle framework can be also used to calculate dephased implied correlation maintained by traders. Withincorrectly this adjust framework for it events. is For important example,same to if tenor EUR/USD are and both AUD/USD ATM raised at due the to an important USD event, the EUR/AUD ATM Dephased Vega ■ 320 ATM VOLATILITY AND CORRELATION o T oaiiyadtecreainbtenERUDadUDCHis EUR/USD to USD/CNH similar and volatility EUR/USD ATM an between has correlation EUR/CNH Therefore, the negative. and small volatility ATM low a 16.13 EXHIBIT 16.12 EXHIBIT 16.11 EXHIBIT httedpae ea ontncsaiysmt qa h rs vega. cross the Note equal will to 16.13. sum Exhibit vega necessarily in not do EUR/CNH shown vegas as dephased of USD/CNH the that than majority rather the EUR/USD into Therefore, dephase EUR/CNH. and EUR/USD 16.12. Exhibit in triangle volatility ATM an in shown as h hrns fteageat angle the of sharpness The Example osdr1rER S,adCHAMvltlt.UDCHhas USD/CNH volatility. ATM CNH and USD, EUR, 1yr Consider : y T mle oaiiytinl otiigER S,adCNH and USD, EUR, containing triangle volatility implied ATM 1yr triangle volatility implied ATM ehsdvgsfo y U/N eaexposure vega EUR/CNH 1yr a from vegas Dephased 𝜃 ′ hw h ihcreain(5)between (95%) correlation high the shows ATM VOLATILITY AND CORRELATION 321 EUR/USD exposures, in effect assuming that USD/CNH as If USD/CAD goes higher in isolation, again, EUR/CAD gets pulled higher and Risk managing cross-gamma positions becomes a multidimensional problem when When managing cross-currency positions, traders often transact delta hedges and Gamma positions in cross-currency pairs produce deltas that are often hedged in the exposures are moved into theexample, major the currency USD/CAD pairs. delta Continuing is thealso no EUR/CAD depends longer on driven EUR/USD only spot. by Therefore,USD/CAD USD/CAD delta it spot; as is a it single no now number. longer It should sufficientgrid be to ideally for view, viewed changes as e.g., a in two-dimensional bothto EUR/USD how multi-asset spot exotic and contracts USD/CAD are spot. risk This managed (see is Chapter very 30). similar EUR/CAD gets longer deltalonger from EUR/USD the delta long and gamma.major longer currency This pairs USD/CAD is to delta rebalance equivalent the so to delta. spot getting can be sold in both the major currency pairsin for cross-currency convenience pairs. Consider or a due longspot to EUR/CAD moves gamma larger higher position. spot in If isolation, bid–offer EUR/USD longer EUR/CAD delta spreads gets from pulled the higher longdelta and gamma. and EUR/CAD This longer gets is USD/CAD equivalent deltato to so rebalance getting spot the longer can deltas. EUR/USD be sold in both major currency pairs then split the risk intois two long separate currency EUR/JPY pairs. delta Forand and split example, it short if into the CHF/JPY short JPY delta, EUR/JPYamounts. trader and they long may CHF/JPY spot sell trades, EUR/CHF with offsetting spot JPY pairs. Sometimes, exposures are moved directly intoexample, more in liquid the currency above pairs. For EUR/USD/CNH example, themight EUR/CNH Greek be exposures viewed doesn’t move (a fairlyUSD/CNH). reasonable assumption given the low implied volatility in It is important that traders understand how to manage exposures in cross-currency Managing Cross-Currency Positions ■

CHAPTER 17

FX Derivatives Market Analysis

raders analyze the FX derivatives market in order to identify relatively cheap and expensive aspects of the volatility surface. This analysis is then used to position Ttrading books and generate trade ideas for clients. The FX derivatives market has many moving parts and there are a correspondingly large number of ways in which 323 it can be investigated.

■ Calculating Breakevens

The simplest FX derivatives analysis involves combining a short-dated vanilla option payoff with its premium to calculate its breakeven. Assuming the option is left unhedged until maturity, if spot moves beyond the breakeven point, the trade will make money. This breakeven point can be compared with expectations of spot movement to determine whether the option is cheap or expensive. For a single vanilla option, the premium (expressed in CCY2 pips) is added onto (for a call option) or taken away from (for a put option) the strike level to determine the breakeven. For example, if current USD/JPY spot is 95.00 and the premium on a 3mth 100.00 USD call/JPY put option is 400 JPY pips, the breakeven on the option is 104.00. This is shown in Exhibit 17.1. As the strike is moved higher, the premium decreases but the breakeven moves further away, as shown in Exhibit 17.2. For a short-dated ATM straddle, the breakeven is calculated by summing the call and put premiums. The breakeven is now two-sided and symmetric, as per Exhibit 17.3. 324 FX DERIVATIVES MARKET ANALYSIS XII 17.2 EXHIBIT 17.1 EXHIBIT aloto breakeven option Call aloto breakevens option Call FX DERIVATIVES MARKET ANALYSIS 325 of spot in, for example, the range on a 1wk ATM straddle. However, care must be breakeven ATM straddle option breakeven Breakeven analysis is particularly popular when the expiry date covers a major Typical trader analysis compares the historic volatility, vanilla optionsrealized should volatility, vanilla be options should sold bebe bought. and effective This if is if applied simple analysis with implied but an it understanding volatility can of its is limitations. lower than The most popular FX derivativesrealized volatility. market The analysis central compares idea is implied that volatility if with implied volatility is higher than realized spot usually needs topositive P&L establish overall to a be generated range from approximately buying a double short-dated ATM theeconomic contract. breakeven event. for The one-daythe forward breakeven overnight over volatility (i.e., theevent). the This event overnight value ATM can can option be be compared expiring with calculated after historic the from spot moves caused by the event. last week versus the taken because these measures arewell not establish directly a range comparable. larger For than80 example, a pips spot 60-pips up may breakeven, and but then iflikely), it 80 money moves, pips will for down, still example, be unless lost the from delta buying has the been straddle. hedged As perfectly a rough (not rule of thumb, EXHIBIT 17.3 Implied versus Realized Analysis ■ 326 FX DERIVATIVES MARKET ANALYSIS XII 17.4 EXHIBIT expensive. is 17.5. volatility Exhibit because implied in that shown volatility signal is a situation implied implied not This case, higher is this volatility In a realized volatility. above realized show volatility dampen holidays usually Year’s will New and January Christmas early in volatility the assessing thing; problem. and another same volatility is the themselves mean Implied events they in volatility. increased when value have reduced compared best usually have are released) usually volatility is realized days data holiday economic and significant volatility which on in days shown (i.e., is This samples. volatility spot Realized historic future. using the calculated 17.4. in is Exhibit expiring it contracts looking; option backward for is pricing the gives it comparable returns. log sample the of volatility annualized the is measure required constructed—additional be to tenor surface benefit. non-market minimal volatility a only Using entire for or market. complication the forward) usually the the require (e.g., is in would strike etc.) quoted expiry another 2mth, directly for 1mth, are volatility (e.g., implied contracts the tenor reference market these a because at used volatility implied ATM the XII 17.5 EXHIBIT o xml,cmaig1t T mle oaiiywt mhrealized 1mth with volatility implied ATM 1mth comparing example, For days Event holidays. and events by impacted are measures volatility both Second, are they but linked are volatility realized and volatility Implied volatility spot (historic) Realized volatility Implied o ubro esn.Frt mle oaiiyi owr looking; forward is volatility implied First, reasons. of number a for elzdvltlt essipidvltlt neryJanuary early in volatility implied versus volatility Realized volatility implied versus volatility Realized sa h oeo h Xdrvtvsmre.Wti analysis Within market. derivatives FX the of core the at is scluae rmso ape.The samples. spot from calculated is o lasdirectly always not FX DERIVATIVES MARKET ANALYSIS 327 . Note the gaps in data P.M 10.5% 7.6% 7.9% = = = samples .samples and the other at 5 P.M. P.M also significantly impact the realized volatility P.M. sample time spot market activity on a given trading venue but noise in the all and High-frequency AUD/USD spot trades In general, using a small numberUsing of higher samples frequency samples causes gives realized better volatility results to up to be a point but realized Using the data from Exhibit 17.6: Sample frequency Therefore, when analyzing volatility at shorter tenors (approximately sub-3mth) Realized volatility calculated using 5 Realized volatility calculated using minute samples Realized volatility calculated using 3 EXHIBIT 17.6 volatility does not necessarily converge‘‘tick data’’ contains as sampling frequency increases. So-called ■ ■ biased low. two daily snapshots, one taken atover 3 the weekends, the outlierinformation samples is in being ignored the if high-frequency only data, daily and samples how are much used. ■ major events. Core implied volatilityvariance can attributed be to calculated major events. bycan At removing be longer the ignored tenors, additional because event they and will holiday not day significantly effects bias the analysis. calculation. Exhibit 17.6 shows one month of market AUD/USD spot trades plus the effect of events and holidays shouldrealized ideally volatility be can removed. be Core calculated (excluding by events) removing log returns at or immediately after 328 FX DERIVATIVES MARKET ANALYSIS h tnadsml elzdvltlt aclto is: calculation volatility realized sample standard The Volatility Spot Realized Calculating used. be could model pricing or from model market parameter surface by volatility volatility impacted a long-term is strike convention-free ATM a the Alternatively, of conventions. positioning the because strike forward the ■ ■ ■ ■ ■ volatility: of source analysis. possible a the introduce into differences error forward versus strike ATM sensible because a years) that ensure to analysis. important the most throughout is used the is It methodology action. for consistent for spot samples and volatility market realized ‘‘correct’’ enough of truly period contain no given is not there a general, will In meaningful. volatility be realized to result daily 1wk using and calculate unnecessary to is volatility data realized 5yr calculate to data the high-frequency volatility. if realized example, increased for point in so, the result traded, not Fundamentally, be should offer? can that and that widens, volatility spread bid measure bid–offer market to a is from calculation derived the of data the is realized or calculate data to used biased be being afterward. to adjusted before is data filtered result tick the often of or is set volatility, full data a tick using Therefore, calculated high. volatility realized cause can data or and where y TFipidvltlt:1.0 mr hn1 eo h T volatility) ATM the below 1% than (more 14.30% volatility: implied strike) ATMF ATM 5yr the above figures five (almost 78.30 forward: 5yr 15.35% volatility: implied ATM 5yr 73.85 strike: ATM 5yr 95.90 spot: AUD/JPY hnaayiglne eosi sotnpeeal oueteipidvltlt of volatility implied the use to preferable often is it tenors longer analyzing When Example two (past tenors longer at volatility implied using when taken be also must Care Using tenor. calculation with aligned be should frequency sample general, In trade it is represent; samples spot the what account into take to important is It = X STDEV.S. stema fthe of mean the is N stenme flgrtrsi h sample, the in returns log of number the is U/P T essAM a-h-oe-owr)implied (at-the-money-forward) ATMF versus ATM AUD/JPY : X i hsfnto a eacse nEclusing Excel in accessed be can function This . 𝜎 Realized = √ ∑ N ( X − i − 1 X X i ) 2 r o returns log are = ln = ( STDEV S S i − i 1 ) , FX DERIVATIVES MARKET ANALYSIS 329 is year N where realized volatility year N √ by Realized 𝜎 252. Going forward, the term √ in realized volatility is also important when comparing realized AUD/USD daily spot samples trends : Calculating AUD/USD realized spot volatility using daily samples: Assessing Example To calculate the annualized volatility, multiply Calculate the standard sample realized volatility offormula. the log returns using the above Calculate annualized realized volatility.volatility must Data be multiplied is by sampledis taken daily to so mean annualized the realized volatility. realized Get daily sampled spot data (shown in ExhibitCalculate 17.7). log returns (shown in Exhibit 17.8). Notes on Calculating Realized Spot Volatility Note 1: The number ofand this trading is days the in factor awhether most there year are often in exactly used the 252 to trading United annualize days States in daily is a sampled usually given realized currency 252, volatility or not. volatility with implied volatility. Set uprealized a volatility rolling time calculation series window using andthe the calculate realized samples a spot within volatility the is window. calculated In using Exhibit a 17.9 rolling 6mth calculation window. ■ ■ the number of samples per year. ■ ■ EXHIBIT 17.7 330 FX DERIVATIVES MARKET ANALYSIS XII 17.8 EXHIBIT XII 17.9 EXHIBIT U/S al ptlgreturns log spot daily AUD/USD U/S mhraie ptvolatility spot realized 6mth AUD/USD FX DERIVATIVES MARKET ANALYSIS 331 ) volatility 1). The weights EWMA < ( 𝜆 2 < 1 − i , EWMA 2 ) 1 i 𝜆𝜎 X ( − + 2 2 parameter (0 N i 1 ∑ X 𝜆 X ) ) 𝜆 𝜆 √ − parameters are shown in Exhibit 17.11. − = 1 1 𝜆 ( ( √ √ Realized 𝜎 = = i 1 , , EWMA EWMA 𝜎 𝜎 0.97 is roughly equivalent (1% tolerance) to using 151 samples. = 𝜆 Exponentially Weighted Moving Average is therefore equivalent to using a larger sample window. A ground- 𝜆 The EWMA calculation builds up over all available samples, not a specific sample Higher Therefore, Note 3: To keep things simple, enough data samples should be used to ensure that Note 2: It sometimes makes sense to remove the mean term from the realized Under EWMA, the impactover of time as the per extreme Exhibit 17.12. spot jump decays away exponentially window. Therefore, for example,rather ‘‘0.97 EWMA than, realized for volatility’’volatility is example, still calculated needs ‘‘6mth to EWMA be annualized realized in the volatility,’’ same although way as EWMA standard realized volatility. exponentially on older samples basedfor on historic a parameters for different breaking J.P. Morgan RiskMetrics paper from 1996shows which that is using available on the internet compare standard realized volatility withjumps. implied A volatility rolling when realized the volatilityexit data of calculation contains an that extreme clearly jump into shows the the sample entrance window and is givencan in be Exhibit used 17.10. instead. Withinsample EWMA, with weights the are most attached recent to samples each log weighted return most in highly the and weights decreasing Exponentially Weighted Moving Average (EWMA) Volatility The standard realized volatilityextreme jumps calculation within the does data: Standard not realizeddata volatility behave drops point sharply well exits as the when the jump there back are of the sample window. This feature makes it difficult to steadily rising the same amount (involatility log-space) when each calculated day with will the havecertainly zero not standard daily equivalent method. realized to However, a that static spot spot within action a is trading position. issues around biased versus unbiased calculations can beof ignored. thumb, As a aim very for rough at rule least 50 data samples within the realized volatility calculation. Tradable volatility depends mainly onis spot trending changes (hence rather a than larger whether mean the term market exists in the calculation). Consider that a spot volatility calculation because it can introduce noise into the calculation, that is, 332 FX DERIVATIVES MARKET ANALYSIS XII 17.11 EXHIBIT 17.10 EXHIBIT mhraie oaiiywt neteeso jump spot extreme an with volatility realized 1mth WAweights EWMA FX DERIVATIVES MARKET ANALYSIS 333 must be made with reference to the sample frequency 𝜆 can make a big difference to the output. Once again, there 𝜆 , although values between 0.90 and 0.99 are most commonly used 0.97 EWMA realized volatility with an extreme spot jump 𝜆 This measure of realized volatility looks intuitively better, but if the extreme Finally, the choice of traded with each day that passes.should This be suggests adjusted that the in realized a volatilityrequires similar calculation substantial manner, using additional different calculation expiry forrarely dates. applied minimal However, in benefit this practice. so this adjustment is Realized Spot Volatility versus Realized Forward Volatility A European vanilla option hasoption delta expiry; exposure i.e., (along a withforward. 1yr all In vanilla other practice option Greeks) this should tomaturity means ideally the throughout that be its if life, delta forward delta hedged deltas hedging to with closer was a and performed closer 1yr maturities to should the be option is no ‘‘correct’’ in practice. The choice of and the approximate number ofEWMA samples calculation. which should have significance within the event is truly an outlier, perhaps itanalysis. would Plus, be better although to the exclude exponential it decay completely functioncertainty from seems the that reasonable, it there is is the no right way to model the market ‘‘forgetting’’ extreme events. EXHIBIT 17.12 334 FX DERIVATIVES MARKET ANALYSIS ■ ■ ■ ■ ■ the in calculated is ■ outright spot: of forward volatility the realized the of as way volatility same realized The samples. daily varied. be both can window now analysis sample volatility and Realized tenor calculated. forward often two-dimensional; is becomes tenor fixed a to outright forward XII 17.13 EXHIBIT neetrt volatility rate Interest correlation rate interest versus Spot calculated is volatility window. realized calculation 6mth outright rolling forward In a 5yr volatility. over and implied of spot series the time 17.14, a generate Exhibit to window calculation rolling a Use frequency. sample on based volatility annualized Calculate 17.13). returns. Exhibit log in Calculate (shown data outright forward 5yr sampled daily Collect ealta ncniuu iespace: time continuous in that Recall by: driven are differences volatility forward realized versus volatility spot Realized Example the of volatility realized the maturities, long-dated analyzing when However, acltn U/S elzd5rfradotih oaiiyfrom volatility outright forward 5yr realized AUD/USD Calculating : U/S al ptad5rfradotih samples outright forward 5yr and spot daily AUD/USD F T = S . e ( rCCY 2 − rCCY 1 ) . T FX DERIVATIVES MARKET ANALYSIS 335 and forward to time ) S ( T realized forward volatility will be realized forward volatility will be as spot moves higher or lower the as spot moves higher or lower the → → → → SwapPoints + S = T F define the difference between spot AUD/USD spot and 5yr forward realized volatility with 6mth rolling : swap points ) T F Plus, if interest rate volatility is zero, realized forward volatility is equal to realized If CCY1 interest rates rise or CCY2Therefore: interest rates fall, swap points go more If CCY1 interest rates fall or CCY2 interest rates rise, swap points go more lower than realized spot volatility. forward moves more in thehigher same than direction realized spot volatility. Positive correlation between spot and CCY1 interestbetween rates or spot negative correlation and CCY2forward interest moves less rates in the same direction Negative correlation between spot and CCY1 interestbetween rates or positive spot correlation and CCY2 interest rates ( spot volatility, whereas if interesteffect), rates then are realized forward volatile volatility (with will no be significant higher correlation than realized spot volatility. ■ positive. If spot is static, the forward moves higher as shownnegative. in If Exhibit spot 17.16. is static, the forward moves lower as shown in Exhibit■ 17.17. T This framework is shown in Exhibit 17.15. EXHIBIT 17.14 calculation window And 336 FX DERIVATIVES MARKET ANALYSIS acltn elzdSo essItrs aeCorrelations Rate Interest versus Spot Realized Calculating 17.17 EXHIBIT 17.16 EXHIBIT 17.15 EXHIBIT orlto samaueo h tegho eainhpbtentovariables. two between relationship of strength the of coefficient Pearson’s measure a is Correlation ptad5rfradfaeokwt ihrforward higher a with framework forward 5yr and Spot framework forward 5yr and Spot ptad5rfradfaeokwt oe forward lower a with framework forward 5yr and Spot 𝜌 stesadr esr fraie correlation: realized of measure standard the is 𝜌 X , Y = cov 𝜎 ( X X 𝜎 , Y Y ) FX DERIVATIVES MARKET ANALYSIS 337 is X 𝜎 Y, and X is the number of data N CORREL. = ,and i X is the covariance between ) Y − i Y )( is the mean of the X N X − i X, X ( ∑ AUD/USD versus USD/JPY spot correlation )= Y , X : Calculating spot versus interest rate correlation using daily samples: ( cov Example It is not always straightforward to determine which interest rate data to use Correlation calculated using a small number of samples can be extremely unstable Get daily sampled spot and interest rate data (shown in Exhibit 17.19). EXHIBIT 17.18 rates in the other currency. These issuesmarket can usually analysis be ignored providing within a FX derivatives consistent measure of interest rates is used. Whether within analysis; different interest rate instrumentsIn are traded liquid at different G10 maturities. currencyare pairs traded, whereas at at shorter longerinstrument. tenors, tenors, In OIS, other interest currencies, rate libor, interest swaps deposits, rateso, are markets for and often are example, the futures less a liquid most USD than liquid rates forwards, curve plus the forward can be used to imply interest variables should be sampled at exactly the same time. ■ the realized volatility of samples. This function can be accessed in Excel using as shown in Exhibit 17.18. As withbe realized linked volatility, to the the calculation sample window frequency; should shortusing time-scale correlations high-frequency should be data calculated plus sample times must be synchronized; that is, both where 338 FX DERIVATIVES MARKET ANALYSIS U/S ptcue owr oaiiyt elwrta ptvolatility. spot than lower be to volatility correlation forward causes negative spot AUD/USD persistent a but spot. AUD/USD spot, and rates AUD/USD interest USD and between rates interest AUD ■ ■ market. though ■ the even in terms, quoted rate actually deposit never in is calculated contract is that rate’’ interest ‘‘5yr the then ments; the as long so change. inconsequential rates becomes interest future market a as or appropriately reacts deposit, rate a swap, a is product the 17.19 EXHIBIT pt(sse nteADUDeapeaoe.Hwvr uhcagsipc the impact changes such and However, rates above). interest interest example CCY2 AUD/USD CCY1 the between between in correlation seen realized correlation (as negative spot realized a own. and positive to spot desirable a major and more rates often are is yield there is higher (unless there with because currency Therefore, a market country) spot the with the problems in stronger relatively becoming aclt h orlto ewe ptlgrtrsaditrs aereturns. rate interest and returns the log spot within between returns correlation log the Calculate than rather go returns can use they since to (i.e., rates, better calculation interest arguably For is appropriate. it as negative, returns or returns log Take aclt elzdcreaintm eisuigarligsml window. sample rolling a using series time correlation realized a Calculate nraigitrs ae napriua urnyotnrslsi htcurrency that in results often currency particular and a rates in interest rates (CCY2) interest USD Increasing between correlation negative the case, this In between correlation strong no is there data the within that shows 17.20 Exhibit instru- different using generated were curves rates interest full analysis, this In U essUD5rdiydpstrt samples rate deposit daily 5yr USD versus AUD X i = r i − r i − 1 where r i sthe is i th neetrt sample). rate interest FX DERIVATIVES MARKET ANALYSIS 339 forward → i , 1 − i EWMA cov , Y 𝜆 i 𝜎 i + , 1 cov i Y Y swap points more negative . . i 1 X X EWMA → . . , ) ) X 𝜆 𝜆 𝜎 − − = 1 1 i , Y , X =( =( , i 1 cov EWMA cov 𝜌 AUD/USD spot versus interest rate correlation It is also possible to calculate EWMA correlation. The calculation is analogous to The realized volatility of the interest rates themselves can also be calculated and 1. Buy or sell an ATM contract and lock in the implied volatility versus realized There are two main ways to trade implied volatility: volatility difference by delta hedgingmost the appropriate gamma at exposure shorter on tenors the where contract. the This main is exposure on ATM contracts is Trading Implied Volatility EWMA volatility: where: spot higher as CCY1 interest ratesmoves higher less than spot. this is shown in Exhibitas 17.21. Interest the rate realized volatility forward reducesnarrows, outright over as the volatility shown same versus in period Exhibit realized 17.14. spot volatility difference EXHIBIT 17.20 forward in the opposite direction and lead to forward volatility below spot volatility: 340 FX DERIVATIVES MARKET ANALYSIS i–fe ped nvltlt wp r ie hntesra neuvln ATM equivalent details. on more spread for the 31 than Chapter wider See are although contracts. swaps moves, spot volatility as on delta ATM spreads no roughly because remain bid–offer exposures attractive the is and This required samples. is spot hedging daily using usually period agreed can an underlying the a on with differences spread bid–offer wide a over-trading with P&L. However, pair expected possible. negative currency as produce a often in as delta hedged be the important should most delta is difference therefore volatility and implied versus volatility realized the be analysis, will contract the hedging overall. profit delta a maturity, generate a option consistently to if spot the expected and example, to volatility For through implied holidays. 8.0% 10.0% for and realizes bought releases be data can contract economic ATM of 1mth impact the for adjusted analysis, this with For compared vega. than rather gamma 17.21 EXHIBIT a ebuh t80 mle oaiiyadte odbc t90 mle volatility contract implied ATM 9.0% at 1yr back a sold then if and example, volatility implied For 8.0% gamma. at exposure than bought main be rather can the vega where is tenors contracts longer ATM at on appropriate most is This levels. volatility .Byo ela T otatadte ae nidtetaea etrimplied better at trade the unwind later then and contract ATM an sell or Buy 2. let oeie iet rd mle oaiiyvru elzdso volatility spot realized versus volatility implied trade to like sometimes Clients this Within consideration. important an also is hedging delta of frequency The bouelevels absolute U n S elzditrs aevolatility rate interest realized USD and AUD oaiiyswap volatility fraie ptvltlt nasmlrtnr ideally tenor, similar a in volatility spot realized of afradcnrc nraie oaiiyover volatility realized on contract forward —a bouelevels absolute fipidvltlt are volatility implied of FX DERIVATIVES MARKET ANALYSIS 341 volatility risk forward volatility contracts for difference on implied volatility. not ). This is a pure exposure to forward implied volatility between in implied volatility and realized volatility since the two measures FVA ( trends . One of the key reasons for this risk premium is the preference of traders Finally, many studies conclude that there is a persistent trend that short-dated Many of the techniques covered in this chapter work best as relative value Clients sometimes like to trade implied volatility using a After trading a long-dated ATM contract, if spot moves, the traded strike is no When trading ATM contracts, although the initial exposures are primarily gamma Trading Implied Correlation To trade implied correlationconstructed using as per ATM Exhibit instruments, 17.22. a volatility triangle can be for the P&L profileover from the long P&L profile gamma from (manyfrom short small a gamma losses job (many preservation versus smalleconomic gains perspective, few data versus particularly large is few on released. gains) large expiry losses) dates when important volatility levels in obscurebut crosses often the suggests spread interesting crossconsidered. trading involved opportunities in actually executing the trade alwaysimplied volatility needs is to too expensive. be Inpremium academic circles this is called the date. See Chapter 31 for more details. tools. Running analysisoutliers that over might work multiple asshould trade currency always ideas pairs be although the taken and bid–offer into spread tenors account. on the Looking can ATM at identify theoretical midmarket implied of the full volatility surface. Whensimply this happens a the function P&L from oflong-dated the ATM vega trade options is and are no longer implied ATM volatility change.agreement Put another way, two dates in the future and therefore requires no delta hedging prior to the first contracts increases if spot stays around thethe strike strike. or reduces Therefore, if the spot realized movesis away P&L not from from simply trading a a functiondependent. delta of hedged realized volatility ATM versus contract implied volatility; it is also path longer ATM and therefore the implied volatility for the strike becomes a function and realized volatility and lookdriving implied for volatility lags changes in or vice the versa? data: Are realized volatilityand changes vega, the exposures changeon as short-dated time vanilla passes or options, the as market time moves. passes For example, the gamma exposure on short-date that will (almost certainly) generate a profit. Forinvestigate this analysis it is most appropriate to often trend in the same direction. Consider the spread between implied volatility 342 FX DERIVATIVES MARKET ANALYSIS elzdVltlt Convexity Volatility Realized period. agreed an over samples a significant using a cost may triangle implied volatility on the of cross. in spread time based level of legs over different amount three P&L changing all a exposures a transacting at involving plus generate issues later apply, though, to ensure unwound volatility, Again hedged or to correlation. implied correlation, implied delta pairs trading realized either currency with versus be major As correlation can exposure. the correlation position in implied the 16) initial Chapter clean (see a vega dephased initial ■ ■ is, that cross, the buy and majors the ■ is, that cross, ■ the sell and majors the Buy CCY1/CCY3: and 17.22 EXHIBIT supin tcnb hw htepce & rmtaigacntn long constant a trading from P&L expected that shown be can it assumptions volatility realized with volatility that implied comparing when is of aware be to point final A u C2CY T contract. ATM CCY2/CCY3 a Buy contracts. ATM CCY1/CCY2 and CCY1/CCY3 Sell contract. ATM CCY2/CCY3 a Sell contracts. ATM CCY1/CCY2 and CCY1/CCY3 Buy let oeie rd mle orlto essraie orlto differences correlation realized versus correlation implied trade sometimes Clients zero is there that such set be should contracts ATM the on notionals The Sell CCY1/CCY3: and CCY1/CCY2 between correlation implied short go To go to that 16 Chapter from Recall xetdPLi o ieri elzdvolatility realized in linear not is P&L expected orlto swap correlation T oaiiytinl o rdn correlation trading for triangle volatility ATM afradcnrc nraie orlto sn daily using correlation realized on contract forward —a ogipidcorrelation implied long ne tlzdBlack-Scholes stylized Under . ewe CCY1/CCY2 between FX DERIVATIVES MARKET ANALYSIS 343 based on the following 2 squared ) is theta. 𝜃 (discussed in the Trading Implied Implied Realized 𝜎 𝜎 ( 𝜃. ]=− Γ . The practical consequence of this is that implied L & P [ E volatility risk premium , the expected P&L from gamma trading will exactly offset Simulation P&L from gamma hedging Implied 𝜎 is the P&L from gamma trading and = is proportional to realized volatility squared as per the above formula Γ ] L Γ L & P Realized & 𝜎 P [ Exhibit 17.23 shows simulation results demonstrating how P&L from gamma If interval is applied the P&LBlack-Scholes formula volatility derivation). is small (this is the essence of the standard E (i.e., the chart is quadratic, not linear). As hedging interval increases, P&L volatility increases. When a very tight hedging EXHIBIT 17.23 trading changes with realized volatility and hedging interval: ■ ■ underperforms implied volatility byrealized volatility the convexity same amount.volatility should In be other higher than words,is realized there volatility another for is cause itVolatility to of section be above). the fair value. This effect where the theta paid.realized But volatility when outperforms long implied gamma, volatility more than money is is lost made if trading realized gamma volatility if gamma position is proportional toequation: realized volatility 344 FX DERIVATIVES MARKET ANALYSIS ■ aktIsrmn Analysis Instrument Market uigtefiaca rssi 08 mle oaiiyi otcrec ar jumped pairs currency example, most in For volatility level. implied 2008, historical in its crisis from financial the changes during are completely moments instrument volatility key financial realized the similar a traders, hence and For time generated. over consistent is fairly be can pair currency curve ATM volatility the realized range. volatility, and historical range implied well-established a below in is still stuck it is is however, spot fine, If range. is historic This its middle. of the the of toward end back consider low move to or to high vital likely the toward more currently is is it it cone, and time) over average long-run its 17.24. plus Exhibit shown, in shown is is cone period volatility and given example lowest An a highlighted. The over is level tenor history. current each its the in versus observed looks volatility curve implied ATM highest current the how visualizing of and reverts, mean often volatility Implied FX markets, Curve rate ATM interest and history. forward, own their spot, against FX assessed also the be can to instruments linked market derivatives being as well As XII 17.24 EXHIBIT mle oaiiyma eet eas h antd fteflw nagiven a in flows the of magnitude the because reverts mean volatility Implied toward back move to tends (i.e., reverts mean volatility implied if is, idea The why U/S y mle oaiiycn nDc 2013 Dec. on cone volatility implied 1yr AUD/USD h urn T uv stwr h xrm o rbottom or top extreme the toward is curve ATM current the oaiiycones volatility eiechanges regime should ena h otmo its of bottom the near be r omnmethod common a are hntelvlof level the when , FX DERIVATIVES MARKET ANALYSIS 345 8.2% 11.5% = = 1yr ATM implied volatility in various currency pairs during the 2008 financial When trading the ATM curve, it is also important to remember that, in practice, AUD/USD 3mth ATM implied volatility AUD/USD 6mth ATM implied volatility fore, buying 3mth ATM against selling 6mth ATM mayThe be a spread good could trading be opportunity. transacted with eitherP&L initially vega-neutral volatility notionals from (to reduce parallelnotionals moves (to in reduce the the P&L ATM volatilitya curve) profit caused if by or the spot initially ATM moves). gamma-neutral curve The ‘‘flattens’’ trade (i.e., will the make 3mth versus 6mth spread decreases). ■ ■ This implies a 3mth in 3mth forward implied11) volatility of (see 14.05%—much the calculation larger in than Chapter the current 3mth ATM implied volatility. There- its range, ATM implied volatility often jumps higher. ATM Curve: Slope If the ATM curve ismay be sharply attractive. upward For or example: downward sloping, calendar spread trades significantly higher, as shown inand Exhibit forwards 17.25 plus caused increased by uncertainty sharp as movements shown in in spots Exhibit 17.25. ATM implied volatility usually falls slower thanimplied it volatility rises. will If tend spot to is slowly range-bound, ATM drift lower over time, but if spot breaks out of EXHIBIT 17.25 crisis 346 FX DERIVATIVES MARKET ANALYSIS T hne ihrs eeslcagsbcuetetoisrmnsotnmove often instruments two compare the to interesting because be reversal/ATM changes also risk can reversal together. It the risk use. so with to ATM skew changes of the ATM measure to cleaner a proportional often often is tenors. is ratio longer reversal at particularly risk flow, the client to Plus due skew negative or positive persistent 17.26. average. Exhibit the in toward shown back is move this likely of more example will An limits historical compared lower volatility be or a upper can from the parameters smile skew volatility or the reversals model. risk surface in market skew either current using history the with data, series time Using Skew Smile: Volatility year. the of times certain year. at lower accounting move their consistently to of volatility implied end cause or can ket start the at at or hedge year to calendar tend the corporates of as start seasonality the exhibit either can ATM tenor hedging, 1yr FX the corporate of around amounts volatility significant are there where pairs currency In Seasonality Curve: ATM though, usual, As conventions. opportunities. market with trading taken imply be may must curve care ATM shape long-dated the the with differences of forward realized versus volatility spot realized Comparing ■ volatility: spot ■ by impacted less relatively and volatility opportunity. rate trading interest a implies this similar perhaps in outlier, shapes an curve is ATM there compare If to pairs. interesting currency also is It history. from different oiint aeapot rdr a h kwi ‘efrig’ rdr ekto seek Traders ‘‘performing.’’ is skew the reversal say risk long traders a profit, cause a moves make surface volatility to and position spot where pairs currency In Skew Value: Analyzing owr oaiiy h T uv ilmr ieyb onadsoiga longer realized at sloping shorter-dated downward be than likely tenors. more lower will curve is ATM the volatility volatility, forward forward realized longer-dated longer at If sloping upward be likely realized more shorter-dated will tenors. than curve higher ATM the is volatility, volatility forward forward realized longer-dated If ihnti nlssi utb eebrdta oecrec ar have pairs currency some that remembered be must it analysis this Within mar- the into flow this and vega sell net structures hedge corporate of majority The by impacted more relatively becomes volatility implied ATM tenors, longer At significantly levels identify to time over assessed be can curve ATM the of slope The If h kwo h oaiiysiema eet,te ut toward quote a then reverts, mean smile volatility the of skew the FX DERIVATIVES MARKET ANALYSIS 347 X of spot log returns with as seen in Chapter 12. ) spot vega covariance 𝜕 𝜕 between spot log returns and ATM ( 1 implies perfect linear dependence while correlation spot and ATM implied volatility are moving =− 𝜌 is dimensionless; it expresses how two variables ( ) 1or how much 𝜌 ( additionally reflects how much the variables are moving. = ) 𝜌 Y they are moving together. Therefore, the spot versus implied .𝜎 X how Historic USD/JPY 1yr 25d risk reversals correlation 𝜌.𝜎 (= 100%. However, the covariance in the second instance will be larger than + ) move together: Y As an aside, One possible skew analysis technique compares market risk reversal quotes with A first idea might be to look at the 100%. Now, if implied volatility moves up 0.02% for every pip that spot moves + up and implied volatility movesthe down correlation 0.02% between for absolute every pipstill spot be that changes spot and moves absolute down, that volatility of changes the will first. and covariance For example, if implied volatility movesand up implied 0.01% volatility for moves every down pipcorrelation that 0.01% spot for between moves every absolute up pip spot that spot changes moves and down, absolute the volatility changes will be implied volatility log returns. However, using correlationit is not is appropriate important because totogether, quantify not just volatility relationship can be modeledimplied volatility using log the returns. where it is not. the realized spot versus volatility relationship. This tiesexposure in in with the the risk fact that reversal the contract main is vanna EXHIBIT 17.26 buy skew in currency pairs where it is performing and sell skew in currency pairs 348 FX DERIVATIVES MARKET ANALYSIS ihhsoy h aktbtefl ntuetcnb sdbtdet h broker the to due compared but be used can be smile can volatility instrument the butterfly of market wings The current history. the with data, series time Using Wings Smile: Volatility delta 25 or delta 10 pairs. how expensive currency see different or to in cheap analysis reversals relatively same risk identify the through to reversals order risk analysis, in delta compare historic 25 they this and all delta interesting 10 is with It both As followed. run blindly managed. to being than or rather in pegged considered intervention be is is must signals spot there trading if if shifts, or looking regime market, are forward spot there the if is incorrect generated strong contract be Plus will data. reversal signals historical is from trading risk calculated This is the covariance value. realized obviously the while perceive Most consider. to to order issues in compared and these plotted 17.27. Exhibit changes, in be volatility demonstrated implied can log versus series changes time spot log of covariance realized covariance changes volatility 17.27 EXHIBIT hw h urgt1dad2dadAMsrkso h U/P oaiiysmiles volatility AUD/JPY the dates. on 17.29 two strikes Exhibit same ATM the while and for 2009, 25d and 1, 10d January outright and the 2008, shows 1, AUD/JPY January shows on 17.28 Exhibit instruments 12. market Chapter butterfly in shown market as versus down wings break can the relationship pairs currency high-skew in conventions butterfly hsaayi ok eta i eos(mht mh u hr r lnyof plenty are there but 6mth) to (1mth tenors mid at best works analysis This the to proportional is reversal risk market delta 10 or delta 25 the Assuming iesre fERUD2 ikrvra ess3-a o ptlgimplied spot/log log 30-day versus reversal risk 2M EUR/USD of series Time FX DERIVATIVES MARKET ANALYSIS 349 can be determined. based on historic ATM ) volga only vega 𝜕𝜎 𝜕 ( 0.2% although the wings of the volatility smile are + 0.25% to + Historic AUD/JPY key strikes on the volatility smile Historic AUD/JPY market instruments the wings of the volatility smile mean revert, then a quote toward the if By hedging away the other elements, the theta from The theta from a broker fly contract comes mainly from volga but there are also One possible wing analysis technique uses the exposures of a long butterfly contract Again, Therefore, rather than using the market butterfly instrument within analysis, Between January 1, 2008, and January 1, 2009, the 1yr 25d broker butterfly The amount that impliedtheta volatility can must therefore be move calculated. per Withinderivative day this of analysis, in price recall order with that volga respect to is tonot make the implied linear second back volatility in and the implied therefore volatility, broker as fly shown P&L in is Exhibit 17.30. implied volatility moves with the theta paid to hold the butterfly contractcontributions over time. from gamma and vanna.shorter-dated The (1wk?) gamma ATM exposure contract cana be and risk hedged reversal the with contract. vanna a exposure can be hedged with make a profit, traders sayin the currency pairs wings where are they ‘‘performing.’’ Traders arethey performing are seek and not. to sell buy wings wings in currency pairs where to compare the P&L generated from the volga upper or lower historical limits will more likely move back toward theAnalyzing average. Value: Wings In currency pairs where volatility surface moves cause a long butterfly position to a convention-free volatility-of-volatility parametercould from be a volatility used surface tostrike model quantify fly the (e.g., wings the of averageless the the of volatility ATM the smile. implied 10 volatility) Alternatively, could delta the be call 10d used. and 10 delta put implied volatility EXHIBIT 17.29 went lower from far higher on the later date. EXHIBIT 17.28 350 FX DERIVATIVES MARKET ANALYSIS aysadr eitosfo h vrg h otrcn apeis: sample recent most the average the from deviations technique. standard a analysis many uses common this a quantifying is of history way standardized its One with quote market current a Comparing Analysis Historical on pairs, Discussion currency different opportunities. over trading practice. value applied relative in identify when captured to be effective deltas to and most value tenors, is the analysis for move order must this in volatility Again, level implied relatively midmarket practice, the addition, in than that, In further means volatility. spreads data implied daily bid-offer using ATM of and large life, volatility its Plus realized over constant data. the stay historical to underestimates from assumed is calculated butterfly are the while moves from looking volga volatility forward of is implied contract plenty daily butterfly with realized the analysis obviously the Most rough-and-ready of. is aware be this to but issues changes volatility implied daily 17.30 EXHIBIT volatility. where h raee al mle oaiiycag a hnb oprdwt historic with compared be then can change volatility implied daily breakeven The X T stems eetsample, recent most the is & rmaln og position volga long a from P&L Z - score X = stesml en and mean, sample the is X T 𝜎 − X Z X -score hc samaueo how of measure a is which , 𝜎 X stesample the is FX DERIVATIVES MARKET ANALYSIS 351 that generate -scores beyond Z client trades -score is a buy signal and a positive Z around current spot because there are fewer the implied volatility for a specific expiry date and long vega 2.5 are significant. + sticky strike analysis Market participants’ vega positions impact the FX implied volatility market. Another way to analyze sticky-strikeness is with a scatter chart of historic ATM Again, this analysis fails during regime changes when the level of aLooking financial only at recent history is another common issue. Even if value is to be For any mean reverting variable, a negative -score is a sell signal. If the variable is normally distributed, Since interbank flow is mostlythe zero-sum net risk market transfer position. it Forstays is example, around if the spot same levels movesconcluded when sharply that it and the would implied market normally volatility is be expected to rise, it can be Market Positioning Understanding how to interpret market positioning isIn a key simple skill that terms, traders acquire. determiningparticipants market react to positioning market involves changes. observing how market impliedvolatilityversusspot.SpothigherorlowerandATMvolatilityhigherorlowershould tie in with the direction (positive or negative)For of example, the risk USD/JPY reversal usually in the has market. avolatility negative risk with reversal, lower implying spot. higher This implied spotvery versus strong ATM during 2007 implied and volatility 2008 relationship as was shown in the scatter plot in Exhibit 17.31. trading opportunity. This analysis isATM essentially implied a volatility cone more and sophisticated againbut version on it of a relies per-expiry the on and implied -strike volatilitytenor. basis mean This rather reverting than analysis in is the appropriate ATMtrade at contract is longer itself vega tenors at and where a the given the analysis main tends exposure to on work the better in high-skew currency pairs. Within strike is tracked over time. Forvolatility example, for if a it September-23 is 2015 currently 1.3050ago, January strike 2015, two can the weeks be implied ago, calculated today, threedata. one weeks week If ago, and the so implied on, volatility using full was sets stable of and historic then market deviates, this may represent a gauged using only quite recent history, itto is instructive determine to the check historic a bounds longer of timeas horizon the sample. a Fundamentally, reference using for historical what data in the mind future that will the look future like may is look sensible very and different convenient indeed. but keep Volatility Smile: Sticky Strike Analysis instrument completely changes from itsgenerate historical strong level. false Regime trading changes signals willshould with often be any considered systematic carefully analysis. before As being always, acted results on. Z approximately –2.5 and 352 FX DERIVATIVES MARKET ANALYSIS ■ ar Trades Carry h aktoe h etwe rs.Ti nomto a esucdfo the a from as sourced available be becoming can regulations. is market expiries information new in on This of expiring data consequence so. increasing strikes or plus big market week the broker next know notionals, interbank the to large over useful with market therefore strikes the is existing It toward strikes.’’ gravitates ‘‘market often called market spot the and positioning round gamma market (i.e., the levels 24). barriers change and spot American 23 dramatically history, Chapters key can (see recent which in When time knocked, stop-loss volatile. first often increased the more are region, for be spot trade values) certain to spot a spot numbered in cause gamma range- may short remain is orders to market spot spot cause the certain may If hedging a bound. delta in when gamma profit long taking are increased participants region, market If hedging. delta participants’ moves. market spot that as Remember change long. will already positioning are vega they since vega buying participants market 2008 and 2007 from samples 17.31 EXHIBIT h aktde o infiatymv) h lsi Xcrytaei rvnby driven is trade carry FX classic The move). (i.e., happens’’ significantly ‘‘nothing if not money make does to market is trade the carry successful a of essence The h aiu am rmvnlaotoscmsfo tie ls omaturity to close strikes from comes options vanilla from gamma maximum The market via markets and spot FX the impact positions gamma participants’ Market S/P y T mle oaiiyvru ptsatrpo hwn daily showing plot scatter spot versus volatility implied ATM 1yr USD/JPY FX DERIVATIVES MARKET ANALYSIS 353 0.9170, money will above AUD/USD spot from 2006 to 2009 In quiet markets carry trade positions build over time; but if there is a market Another carry trade variation is achieved within a trading position by holding EXHIBIT 17.32 rise was due toAUD increasing versus carry short trade USD positioning.from within Each the carry interest time trades, rate investors differential at went but maturity also long they the currency had appreciation. gained not only in currency pairs that contain onelow-yielding high-yielding G10 emerging currency. markets currency and one shock, the positions can rapidly unwindtime. as The everyone exits longer the the same carry trade tradeit at occurs. builds the Exhibit up, same 17.32 the shows more AUD/USD dramatic spot the from 2006 unwinding to when 2009. Part of this spot be made from the strategy. long cash balances in higherin lower yielding yielding currencies currencies. versus As holding described,their cash yield short balances and cash the generate balances net income money basedbe earned on thought from of the as long carry. Interest and rate short differential cash carry balances trades can are also particularly popular interest rate differentials in the universedifferential of G10 by currency pairs historical (although it standards).interest is If a rate small is AUD/USD 3%, spotforward and is will the 0.9400, be 1yr approximately the USDholding 0.9170. 1yr interest it By AUD rate to buying maturity, is the if 0.5%, 1yr AUD/USD then forward spot AUD/USD at at 1yr 0.9170 maturity and is the interest rate differentials incurrency a and given selling currency the pair—buyingtrading lower the yielding the higher currency. forward. yielding This can For be example, executed AUD/USD simply by currently has one of the largest 354 FX DERIVATIVES MARKET ANALYSIS hte h owr saoeo eo pt h w ae r hw nExhibits in shown are cases two The on spot. depending below spread, call or a above or is spread put forward a the either whether be can This spread. forward static. ATM stays market volatility the the providing positively, curve, decay factor ATM strategies common The carry the three. all all that on of is combinations based or differential, strategies rate interest carry the smile, be can there derivatives differentials rate interest are in volatility. high trades implied increase relatively carry low with FX big but Therefore pairs a reversal. currency cause risk in the and popular of particularly rapidly side high unwind the can to volatility which implied positioning, carry trade pure carry by a low-yielding on the based sell down. strategies shut versus that permanently incurred currencies trading be bias to high-yielding losses currencies) systematic the though, long-term any (buy practice, a methodology In cause trade is P&L. would losses expected there large 2008 positive few that in a versus conclude trades gains carry small often gives many studies of that academic is although trades carry from distribution 17.34 EXHIBIT 17.33 EXHIBIT aiu aof aisoe . r fe huh fa neetn.Ti carry This smile. volatility interesting. the potential as and the drift of over forward thought spread the of the often function of are a cost is 2.5 the trade over of ratio Ratios a payoff. as maximum analyzed be can strategy This 17.34. and 17.33 gi,teie sta h pedwl a u fso ean tiscretlevel. current its at remains spot if out pay will spread the that is idea the Again, versus spot ATM vanilla a trading involves trade carry derivatives FX simple One FX vanilla In possible. are theme trade carry the on variations of multitude A driven partly is reversal risk the and differentials rate interest between link The P&L The wrong. spectacularly went it until well really worked strategy This u pedcrytrade carry spread Put trade carry spread Call PART IV

EXOTIC FX DERIVATIVES

hen an FX derivative contract contains additional features above and beyond Wthe basic vanilla option it becomes an exotic FX derivative contract. There are a staggering range of features that can be added: barriers, averages, variable notionals, payoffs returned in a third currency, and so on. Exotic options can also be combined to form popular structures used to hedge FX exposures. Clients like trading exotic options because they can reflect more precise market views. For example, introducing a barrier that causes the contract to expire if a specified spot level ever trades can make a vanilla option payoff significantly cheaper. By adding a downside knock-out barrier to a vanilla call option the market view expressed by the trade (in isolation) evolves from ‘‘spot will be higher at maturity’’ to ‘‘spot will be higher at maturity without having first gone below the barrier level.’’ As shown previously, the market price for vanilla options is determined using the volatility surface. Exotic options are priced differently because they cannot be successfully priced using a single adjusted volatility. Instead, a reference price for the exotic contract is generated using the Black-Scholes framework and an adjustment is then calculated. The adjustment takes into account the volatility smile plus other 356 EXOTIC FX DERIVATIVES i fti prahi htb xliigtaigrssfo rtpicpe,risk principles, how matter first no from product, risks exotic any trading to explaining complicated. generically applied by The be models. that can pricing rules is and management of approach each profiles management Greek this For risk to of introduced. and reference aim pricing are with the explained products and is exotic explored product are of the risks range trading wide key the a product and models pricing of particularly large levels. get barrier can and near differently evolve options exotic from exposures Greek in adjustment. (covered this models generate to Pricing used framework. usually are Black-Scholes 19) the Chapter in included not factors ihnPr V xtcF eiaie rcn sotie n hnpplrclasses popular then and outlined is pricing derivatives FX exotic IV, Part Within The options. vanilla than manage risk to difficult more often are options Exotic CHAPTER 18

Exotic FX Derivatives Pricing

anilla option contracts are priced using a volatility surface that returns a Vmidmarket implied volatility for a specific maturity and strike. A bid–offer spread is then applied around the mid-rate to get a two-way price quoted in implied 357 volatility terms. Exotic option pricing works differently. Exotic contracts cannot be priced directly off a volatility surface because they have additional parameters (e.g., barrier levels) and therefore a more generic approach is required. Exotic option contracts are priced in premium terms and the pricing is anchored by Theoretical Value (TV)—the CCY1% value of the exotic contract under Black-Scholes assumptions, specifically:

■ A single, static volatility

■ A single, static interest rate in each currency

■ No relationship between spot, volatility, and interest rates The ATM volatility to the final expiry date is used for calculating TV on exotic contracts. This volatility is often taken directly from the desk volatility surface. When calculating TV on an exotic contract it is vital that the correct ATM volatility is used. In practice this means the exotic trader checking the validity of the desk ATM curve with the appropriate vanilla trader or with interbank brokers prior to pricing. After calculating TV, the market price is then quoted as an adjustment to TV that takes into account all relevant factors not included within the Black-Scholes 358 EXOTIC FX DERIVATIVES PRICING res(.. xoue acltduigteBakShlsfaeok.Ti keeps This framework). Black-Scholes the using these calculated exposures in (i.e., and Greeks contract a adjusted. on manually recognize risks be also to significant need must capture will Traders pricing not situations price. will midmarket models a pricing obtain when to model popular pricing of suitable 19. These selection Chapter in A surface. introduced contracts. are volatility dynamics exotic various the price their by and to models used generated pricing then prices are match models model calibrated options the vanilla example, using for that, priced such calibrated framework are Black-Scholes that the parameters of additional extensions with are models Pricing quants. by developed adjustment. TV value of the the in case none included this be correlation, In must rate framework. effects Black-Scholes these interest the of versus within for spot accounted or are TV which gamma, the rate in interest included structure, be must structure term In curve itself. maturity ATM option the the of adjustment. at value volatility ATM the the case to this just not maturity, 18.1. option Exhibit the in shown is This the details). be for would 14 adjustment Chapter TV (see price the option vanilla and vanilla the volatility be ATM would the TV the using framework, calculated exotics this under priced were option two-way terms. a notional obtain of to CCY1% price in midmarket quoted the price around market applied then is spread bid–offer A that: such the terms called is This framework. XII 18.1 EXHIBIT hogotti eto,i o pcfidohrie re xoue r TV are exposures Greek otherwise, specified not if section, this Throughout most the use traders contract, exotic an on risks trading main the understanding By models pricing using contracts exotic on adjustments TV generate Traders term rate interest to exposures significant have contracts options exotic Some to up curve ATM whole the to exposures have contracts options exotic Many vanilla a If smile. volatility the usually is adjustment TV the within factor main The ail n xtcpiigmethodologies pricing exotic and Vanilla TV + VAdjustment TV Vadjustment TV = imre Price Midmarket ti utdi C1 premium CCY1% in quoted is It . zeta fthe of EXOTIC FX DERIVATIVES PRICING 359 ). . ) (–55/0.30 3mth 30 oct-1 nov 13 3mth 30 oct-1 nov 13 tky/0.8100 aud/usd ). ) ). A$1-2 vol 11.4 ): A one-touch contract pays out cash at maturity if aud/usd ot ) : 0.8100. : 11.4% ( : –55/USD deposit rate: 0.30% ). spot 0.9080 One-touch option contract details tky : one-touch ( : AUD/USD ( : AUD1m to AUD2m ( : 3mth, and the exact dates are specified for clarity: expiry date: October : 0.9080 ( : Tokyo ( Reference AUD/USD market data is also given: Translating, that is: Spot 3mth swap points 3mth ATM volatility Notional Expiry 30, 2013/delivery date: November 1, 2013 ( Cut One-touch barrier level Contract type the barrier level trades prior to maturity. Currency pair EXHIBIT 18.2 ■ ■ ■ ■ This is enough information to enterin the Exhibit contract 18.2. details into a pricing tool as shown ■ ■ ■ ot/spot 0.9080/tv 5.35% vol 11.4/del A$2.1/–55/0.30/A$1-2 vh ■ ■ Here are some exoticinterbank contract exotics broker details on quoted July 30, on 2013: a chat between a trader and an the profiles clean butmodels since in these practice exposures better tradersmoves. reflect often expected changes use in P&L Greeks as generated the market by pricing Exotic Pricing Example ■ 360 EXOTIC FX DERIVATIVES PRICING ■ rcn h oaiiySmile Volatility the Pricing r hrfr:Wa osteT dutetrpeet n o si calculated? it is how questions and key represent, The adjustment adjustment. TV TV the two-way does a What of therefore: form are the in price a quotes ( trader details the broker the matches also which the in whichmatchesthebrokerdetails( trading it. use when also occurs clients additional institutional contracts this some exotic plus is contracts on market contract broker exotic matching interbank correct within TV the important. parameters that is extra confirm safety the helps Given matching priced. TV being This reference. a against generated. be would TV same the and contract data by exotic priced these Therefore, be volatility. could the details and rate, interest one forward, 18.4 spot, Exhibit and time tool the pricing at the outputs. market in tool data the pricing market in the the observed shows shows values 18.3 mid Exhibit to pricing. close of be should data market This 18.4 EXHIBIT 18.3 EXHIBIT h rmr lmn ihnteT duteti sal h oaiiysmile. contract. volatility given a the estimate for to smile usually how volatility knowing the is is of pricing impact adjustment exotic the understanding TV in step the first the within Therefore, element primary The on akt h xml rd,teT hw ntepiigto s53 AUD%, 5.35 is tool pricing the in shown TV the trade, example the to back Going checked be can TV resultant the data, market and details trade the entering By using defined fully is date expiry given a to data market Black-Scholes, Under n-oc pinpiigoutputs pricing option One-touch data market option One-touch n rdn desk trading any tv5.35% e A$2.1 del ).Thedeltainthepricingtoolis214AUD% ntemre sn h upidmarket supplied the using market the in .Oc h Vhsbe matched, been has TV the Once ). EXOTIC FX DERIVATIVES PRICING 361 Long risk reversal vega profile Just as the volatility smile can be split into the wings and the skew (as described In the context of the volatility smile, long smile exposures (i.e., long-skew Starting from first principles, exotic contracts with ‘‘good’’ features compared to EXHIBIT 18.5 reversal position gets longer to theExhibit topside 18.5. and shorter to the downside as shown in A vega versus spotsmile profile exposures and of hence an whether a exotic contract contract will trade is over a or under simplePricing TV. method the Skew for assessing In a currency pair with a topside risk reversal, the vega profile of a long risk it will have asmile exposures, positive in TV which case adjustment itunder will and TV.’’ have a will negative ‘‘trade TV adjustment over and TV,’’ will ‘‘trade or itin has Chapter short 12), exposureselements: exposure to to the the wings volatility of the smile smile can and exposure be to the split skew into of the the smile. same two cost less than TV to buy (a negative TV adjustment). or long-wing exposures) areshort-skew ‘‘good’’ or features short-wing whilecontract exposures) short is are assessed smile to ‘‘bad’’ exposures determine features. whether (i.e., it Therefore, has long the smile exposures, exotic in which case the Black-Scholes model will costand more exotic than contracts TV with to ‘‘bad’’ features buy compared (a to positive the TV Black-Scholes model adjustment) will 362 EXOTIC FX DERIVATIVES PRICING tmtrt rvdn pthstuhdtebrirlvltruhu h ieo the of life the throughout level barrier the cash touched of amount has fixed spot a providing out pays maturity contract at one-touch a to previously, stated said As is contract. exotic the smile, reversal.’’ volatility risk the ‘‘long of be side to reversal.’’ lower risk said the ‘‘short is on be exotic longer gets the vega smile, the volatility If the of side higher the or reversal’’ risk ‘‘short be the to to said skew.’’ longer is gets exotic vega the ‘‘short the topside, risk the ‘‘long if gets to be pair, shorter to contract currency said and exotic same is downside the exotic an In the on skew.’’ downside, ‘‘long exposure the or to vega reversal’’ shorter the and topside if the topside, to for longer reversal risk the with risk a buying definition, negative. By be positive. well is position. smile may reversal long which a risk in strike, this results downside reversal on short zeta the net of Therefore, zeta the than greater h n-oc eapol nEhbt1. s‘ln ikrvra’ eas eagets vega 90.00, because reversal’’ at risk spot ‘‘long With is 18.6 likely. Exhibit more in payout, profile the vega hence one-touch the and knock, barrier the makes a assumed contract, is exotic single option a the on in adjustment TV a calculating When option. nutvl,teln n-oc pini ogvg eas ihrso volatility spot higher because vega long is option one-touch long the Intuitively, one-touch 130.00 1yr long a from profile spot versus vega on the shows 18.6 longer Exhibit gets contract exotic an on exposure vega the if generally, More pair currency a In contracts. exotic to applied be also can methodology similar A zeta downside reversal short risk and 25d strike Long topside long is reversal risk 25d long pair, currency this In zeta strike Topside volatility smile strike Topside volatility ATM strike Topside 101.95 strike: call 25d Topside zeta strike Downside volatility smile strike Downside volatility ATM strike Downside 82.95 strike: put 25d Downside 89.50 Forward: 90.00 Spot: 1yr Tenor: and positive be will reversal risk this within strike topside long the on zeta The strike. =+ . = xml:Ln ikRvra Zeta Reversal Risk Long Example: .5CCY1% 1.35 11 CCY1% –1.10 = = =+ = = 19.40%/premium 15.10%/premium 11.60%/premium 15.10%/premium 1.35% − –1.10% = = = = .0CCY1% 3.10 .5CCY1% 1.75 =+ .0CCY1% 1.70 .0CCY1% 2.80 .5CCY1% 2.45 ogposition long EXOTIC FX DERIVATIVES PRICING 363 (i.e., it will have a positive TV adjustment). Example: Topside One-Touch over TV 6.5 CCY1% + Long topside one-touch vega profile TV adjustment: One-touch midmarket price: 8.5 CCY1% Buying this one-touch contract makes a trading position longer topside vega. To Tenor: 1yr Spot: 90.00 Forward: 89.50 One-touch up barrier: 130.00 ATM volatility: 15.1% TV: 2.0 CCY1% Therefore, in this example, with spot at 90.00 and a risk reversal for topside, the topside vanilla options trade atrisk a reversal higher being implied for volatility topside. thanATM Selling the earns vanilla ATM zeta. options due Therefore, at to theto a the exotic must higher buy cost volatility the more than one-touch than the underover TV: TV, TV If this it and would were then be possible sell a (very topside weak) vanilla form options of arbitrage. on the hedge hedge this vega exposure, vanilla options with topside strikes must be sold. These one-touch will trade EXHIBIT 18.6 longer to the topside andposition shorter in to this currency the pair downside, with exactly topside like skew. a long risk reversal 364 EXOTIC FX DERIVATIVES PRICING ihnteeoisfaeok ol aeapstv Vadjustment. TV positive a have would framework, exotics the within butterfly the within strikes same-delta downside positive. and be topside will the on zetas the 18.7. of Exhibit sum in contract shown butterfly as long downside, a and on topside pricing. vega the within The both considered contract. to butterfly longer be gets the also using must done contract be exotic can This the of exposure wing The Wings sold. the be Pricing should value smile of amount similar a hedge, vanilla USD1m the in On bought value purchased. was of smile contract adjustment the one-touch TV the to a if negative) at example, (but For option. equal exotic approximately the be of should hedge vanilla the 18.7 EXHIBIT owr:89.50 Forward: 90.00 Spot: 1yr Tenor: priced if that, position smile long a in results butterfly a buying definition, By the and wings positive have smiles volatility complications, fly broker Ignoring of value smile the options, vanilla with contract one-touch the hedging When ogbtefl eaprofile vega butterfly Long xml:Ln rkrBtefl Zeta Butterfly Broker Long Example: + S65,ta implies that USD6.5%, + S6ko ml au a been has value smile of USD65k EXOTIC FX DERIVATIVES PRICING 365 0.15 CCY1% =+ 2.15 CCY1% 1.05 CCY1% 2.35 CCY1% 3.60 CCY1% = = = = 1.25% + –1.10% = 15.10%/premium 11.05%/premium 15.10%/premium 18.65%/premium = = = = –1.10 CCY1% 1.25 CCY1% = =+ Vega profile of long DNT because ATM contracts have zero zeta. Exhibit 18.8 shows the vega versus spot profile from a long 1yr 80.00/100.00 Note that the ATM strikes within the butterfly areLong 25d ignored broker when butterfly assessing zeta the zeta Assuming the exotic contract is vega hedged (which a butterfly is by construction), Downside strike zeta Topside 25d call broker butterfly strike:Topside 99.20 strike ATM volatility Topside strike smile volatility Topside strike zeta Long 25d butterfly is long topside strike and long downside strike. Downside 25d call broker butterfly strike:Downside 82.95 strike ATM volatility Downside strike smile volatility EXHIBIT 18.8 life of thelower option. spot Therefore, volatility the makesthe long it payout. more double-no-touch likely contract that is spot short stays vega; within the range to get if the vega gets longer into the be wings ‘‘long (i.e., wings’’ away or fromexotic ‘‘long current fly.’’ is spot), If said the to the exotic be exotic is ‘‘short vega wings’’ said or gets ‘‘short shorter fly.’’ in thedouble-no-touch wings, (DNT). the A double-no-touchproviding contract spot pays hasn’t out touched cash either at of maturity the two barrier levels throughout the 366 EXOTIC FX DERIVATIVES PRICING vrTV over long a like exactly wings, the in position. vega butterfly longer gets double-no-touch long hedged vega the shows double-no-touch 18.10 the Exhibit from while profile separately vega the aggregate hedge vega affect ATM not its and does double-no-touch ATM the option hedge. with plus ATM hedging contract an exotic since with the valid spot of position current is smile at This vega clarity. the more Hedging gives wings. short or wings long 18.9 EXHIBIT oben-oc imre rc:2.5CCY1% 20.75 price: midmarket Double-no-touch adjustment: TV CCY1% 12.75 TV: 15.10% volatility: ATM 89.50 Forward: 90.00 Spot: 80.00/100.00 barriers: Double-no-touch 1yr Tenor: trade will double-no-touch the 90.00, at spot with example, the this in since wings’’ Therefore, ‘‘long be to said is double-no-touch the 90.00, at spot With 80.00/100.00 1yr long a from profiles spot versus vega the shows 18.9 Exhibit is contract double-no-touch the whether 18.8 Exhibit from judge to hard is It .. twl aeapstv Vadjustment. TV positive a have will it i.e., , eapolso ogDTadAMvg hedge vega ATM and DNT long of profiles Vega + . CCY1% 8.0 xml:Double-No-Touch Example: plus h T eahedge. vega ATM the EXOTIC FX DERIVATIVES PRICING 367 Aggregate vega profile of long DNT plus ATM vega hedge As in the risk reversal case, when hedging the double-no-touch contract with Buying this double-no-touch contract with vega hedge makes the position longer at the same maturity.assessed Exposure with to reference the toexamples wings the given of butterfly in contract theoption this at volatility contracts the section smile have same is neatly bothadjustment. maturity. primarily separate exposures, The the two which two must effects be but combined most within exotic the TV Summary For a given exoticsplit contract, into the two TVvolatility separate adjustment smile from effects: is the skew primarily volatility and assessed smile wings. with can reference be Exposure to to the the risk skew reversal of contract the and then sell wing vanilla optionsof on arbitrage. the hedge over TV, again, this would be a form vanilla options, the smile value(but of negative) the to vanilla the smile hedge value should of be the approximately exotic equal option. vega to both the downsideand and downside topside. wing To vanilla hedge options thistrade wing must at vega be a exposure, sold. higher topside On impliedsmile. average, volatility Selling wing than vanilla vanilla the options options ATM atmust a due cost higher to more volatility the than earns shape TV zeta. becauseto of Therefore, buying be the the it sold volatility allows exotic on vanillas the that trade hedge. higher If than it TV were possible to buy the double-no-touch under TV EXHIBIT 18.10 368 EXOTIC FX DERIVATIVES PRICING ■ V rcn xml:Pr 1 Part Example: Pricing VVV oio hl xii 81 hw h eapol rmteoetuhoption. one-touch the deal from the profile on downside vega smile the a shows volatility has 18.12 3mth one-touch Exhibit AUD/USD the while the horizon and shows downside 18.11 for chapter. Exhibit is the also. in barrier reversal earlier risk introduced the was AUD/USD that In contract barrier 0.8100 with (explained one-touch time stopping contract. the exotic by the smile’’ cost of the chapter) this ‘‘on this replication weighting in vanilla and later the zeta) of cumulative the cost of its adjustment the (i.e., TV calculating The by exotic. the estimated as is and maturity call, exotic same 25d the ATM, to using options contract vanilla exotic put the 25d in exposures vanna and volga, vega, the ■ and skew the ■ to has contract exotic vega smile: the volatility Second-order exposure the the of approach. wings measure to this used formalizes are Greeks pricing (vega/volga/vanna) VVV XII 18.11 EXHIBIT oaiiysmile. volatility Volga smile. Vanna hsmtoooycnb ple oteeapeboe U/S 3mth AUD/USD broker example the to applied be can methodology This replicates implementation possible One model. a than rather heuristic a is VVV ( ( 𝜕 𝜕 𝜕 vega 𝜕𝜎 spot vega ) ) U/S mhvltlt smile volatility 3mth AUD/USD ie h xoueo h xtccnrc othe to contract exotic the of exposure the gives ie h xoueo h xtccnrc othe to contract exotic the of exposure the gives skew ftevolatility the of wings fthe of EXOTIC FX DERIVATIVES PRICING 369 Vega profile of long AUD/USD downside one-touch contract The AUD/USD 3mth 25d put vanilla has a 0.8660 strike and: The AUD/USD 3mth 25d call vanilla has a 0.9360 strike and: Under Black-Scholes, and with all exposures quoted in AUD% terms, the long Vega: 0.115% Vanna: –1.85% Volga: 0.007% Vega: 0.165% Vanna: 1.95% Volga: 0.006% Vega: 1.99% Vanna: –47.1% Volga: 0.29% ■ ■ ■ ■ ■ ■ The signs of these exposuresspot should volatility increases not the be chance of avega the surprise: barrier gets Vega touching, longer vanna is to is long the negative downsideoverall because because at gets higher longer current in spot, the and wings. volga is positive because vega one-touch contract has: ■ ■ ■ EXHIBIT 18.12 370 EXOTIC FX DERIVATIVES PRICING ■ tpigTime Stopping pini 0%sneteoto laslvsrgtu oexpiry. vanilla to European up the a right 50%, of lives is time always stopping option option The the exotic life. since an its 100% of of is half option time for live stopping to the expected if is option Therefore, option. the management. of risk life and pricing trade option the exotic on within risk measure the important about an information is useful is it This and alive. stay will option exotic an as known (also time Stopping contract. downside. for the also of is AUD/USD vega in since reversal risk expected the as and exposures, downside smile the to long longer has gets option one-touch the equates that which implies volatility, implied 5.8 12.95% to and at 11.4%), marked is volatility is ATM to vanilla the downside that (recall 25d zeta the AUD% –0.14 to profile. equates vega which the hedge to sold and be options could put option 25d call of 25d this of of AUD60m AUD10m AUD2m approximately if bought, Therefore, were options. contract vanilla one-touch topside 25d of notional one-touch the 31.6 buying by give: to solves which respectively, options vanilla downside notionals and unknowns—the two topside with the equations of linear two of system a is remains What oelkl otigrerir xii 81 hw o h tpigtm reduces time stopping increases. the volatility how as barrier, shows sooner) are knock 18.13 the barriers barrier Exhibit from since (expected earlier. far time trigger stopping is the to spot reduces likely If volatility more low. Higher is high. time is time stopping stopping barrier, American an (called to barriers close monitored continuously contains it that h otcmo esna xtcoto ih o ietruht xiyis expiry to through live not might option exotic an reason common most The the of percent a as expressed 100%, and 0% between value a takes time Stopping volatility, implied 10.45% at marked is vanilla topside 25d 3mth the smile, the On replicated approximately be can one-touch the from profile vega the Therefore, exposures. smile the impact not does it since ignored be can component vega The h nlse nteVVmtoooyi owih hscs ytesopn time stopping the by cost this weight to is methodology VVV the in step final The + .5AD ea hrfr,terpiainhsacs ntesieequal smile the on cost a has replication the Therefore, zeta. AUD% 0.25 ×− 0.14 × %+ h n-oc oinlo 5 onievnlaotosad5.8 and options vanilla downside 25d of notional one-touch the n 31.6 call n call . . 1.95 ×+ 0.006 rtei time exit first %+ 0.25 %+ %=+ n n n put put call n put = . or = − 7.1 . 31.6 xetdlife expected 5.8 0.007 1.85 AUD% %=− %= steepce egho time of length expected the is ) h oiieT adjustment TV positive The . 0.29 mrcnbarriers American 47.1 n call % % and n put : .I ptis spot If ). × EXOTIC FX DERIVATIVES PRICING 371 × 7.1 AUD% =+ prior to expiry. However, the no-touch value do not touch , although given the market bid–offer spread for this contract Stopping time of a EUR/USD 1yr 1.2500 American barrier at different implied AUD%. This VVV TV adjustment5.35% is fairly close to the market TV + contract (see Chapter 23) that pays out a fixed amount of cash at 7.0 =+ no-touch Stopping time is also an important measure forWhen target stopping redemption time options is (see displayed in a pricing tool it is important to understand Stopping time on American barrier options is conceptually similar to the valuation It is important to appreciate that stopping time doesn’t depend on the option Back to the VVV3mth pricing 0.8100 example: one-touch The contract is stopping 98.6%—veryfar time from close of current to spot the 100% (given since example its the maturity).zeta AUD/USD barrier Applying from this is weighting the to replication the cumulative 98.6% has minimal impact: TVadjustment Adjustment of rates used or is the full interest term structure used? compares stopping time with the TV of an equivalent no-touch contract. Chapter 28) where the option expires based on a target. what methodology is used tofull calculate ATM it: curve but Is no a smile single or ATM the volatility full to volatility expiry, surface the used? Are single interest payoff, only the relative positioning of barriers within the contract. of a maturity if barrier levels will be lower than stoppingno-touch time has because zero if value the where barrier realized trades stopping prior time to expiry, is the non-zero. Exhibit 18.14 EXHIBIT 18.13 volatility levels VVV Pricing Example: Part 2 ■ 372 EXOTIC FX DERIVATIVES PRICING airtdt h hl oaiiysraerte hnaVVbsdapproach. VVV-based a models than pricing rather using option surface to exotic volatility moved whole for now the However, have to desks used. calibrated trading were were derivative caps FX options and most vanilla pricing, floors (e.g., or delta exposures methodology weightings advanced different VVV different more or example, the replication, itself. For to the TV deficiencies. fixes within than its applying used negatively into for larger large, put adjust adjustments is was to TV smile include effort volatility VVV problems the much generate within exposures Other Historically, to skew that ignored. possible the fact is is if the levels and it spot and barriers, different near price jumps at the valuation or within time used over are change spot current at exposures 30 selling by options. hedged vanilla current be downside at 25d can of risk wings notional smile short one-touch the AUD/USD or of the majority In long strategy. the and hedging example, vanilla skew one-touch appropriate short to an possible or suggests is VVV long it Also, exposures is spot. volga exotic and an vanna analyzing whether By judge trade: the on risks the about used be to enough accurate not is adjustment TV practice. VVV in the 2.0%, around be would 18.14 EXHIBIT ml xoueicnitnishv enbuhdudrtecre;frms traders versus most concerns. exposure for important carpet; TV aren’t the these under and brushed been currencies have premium inconsistencies exposure different smile with contracts ATM from ial,nt htti ssyie nlss susrgrigdfeigvnaexposures vanna differing regarding Issues analysis. stylized is this that note Finally, only because market the match consistently not do prices VVV However, intuition gives it and calculate to quick very is it that are VVV of advantages The tpigtm essn-oc Vo y .50ERUDAeia barrier American EUR/USD 1.2500 1yr a of TV no-touch versus time Stopping 𝜕 𝜕 vanna spot eeadded, were ) × the EXOTIC FX DERIVATIVES PRICING 373 and delta . Path dependence means that spot takes over the life of the path the chance of the window barrier knocking. path dependence overestimate the option expiry. Although note that when a vanilla option is delta at is highly path dependent. Within exotic contracts, the presence of barriers, interest rate term structure and volatility surface (up to the option maturity) Traders learn which exotic option contracts have significant path dependence. For pricing and risk management the consequence of path dependence is that This is worth restating to be as clear as possible: Trading desks use forward curves, For example, consider a windowbarrier barrier that option is only (see active Chapter for 26) thesloping first with (short-date month a of ATM knock-out the volatility trade. lower Ifvolatility than the ATM TV long-date curve calculation ATM is will volatility), upward the single interest rate curves, and the volatilityvanilla surface option, for only valuing the derivative forward, contracts. intereststrike For rate, and a and expiry implied date volatility for isentire the used specific for pricing. Formust a be path-dependent exotic incorporated option, intorequired. the the pricing, hence a different pricing methodology is averages, or targets makes the option path dependent. the full ATM curvethese factors and must be interest included within ratetrading the desks curve TV use adjustment. two must Alternatively, TV values: some be an bank used ATM TV to and a value term structure options TV. and that the exotic option payoff isoption. affected by Vanilla the options are paththe independent spot because level their payoffhedged depends infrequently only on within a tradinghedges portfolio, the P&L from the option A key feature of exotic options is their Path Dependence ■

CHAPTER 19

FX Derivatives Pricing Models

X derivatives trading desks use pricing models to value exotic contracts. Pricing Fmodels extend the Black-Scholes framework by adding new elements into the model dynamics. Different pricing models have different spot, volatility, and interest 375 rate dynamics, which in turn generates different prices on exotic contracts. When using any pricing model it is vital to understand the model dynamic and how this dynamic impacts pricing. Exhibit 19.1 shows the high-level connections between vanilla options, exotic options, probability density functions, and exotic pricing models. Exotic pricing models are split into two main categories: 1. Smile models incorporate the volatility surface. All smile models are calibrated to the volatility surface, plus some smile models have additional calibration to exotic contracts. Common smile models are stochastic volatility, local volatility, mixed volatility, and jump diffusion. Smile models often have static or deterministic interest rates. 2. Interest rate models incorporate stochastic (i.e., randomly moving) interest rates in order to correctly value the effects of interest rate volatility and spot versus interest rate correlation. These effects are particularly important on long-dated contracts. Interest rate models often have static or deterministic volatility of the underlying’s returns. Models exist that combine both the volatility surface and stochastic interest rates but it is important to understand that more features within a model does not necessarily make it better. The more complex a model, the harder it is to keep 376 FX DERIVATIVES PRICING MODELS xoue rmtemdlalwte oscesul eg hi xoue over exposures their hedge successfully P&L. to lock-in them and allow managing time model risk the the whether option By discover from traders exotic models. exposures models, pricing of pricing different different types by generated under which exposures using prices learn market therefore match They contracts prices. market broker that exotic price. available market the model and pricing price model simplest the the between difference choosing minimal model, gives pricing the of dynamic the challenging sharply. generate extremely moves to becomes market hours management the five involves risk when takes portfolio, position it trading derivatives If a very FX portfolio. for the an exposures is in managing Speed deals to risk all price. need and revaluing contracts a quickly periodically exotic generate clients on for to Prices made management. takes be risk it and longer pricing the within important and calibrated correctly it 19.1 EXHIBIT ipya es hc,taessmtmsas efr aulrs nlss As analysis: risk or manual products, a complex perform more also as sometimes On of traders contract. thought check, the sense be within a can risk as models model simply the the by of generated measure prices a of range The models. pricing tcnb ntutv opietesm xtccnrc ihmlil different multiple with contract exotic same the price to instructive be can It interbank with prices model compare traders the models, with pricing contract assessing exotic When the of features main the match traders practice, In Xdrvtvsvlainframework valuation derivatives FX FX DERIVATIVES PRICING MODELS 377 , ) 𝜆 ( ) t , 2 dW ) 2 t , 𝜌 1 . The process for variance − ) t dW t 1 v v ( ( √ √ + + models (see Chapter 14). t , , the speed of mean reversion dt 1 ) ) 𝛼 1 ( dW is applied to the two independent Wiener 𝜌 ) ( 𝜌 t rCCY ( v , although this parameter is often referred to sticky delta ) − is increased, the volatility smile tilts such that √ 𝛽 2 ( 𝛽 ) is increased, the wings of the volatility smile move 𝜌 ) ( + 𝛽 rCCY dt ( ) t v =( t − t S dS . The first Wiener process drives spot and the two together 𝛼 t , ( 2 𝜆 W = t and dv t , 1 W If the correlation parameter If the vol-of-vol parameter Pure stochastic volatility models do not depend on the level of spot; an equivalent The parameters within the model are calibrated such that the vanilla volatility topside strikes have highershown implied in volatility than Exhibitvolatility same 19.4. smile. delta This downside is strikes as equivalent to a larger topside skew within the and long run volatilityvolatility smile is produced by set the model to is 10%, a flat with 10% as zerohigher shown as vol-of-vol in shown Exhibit in and 19.2. Exhibit 19.3. correlation,too, Note unlike the that the in standard this volatility instance, smile the construction. ATM level increases which is an issue within the Heston model. volatility surface will be generatedvolatility no models matter can the be initial thought of spot. as Therefore, stochastic surface is (approximately) reproduced within the model. For example, if the initial as the and ‘‘vol-of-vol,’’ the correlation processes drive variance. The Ornstein-UhlenbeckUhlenbeck (OU) and SABR model models and are other ExponentialWithin commonly used Ornstein- the stochastic Exponential volatility models. Ornstein-Uhlenbeck model, volatility cannot go to zero, The first line ofbeen the replaced with model the is square rootincludes identical of parameters to a variance for Black-Scholes term long-run exceptvolatility variance that of instantaneous volatility variance has has intuitive parameters that mirrora the closed-form risk reversal expression and for butterfly(SDE) vanilla instruments of options. the and Heston The model stochastic are: differential equations Within stochastic volatility models (sometimesvolatility shortened has its to own ‘‘stoch vol process.but There models’’), one are of many different the stochastic original volatility and models best known is the Heston model from 1993 because it shown in Chapter 18, a vegaTV versus adjustment spot from profile the can volatility be smile used should to estimate be whether positive the or negative. Stochastic Volatility Models ■ 378 FX DERIVATIVES PRICING MODELS parameters 19.2 EXHIBIT o-fvlparameters vol-of-vol 19.3 EXHIBIT oaiiysiefo etnmdlwt eovlo-o n correlation and vol-of-vol zero with model Heston from smile Volatility oaiiysiefo etnmdlwt eocreainprmtradpositive and parameter correlation zero with model Heston from smile Volatility FX DERIVATIVES PRICING MODELS 379 ) t , 2 . ) dW 𝜌 ) ( 2 ) t ( 𝝆 − 1 ( √ + t , 1 dW ) t ( 𝝆 ( t v √ ) t ( volatility convexity is overvalued by stochastic volatility 𝜷 + dt ) t v − 𝛼 )( t Volatility smile from Heston model with positive correlation and vol-of-vol ( 𝝀 = t dv . For this reason, stochastic volatility models do not consistently match prices In practice, in order to successfully calibrate to vanilla contracts, stochastic These Heston parameters can attempt to match a volatility smile at a single tenor Parameter calibration within the stochastic volatility model is an automatic parameters should evolve smoothly rather than jumping around. volatility models often have ain higher the volatility market. of Put implied another volatilitymodels way, than is observed In other words, the variance parameters within the SDE become functions of time: When calibrating models with parameter sets that are functions of time, the When using anyobserve model the parameters with over time calibrated andchanges ensure parameters in they the are it stable volatility and surface; is respond thatshould sensibly is, be important to changes mainly in reflected that the by skew changes traders of in the the volatility correlation surface parameter but in order to match the entire volatility surface, parameters must change over time. process that usesaccurate vanilla approximations for option vanilla option prices. pricesimportant; under This it a enables given is the pricing model model why to aremultiple be so closed-form calibrated different quickly. expressions In parameter practice, or there sets are that often generate near-identical volatility surfaces. EXHIBIT 19.4 parameters 380 FX DERIVATIVES PRICING MODELS ■ oa oaiiyModels Volatility Local lhuhi smr tcysrk hnsik et.The delta. sticky than strike sticky more risk is downside it small although a respectively. 19.7 with reversal and risk 19.6 pair topside Exhibits large currency in a shown with (a are pair currently) currency USD/JPY (a USD/BRL volatility in and local currently) and tenor reversal volatility Implied one 19.5. volatilities at forward Exhibit of in grid smiles shown a as as time of in thought out be spanning can surface volatility local The surface. a is stochastic constant, Black-Scholes being the than Therefore, to: rather time. extended volatility, is and equation that differential spot is of model function the deterministic Bruno to quant superstar key by mid-1990s The the Dupire. in developed was model volatility local The the produce to maturities short at vol-of-vol smile. enough volatility have correct to volatility struggle stochastic some can plus reversion, models mean volatility overvalued. the be by overwhelmed to get smiles can forward causes also effect model same volatility The high. stochastic significant too the with be which will contracts price for for options, particularly double-no-touch market, like exotics convexity, broker interbank the in XII 19.5 EXHIBIT h oa oaiiymdli ete tcysrk o tcydlamodel, delta sticky a nor strike sticky a neither is model volatility local The volatility full the using model the within generated is surface volatility local A smile volatility the within skew the models volatility stochastic under addition, In oa oaiiysraeconstruction surface volatility Local dS S t t =( rCCY 2 − rCCY 1 ) dt + 𝜎 ( S t , t local ) dW t oaiiydpnson depends volatility FX DERIVATIVES PRICING MODELS 381 USD/BRL implied volatility and local volatility smiles USD/JPY implied volatility and local volatility smiles EXHIBIT 19.7 EXHIBIT 19.6 382 FX DERIVATIVES PRICING MODELS ■ ie oaiiyModels Volatility Mixed oee,frcnrcswt infiatcneiyo owr kwepsrs local prices. exposures, inaccurate skew gives forward dependence. often or volatility path convexity minimal significant with with contracts options for However, for particularly exposures, forward and price valuations to volatility. implied used of value nature be option random where not the 31), on Chapter should primarily deterministic (see depends options a it start is forward models or volatility agreements volatility since volatility Also, local smile. within volatility this 1yr function and pairs current wings currency lower the significantly most than have In for will skew model. volatility querying the local by from under checked smile smile volatility be volatility forward can forward This 1yr volatility. in 1yr implied the of volatility low by low. caused too like be will convexity, interbank model significant volatility the local with the in which contracts prices for for options, match double-no-touch particularly consistently Put market, not reality. exotics do than broker models lower volatility is local function reason, volatility way, local the another from generated volatility dependence. path no/minimal with pricing contracts good exotic a for is use model to volatility model local Dupire the required, is calibration consuming form functional the within parameters the calibrated. arbitrage-free because be an slower must is guaranteeing it of but advantage surface the volatility has approach identify This to form. functional used be can feature surface. This volatility volatility. the with local will problems calendar model undefined potential or the to maturity), strikes) due longer higher unstable at value at be strike decreasing value either with contains increasing (options surface arbitrage with volatility spread options vanilla the (call if arbitrage Plus, spread price. and setup to a quick for volatility implied the moves. that spot imply as necessarily fixed not is does strike that specific but spot of level the oaiiycneiy n oa oaiiymdl(hc nevle volatility undervalues overvalues (which (which model model volatility volatility stochastic local a a convexity). of and combination convexity) a volatility sounds, called it (sometimes as models volatility Mixed nsmay oa oaiiymdl r tbe uc,adgv aryaccurate fairly give and quick, stable, are models volatility local summary, In again, undervalued, be can smile forward that is model the of feature Another implied of volatility the that is models volatility time local no with problem and main model The the within priced correctly are options vanilla Because defined a using surface volatility local the build to is approach possible Another is model the and calibrate to parameters no has model volatility local Dupire The oaiiycneiyi nevle ylclvltlt models volatility local by undervalued is convexity volatility tcatclclvolatility local stochastic oes are, models) o this For . FX DERIVATIVES PRICING MODELS 383 : ) t (ξ) , 2 dW ) t 2 ) is log-normally t ( dN ) 𝜌 ) t 1 , 1 1 − − − 1 t dW ( 𝐭 y ) ( 𝐲 t √ , t +( S + plus a mixing weight ( t and t , ) ̂ 1 𝝈 ̂ t 𝜎 λ dW v ( dW 𝜎 √ ) + t + ( dt 𝜌 dt ) ( t 1 v −λκ) √ ) rCCY 1 t ( − 𝛽 ) 2 t rCCY ( 𝜉 − rCCY 2 + . Note that the introduction of jumps also requires an κ dt =( ) t rCCY v t t S dS − =( 𝛼 t t )( S t dS ( 𝜆 is a poisson process with intensity t = N t dv Variations on the mixed volatility model are used by the vast majority of FX The behavior of the model can be driven by a single mixing weight or a term Mixing weights define how much stochastic volatility is applied at each tenor. There are many different ways in which the stochastic and local volatility models where distributed with mean additional correction to the drift. There is a lot oforiginal jump evidence diffusion of models jumps is inBlack-Scholes the SDE the Merton by history model adding of from a 1976, financial jump which term: markets. extends One the of the most currency pairs. It is somodels successful on that bank FX development derivatives of trading new desks single has asset virtually pricing stopped. significant volga exposures. Into this increase case, the mixing stochastic weightsobscure volatility should element pairs be within with moved thesimilar, few higher mixed more liquid volatility quotes currency model. pairs on In can interbank be used. broker exotics,derivatives mixing trading weights desks in to price exotic contracts out to two-year maturities in structure of mixing weights set bymarket. traders to Alternatively, match exotic the contracts model incontracts. the could In interbank general, be if convexity calibrated ismodel, being directly the more model to highly price valued will prices in be the of lower market than than exotic the the market price for exotic contracts with Conceptually, the stochastic volatilityvolatility model component can can be thenis calibrated be generated first used and after to the theweight. ensure local Within stochastic this that volatility formulation, the a componentand mixing correct is a weight mixing vanilla of weighted weight 0% surface of by implies 100% full the implies local full mixing volatility stochastic volatility. can be combined. Onevolatility possible model approach with would a local be volatility to component extend a Heston stochastic Jump Diffusion Models ■ 384 FX DERIVATIVES PRICING MODELS ■ tcatcItrs aeModels Rate Interest Stochastic eue o o xml,ipya xetdpoaiiyadsz fso jump spot of size identify can and traders surface. probability intuition volatility their the expected in with value this also an relative comparing can By imply models smile. volatility example, Jump the dynamic. for liquid from market to, for real used the market where doesn’t matches be pairs interbank best currency pricing pegged dynamic or their the model managed but in in the best smiles work observed models volatility Jump prices pairs. of currency exotics range diffusion match wide Jump surface. a consistently volatility generate the of can wings in and models intuitive skew has skew it the Heston, mirror and like that and parameters calibration, wings quick for the allowing hence generating options, vanilla jumps down or up surface. of volatility the size and probability 19.8 EXHIBIT ae aeavltlt fteronadte a oei orltdmne with introducing manner by correlated SDE a Black-Scholes in the move extend can interest models they rate practice, and interest or In own Stochastic static their manner). spot. are of predetermined that volatility a rates a have interest in rates have move all they chapter (i.e., this deterministic in reviewed models smile The etni oua oe eas thsasm-lsdfr xrsinfor expression semi-closed-form a has it because the model popular with a model, is Merton Merton a from generated are 19.8 Exhibit like paths Spot apeMro oe ptpath spot model Merton Sample FX DERIVATIVES PRICING MODELS 385 t dW t t , , 𝜎 1 2 + dW dW 1 2 dt b b ) t 1 + + dt dt ) ) t t rCCY ( ( 1 2 − a a t 2 = = t t 2 1 rCCY =( drCCY drCCY t t S dS The effect of stochastic interest rates is particularly important on long-dated Finally, it is important to note that generating TV adjustments with a smile model always a valid approach.other. The Local smile volatility and is often interest addedvalue rates into the may stochastic volatility well interest surface rate interact in modelsbe with a in calibrated. each manner order to which requires no additional parameters to (approximately past two-year)sensitivity to exotic interest rates, contracts. theand effect When included of within stochastic a the interest TV rates contract adjustment. must be has quantified aand large an interest rate model separately and then summing the adjustments is not processes for the two interestso-called ‘‘short rates rates’’ are (or being perhaps modeled: their spread). For example, if the

CHAPTER 20

Exotic FX Derivatives Product Classification

387 xotic products are typically split into different generations, indicating how long Ethe product has been traded in the market. When reviewing the exotic product classification it is important to understand that risk management does not get more complicated for higher-generation options. In fact, the opposite is often true as features like averages, accruing notionals, or targets can reduce risk management complexity. In general, exotic products which are conceptually simple are often harder to risk manage while exotic products which are conceptually more involved are often easier to risk manage. The following list of exotic FX derivative product types is by no means exhaustive; it primarily aims to introduce the main exotic option types. Exotic features can be combined and extended in many different ways. Over time, different products and structures come in and out of fashion, plus new structures are developed by innovative trading and structuring desks to meet client requirements.

■ First-Generation Exotics

First-generation exotic products are the fundamental building blocks of exotic risk. The two primary exotic features are European barriers and American barriers. European digitals and touch options are the simplest exotic contracts, followed by European and American barrier options. 388 EXOTIC FX DERIVATIVES PRODUCT CLASSIFICATION nbetepyf) hr r w antpso are:Erpa n American. (i.e., and in European option barrier: have the of knock they types or main Additionally, it) two are expire maturity. There (i.e., at payoff). out the option option enable the vanilla knock a can that like barriers off pay options Barrier Options Barrier is price the if so payout, the USD105k. of cost % will a contract the as USD%, quoted 10.5 are options touch digitals, European options. one-touch double-no-touch as as touches touches barrier barrier two single and maturity trade at to payout is with options convention market The hit. is barrier options, touch below. be must barrier one and spot above ranges a or no-touch option. the of life called the are throughout active options are barrier touch they specified because (no-touch) within touched barriers never The or (one-touch) levels. touched ever spot if rity called also options, Touch Options Touch 25.0 is options offer cash-or-nothing 50%. the options, above trade be if to rarely so will common prices most payout, digital is the European it so of vanillas, side Like out-of-the-money % EUR250k. the a cost will as contract quoted the EUR%, are payment options cash Digital the happens, payout. it If maturity. date. at delivery the level on digital occurs actually the put) digital (CCY1 digital European Options Digital European utteeoeb oiindi-h-oe esstepyf;ohriete have they otherwise payoff; the impact. versus no in-the-money positioned be therefore must applicable only are barriers European Options Barrier European oc pin r oee nCatr23. Chapter in covered are options Touch example One-touch also are There date. delivery the on out pay options touch Standard a called is it barrier, single a has option touch the If including names, other various by known are options digital European example digital European double-no-touch ihndul are oc pin,a neto,oebrirms be must barrier one inception, at options, touch barrier double Within . N) fi a w ares ti alda called is it barriers, two has it If (NT). hc nta a u w uiesdy vleso)atrthe after spot) (value days business two out pay instead which U/S y .00oetuhwt S1 aot Like payout. USD1m with one-touch 1.3000 1yr EUR/USD : pin aotcs fso saoe(C1dgtlcl)o below or call) digital (CCY1 above is spot if cash payout options DT,pu oben-oc pin r lokonas known also are options double-no-touch plus (DNT), rebates U/S y .00ERdgtlcl nEUR1m in call digital EUR 1.3000 1yr EUR/USD : ,or or iia bets digital mrcndigitals American tteoto maturity option the at hyaecvrdi hpe 21. Chapter in covered are They . eeaeacs aota matu- at payout cash a generate , double-one-touch American uoenbrirlevels barrier European . one-touch or continuous O)o a or (OT) instant (DOT) binary EXOTIC FX DERIVATIVES PRODUCT CLASSIFICATION 389 European European throughout the life of 1.4500, the payoff of this Wellington time (sometimes 1.4500, the option will have no below continuously A.M. below New York time (sometimes called ‘‘market 1.4500, the EKO will have no payoff because P.M. above : EUR/USD 1yr 1.3000 EUR call/USD put : EUR/USD 1yr 1.3000 EUR call/USD put trades through the barrier level prior to expiry, the structure ever European knock-out call option payoff at maturity (EKO) 1.4500. If spot at maturity is (EKI) 1.4500. If spot at maturity is 1.4500, the payoff of the option will be the same as a EUR call/USD put To clarify what is meant by ‘‘ever’’ in the previous paragraph: In liquid G10 European barrier options are covered in Chapter 22. European knock-in example European knock-out example above currency pairs, for thethrough barrier the to barrier havecalled level ‘‘market triggered, open’’) and between spot Friday 5 Monday mustclose’’) 9 usually in ‘‘market have size,’’ generally traded aroundpairs, USD5m. In the emerging-market spot currency market usually has to be officially open for a barrier knock to American knock-out and knock-in barriersthe exist option. If spot either knocks in (comes alive)the or product, knocks barriers out within (expires). the In structure the are standard either variations all of knock-out or all knock-in. payoff because spot is not through theis knock-in barrier. However, if spot atvanilla maturity option with the same strike. This is shown in Exhibit 20.2. American Barrier Options option will be the same asHowever, a if EUR spot call/USD at put maturity vanillaspot is option is with through the the same knock-out strike. barrier. This is shown in Exhibitknock-in 20.1. EXHIBIT 20.1 knock-out 390 EXOTIC FX DERIVATIVES PRODUCT CLASSIFICATION nc-u .00 hsoto ilpyoflk .00ERcl/S u vanilla put call/USD EUR 1.3000 a like off optionatmaturityunlessspotevertradesthrough(above)1.4000duringthelifeofthe pay will option This 1.4000. knock-out in a 20.3. is option, Exhibit option in the the shown payoff, of is This life maturity. the at payoff during no maturity be 1.2000 at will (below) option there vanilla case through put which trades call/USD EUR ever 1.3000 spot a like unless off pay will option This called simply a barrier is American or option single the a payoff, has the option versus the out-of-the-money If positioned called. is option barrier American occurs. an disagreement what a if knock barrier a a been has is there there whether within on option decision described each are for knock agent barrier plus determining a documents, (‘‘confo’’) constitutes what confirmation of the details exact The occur. 20.2 EXHIBIT nc-u .00140.Ti pinwl a f iea130 U call/USD EUR 1.3000 a like off pay will option This 1.2000/1.4000. knock-out a is it level, spot ThisisshowninExhibit20.4. inwhichcasetherewillbenopayoffatmaturity. option, obekokotexample inception knock-out the of Double side either one barriers, American two has structure option an If the versus example in-the-money knock-out positioned Reverse barrier single a has structure option the If example Knock-out determines option the within barrier and strike the of positioning relative The eua knock-in regular knock-out uoenkoki aloto aofa maturity at payoff option call knock-in European obeknock-out double U/S y .00ERcl/S u nc-u 1.2000. knock-out put call/USD EUR 1.3000 1yr EUR/USD : ees knock-out reverse oeo h pincutrate h ae arfinal fair a makes who counterparties option the of —one fe h ‘eua’ sdopdadteecnrcsare contracts these and dropped is ‘‘regular’’ the Often . K)or (KO) U/S y .00ERcl/S u reverse put call/USD EUR 1.3000 1yr EUR/USD : U/S y .00ERcl/S u double put call/USD EUR 1.3000 1yr EUR/USD : knock-in DO or (DKO) RO or (RKO) K)options. (KI) obeknock-in double ees knock-in reverse eua knock-out regular (DKI). barrier (RKI). EXOTIC FX DERIVATIVES PRODUCT CLASSIFICATION 391 Reverse knock-out option structure Knock-out option structure Payoff, valuation, and exposure charts for options with American barriers assume (above) 1.4000 during the life ofmaturity. the This option, is in shown which in case Exhibit there 20.5. will be no payoff at that barriers have notbarrier knocked. level, exposures However, change. once For spot a goes knock-out through American barrier, an all American exposures EXHIBIT 20.4 put vanilla option at maturity unless spot ever trades through (below) 1.2000 or EXHIBIT 20.3 392 EXOTIC FX DERIVATIVES PRODUCT CLASSIFICATION aeso a ee otruhtesrk n ec h tiepoue ooptionality. no produces strike the knock-out hence ITM and strike the the through In go barriers. never can American spot case regular from different are options barrier or barrier) knock-out a maturity. (below) at payoff 20.6. through no Exhibit be trades will in there shown ever case is which spot EUR in This option, 1.1000 unless the a of maturity life like the off at during 1.2000 pay option will vanilla option This put 1.3000. call/USD at spot with and 1.2000 knock-out payoff the versus in-the-money option barrier positioned (ITM) is in-the-money is spot barrier inception American the single a If Options Barrier In-the-Money the barrier, American knock-in a option. vanilla for standard a whereas of hits, that become barrier exposures the when disappear 20.5 EXHIBIT ihrkokoto nc-nbttems omncngrto a knock-in a has barrier. configuration American common knock-out most a and the separately barrier be but European can knock-in barriers Both or name). knock-out the either (hence barrier European one and barrier options barrier Transatlantic Options Barrier Transatlantic ftesrk n are r ttesm ee,teoto scle a called is option the level, same the at are barrier and strike the If example knock-out In-the-money 24. Chapter in shown are barrier American of types Different obekokototo structure option knock-out Double vnfurther even strike-in U/S y .00ERcl/S u ITM put call/USD EUR 1.1000 1yr EUR/USD : aeavnlapyf tmtrt lsoeAmerican one plus maturity at payoff vanilla a have frakoki are) h rdn ik nITM on risks trading The barrier). knock-in a (for ntemny hsi pca aecle an called case special a is this in-the-money, . strike-out (for EXOTIC FX DERIVATIVES PRODUCT CLASSIFICATION 393 : EUR/USD 1yr 1.3000 EUR call/USD put European knock- Transatlantic option structure ITM knock-out option structure Transatlantic example EXHIBIT 20.7 call/USD put European knock-in 1.4500 at maturity(below) unless 1.2500 spot ever during trades the through lifeat of maturity. the This option, is shown in in which Exhibit case 20.7. there will be no payoff EXHIBIT 20.6 in 1.4500/American knock-out 1.2500. This option will pay off like a 1.3000 EUR 394 EXOTIC FX DERIVATIVES PRODUCT CLASSIFICATION unoOptions Quanto a not Definitely exercise: American with knock-out reverse a product. in standard price a for asked buyer option the by exercised be only expiry can at buyer options option vanilla the European by where except expiry exercised options to be vanilla can European options vanilla standard American to that identical are options vanilla American Options Vanilla American barriers. European or at American standard market spot the from daily sampled after. is shortly fix ECB37 2.15 The around fix. ECB37 on 1.4500 market. spot the against monitored continuously being than rather options barrier Discrete Options Barrier Discrete 2. variations: possible two 1. are there that The fact knock-out. the is from other comes the product this and with knock-in subtlety is barrier One barriers: American two plus Knock-in/Knock-out Options is Barrier barrier Knock-in/Knock-out American the and impact) payoff. the no versus has out-of-the-money it positioned usually (otherwise payoff the versus money sn h tierte hnteso tepr,ti sa CCY1 is this a expiry, becomes at spot this the than If rather strike payout. the CCY2 using natural and the notional I, Part CCY1 notional in is seen case As payoff currencies. vanilla payoff nonstandard have options Quanto mrcnvnlaotosaecvrdi hpe 27. Chapter in covered are options vanilla American once client A options. vanilla to applicable mainly is feature exercise early This 26. Chapter in to covered are challenges barriers Discrete management risk additional creates monitoring barrier Discrete example barrier Discrete in-the- positioned always is barrier European the option, transatlantic the Within ekokdotadteeoeteoto a urnedpyf tmaturity. at payoff guaranteed a has option the therefore and out knocked be in knock later. until trades Knock-out level barrier knock-out the if out expiry until Knock-out . and C1pyu,wt ovrinfo C2pyu oCY payout CCY1 to payout CCY2 from conversion with payout, CCY1 P.M. eta uoentm CT n hnpbihdt website a to published then and (CET) time European Central ftekoki are isfis,teoto ilsilknock still will option the first, hits barrier knock-in the If . U/S y .00ERcl/S u nc-u at knock-out put call/USD EUR 1.3000 1yr EUR/USD : ftekoki are isfis,teoto antthen cannot option the first, hits barrier knock-in the If . KK)brirotoshv ail aofa maturity at payoff vanilla a have options barrier (KIKO) r oioe gis pcfid(sal al)fix daily) (usually specified a against monitored are self-quanto taytime any at pin covered option, prior EXOTIC FX DERIVATIVES PRODUCT CLASSIFICATION 395 option. If barrier observation ). At each fixing, the client transacts a fixed option, covered in Chapter 30. TARF ( front window barrier option. Like American barrier options there are up barrier, . At maturity, the client buys or sells the accrued notional at third currency quanto The most common option structure involving a target redemption feature is the Target redemption options are covered in Chapter 28. The most commonly traded option structure involving an accrual feature is the Accrual options are covered in Chapter 28. Window barrier options are covered in Chapter 26. expiry dates at which the client has profited. target redemption forward notional, providing the target has not been reached. Target Redemption Options Within target redemption options,leveraged the forwards payoff that is knockspecified generally in out a different if ways, strip some for of example, target forwards cumulative is client or reached. profit or This the target number of can be 1.3500 the option notional will not increase. accrual forward the strike. Accrual Options Within accrual options, the option notionalup) is over not the life fixed of but the ratherbe option it that depending accrues each on (builds time how there spot isoption moves. a notional For daily increases example, fix by it where USD100k, could spot whereas lies if between spot 1.2500 fixes below and 1.2500 1.3500 or the above to expiry, the optionstarts is at a some daterear window after barrier horizon anddown barrier, ends knock-in, at knock-out, and the double barrier option variations. expiry, the option is a options each extend standard forward or European vanilla payoffs. Window Barrier Options Window barrier options existbarrier for observation some starts (but at not the all) option of inception the and life goes of until the some option. date If prior Second-generation exotics addor additional first-generation features exotic toextensions of products. forwards, American barrier For vanilla options while example, options, accrual, target window redemption, and barrier Asian options are in Chapter 27. Alternatively, if payoutthisisa is in CCY3 (i.e., neither CCY1 nor CCY2), Second-Generation Exotics ■ 396 EXOTIC FX DERIVATIVES PRODUCT CLASSIFICATION ■ hr-eeainExotics Third-Generation eto/os-fOptions in Best-of/Worst-of the covered is maturity product This At strike. basket. basket a the and spot in 30. basket Chapter pairs the of currency function a the is payoff across changes spot averaging options Basket Options Basket due books trading separate in complexity. managed their risk to often are options corre- These and products. products, lation volatility products, multi-asset cover exotics Third-generation later. months three expiring 30. option call Chapter a in three of covered after strike is example, the product set For This to future. used the is in fix date spot specified a a months at fix spot on a levels) from barrier determined and strike (e.g., details contract The 31. Options Chapter a Start in to Forward covered option is vanilla product specified This a premium. sell fixed or a buy for to date date expiry expiry second first a at opportunity the has options Compound Options Compound 3. 2. three are 1. There fixings. spot from calculated variations: average common an on depends payoff as option known also are options Asian Options Asian urnypis hspouti oee nCatr30. between Chapter correlation in the covered on is dependent pairs product highly This is currency pairs. payoff multiple currency The in date. expiry options same vanilla the from to resulting payoff (minimum) worst Best-of sa pin r oee nCatr29. Chapter in covered are options Asian vrgso ptfie ae vrdfeetprosdrn h ieo h option. the of life the during periods different over taken fixes spot of averages option rate average Double option strike Average option rate Average B)and (BO) Worst-of aeapyf ae nabse ptwihi acltdby calculated is which spot basket a on based payoff a have pta auiyi elcdwt naeaeo ptfixings. spot of average an with replaced is maturity at spot : h pinsrk srpae iha vrg fso fixings. spot of average an with replaced is strike option the : r an are ohso tmtrt n h tieaerpae by replaced are strike the and maturity at spot both : pino noption an on option W)otospyotete h et(aiu)or (maximum) best the either out pay options (WO) vrg aeoptions rate average h we ftecmon option compound the of owner The . owr tr options start forward oeapc fteAsian the of aspect Some . are EXOTIC FX DERIVATIVES PRODUCT CLASSIFICATION 397 pay out based on realized spot pay out based on the future level of implied digitals are in-the-money at maturity, the option Variance swaps both have two European digital payoffs in different currency pairs pay out based on realized spot correlation between two pay out based on realized spot volatility compared to a ‘‘strike’’ volatility. For example, a 3mth involatility 3mth FVA between is the a 3mth trade on expirycovered the date in forward and Chapter ATM 31. implied the 6mth expiry date. This product is Correlation swaps currency pairs compared to a ‘‘strike’’ expressed in correlation terms. Forward Volatility Agreements (FVA) Forward volatility agreements expressed in volatilityvariance terms. compared tocovered a in Chapter ‘‘strike’’ 31. expressed in variance terms.Correlation This Swaps product is to the same expirygenerates date. a If cash payout. This product is covered in Chapter 30. Volatility (Vol) and Variance (Var) Swaps Volatility swaps Dual Digital Options Dual digital options

CHAPTER 21

European Digital Options

uropean digital options are conceptually one of the simplest exotic products; at Ematurity the option either pays out a fixed cash amount or nothing, depending on whether spot is above or below a specified digital level. However, the risk 399 management of these options, particularly at expiry, can be challenging due to the binary nature of the payout (either receive all the cash or none) over a one-pip spot difference. A European digital call option pays out cash on the delivery date if spot at maturity is at or above the digital level, as shown in Exhibit 21.1. While a European digital put option pays out cash on the delivery date if spot at maturity is below the digital level. As a technical aside, the European digital call here is defined as paying out at the digital level while the European digital put does not. This is not necessarily the case but it is important that European digital put + European digital call = guaranteed payout at maturity. Put another way, transacting both the digital put and digital call with all other details the same should not result in both contracts paying out. In practice these issues are dealt with via the confirmation documents agreed when trading the contract. European digital options prices are quoted as a percent of the payout amount, with prices generally rounded to the nearest 0.05% for customers or 0.25% in the interbank broker market. For example, a trader might make a rate of 23.5/24.5 USD% on USD/JPY 3mth 100.00 USD European digital call in USD1m payout (also called USD1m notional). If the client wishes to buy, they must pay USD245k value 400 EUROPEAN DIGITAL OPTIONS ■ uoenDgtlReplication Digital European hsintpsil npatc u otentoasrqie,pu tielvl are market. levels interbank strike the in plus particularly required, pips, ten notionals or the five nearest to the due to replication rounded practice tightest usually in The possible However, 21.4. notionals. isn’t Exhibit enormous with this in spread one-pip shown a trading as involve would payout, possible digital the replicate 21.3. to Exhibit spread in call shown vanilla is The call digital spreads. European put 100.00 vanilla a tight European for using and replication spreads replicated call be vanilla can tight puts using digital replicated be can calls digital European 50%. around be 21.2. will Exhibit value in shown option are digital time over European prices the call digital level, European digital the to close have not does option the because currencies currency; the non-payout between on the switch astrike. USD1m of to receive percent way a will no as client is quoted the there be 100.00, cannot above Prices is date. rate delivery spot the the if expiry, At spot. 21.1 EXHIBIT aual eeaePLi h poiecrec otenotional. the options to vanilla currency that opposite I the Part in from Recall P&L digital. generate the naturally of currency payout the on depends h aclto o eemnn h oinl eurdwti h replication the within required notionals the determining for calculation The required notional the larger the replication, the in strikes the tighter The is forward the If 100%. and 0% between generally are prices digital European 0.0Erpa iia alpyf tmaturity at payoff call digital European 100.00 EUROPEAN DIGITAL OPTIONS 401 100.00 European digital call vanilla call spread replication 100.00 European digital call value over time EXHIBIT 21.3 EXHIBIT 21.2 402 EUROPEAN DIGITAL OPTIONS oinl u ifrn C2ntoas hc nefc eeae h payout. the generates effect in which notionals, CCY2 different but put notionals USD the so (CCY2) to CNH1m is level digital digital the the notional of spread around payout The constructed payout. is the spread replicate put vanilla USD 6.1250/6.0750 notional. required closer the expected, As is, that at around strike long with are call strikes replication digital European a Given Replication Payout CCY2 with Digital European 21.4 EXHIBIT srqie,aCY oinlwl aual ecluae ihntereplication. the within terms. calculated reciprocal into be flipped naturally be must will strikes notional the Therefore, CCY2 a required, with is call digital European a Given Replication Payout CCY1 with Digital European K Example hntedgtlpyu si C2 h elcto pedhseulCCY1 equal has spread replication the CCY2, in is payout digital the When 2 Therefore: . K ihCY notional CCY1 with ) elctn S/N y .00UDdgtlpt 1yr A put. digital USD 6.1000 1yr USD/CNH a Replicating : 0.0Erpa iia alvnlacl pedrpiainwt ihe strikes tighter with replication spread call vanilla call digital European 100.00 = CNH1 K m∕( 2 and 6.1250 X N K =( C2pyu of payout CCY2 a , K 1 N CCY1 1 r together are CCY2 = strike short and K − 2 K − payout 6.0750 2 − payout K X 1 )× K 1 X )= ( N smaller tlevel at X USD20 tlevel at K ( K 2 2 smerclypositioned (symmetrically − K K 2 ic C1payout CCY1 a since , m K − 1 K )× e leg. per ftecl spread call the if , K 1 ) h agrthe larger the , N sgenerated is EUROPEAN DIGITAL OPTIONS 403 (the with is the 2 2 ) K K X . ( T . per leg. N ) 2 2 𝜎 m − 1 2 are continuously rCCY T − √ 2 is generated at 𝜎 is volatility, rCCY N rCCY and short strike CNH744.2 𝜎 ( 1 ) ) + K 2 1 ) d d S K )= )× ( ( 1and ( 2 . There is a strong link between N N ln K ) ) N T T 2 . . . X d 2 1 ∕ ) = rCCY ( ( 1 2 ) 2 1 6.1250 N N K 2 rCCY rCCY d at maturity is given by ∕ 1 − − − K of the European vanilla option with the 1 e e X K − 1 )= − K 1 = = − ≥ 1 K K 1 ∕ 1 delta S K 1 ≥ ( ( (measured in years), ( S . t ( T = . ) = Digital Call Digital Call 2 6.0750 N 𝜎 prob X % % 1 2 ∕ 1 2 1 + 1 CCY CCY P P t rCCY m∕( due to a change of numeraire plus the discounting back to − √ 2 𝜎 1 d rCCY is the strike or digital level, USD1 ( to + , a CCY1 payout of K 2 ) = N d S K ( ln : Replicating a USD/CNH 1yr 6.1000 USD digital call. A 1yr = is spot, 1 S d The CCY2% European digital call price under Black-Scholes discounts this value The CCY1% European digital call price under Black-Scholes is calculated by Under Black-Scholes, the chance of This time, when the digital payout is in CCY1, the replication spread has equal Example If the call spread replication strikes are long strike adjusting from the horizon is changed to CCY1 also: where back to the horizon in the payout currency: where compounded interest rates to time cumulative normal distribution function, and Prices for European digitalusing options cumulative under normal Black-Scholes distribution function canEuropean be digital calculated option directly prices andsame the maturity and the vanilla strike set to the digital level. CCY2 notionals, but different CCY1 notionals. 6.0750/6.1250 USD call spread canthe be constructed payout. around the Thespread digital notional to payout replicate of the digital is USD1m (CCY1), so the USD put that is, CCY1 notional strike order in the calculation gets flipped when the reciprocal is taken). Therefore: European Digital Pricing ■ 404 EUROPEAN DIGITAL OPTIONS ■ to reason no details: contract is there market. pair, the match currency will pegged price a smile-on in digital example, the for that case, expect the level. digital not the is around this specifically market, If the for matches maturity prices option the mid at smile smile-on accurate can obtain spreads to put order and options. in digital call European code vanilla within attainable) constructed practically be (not tight very Therefore, ■ ■ market: ■ vanilla direct the more from prices a digital midmarket are European obtaining above for of examined price method strategies midmarket replication a the generate Alternatively, to options. used be adjustment therefore TV could their model and on dependence only path depend no payoffs have comes they digital Hence European maturity. price. at market spot the a to (TV) value theoretical included. is hold smile they volatility that the note once but not and product useful only digital are Black-Scholes European relationships under the spot These of level. understanding to strike/digital building and respect for expiry with same the derivative and with vanilla payout next with the the (and European that a level Taking implies of strike/digital aligned). delta and the conventions expiry to same equal quoting the is option with digital option European vanilla a of value CCY1% The ■ eto h iia.Tetgtrterpiain h oeacrt h valuation. the accurate more the replication, the adjust- tighter TV The the digital. approximates the therefore of values ment two these smile, digital. between volatility the difference the of The price on smile priced midmarket are the spread approximates replicating price replication the the in options vanilla the digital. If the volatility, of ATM TV the the using approximates priced price are spread replication replicating the the in options vanilla the If urnedpyu tmaturity. at payout guaranteed price midmarket put digital European maturity. TV put digital European h olwn auto dniisapyt uoendgtlotoswt h same the with options digital European to apply identities valuation following The volatility the assume prices digital ‘‘smile-on’’ these that note to important is It Black-Scholes the from adjustment the calculating by options exotic price Traders is: option call vanilla European a of delta the that C Practical from Recall only rmtevltlt ml.Aywl-airtdvltlt ml pricing smile volatility well-calibrated Any smile. volatility the from delta faErpa iia pini the is option digital European a of Δ call + = uoendgtlcl TV call digital European 𝜕 𝜕 P call S = + e uoendgtlcl imre price midmarket call digital European − rCCY 1 . T N ( d 1 ) = gamma urnedpyu at payout guaranteed faEuropean a of = EUROPEAN DIGITAL OPTIONS 405 Typical EUR/USD European Digital Bid–Offer Spreads In practice, trading desks more often maintain grids of digital bid–offer spreads for the barrier level in order to calculate bid or offer prices directly. EXHIBIT 21.5 3mth6mth1yr 2% 2% 2% TenorO/N1wk2wk1mth2mth Bid–Offer Spread (CCY1%) 20% 10% 7% 4% 3% Many of theexposures Greeks on arising vanillamore from options. challenging However, if European spot risk ends digital up management near options the becomes digital are level significantly at similar expiry. to the wider but the relative shapeoften of remains unchanged. the bid–offer spread function over different tenors vanilla spreads can be constructed suchat that e.g., the full digital payout is generated different maturities. Bid–offer spreads for Europeanpairs digitals often in have liquid a G10 term currency structurecurrency pair that and looks EM similar currency to pair digital Exhibit bid–offer 21.5. spreads Less will liquid typically G10 be quoted The vanilla replication of thebid–offer European digital spread by can multiplying also the be vanillawithin used the option to replication bid–offer calculate for spread a strikes by set digital the at some notionals fixed width. Alternatively, conservative be the same ascontract the details TV the adjustment same. on Indescribed the a as same short ‘‘strikes’’ way within European a that digital trading call position,in put and European terms put option digitals of vanilla with are their options spoken payout all are about direction both and the higher P&L side at maturity. This means that thenegative TV to the adjustment TV on adjustmentthis a on a European the step equivalent digital further, European call the digital will TV put. adjustment be Plus on taking equal a and long European digital call option will European Digital Greeks European Digital Bid–Offer Spread ■ ■ 406 EUROPEAN DIGITAL OPTIONS h Vajsmn eeae rmbyn tie bv h iia ee n selling and level digital level. as the digital direction above the same higher strikes below the buying strikes in the from be if will generated adjustment Alternatively, adjustment TV TV level. the the downside, digital the the to is above the be scenario below strikes will P&L strikes selling buying adjustment from and TV generated the level adjustment topside, TV digital the the as to direction is same the scenario in P&L higher the if Therefore, ■ ■ the on sharply up maturity. vega pick option versus will the vega gamma into gamma that option peak the implies digital the European this about how option, vanilla introduced 21.6 Thinking a Exhibit time. on strategies in relationship over vega Notice replication constant reversal shock. the approximately a risk be stays Recalling a not to time. should identical this over is previously, tightens spot against that profile profile vega digital European The Risk Vega XII 21.6 EXHIBIT ogvg adln am)we h owr so h oe & ieof side P&L the lower to moving the of on chance increased is area. an P&L forward means higher the volatility Higher when barrier: gamma) digital the long (and the of vega in side staying Long P&L of higher chance the increased on an area. means is P&L higher volatility forward Lower the barrier: when digital gamma) the short (and vega Short uoendgtlotosare: options digital European 0.0Erpa iia alvg vrtime over vega call digital European 100.00 EUROPEAN DIGITAL OPTIONS 407 It’s a digital call so the payoff is to topside, so the vega . European digital call TV adjustment and volatility smile in a currency pair with a of the volatility smile at the digital level to the expiry date. The thought process The wider the strikes in the vanilla hedge, the quicker the hedge breaks down. Hedging European digital vega risk with a vanilla spread or risk reversal works The important decision is how wide apart to place the strikes around the digital Exhibit 21.7 shows TV adjustments for a European digital call with different This implies that the TV adjustment on a European digital option depends on the EXHIBIT 21.7 downside risk reversal level and thereforeEUR/USD, in if the what digital notionalsspread is in at to EUR10m 1.4000 or transact in areplicate USD1m the 1.3900/1.4100 the payout, spread size spread. in either of EUR50m the For a could payout 1.3500/1.4500 be and example, therefore transacted the in to vega profile. As a rule of thumb, the hedges will stay relevant until they go beyond 25 delta well when there is a fairOn amount the of time hedge, until a expiry vanilla (approximatelyin over spread the a to positive month). P&L the region same forP&L expiry the region as bank for the the the strike digital bank will the will be strike be bought will transacted; and be in sold. the negative spot) is therefore something like: gets longer to the downside.to That’s trade a over/under long/short TV risk reversal position, so I expect thisdigital option levels in a currency pair with a downside risk reversal. slope for traders when asked to price a digital option (with a digital level around current 408 EUROPEAN DIGITAL OPTIONS roto-h-oe nt h iiaiiswt et le nvnlaoptions). vanilla in-the-money on value is bleed the spot delta expiry, whether with on at depending similarities Then 100% the 50%. or (note around 0% out-of-the-money be either or will is digital digital the the of of the value date the expiry 21.9. the expiry, Exhibit into in shown Overnight is level. call digital digital the 100.00 USD1m to a close from volatility is theta P&L spot significant if generate date options expiry digital European date. expiry the the the digital. on the to of options exposures side buying P&L by negative in the hedged on the is options shown of position selling rest and this is the side Again, digital overwhelm P&L time position. can the positive exposures trading near over theta a is and in spot digital gamma risk if the European as challenges expiry management a near risk level toward presents for dramatically This profile increases 21.8. digital Exhibit gamma European The a from maturity. theta) hence (and Gamma Risk required. Pin longer and no is Gamma hedge the level, valid. digital stays the hedge from the away time notionals moves of length (larger spot the if costs and that hedge) Note transaction the between establish until to month balance cost one higher best mean has levels, the delta option the 25 seek the the to until Traders around set approximately strikes expiry. relevant the spot with remain with up will options set hedge is put the hedge and vanilla the call If 25d level. 1mth digital price Therefore, levels. strike XII 21.8 EXHIBIT osdrti ht rfie fso saon h iia ee n a ro to prior day one level digital the around is spot If profile: theta this Consider gamma and vega managing from turns attention management risk point, some At 0.0Erpa iia algmaoe time over gamma call digital European 100.00 i risk pin h hneo ptedn ertedgtllvlon level digital the near ending spot of chance The ; on the EUROPEAN DIGITAL OPTIONS 409 the payout amount. half USD1m 100.00 European digital call theta into expiry date equal to the payout through the digital level. equal to the payout through the digital level. A large delta exposure can loss gain In general, it is easier to risk manage a trading position that starts with positive If spot is on the positive P&L side of the digital coming up to expiry, risk If spot is on the negative P&L side of the digital coming up to expiry, risk When risk managing a European digital at expiry, traders often work to reduce P&L (the theta) but will lose P&Lposition if that a starts certain with spot negative move occurs,move P&L rather (the occurs. than theta) a Even but trading experienced, will well-regardedat gain traders a P&L spot if have rate a been on certain known a spot to screen to shout encourage it to move. management is easier. Into theP&L expiry date, theta will betherefore earned be and used there to will generate be(and additional a hence P&L toward if the spot negative moves P&L toward side). the digital level short strikes generatedelta larger negative exposure P&L andachieved. in buy the wing wings, strikes so until traders the adjust bestmanagement their possible is P&L difficult. Into distribution theP&L is expiry date, theta will be paid and there will be a the worst-case final-day negative P&Lscenario to most often an occurs acceptable with level.negative spot The P&L one side. worst-case pip To P&L away hedge this(to through risk, the the strikes digital same can level be expiry) on sold the and at or this near earns the digital additional level theta around that level. However, EXHIBIT 21.9 Therefore, the daily theta intoThe the practical expiry consequence date of is thisP&L at is from most that, a European 50% ignoring digital of hedges, is the the worst-case digital final-day value. 410 EUROPEAN DIGITAL OPTIONS ■ uoenDgtlRange Digital European XII 21.10 EXHIBIT European a of maturity if at out payoff pays The and levels. barriers barrier digital the two between has is a that maturity called spread) at (also bet spot range a digital or European spread a digital is European product digital European popular Another XII 21.11 EXHIBIT ogUDm9.0150 uoendgtlrnepyf tmaturity at payoff range digital European 95.00/105.00 USD1m Long ogUDm9.0150 uoendgtlrnevg profile vega range digital European 95.00/105.00 USD1m Long EUROPEAN DIGITAL OPTIONS 411 Long USD1m 95.00/105.00 European digital range volga profile This long-wings vega profile implies a long volga and long convexity exposure European digital ranges can be used to estimate the probability of spot being The vega profile of this European digital range is shown in Exhibit 21.11. With within a certain range onthat significant a risk given management date challenges will in1.6350 occur the at if future. spot the is For end between example, 1.6250date of it and and February is spot next anticipated levels year. inthe A question approximate can European probability be digital of constructed. spot range The being for price within the of that expiry range. the contract gives With spot outside the barriers, the optionsince increases there in will value be if more implied volatility chance rises of spot moving into thethat range, is hence confirmed long vega. inpositive Exhibit TV 21.12. adjustments. Therefore, European digital ranges often have digital range is shownEuropean in digital Exhibit calls or 21.10. two This European digital product puts: can one be bought and replicatedspot one with between sold. two the barriers,since the there will option be more increases chance of in spot ending value up inside if the implied range, hence volatility short vega. falls EXHIBIT 21.12

CHAPTER 22

European Barrier Options

uropean barrier options have a vanilla payoff at expiry plus they also have a Esingle European barrier. For a European knock-out (EKO) barrier option, if spot at maturity is beyond the barrier level, the contract expires worthless despite being in-the-money. The payoff at maturity of a long European knock-out call with 413 1.3000 strike and 1.4500 European knock-out barrier is shown in Exhibit 22.1. European barriers must be positioned in-the-money, otherwise they have no impact. European knock-in (EKI) barrier options have a vanilla payoff at expiry only if spot at maturity is beyond the barrier level. The payoff at maturity of a long European knock-in call with 1.3000 strike and 1.4500 European knock-in barrier is shown in Exhibit 22.2. In the European knock-out barrier case, the curtailing of the payoff beyond the barrier can significantly reduce the cost of the European barrier option compared to the equivalent European vanilla option. This is shown in Exhibit 22.3. Beyond the barrier, the European knock-out contract still has value prior to expiry because there is time remaining for spot to move back inside the barrier. However, as time passes and the probability of such a move occurring reduces, the value of the European knock-out option with spot beyond the barrier level decreases. This is shown in Exhibit 22.4. 414 EUROPEAN BARRIER OPTIONS XII 22.2 EXHIBIT 22.1 EXHIBIT uoenkokotpyf tmaturity at payoff knock-out European uoenkoki aofa maturity at payoff knock-in European EUROPEAN BARRIER OPTIONS 415 European knock-out barrier value over time European knock-out barrier versus European vanilla value profiles EXHIBIT 22.4 EXHIBIT 22.3 416 EUROPEAN BARRIER OPTIONS ■ uoenKokotReplication Knock-out European XII 22.5 EXHIBIT notionals with spread call B long a using replicated perfectly options, be can digital products. European simpler dependence: of like combinations path Also using replicated maturity. no be at can has options spot barrier knock-out on European only European depends a payoff options, The digital European Like XII 22.6 EXHIBIT soni xii 25 lsasotErpa iia at digital European short a plus 22.5) Exhibit in (shown ogErpa nc-u aloto ihnotional with option call knock-out European long A ogvnlacl pedpyf tmaturity at payoff spread call vanilla Long hr uoendgtlpyf tmaturity at payoff digital European Short B N soni xii 22.6). Exhibit in (shown ,strike N n strikes and K n barrier and , K and B EUROPEAN BARRIER OPTIONS 417 the at | | USD 16.7m. There- Strike Strike = | − − Barrier Strike − Barrier Barrier | | Barrier × × USD 16.7% 1 1 = Barrier | CCY CCY 120 = % )∕ 1 Notional Notional CCY 100 = = − 2 1 CCY CCY 120 =( Intrinsic Value . Intrinsic value measures the payoff at maturity at the barrier level, Intrinsic Value Intrinsic value in a European knock-out barrier option Intrinsic Value : USD/JPY USD100m 1yr 100.00 USD Call/JPY Put European knock-out Example The CCY2 intrinsic value is divided by the barrier level to get it into CCY1 terms Therefore: Note that the digital in Exhibit 22.6 could itself be replicated using vanilla options EXHIBIT 22.7 120.00. Intrinsic value rather than current spot because forbarrier the level. P&L jump to be realized spot must be the barrier level or theoption other. is The highlighted intrinsic in Exhibit value 22.7. on a European knock-out barrier The amount of Europeanintrinsic digital value risk in aand European therefore barrier how much option P&L is change given occurs by at maturity the if spot ends up one side of so a European knock-out can theoretically be replicated using vanilla options only. Intrinsic Value ■ 418 EUROPEAN BARRIER OPTIONS ■ uoenKoki Replication Knock-in European pin ttebrirwti h elctoscne ahohrout. other each cancel replications the within barrier the at options European a knock-out on European value intrinsic 22.10. Exhibit The in same highlighted case. the is using option knock-out determined barrier European knock-in be can the value as Intrinsic calculation knock-in. European the of value notional with option call vanilla 22.9). Exhibit long a notional using with strike replicated option perfectly call be knock-in can European long A management. risk and pricing derivatives FX exotic within often to returned risk digital European of USD16.7m 120.00. contains at option knock-out European the fore, XII 22.8 EXHIBIT h elctoscnr ht sepce,ln uoenknock-in European long expected, as that, confirm replications intrinsic The the is replication the in digital European the of notional the Again, called also is value Intrinsic B soni xii 28 lsaln uoendgtlat digital European long a plus 22.8) Exhibit in (shown ogvnlacl aofa maturity at payoff call vanilla Long = ogErpa ail ic h iia pin n vanilla and options digital the since vanilla European long parity , cliff ,or spike n ti e ocp htis that concept key a is it and N ,strike K n barrier and , B sonin (shown + N long and B EUROPEAN BARRIER OPTIONS 419 Intrinsic value in a European knock-in barrier option Long European digital payoff at maturity European barrier option Greeksshown exposures previously. can be understood via the replications EXHIBIT 22.10 EXHIBIT 22.9 European Barrier Greeks ■ 420 EUROPEAN BARRIER OPTIONS are ee n ogvg xoue ntengtv & ieo h are level. barrier the of side 22.13. P&L the Exhibit negative in of the shown side on is P&L European exposures This positive digital, vega the long European on and a exposures of level like vega barrier chance short hence, the have and options increases barrier barrier volatility the knock-in inside higher back level, moving barrier spot European the beyond spot between barrier. negative barrier. the and the beyond positive and moving both strike spot go the of contract the chance the barrier, the from increases exposures the also vega and Therefore, it but strike strike more the the used from optionality between be the to is allows volatility spot implied spot Higher If of conflicted: is barrier. chance contract the the increases inside volatility back higher because moving long is exposure exposure. vega vega barrier, and value option lower the causes Also, barrier 22.12. and Exhibit strike in the shown the of as proximity together while merge profile barrier European vega the and vanilla strike profile. standard vega reversal a risk or is spread as there vanilla profile a knock-out strike produces vega European the barrier the a European At within within separately 22.11. apart observed Exhibit enough be in can far element are each levels option, barrier barrier and strike the If Risk Vega XII 22.11 EXHIBIT h uoenkoki eapol lomthsteitiino h aof With payoff. the of intuition the matches also profile vega knock-in European The the beyond spot With payoff: the of intuition the match profiles vega the These from exposures vega the together, closer are levels barrier and strike the If uoenkokotbrirvg rfiewt tieadbrirfrapart far barrier and strike with profile vega barrier knock-out European EUROPEAN BARRIER OPTIONS 421 European knock-in barrier versus vanilla vega profiles European knock-out barrier vega profile with strike and barrier close together EXHIBIT 22.13 EXHIBIT 22.12 422 EUROPEAN BARRIER OPTIONS ■ ■ uoenBrirBdOfrSpread Bid–Offer Barrier European Pricing Barrier European uoenbriroto s2,teErpa are i–fe pedwl be will spread bid–offer barrier value. European intrinsic the the by 2%, multiplied 2% is approximately option barrier bid-offer. digital European barrier bid–offer the take and bid-offer to strike appropriate the only both of usually to larger is close the it be from maturity cannot spread spread) at spot strike wider Since the vanilla the barrier. and the hence of barrier at (and the risk consists risk digital main knock-out the barrier from The European comes European risk. usually a digital on European replication, spread plus the bid–offer risk spread from the calculate seen to As used options. also is value Intrinsic any distribution, spot terminal used. Alternatively, be the could replication. the on model option pricing only incorporating smile depends vanilla well-calibrated options value using barrier barrier generated European European be since can for smile prices volatility digitals, European Like as that value. managed is intrinsic the risk risk to trading be equal the size can barrier, in and European option the digital option the European to barrier, vanilla a close knock-out of regular is European a spot a if of of Alternatively, that strike such. simply the is to risk close is trading spot if expiry, Toward Risk Pin and Gamma hrfr,i h i–fe pedo h iia ikebde ihnthe within embedded risk digital the on spread bid–offer the if Therefore, CHAPTER 23

Touch Options

f FX derivative contracts contain a barrier that is monitored continuously against Ispot, the barrier is described as American or continuous. The simplest American barrier products are touch options. There are two main kinds of touch option: One-touch options pay out a fixed amount of cash on the delivery date if spot trades through a specified barrier level at any time between horizon and expiry. No-touch options pay out a fixed amount of cash on the delivery date if spot does 423 not trade through a specified barrier level. Touch options are the American barrier version of European digitals and there are many similarities. Prices on touch options are quoted as a percentage of the payout notional, and like European digitals, prices cannot be quoted in the non-payout currency because there is no strike to switch between notional currencies. The Greek exposures on touch options can be thought of as being ‘‘caused by’’ the American barrier, just as Greek exposures on vanilla options are ‘‘caused by’’ the strike. The theoretical value (TV) of a one-touch usually lies between 0% and 100%, as shown in Exhibit 23.1. Note that one-touch TV doesn’t always lie between 0% and 100% because interest rates could be negative. It is important to understand that prior to expiry, in a delta hedged trading position, there is no large P&L gain or loss generated from the touch option triggering. As spot nears the barrier, the mark-to-market value of the one-touch option rises (ignoring discounting), e.g., 98% to 99% to 100% (barrier touched), and the trading risks prior to expiry can be hedged using standard Greek exposures. On the expiry date itself, if risk is viewed in discrete daily steps (see Chapter 11) and the one-touch has not previously triggered, its value is 0%. If it then triggers, 424 TOUCH OPTIONS ■ et Risk Delta ogbriro oadasotbrir o xml,o h a eoeepr,an expiry, whereas a before exposure, from day delta long the away a on have bleed example, will delta option For one-touch large barrier. topside short long a untriggered a causes toward potentially or barrier This long exposure. delta no TV. spot with 30% has in option changes one-touch occurs the to context which this sensitivity at in level near the spot and far the as between around barrier point inflection the is The time. spot toward over If increases bleeds time. over delta decreases the spot barrier, in changes 23.2. to Exhibit to sensitivity in closer the shown spot is time with barrier over increases topside option value one-touch a option a into on since long Delta barrier barrier. be downside the will a contract into one-touch short a and from exposure delta The manage, risk to situation large. challenging P&L is a instantaneous notional be one-touch an the can as if This shown particularly level. is barrier this the systems through trading jump In 100%. becomes value its 23.1 EXHIBIT nteepr aeisl twl aezr et.Teeoe hr ilb significant be will there Therefore, delta. zero have will it short itself date expiry the on nteepr ae h n-oc pini iwda aho-ohn payout cash-or-nothing a as viewed is option one-touch the date, expiry the On is spot If et le noteepr date. expiry the into bleed delta far 0.0oetuhT vrtime over TV one-touch 100.00 rmteoetuhbrir et lesaa rmtebriras barrier the from away bleeds delta barrier, one-touch the from near h one-touch the TOUCH OPTIONS 425 100.00 one-touch delta over time Therefore, if a one-touch option is hedged with vanilla options, no static vanilla The vega exposure on a one-touch can be hedged with a vanilla option since the The maximum sensitivity to implied volatility comes at the point where the barrier Far from the barrier, spot has a very small chance of trading through the barrier hedge will remain valid forbe the better life now of or the better option. later The as vanilla the vega exposures hedge evolve. will either in the delta risk section, theone-touch point is is around located 30%. at the spot level for which the TV oftwo the vega profiles takedifferently similar over shapes. time. Vanilla However, vega vegaExhibit reduces 23.4. on and drifts a toward vanilla the option strike as evolves shown in level and therefore the sensitivityto to the implied barrier, volatility is spotimplied low. will volatility Similarly, is almost very low. close certainly touch the barrier levelknock is so in the sensitivity balance. This to point moves closer to the barrier over time. As discussed One-touch options are generallyis long more vega chance since ofone-touch if vega spot spot moves moving volatility toward and the increasesroughly barrier hitting there constant over the and time. there The barrier. is peakbecause Exhibit vega no the 23.3 exposure exposure contract stays has to shows knocked implied how and volatility is through now the a barrier guaranteed cash payout at maturity. EXHIBIT 23.2 Vega Risk ■ 426 TOUCH OPTIONS XII 23.4 EXHIBIT 23.3 EXHIBIT h ail eg trst okmr ieteoetuhol.A hspit the point, this than rather At barrier only. the from one-touch coming the gamma vega. like the the to more switches look focus management to risk starts be hedge can vanilla there 23.5. the Exhibit then, in Even shown initially. as barrier, least the at at time, particularly risk, over residual better substantial becomes that place in stevnlavg isaa,tecmoievg oiino h n-oc plus one-touch the of position vega composite the away, dies vega vanilla the hedge As vega vanilla a put to preferable usually is it costs rehedging minimize To 3.0oetuhvg vrtime over vega one-touch 130.00 2.0vnlavg vrtime over vega vanilla 120.00 TOUCH OPTIONS 427 AUD/USD 5yr 0.8000 one-touch vega Vega of 130.00 one-touch hedged with 120.00 vanilla in proportions such that than the expiry date. It is important to always view bucketed vega An important subtlety is that one-touch vega, unlike vanilla option vega, is not In addition, when the touch barrier is closer to spot, stopping time will reduce; EXHIBIT 23.6 closer maturities Greeks when risk managing contracts with American barriers. always positive. In high-interest-rateExhibit 23.6 differential can be pairs generated the for a vega long downside profile one-touch shown option. in EXHIBIT 23.5 vega risk at 6mth tenor is minimized the option is not expected to live to expiry and therefore vega exposures move to 428 TOUCH OPTIONS ■ am n i Risk Pin and Gamma nfoto)tebrirlvlt h aeepr ae hsofesteincreased the offsets This date. slightly expiry (or the at same therefore vanillas the and sell to to sizeable level often be barrier is the can expiry of) into theta system one-touch front This management long in risk 0%. a the for to in hedge 100% value best the almost as from one-touch the drops of value remaining being the into risk gamma date. being expiry from the changes on option risk one-touch P&L cash-or-nothing a date, expiry 23.7. the the Exhibit on toward in shown moves as gamma expiry into the sharply and increases and gamma time over long barrier generally are options One-touch spot that means it static; high-interest-rate is pairs. in currency spot particularly pegged relevant the or is described differential mean This first path. not forward framework does the follows Black-Scholes volatility barrier. perfectly the the Zero touch 5: of to Chapter feature likely in less to important is likely an hence is and spot recalls rises, path This volatility forward implied the If from barrier probability. further the the high diverge expects 0.9000, very model example, with the touched for therefore be and at, to barrier spot one-touch With the below and negative. is rates large forward interest CCY2 are than points larger far swap are therefore rates interest CCY1 when occurs barrier XII 23.7 EXHIBIT fso scoet h are noteepr ae h ht ilb qa to equal be will theta the date, expiry the into barrier the to risk close strike is being spot into If risk gamma being from changes option vanilla a as Just downside a with option one-touch long a on vega negative counterintuitive This 0.0oetuhgmaoe time over gamma one-touch 100.00 TOUCH OPTIONS 429 stop-loss order. stop-loss order. in order to replace the delta that has been lost. As described in Chapter 3, bought Within risk management systems it is usually possible to view spot ladders In G10 currency pairs, spot orders to hedge barrier delta gaps in the position The following rules apply to all American barrier products: For example, if a long topside one-touch barrier knocks, the delta from the Traders often find themselves long one-touch barriers and short vanilla hedges Short the barrier (i.e.,take-profit don’t want the barrier to touch) naturally generates a Long the barrier (i.e., want theorder. barrier to touch) naturally generates a becomes cash-or-nothing risk, theexists delta on the disappears expiry and date of hence options no with American delta barriers. gapassuming order delta gap spot ordersTraders from in American barriers G10 are currency either pairs executed usually or view not. their trading positions assuming delta information on how large spot orders can cause negative P&L. are usually calculated ahead ofthe time delta exposure and within placed the in tradingNote an position that remains order when unchanged management as risk system barriers is knock. so viewed in discrete daily steps, on the expiry date, when risk ■ Executing these spot orders canmoves result against in the ‘‘slippage,’’ the spot riskP&L. order that This as is the particularly it spot a is risk market for being larger-sized executed spot orders. and See hence Chapter 25 results for more in negative be buying spot when it moves higher is called a ■ If a one-touch barrier knocksto prior zero. to This expiry, potentially the causesbe delta a risk from managed. delta the jump option (also disappears known as a deltaone-touch gap) jumps that from must long deltaknocks. to This flat causes delta the (no trading further position exposure to to get spot) shorter once delta. it Therefore, spot must better to increase vanilla hedges overhedge time notional if up possible, front. rather than putting on the full at/near the barrier levelexpiry, to the the exposures same fromoptionality the expiry from one-touch date. the disappear If vanilla and hedges.the the the vanilla If barrier hedges position the must knocks is position be prior bought is leftimplied back volatility now to short just has too as often short spot risen has gamma/vega, due broken to into increased a uncertainty. new Therefore, range it and is often gamma risk on theitself. last few days of the option and the theta risk into the last day Touch Barrier Delta Gap ■ 430 TOUCH OPTIONS ■ n-oc Pricing One-Touch one-touch. TV the higher to and adjustment. adjustment direction TV TV opposite positive negative a the a have have will in will one-touches will TV is one-touches negative one-touches reversal a TV TV have risk lower lower will barrier, the barrier, one-touches if TV the higher Alternatively, as and adjustment. adjustment direction TV same positive the a have in is reversal risk profile 23.9. vanna Exhibit the of TV one-touch, negative versus barrier the and downside vanna simply a estimated typical is For a be one-touch. shows topside can 23.9 a a adjustment for Exhibit knowing profile TV price. by the model simply a contract, that versus one-touch means sense-checked a this about practice details In pairs. few currency and tenors across tenors across 23.8. consistent Exhibit fairly in is shown as profile pairs, TV currency versus and vega the options, one-touch For must positions trading within gaps time. delta over executed. of monitored are sizes and orders and calculated currency levels gap be market the delta pairs, barrier emerging currency no in all assuming In traders positions but their view executed, often are pairs barriers from orders gap XII 23.8 EXHIBIT xii 31 hw yia og essT rfiefratpieo downside or topside a for profile TV versus volga typical a shows 23.10 Exhibit the if only, exposure skew of terms in that implies 23.9 Exhibit in vanna The consistent fairly are profiles TV versus vanna and TV versus volga Similarly, n-oc eavru TV versus vega One-touch TOUCH OPTIONS 431 One-touch volga versus TV One-touch vanna versus TV (topside barrier) This profile implies that in terms of the wing exposureHow only, these lower TV vanna one- and volga exposures are reflected within a TV adjustment touches will have a positive TVnegative adjustment TV while adjustment. higher TV one-touches will have a depends on the volatility surface in a particular currency pair. EXHIBIT 23.10 EXHIBIT 23.9 432 TOUCH OPTIONS oesfra1rEUR/USD 1yr downside a for models these from in seen Recall be between the can expected. and this prices convexity convexity; results. as gives volatility volatility undervalues overvalues models, model model model volatility volatility volatility volatility stochastic local mixed the stochastic that The and 19 Chapter volga. volatility short local the the to due the TV TV exposures. high volga at short and low the exposure at to vanna then due the negative exposure, to stays volga due adjustment long negative TV the goes to adjustment due TV contracts the TV TV low very for positive are EUR/USD 1yr downside a for models pricing models 23.11 EXHIBIT n nevle h kwcomponent. skew the undervalues convexity volatility and overvalues model volatility stochastic volga The well. short component skew the and vanna flipped the to due adjustments negative TV exposures. exposures. are volga one-touches long TV the and high vanna on the to due positive are touches gi,telclvltlt oe nevle oaiiycneiy u austhe values but convexity, volatility undervalues model volatility local the Again, one- TV low on adjustments TV profiles, volga and vanna the by predicted As pricing various by generated adjustments TV one-touch the shows 23.12 Exhibit 40% around adjustment negative most their have VVV except models All one-touches these on adjustments TV profiles, volga and vanna the by predicted As smile various by generated adjustments TV one-touch the shows 23.11 Exhibit . . y U/S osd n-oc Vajsmn ne aiu ml pricing smile various under adjustment TV one-touch topside EUR/USD 1yr downside topside n-oc ihars eeslas for also reversal risk a with one-touch n-oc ihters eeslfor reversal risk the with one-touch TOUCH OPTIONS 433 forward volatility. This is obviously not ≈ the expected spot volatility. Therefore, when pricing a not . Higher interest rate volatility increases the forward volatility 1yr EUR/USD downside one-touch TV adjustment under various smile pricing Interest rate volatility and hence reduces the market value of one-touch options. This interest rate impact can be quantified using a stochastic interest rate pricing The pricing models examined in this chapter all assume that interest rates are In general, the mixed volatility model gives good pricing for one-touch contracts likely to knock the barrier than the model suggests. If the volatility ofvalue spot of is the higher one-touch thanmore will likely the to be volatility knock higher the of than barrier the than the forward, theIf smile model the the suggests. model market volatility price of sincevalue spot of spot the is is one-touch lower will be than lower the than the volatility smile of model the price since forward, spot the is less market model. There are two main elements to consider1. within such a model: ■ ■ constant and therefore spotthe volatility case inAmerican practice barriers knock (see out ChapterATM on volatility, 17 spot, which but is for areone-touch: more usually priced details). and The valued using key the point is that main situations in which thethe mixed market: volatility model may fail to1. give prices Contracts close for to which2. there is a Currency large pairs exposure in to which interest there rates is a spot jump dynamic. EXHIBIT 23.12 models in liquid currency pairs provided the model is correctly calibrated. There are two 434 TOUCH OPTIONS ■ n-oc Variations One-Touch n-oc pin,aCY aotwl aerltvl ihrvlainta CCY2 than valuation higher For relatively maturity. have at will cash payout CCY2 CCY1 or a CCY1 options, either one-touch out pay can options One-touch Payout CCY2 versus CCY1 from variations subtle have that contract. options one-touch one-touch standard the on prices request often Clients often are width. contract, they the digital European of expiry, equivalent two-thirds expiry, the at perhaps than to challenges spread prior bid–offer management tighter out a risk with biggest knock quoted the can avoid contracts hence one-touch and maturities Since different traders. at spreads by European bid–offer of for maintained grids spreads using bid-offer generated like usually options, are touch digitals, for spreads bid-offer practice, In Spread Bid–Offer One-Touch case. the is is it this the valuation, why about one-touch investigate the intuition to than from important trader’s different rather significantly the models is if probability pricing knock However, the barrier using probabilities. of priced limitations knock be the barrier should for guessing options adjusting Touch and volatility assumptions. models static rates pricing single interest under static priced single touching and barrier the of discounted, the probability is TV risk-neutral one-touch Specifically, touching. barrier one-touch the of pricing. bility within quantified cases be always these should In rates pairs. interest stochastic currency of pegged/managed impact tenors in the longer or at 2yr) important is beyond difference (approximately volatility forward versus volatility spot This 2. h V ofl.Ehbt2.3sosa xml fthis. of cause example will an one-touch shows 23.13 topside Exhibit a fall. on level. to payout barrier TV% CCY2 the the to at higher CCY1 spot from with flipping more Therefore, worth relatively is CCY1 because payout ial,i swrhntn htoetuhvlei nutvl iia oteproba- the to similar intuitively is value one-touch that noting worth is it Finally, ftefradadhnedcesstemre au foetuhoptions. one-touch of value market the decreases hence volatility and CCY2 the forward increases and the rates CCY1 spot of and spot between between correlation correlation negative positive or value rates Similarly, market the options. CCY2 increases hence and one-touch and spot forward of between the of correlation volatility negative the depresses or rates rates CCY1 and spot between rates interest and spot between Correlation sse nCatr1,pstv correlation positive 17, Chapter in seen As . topside TOUCH OPTIONS 435 lower, there is or (value spot). These are when the barrier touches one-touch options. one-touch options, a CCY2 payout will have relatively instant ,or downside CCY1 versus CCY2 one-touch options in pricing tool pay-at-touch Since the standard one-touch pays out at maturity, the effect of discounting For two-sided touch options that could knock with spot higher Similarly, for Pay-at-Maturity versus Pay-at-Touch Standard one-touch contracts pay aton maturity, one-touch but clients contracts sometimes that request paycalled prices out must also be takenthe into interest account. rates Forsince in the a the payout standard payout at one-touch maturityChapter currency, must 10). contract, the be the present relatively valued higher lower to the the barrier option knock date value (see higher valuation than CCY1 payout because CCY2 islower. relatively Therefore, worth more flipping with spot from CCY1will cause to the CCY2 TV% payout to on rise. a downside one-touch usually minimal valuation difference between CCY1 payout and CCY2 payout. EXHIBIT 23.13 436 TOUCH OPTIONS ■ oTuhOptions No-Touch eafis nrae stevleo h oben-oc ie;te tsat osplit to starts time. it over then barrier rises; each from double-no-touch The exposures 23.15. the vega Exhibit of separate in value two shown into the is as this increases and first clear vega become being barrier payout each 23.14. the from Exhibit exposures hence in shown and is barrier option double-no-touch either a touching of profile not vega spot The generated. of chance greater on a more and volatility implied and volatility. implied spot of pricing between volatility spot, relationship the current the of side on either less barriers depends with sense; broker makes this interbank Intuitively volga. the in product exotic clients. standard institutional with a and market are double- options therefore, options; (DNT) no-touch as no-touch contracts touch double-barrier and options (negative) opposite Greeks and equal The option. the one-touch maturity. are equivalent contract at the no-touch payout of a guaranteed on adjustment a TV the in the results and and therefore date same horizon the details the between time any does spot at if level maturity date. at barrier expiry cash specified of amount a fixed through a out pay options No-touch the of value version. market standard the the than and higher barrier, adjustment be one-touch payout should TV topside one-touch the model instant a in smile toward and rates higher between spot interest heads differential between spot if the correlation as example, strong increase For to a currency. tend in is payout currency rates there the interest if in if or rates particularly high interest difference, are price currency significant payout a the be can received there be expiries will maturity). cash (at payout later than of rather amount knocks) fixed barrier the a (when because sooner version standard the than stm ass(ri h aresaepae ie pr)tesprt vega separate the apart) wider placed are barriers the if (or passes time gives As volatility lower since exposure vega short a has double-no-touch usually long is A double-no-touch a on exposure main the 18, Chapter in seen As one-touch as contracts touch single-barrier trade to is convention market The contract other all with option no-touch long a plus option one-touch long A longer for but negligible, usually is difference value this contracts, short-dated For expensive more be will one-touch instant the positive, are rates interest When not trade TOUCH OPTIONS 437 Double-no-touch vega profile over time Double-no-touch vega profile EXHIBIT 23.15 EXHIBIT 23.14 438 TOUCH OPTIONS e omtrt ihu aigtigrdtebrirfran-oc pint a out. pay to option no-touch a for barrier the triggered having without maturity to get expiry. that at barriers unlikely both is at it issues since management options risk touch encounter will single-barrier traders on shown spread bid–offer the increases. TV a hence on and exposure widen volga barriers the the how as shows changes 23.16 double-no-touch Exhibit symmetric volga. negative have can touches 23.16 EXHIBIT ial,nt htisatvrin fn-oc pin r o osbe ptmust Spot possible: not are options no-touch of versions instant that note Finally, to similar usually is options touch double-barrier on shown spread bid–offer no- The double TV higher but volga, long are options double-no-touch TV Lower ymti-are oben-oc og gis hoeia au profile value theoretical against volga double-no-touch Symmetric-barrier CHAPTER 24

American Barrier Options

tandard American barrier options are one of the most frequently traded exotic FX derivative contracts. American barrier options have a vanilla payoff at expiry Splus they also have a single American-style barrier. There are two main variations: Regular barrier options have the American barrier positioned out-of-the-money 439 compared to the option payoff and reverse barrier options have the American barrier positioned in-the-money compared to the option payoff. As described in Chapter 20, American barriers are monitored continuously against the spot level in the market.

■ Regular American Barrier Options

A regular American knock-out (KO) call option structure is shown in Exhibit 24.1. Note that the barrier is positioned out-of-the-money. Consider a CCY1 call knock-out barrier option with the barrier positioned so far below the strike that there is zero chance of spot hitting the barrier and then ending up back in-the-money at expiry. The pricing and risks of this knock-out option will be identical to the equivalent vanilla option:

■ Far barrier knock-out option TV = vanilla priced using ATM volatility

■ Far barrier knock-out option vega = vanilla vega

■ Far barrier knock-out option TV adjustment = vanilla zeta 440 AMERICAN BARRIER OPTIONS XII 24.2 EXHIBIT 24.1 EXHIBIT n h nc-u Vcnegst h ail V fso osdw hog the touched through down be goes worthless. to becomes spot and likely If out TV. less knocks vanilla option is the barrier barrier the to barrier, converges downside TV the knock-out higher, the and goes spot as (1.3000), sEhbt2. hw,ee ihabrir(.00 uhcoe otestrike the to closer much (1.2000) barrier a with even shows, 24.2 Exhibit As C1cl nc-u are pinstructure option barrier knock-out call CCY1 nc-u are n ail Vprofiles TV vanilla and barrier Knock-out AMERICAN BARRIER OPTIONS 441 TV of knock-out barrier option with different barrier levels Knock-out barrier and vanilla delta profiles Contract details: EUR/USD 1yr 1.3000 EURSpot: call/USD 1.3000. put knock-out option. As spot goes higher, the knock-out delta converges to the vanilla delta. If spot Exhibit 24.4 shows how the theoretical value of a knock-out barrier option The delta profile of the knock-out option versus the equivalent vanilla option is EXHIBIT 24.4 1.00001.10001.20001.25001.2900 99.94% 98.18% 80.52% 52.38% 12.85% 3.781% 3.781% 3.638% 2.782% 0.762% Barrier Level0.5000 Stopping Time Knock-out 100.00% Barrier TV (CCY1%) 3.781% changes as the barrier levelknock-out changes. barrier As TV the reduces barrierknocking level because prior moves there to closer maturity. is to an spot, increased the chance of the barrier shown in Exhibit 24.3. goes down through the barrier,delta the exposure knock-out or barrier indeed has any triggeredoptions other and have Greek there barrier exposures. is delta Therefore, gaps no American like barrier touch options (see Chapter 23). EXHIBIT 24.3 442 AMERICAN BARRIER OPTIONS lsrt pt og esgnrlyls oiieadeetal rae narea an creates eventually and general, moves positive In barrier less volga knock-out volga. generally the negative gets As volga of expected. as spot, profile, to volga closer vanilla the like looks 24.6. Exhibit in shown as higher, moves barrier the to are exposures vega American with contracts therefore view option to any important and barriers. managing is risk it expiry when Again, exposures expiry. to Greek the bucketed than live closer maturities to at volatility expected implied not is option level. vanna barrier the increased into causes sharply falls also vega because barrier spot current closer exposure The vega the reduce. and higher premium to the shifts causes also out vega knocking peak of the chance increasing higher, the moves and barrier the As levels. barrier different 24.5 EXHIBIT ■ ■ is: this oaiiylwrrdcsteepce pinpayout. option expected the out. reduces pay lower to Volatility likely less spot and knock to to closer likely relatively more barrier is knock-out the brings higher Volatility ihbrir a rmso 05 n .0 h nc-u are og profile volga barrier knock-out the 1.00) and (0.50 spot from far barriers With volga The the spot, to close are barriers American when 23, Chapter in mentioned As these with options barrier knock-out for profiles vega the shows 24.5 Exhibit hc ed omr eaieT dutet.Itiiey h esnfor reason the Intuitively, adjustments. TV negative more to leads which , ( 𝜕 eapolso nc-u are pinwt ifrn are levels barrier different with option barrier knock-out of profiles Vega 𝜕𝜎 vega ) rfieo nc-u are pincagsdaaial as dramatically changes option barrier knock-out a of profile eua mrcnkokotbrir rdc negative produce barriers knock-out American regular ( 𝜕 𝜕 vega spot ) xoue at exposures → option AMERICAN BARRIER OPTIONS 443 vanilla = triggered prior to has vanilla option = knock-in barrier TV adjustment + vanilla TV = knock-in barrier + knock-in TV + Volga profiles of knock-out barrier option with different barrier levels triggered prior to expiry, whereas knock-in (KI) barrier options have not Looking at the vega profile, it is clear that the knock-in barrier option will be Exhibit 24.7 shows the vega of a knock-in barrier option with a 1.2000 barrier. Providing all contract details (expiry, strike, barrier, cut, and notional) are the Knock-out TV Knock-out barrier TV adjustment zeta Knock-out barrier Note the vega symmetry around thevega barrier beyond with the peak knock-in vega barrier at must the be barrier equal level. to The the vanillalonger vega. volga around 1.3000 spot since vegain increases Exhibit to 24.8. the Again, downside. note This the is shown symmetry around the barrier level. ■ ■ expiry. same: ■ It follows that: Regular Knock-in Barrier Options Knock-out barrier options havebarrier a has vanilla payoff ata expiry vanilla providing payoff the at American expiry, providing the American barrier EXHIBIT 24.6 It is therefore preferablea that negative volga implied exposure. volatility does not move, hence producing 444 AMERICAN BARRIER OPTIONS XII 24.8 EXHIBIT strike 24.7 EXHIBIT strike eapol favnlaoto essakoki are pinwt h same the with option barrier knock-in a versus option vanilla a of profile Vega og rfieo ail pinvre nc-nbriroto ihtesame the with option barrier knock-in a verses option vanilla a of profile Volga AMERICAN BARRIER OPTIONS 445 , one-touch risk is effectively bought stopping time (i.e., the cost of the strike on × because P&L changes negatively as spot goes through the barrier level. Exhibit 24.10 shows the TV profiles of a reverse knock-out barrier option versus Reverse barrier options have additional trading risks that arise from the in-the- Finally, barrier delta gaps must be considered when pricing knock-out and In terms of bid–offer spread, knock-outs and knock-ins are generally spread As a quick sense-check for knock-out TV adjustment, it will usually be lower Local volatility pricing models will generate a price above the market price nothing if spot trades throughoption. the When barrier the level: reverse the knock-outsold same option P&L is dynamic as a touch the equivalent vanilla option. payoff at expiry plus a single Americanthe barrier that payoff, is as positioned shown in-the-money in versus Exhibit 24.9. money barrier. At expiry,intrinsic a reverse value knock-out (see option Chapter goes 22) from with being spot worth just the before the barrier to being worth is covered in more detail in Chapter 25. Reverse knock-out (RKO) and reverse knock-in (RKI) barrier options have a vanilla knock-in options. Mostorders often, since they market represent participantsof a larger regular do P&L American not risk barrierbecome than want relatively options take-profit more stop-loss orders. reflects attractive The to spot thatoptions pricing buy become preference: and relatively less less Knock-out attractive attractive to options to buy and sell, more while attractive knock-in to sell. This idea (due to the volga) thanthe vanilla smile multiplied zeta by the length of time the option willslightly wider be than alive). the equivalent vanilla because barrierhigher options have trading only risks marginally than the equivalent vanilla options. for knock-out options andlocal volatility below undervaluing the (short) marketmodel convexity. will price generally A give for good well-calibrated pricing knock-incurrency mixed-volatility for pairs, options knock-out but and as due knock-in always, options the to on in interest longer-dated liquid rate trades. effect must be additionally quantified exposures, which can be hedgedout with vanilla option options. triggers Over time, ortriggering either the decreases. the For risk the knock- knock-in becomes option, the more optiona either vanilla vanilla triggers and option as becomes or the theAmerican chance risk barrier dies of options away the therefore as become barrier the easier chance to of risk triggering manage over decreases. time. Regular Regular Barrier Option Pricing The main trading risks on knock-out and knock-in options are vega and gamma Reverse American Barrier Options ■ 446 AMERICAN BARRIER OPTIONS XII 24.10 EXHIBIT 24.9 EXHIBIT rnatdlv ie,n et eg) n hnlf ni xiyt oeul generate hopefully to cheaply, expiry until bought left be payoff. then a and can hedge), institutional They delta For no product: far. (i.e., very attractive live not transacted an but are direction RKOs certain a short-dated express in clients, to move way will effective cheaper spot an that therefore significantly view are often they the and are options options vanilla equivalent knock-out the reverse than reason, this For barrier. h niie piefo h ail pini utie ytein-the-money the by curtailed is option vanilla the from upside unlimited The C1ptrvrekokotstructure knock-out reverse put CCY1 ail aladrvrekokotT profiles TV knock-out reverse and call Vanilla AMERICAN BARRIER OPTIONS 447 and K with strike N USD3.06m of touch risk = B K 95.00)/98.00 − with strike with barrier can be approximately decomposed into these two N (98.00 IV B × ). For example, USD100m of USD/JPY 1mth 95.00 reverse of the option. IV Long 1yr 1.3000 strike/1.5000 barrier CCY1 call reverse knock-out vega leverage Specifically, a long reverse knock-out option in notional The trading risk on a long reverse knock-out option can be decomposed into two Within a delta hedged options portfolio, reverse knock-out options with large Traders and clients often compare the value of the reverse knock-out option EXHIBIT 24.11 profile elements: 1. Long vanilla call in2. notional Short one-touch in payout parts: the long strike and thealways short traded one-touch in (recall the that market single viaknock-out barrier a shown touch one-touch in risk contract). Exhibit is The 24.11 vega confirms profile this. of a reverse American knock-out barrier notionals can be challengingAs to with risk European manage barriers duethe (see intrinsic to Chapter value the 22), ( touch theknock-out risk size 98.00 at has of 100m the the barrier. embedded risk within at it. the barrier is equivalent European knock-out option to ascertain theof value of the the ‘‘American-ness’’ barrier. Anotheroption popular (i.e., analysis spot just compares inside the thecalled knock-out the maximum barrier) payoff with the from cost. the This is sometimes with the equivalentbarrier vanilla provides. Plus option they compare to the value get of a a reverse knock-out measure option with of the how much discount the 448 AMERICAN BARRIER OPTIONS ie uc udneo hte h ees nc-u iltaeoe rudrTV under or over trade direction. will opposite the knock-out the at reverse in one-touch the TV under whether a or on of over guidance trades TV quick option equivalent gives one-touch the the whether and knowing barrier Then, level. the barrier at value intrinsic the know tool. pricing a within options these shows 24.12 Exhibit + (0.02%). strike the than rather the (-0.19%) of barrier the majority from the comes that suggest results These ■ ■ ■ ■ ■ CCY1% in point ■ basis nearest the to Working EUR%. terms: 0.73 TV RKO and 8.15% ■ ■ calculating: by obtained h K tieadRObrirlvlt h K xiydt sotna effective an often is date expiry vega. at RKO knock-out strikes the reverse with to hedge spread level to vanilla barrier way a RKO Trading vega and reversal. a strike risk creates RKO a spread the to risk similar one-touch one-touch quite short and is the versus risk that strike At profile strike long 24.13. into This barrier. splitting Exhibit the noticeably in at started risk behavior has similar risk the shows expiry, exposure 1mth vega RKO An reduce. .1 scoet h ie oaiiymdlT dutetof adjustment TV model volatility mixed the to close is 0.21% nrni Value Intrinsic Adjustment TV One-touch Value Intrinsic Zeta Strike Time Stopping ATM) the to close is strike vanilla the 1.3500 6mth EUR/USD from Zeta Strike a long because subtracted is risk. quantity one-touch This short gives level. barrier barrier knock-out reverse the at risk one-touch for. exist ‘‘ to The likely is option the long how by weighted ‘‘ The ealta n-oc eamvstwr h are vrtm u osnot does but time over barrier the toward moves vega one-touch that Recall to important is it knock-out reverse a pricing when that means this practice, In 0.02% of approximation adjustment TV knock-out reverse The Example be therefore can adjustment TV knock-out reverse the for sense-check quick A K TV RKO nrni Value Intrinsic tieZeta Strike U/S mh130 K .50wt pt133,AMvolatility ATM 1.3430, spot with 1.4500 RKO 1.3500 6mth EUR/USD : × tpigTime Stopping Adjustment × = = n-oc VAdjustment TV One-touch (1.45 93% × × tpigTime Stopping n-oc VAdjustment TV One-touch − 1.35)/1.45 − = = = nrni Value Intrinsic tieZeta Strike –2.85% 0.02% ’gvstecs ftesrk ntesmile, the on strike the of cost the gives ’’ = 6.9% × tpigTime Stopping × ml risk smile = ’gvsteT duteto the on adjustment TV the gives ’’ n-ocT Adjustment One-touchTV –0.19% =+ .2 asalvlesince value small (a 0.02% nti ees knock-out reverse this on − ( − 0.19%) + 0.19%. = AMERICAN BARRIER OPTIONS 449 Long 1.3000 strike/1.5000 barrier CCY1 call reverse knock-out vega profile Reverse knock-out ‘‘replication’’ in pricing tool EXHIBIT 24.13 over time EXHIBIT 24.12 450 AMERICAN BARRIER OPTIONS sal iervrebriroto rcsta ac h aktwl,especially well, market the match that prices option barrier reverse give hedged usually be can this and risk major barrier. option. the the one-touch becomes standard from barrier a comes the like risk volatility at vega the risk main in touch skew the time, the option, Over The by knock-in impacted spread. reverse heavily vega a be barrier will For smile versus a surface. the strike For on a risk vanillas. is this with options of risk hedged pricing main barrier approximately the reverse be option, on can knock-out risks which reverse Greeks, major vega the initially perspective, are management risk a From Pricing Option Barrier the Reverse so vanilla a point. that becomes at knock-in intersect reverse profiles vega the vanilla barrier, and the knock-in At reverse barrier. the of front vega the in shown is This 24.14. apart. Exhibit enough in far are profiles barrier and strike the providing level, same: ■ the contract are all notional) Providing and options. cut, barrier, barrier strike, reverse (expiry, for details exists identity pricing familiar A Options Barrier Knock-in Reverse XII 24.14 EXHIBIT ees knock-out Reverse nfel otn urnypis elclbae ie oaiiymdlwill model volatility mixed well-calibrated a pairs, currency floating freely In in similar very are profiles vega one-touch equivalent and knock-in reverse The barrier the at one-touch long a to risk trading similar has knock-in reverse long A ogrvrekoki n ogoetuhvg profiles vega one-touch long and knock-in reverse Long + ees knock-in reverse = vanilla AMERICAN BARRIER OPTIONS 451 equivalent one-touch × intrinsic value = Reverse knock-out and equivalent one-touch option within a pricing tool However, a reverse knock-out generally has less vega than a one-touch (due to the Reverse barrier bid–offer spread In terms of bid–offer spread, reverse knock-out and reverse knock-in options EXHIBIT 24.15 bid–offer spread. strike versus barrier risk offset),slightly so tighter. the reverse knock-out may be quoted relatively usually derive the majoritythat of European their barrier options spread derive the fromdigital majority their risk. of Exhibit the touch 24.15 spread shows risk from a their in reversewithin European knock-in a the and pricing equivalent same tool. one-touch way option In approximate terms: since reverse barrier options are generallyrate quite risk short-dated and hence have minimal low interest premium hence low vega exposures. 452 AMERICAN BARRIER OPTIONS ■ obeAeia are Options Barrier American Double sdfrdul nc-u pin srvrekokotoptions. knock-out reverse are as options methodologies knock-out spreading double and for used pricing similar Therefore, barrier. in-the-money shown as payoff the versus in-the-money out-of-the-money 24.16. positioned Exhibit positioned in barrier one one and barriers: payoff the American vanilla versus two a plus have options expiry barrier at (DKI) payoff knock-in double and (DKO) knock-out Double the Therefore, lower. suggests. correspondingly model priced pricing be a should than knock-in higher vanilla reverse trade since often would However, identical and barrier. virtually TV low no a with have option options knock-out vanilla These reverse the strike. to the value to theoretical an close showed barriers ‘‘I with than options useful more far AUD/USD is an wide.’’ one-touch’’ 0.20% showed the RKO ‘‘I 6mth on Saying AUD/USD wide bid-offer. 1.5% spreads one-touch RKO bid-offer embedded barrier 6mth the reverse of describe to terms tend in traders exotics so market broker XII 24.16 EXHIBIT h rmr iko obeAeia are pin sal oe rmthe from comes usually options barrier American double on risk primary The knock-in reverse in prices request clients when careful be to need traders Finally, interbank the in observed directly and monitored are spreads bid–offer One-touch C1cl obekokotstructure knock-out double call CCY1 + ees nc-n h qiaetrvrekokoti very is knock-out reverse equivalent the knock-in, reverse = AMERICAN BARRIER OPTIONS 453 ) K > B1 ( B1 , the knock-out B2 first: The knock-out and the B2 first: The short double knock-out B2 B1 and B1 ) can be replicated with the following standard K B2 < B2 ( B2 Therefore this variation is called ‘‘knock-out until expiry.’’ Exhibit 24.17 shows Thinking through the possible scenarios: A long knock-out until expiry KIKO option with knock-in barrier Finally, the presence of two barriers can sometimes cause double knock-out Traders often simplify the product in order to identify the most important risks double knock-out both expire and there is noSpot payoff touches the at KIKO maturity. knock-in barrier level expires and only the long knock-out risk remains;in.’’ the If structure has spot been ‘‘knocked thenoption touches will the expire; otherwise KIKO the knock-out option barrier has a level vanilla payoff at maturity. Spot gets to maturityknock-out and without double having knock-out payoffs touched withinno either the payout. replication KIKO offset and barrier there level: is The Spot touches the KIKO knock-out barrier level Long knock-out with barrier Short double knock-out with barriers the legs of the replication. Note that this replication works no matter the payoff. ■ ■ ■ and knock-out barrier American barrier options with the same strike, payoff,■ and notional: ■ the-money. One of the barriers istwo knock-out variations while of the knock-in/knock-out other options: is knock-in. There1. are Knock-out until expiry 2. Knock-out until knock-in Knock-in/knock-out (KIKO) optionsAmerican have barriers: one a barrier positioned vanilla in-the-money and payoff one positioned at out-of- expiry plus two Therefore, pricing and risk management can beelement focused of around the the most structure. appropriate options to havelarger trading convexity risks exposures. initially similar to a double-no-touch, specifically, on a double-barrier option.implies By an removing unimportant each barrier and barrier a in large turn, TV change a implies small an important TV barrier. change Knock-in/Knock-out Option Replication ■ 454 AMERICAN BARRIER OPTIONS ■ tieotOptions Strike-out ont sta hr sn pinlt oigfo h tiedet h barrier. the to due strike the from coming optionality no the is in thing there position important The that spot trades. long is level a barrier note the to once to equivalent out closed is is trade that This amount notional barrier). the (above spots all and 0.8000)/0.9000 zero are rates interest USD and AUD that spot knock-out hence initially American Assume with 0.8000. put call/USD at AUD barrier 0.8000 1yr AUD/USD strike an that Consider such options barrier knock-out American are options Strike-out 24.17 EXHIBIT o73% motnl h pinvlei o qa oteitiscaymore: any drops intrinsic value the option to the equal and not 0.8571) is (to value left option the the to Importantly moves 7.31%. forward to The 0%. at tool. pricing a in option this shows 24.19 Exhibit U CY)rtsaenwicesdt %adUD(C2 ae r kept are rates (CCY2) USD and 5% to increased now are rates (CCY1) AUD (0.9000 is, that intrinsic, the simply is option this of TV the 0.9000, at spot With = owr ssoni xii 24.18. Exhibit in shown as forward nc-nkokotrpiaini rcn tool pricing in replication Knock-in/knock-out = 11% hr s10 et,zr ea n eogmaover gamma zero and vega, zero delta, 100% is There 11.11%. = barrier. − AMERICAN BARRIER OPTIONS 455 AUD/USD strike-out option with 0% AUD rates and 0% USD rates AUD/USD strike-out option structure with 0% AUD rates and 0% USD rates EXHIBIT 24.19 EXHIBIT 24.18 456 AMERICAN BARRIER OPTIONS ec h eg a euwud hsepan h h ihrAD(C1 rates (CCY1) and time. AUD higher possible over the as why money quickly explains cost This as unwound. will level CCY2 be barrier can hedge than hedge the larger the the through are hence on trades rates position CCY1 spot cash If ideally barrier. Therefore, short the the through holding or at rates, trades spot if back the is options strike-out within element key The from? coming higher 24.25. with Exhibit strike-out in the with shown of new priced is profile this shown rates vega interest The is using 24.24. to CCY2 option payoff Exhibit strike-out increases in option The data TV The market 24.23. new option short. Exhibit this in the now shown is 0.9465), is vega (to data and right market 110%, the is to delta moves 15.07%, forward The 5%. to shown is rates interest CCY1 higher mechanism with 24.22. another strike-out Exhibit be the in must of there option profile so the vega delta, 24.21, The 80.52% Exhibit work. has in only at it tool and pricing vega, the long in now shown is As 24.20. Exhibit in shown is (0.8571 24.20 EXHIBIT aei ogvg;hge oaiiywl nraetepoaiiyo ncigout knocking of probability the increase will volatility higher earlier. vega; long is case nutvl,tehdefrti tieototo st elso o n u it buy and now spot sell to is option strike-out this for hedge exposure the vega Intuitively, this is where strike, the from coming optionality no is there Given increased are rates (CCY2) USD and 0% to back put are rates (CCY1) AUD Now, − 0.8000)/0.8571 U/S tieototo aofwt %ADrtsad0 S rates USD 0% and rates AUD 5% with payoff option strike-out AUD/USD = .6.Teoto aofuigti e aktdata market new this using payoff option The 6.66%. neetrt carry rate interest . AMERICAN BARRIER OPTIONS 457 0%) = 5%/USD rates = AUD/USD 0.8000 strike-out vega profile (AUD rates AUD/USD strike-out option with 5% AUD rates and 0% USD rates EXHIBIT 24.22 EXHIBIT 24.21 458 AMERICAN BARRIER OPTIONS XII 24.23 EXHIBIT XII 24.24 EXHIBIT U/S tieototo aofwt %ADrtsad5 S rates USD 5% and rates AUD 0% with payoff option strike-out AUD/USD U/S tieototo ih0 U ae n %UDrates USD 5% and rates AUD 0% with option strike-out AUD/USD AMERICAN BARRIER OPTIONS 459 5%) = 0%/USD = 0%/USD rates = . The single interest rate to expiry may be AUD/USD 0.8000 strike-out vega over time (AUD rates AUD/USD 0.8000 strike-out vega (AUD rates 5%) = When pricing and hedging strike-out options, it is important that: Alternatively, if CCY1 rates are lower than CCY2 rates, holding a short cash The full interestdrastically rate curves different are frompositive used the or negative short-dated return from interest holding the rates, spot hedge. which generate the EXHIBIT 24.26 rates increase the probability of knocking out earlier. ■ EXHIBIT 24.25 position on the hedge will earntrades money through over the time. barrier It level is as thereforewhy late preferable the as that possible, spot higher or USD ideally not (CCY2) at rates all. This case explains is short vega; again, higher volatility will 460 AMERICAN BARRIER OPTIONS ■ ■ ssoni xii 42.Teeoe n ail egspto oofe the offset to on put time. over hedges rebalanced vanilla be to any need Therefore, will vega 24.26. strike-out Exhibit interest differential in (since rate shown time interest as with linearly is away TV carry the decays on rate vega impact the significant account a time, into have Over will adjustment. taken hedge on are earned carry on correlations versus impact rate probability significant interest a versus have Spot could this short, is Greeks. and time pricing stopping the used and is sloping curve ATM full The fteAMcrei tel pado downward or upward steeply is curve ATM the If . × ie n oe oadtebarrier the toward moves and time) hne nteknock the in Changes . CHAPTER 25

Exotic FX Derivatives Trading Topics

461 he following topics cover common situations that exotic FX derivative traders come across. Issues around risk management and pricing are investigated, along Twith how exotic derivatives are used within FX hedging and investment strategies.

■ Exotic Risk Management Overview

Exotic FX derivative contracts are primarily risk managed using the same Greek exposures (delta, gamma, vega, etc.) as vanilla FX derivatives contracts. Therefore, exotic and vanilla FX derivatives in the same currency pair are often risk managed within the same trading position. When vanilla and exotic contracts are risk managed together it is important that their valuation and Greeks are aligned as closely as possible. Where possible, exactly the same volatility surface (including, e.g., ATM event weights) should be used for all options. If it isn’t, risk management becomes more challenging, particularly when exotic risk is hedged with vanilla options close to maturity. The main additional complication when risk managing exotic FX derivatives comes from barriers (both European and American) close to maturity. For exotic FX 462 EXOTIC FX DERIVATIVES TRADING TOPICS ■ xtcBdOfrSpreading Bid–Offer Exotic ecagdt oe hsrs,prasuigepce lpaemlile ythe by multiplied slippage expected using perhaps risk, therefore must this spread cover Additional can to volatility. gaps charged P&L delta barrier be increases 23, which Chapter slippage, in to discussed As lead spread. bid–offer the spot). in from included far very barriers on taken be managed should risk spread be to bid-offer needing less barrier one- (i.e., the of equivalent spread probability bid–offer the the and digital value), barriers), European European or intrinsic (for barriers) (using American (for size spread barrier bid–offer the touch of terms 18). in vanna and considered and 12 Chapters volga (see or methodology exposures VVV-esque a sega via and volatility used option rega the be but digital to using could exposures exposures quantified spot European from be current a spread either Bid-offer at example, could forward). smile vega (for the at zero wings level have the digital with in may exposures contracts vega exotic significant that note but exotic the spreads, on vega from spread ATM bid–offer spread CCY1%. 1yr 0.08 the the bid–offer be 0.4%, and would vega exotic is CCY1% spread 0.20 the has bid–offer obtain option volatility barrier to knock-out 1yr expiry a if deal example, the to volatility spread important most The primary vega. barriers. two often from is and exposure comes exposures surface contracts surface derivative FX volatility exotic factors: on spread bid–offer The risk is barrier the spot hedge and hedges. and vanilla large try large to is increasingly preferable risk transacting be than barrier may rather it the itself level if can barrier the expiry, hedges to to vanilla close up smoothly, fairly fairly coming evolve However, Greeks well. and work vega-based mainly is exotic options. exotic with Hedge options. 2. vanilla with Hedge 1. There risk. barrier hedge approaches: to possible how two essentially around are come decisions key the traders derivatives hti ie hncnb hw npatc.Frti esn eti lmnsare calculation. elements the of certain end reason, the this at applied For spread is practice. factor bid–offer in scaling a a shown in or be omitted results can often usually than trade wider exotic is an that of elements different these all size. gap delta barrier aigsi l hs rdr ucl er htsmigtebdofrsra from spread bid–offer the summing that learn quickly traders this, all said Having be also should that barrier, American an at gap delta a contains option exotic the If individually be must options exotic within risk barrier European and American bid–offer vanilla and exotic between link important an keeps approach This volatility ATM the by multiplied be can options exotic on exposure vega The the on risk the when expiry, to Prior approaches. both use traders practice, In rmvega from For . EXOTIC FX DERIVATIVES TRADING TOPICS 463 vega , denoted , denoted ‘‘rh.’’ (i.e., barrier) risk exotic delta exchange rho hedge forward hedge) such that the initial rho or a spot hedge either . Within the AUD/USD broker details in Chapter 18, the letters ‘‘vh’’ Some exotic contracts are priced and traded with Some exotic contracts are priced and traded with a Exotic contracts in the interbank broker market are generally traded For a trader it is most important to understand the exotic bid–offer spreading FX derivatives contracts are often combinedtheir into FX strategies exposures. that clients use to hedge ‘‘dx.’’ This involves the two counterpartiesagreeing within that when an the American barrier level barrier hits transaction theyat will the exchange an barrier agreed notional level. of This spot to agreement transact reduces in the the amount marketexchange is of and particularly spot relevant therefore on both reduces exotic options parties slippage with need and significant barrier P&L delta gaps. volatility. Delta This involves trading different notionals ofthe spot trade and (rather forward contract than toexposure delta hedge on the transaction isthe zero. contract. Again, Rho this hedges is are done particularly to relevant isolate on the long-dated exotic exotic options. risk on (priced at the implied volatilitythat level used the to initial generate vega thein on exotic the contract the TV) ATM transaction such implied is volatility,position. zero. no Hedging This significant means the P&L that, vega willwithin for the in be small contract. generated this changes in way the helps trading isolate the systems and by voice. hedged indicate that the contractexotic option, is an appropriate traded notional of with ATM to a the vega option expiry hedge; is also that transacted is, when trading the derivatives because there is noother direct for market prices (i.e., on traders exotic doto contracts). not get In prices directly on the call almost interbank each any brokerare exotic knock-outs, market, contract reverse but it knock-outs, the one-touches, is most and possible commonly double-no-touches.swaps, Volatility traded variance contracts swaps, and correlation swapsmajority are of also quoted trading fairly in frequently. The the exotic interbank broker market takes place over chat the spreading methodology remains appropriate as market conditions change. The interbank broker market is the primary source of liquidity in exotic FX methodology used by the desktouch pricing or tool, European keep digital any spread input grids) parameters updated, (e.g., one- and work with quants to ensure Structured FX Hedging Strategies Exotic Interbank Broker Market ■ ■ 464 EXOTIC FX DERIVATIVES TRADING TOPICS sprilyofe.Ti U/B owr xr snet is extra forward EUR/GBP This offset. partially barrier. is (lower) better the a more to the down rate, movement hedge spot guaranteed favorable the from (higher) benefit worse can the client case this In struck. be to plus for transacted exposure FX 25.3. client’s Exhibit the in combined and are 25.2 strategy Exhibit extra in forward the shown is strategy extra forward 0.7500 if 0.8000 and to 0.7500 order between in rate a trades. market 0.7900) accepting never at versus be currency (0.8000 would the forward client buy the the potentially than example, worse this rate In hedge 0.7900. worst-case at forward 0.8000. at EUR/GBP spot rate. a buy prevailing again, the will, at client never market the has the knocked, spot in has barrier (and spot 0.7500 buy 0.8000 the can below If client is the maturity 0.7500), at below spot traded If 0.8000. knock-in at EUR/GBP American buy reverse a with 0.8000 at call put/GBP EUR 0.8000. sells at Client vanilla put call/GBP 2. EUR buys Client 1. notionals: equal obligation. with an contracts becomes the derivatives right in the time, currency barrier any a a at touched buy trades hasn’t level to spot barrier obligation, providing the If strike the level. pre-agreed not the but at maturity) right, (at the future has client a the is strategies contract simplest the of One 2006). book: Wystrup’s Uwe additional in some for return in forward the risk, than additional rate some worse upside. accept a potential must accept they to case willing which are in or rate, either forward clients the general, beat In requirements. to client want the match strategies closely to hedging customized options, be FX can exotic particularly options, using by However, flows. t080 eoe nolgto ie,i sls ieyta ptht h barrier). the buy hits to spot right that likely the less that is likely it less (i.e., it obligation makes an this market becomes because 0.8000 current at spot the the volatile Intuitively, that less occuring. a believe scenario wants particular to client any clients than many rather continue, for will easier situation is It strategies. hedging hence tutrdfrad fe aeoebuh e n n odlg otevg risk vega the so leg, sold one and leg bought one have often forwards Structured is package the the that such set of often are strategy payoff the within levels The barrier and strike 25.1. The Exhibit in shown is exposure FX short client’s The into entering than rather strategy extra forward this enter to choose can Clients to option call EUR their use will client the 0.8000, above is maturity at spot If separate two using constructed is GBP sell and EUR buy to extra forward A covered are which of many available, strategies hedging FX numerous are There FX future their hedge to contracts forward use simply could clients Corporate are t0.7500. at barrier ogvega long eopremium zero o h rdn ek hsi omn let fe e elvg within vega sell net often clients common; is This desk. trading the for XOtosadSrcue Products Structured and Options FX ihnhdigsrtge hr sivral balance a invariably is there strategies hedging Within . owr extra forward hr vega short ihnafradextra forward a Within . Jh ie Sons, & Wiley (John o h letand client the for EXOTIC FX DERIVATIVES TRADING TOPICS 465 , in which the client structured deposit Forward extra payoff Underlying client FX exposure FX derivatives contracts aresimplest also investment often instrument combined is into investment a products. The EXHIBIT 25.2 EXHIBIT 25.1 FX Derivatives Investment Products ■ 466 EXOTIC FX DERIVATIVES TRADING TOPICS ■ hdwBarriers Shadow hdwn aresi ikmngmn ehiu hti ocpulysmlrto similar conceptually is that technique management risk a is barriers Shadowing rate. spot initial the than higher is coupon 1yr maturity 10% a at approximately spot purchase (i.e., if to AUD100k deposit) used the out and on pays valued that present approximately option is earn digital amount should European this rate but interest maturity AUD at 5% AUD50k the deposit, the risk, Within at deposit. put is principal purchased. they of be interest can amount The notionals the increasing maturity. option forgone an large at As have increasingly more earned. they out have pay although otherwise may investment, would that their options lose purchase cannot to client it using and front up be to derivative FX embedded an by money generated the instead on contract. is based coupon coupon the guaranteed rate, a interest receiving market than rather and money deposits 25.3 EXHIBIT S1 iha100 are.Ti are a be can barrier This barrier. 1.0500 windfalls. a potential with USD1m large creates but money as of known amount also small is a position technique This trading costs the that within way levels a barrier in moving involves It risk. vanilla writing-off Example FX structured a within year one for AUD1m deposits client a example, For deposit the for common is it currency, deposit the in high are rates interest When rnia protected principal rdn oki ogadwsd S/A n-oc pinin option one-touch USD/CAD downside a long is book trading A : owr xr n letF xouentposition net exposure FX client and extra Forward hsivle aigteitrs ando deposit on earned interest the taking involves This . smoothing or bending shadowed barriers. ie,mvd o1.0450, to moved) (i.e., EXOTIC FX DERIVATIVES TRADING TOPICS 467 A similar barrier shifting approach can also be used to generate bid–offer spreads Like writing-off vanilla risk, shadow barriers work best when traders are engaged The 1.0450 one-touch barrier can now be risk managed as usual, but a windfall To be clear, the barrier on the client transaction is not changed, but for risk barriers on both sides. When spot moves significantly higher, distant downsideat barriers minimal can cost. be shadowed When spot moves significantly lower,minimal distant cost. topside strikes can be shadowed at When implied volatility drops, there may be additional opportunities to shadow broker market. For example,6mth ATM. a The current trader midmarket level maystarting is 10.2%, rate) want the two-way ‘‘support’’ (i.e., volatility to the in generic the sellif interbank the broker USD/MXN trader market sells USD100m is within 10.0/10.4%, the and broker market, the expected transaction level is 10.1%. Traders on bank trading desks have the abilityto to complete access transactions the at bank’s better client levels base than in order would be possible within the interbank for barrier products by lookingand at the down value a change produced certaincontaining by amount. many flexing different This barriers barriers. up approach works particularly well in structures ■ ■ ■ barrier. Traders sometimes use parttransactions of to the establish initial bid–offer shadowshadow spread barriers barriers further cross at away on gradually the exotic over deal time. inception, and/orwith they their may position: sell the 1.0500 barrier andshadow buy barrier the trading 1.0450 book barrier should at beshadow zero left is cost. untouched unwound. The until risk either maturity within or the if the equal to the barrierspot notional touches the is real generated long in 1.0500 the barrier shadow but barrier does not trading touch book the if 1.0450 shadow management purposes the barriernot level available within has the beenmanually risk changed. by management booking If system, a the this barrier samebarrier spread functionality trading between effect book. is the can Within main the be trading USD/CAD achieved example, book the and main a trading shadow book would which will cause a lossHowever, in if the the trading one-touch has bookthe a as barrier low the a value long value way. initially, of it the may option not has cost reduced. much to move Recycling Exotic Risk ■ 468 EXOTIC FX DERIVATIVES TRADING TOPICS ■ mrcnBarriers Sell American to Prefer Participants Market Why oiin a eotie y .. eln n-oc,byn double-no-touch, a buying one-touch, a barrier selling short e.g., Likewise, by, level. obtained barrier be the can through positions trading spot from a results selling change one-touch, a buying knock-out. e.g., reverse a ways, positions selling different barrier knock-out, of a Long number selling level. double-no-touch, a barrier in the obtained through be trading can spot from results change Buying barrier European on options. risk digital digital European using the recycled way, reverse exotic be same selling can the the by options if In hedged Alternatively, clients. be to rate. could options improved they risk knock-out an options, one-touch at one-touch This clients short 24. contains to Chapter position out in offered described be as then level can barrier the at risk been has one-touch option knock-out originally reverse a maturity if and strike exact the than rather clients. by used traded be often must contracts called sometimes potentially and client, the with relationship sales too. their and their commission, midmarket, improved earned about at deal, deal anything possible the a revealed be transacted executed not has probably have client have the would they market, the plus than to market, levels positioning win-win-win broker a better interbank creates at it the works transacts in it when trader but showing The applicable, Therefore, always situation. contract). not of is types clients same to the axes sell or buy to want simultaneously transaction direction. the specific completing in a interested in especially are they because showing is trader contract ATM the purchasing clients in to interested level. shown be midmarket be may a first who at could desks) sales ATM the 6mth via USD100m (usually for offer 10.2% a However, elAeia aresfranme freasons: to prefer of generally number participants a Market for place. barriers in American are sell which hedges the to due money knock-out. reverse a buying knock-out, a buying motnl hsi o h aea h hl rdn oiinmkn rlosing or making position trading whole the as same the not is this Importantly Selling example, For ways. complex more in recycled be can risk options, exotic Within is flows client the offset to prices out axing of process the options, vanilla For participants market all (i.e., way’’ ‘‘one get often can market derivatives FX The an called is offer 8.2% The or or long short h mrcnbrirmasta ntebrirdas oiieP&L positive deals, barrier the on that means barrier American the h mrcnbrirmasta ntebrirdas eaieP&L negative deals, barrier the on that means barrier American the recycling ik hsms edn ihcr.Freape generic example, For care. with done be must This risk. axe: pca,bte-hnuulbdo fe htthe that offer or bid better-than-usual special, a bought yacin,tetae slf with left is trader the client, a by long EXOTIC FX DERIVATIVES TRADING TOPICS 469 and it can slippage : Long an AUD/USD 1.1000 topside one-touch generates a stop-loss Example The same preference is observed in the vanilla market. In liquid currency pairs This effect is particularly important in pegged or managed currency pairs, where Consider what happensmarket in open emerging hours. When market theand currency spot it market is pairs is possible closed, to with accurately the imply restricted NDF a market spot spot rate is from still the active level of the NDF. The NDF order to buy AUD150m versus USD.higher Spot trades before through all the barrier thetransacted and spot at then even an can average be of 1.1012—a bought painful in loss of the USD180k. market.Limited In Spot the Market Open end, Hours the order is orders than take-profit orders.through With the a barrier stop-loss and order, continuesThe before the difference the risk between stop-loss is the order order that can levelcause spot be and large fully trades the negative executed. execution P&L. level is Therefore,than long short American American barriers. barriers are more attractive Delta Gap Orders As described in Chapter 23,barriers within produce delta a hedged stop-loss trading spot portfolios, orderprofit and long spot short American American order. barriers There produce is a take- usually a larger chance of losing money from stop-loss slightly below the lowerprice band. will The be far TV lower. of this one-touch is 92%,there but is the common market marketvanilla options preference away from to current sell spot). short-dated wings (i.e., short-dated time. there is often anexotic extreme contracts jump with dynamic American andtouch barriers. traders option. Consider must USD/HKD a be spot USD/HKD very isthe 1yr careful currently HKMA 7.7490 kept pricing one- within (Hong a Kong 7.7500/7.8500 Monetary range Authority) by so the down barrier is positioned low volatility but itoccur. This jumps creates a as fat-tailed distribution economic that isbut data reflected importantly within is the it volatility is released smile, specifically or thesell spot American other jump barriers. external dynamic Selling that high events leads TVan (i.e., to effective expensive/close) a strategy one-touch preference when barriers to spot is followslow a and jump the dynamic. barrier If does realized not spot trigger, volatility one-touch is options can lose value rapidly over Spot Dynamic In some currency pairs, spot often follows a dynamic whereby it usually moves with 470 EXOTIC FX DERIVATIVES TRADING TOPICS hrfr tutrswt mrcnbrir,priual tlne eos ilbe will tenors, longer stochastic at a particularly have barriers, mispriced. and not American probability does knock with barrier that structures the model capture therefore correctly pricing not any will component spot, rate than interest volatile less maturity), or changing more a (with forward the of but volatility the represents curve ATM The Volatility Forward versus Volatility Spot if damned preference market and strong hedge particularly a with barriers. the is American pairs transact sell there currency to hours in they open Therefore, if market manage. spot risk damned restricted to be situation difficult to don’t—a traders they for possible will is trade NDF the unwinding and triggered below the loss. retraces have and P&L then not a barrier market cause will the the barrier the but through the NDF, when level barrier, an loss spot trading the P&L a by a executed implies cause is NDF order hence the stop-loss and If level have reopens. barrier might market order the spot spot through stop-loss far the executed higher, be continues spot to if executed, not is the order reopens. but market trigger spot would the barrier once American knocked be the officially which only at can rate barrier spot a imply may market XII 25.4 EXHIBIT ihso-osodr ncrec ar hc r o osatytaal it tradable constantly not are which pairs currency in orders stop-loss With stop-loss the but level barrier topside a through rate spot a implies NDF the If mrcnbrir nc u nspot on out knock barriers American S/P T oaiiyrun volatility ATM USD/JPY hrfr,i h owr ssignificantly is forward the if Therefore, . EXOTIC FX DERIVATIVES TRADING TOPICS 471 Generally, forward volatility is higher than spot volatility and therefore the Often the ATM curve rises at longer tenors but that does not imply that the long-dated American barrier options. An example long-datedis USD/JPY shown ATM in curve Exhibit 25.4. market price for one-touch options will bepricing lower model than without prices stochastic generated interest using rates. a smile market expects spot to getsignificant more interest volatile rate over component time; that rather it must means be that taken there into is account a when trading

CHAPTER 26

Window Barrier and Discrete Barrier Options

473 indow barrier options are extensions of American barrier options. The Wdifference being that window barriers are active only for a subsection of the life of the option. The two main window barrier variations are front-window barriers (barriers active from the horizon until a specified date prior to expiry) and rear-window barriers (barriers active from a specified date after the horizon until expiry).

■ Front-Window Barrier Options

Front-window barrier options, also called early ending barriers, have a vanilla payoff at expiry plus single (one) or double (two) American barriers that are active from the horizon but cease to be active on the barrier end date. The barrier end date occurs prior to the expiry date and the window barriers are either all knock-out or all knock-in. Exhibit 26.1 shows a typical front-window double knock-out barrier contract. Within a front-window knock-out barrier option:

■ If spot trades at or beyond (touches) a barrier prior to the barrier end date, the whole contract expires.

■ If spot does not touch a knock-out barrier prior to the barrier end date, at expiry there is a vanilla payoff. 474 WINDOW BARRIER AND DISCRETE BARRIER OPTIONS sa,teitiscvlegvstentoa ftuhoto htms etransacted be must that the option of touch value of notional the the the is gives and as it value date of end rather intrinsic thought barrier the be option; the usual, can to touch level set a barrier horizon the with for payoff at is the spot ‘‘payment’’ of it value context, as The this expiry. amount at In cash payoff trade. fixed do a levels not to barrier equivalent is the is if barriers the occurs ‘‘Payment’’ within risk risk: the knock-in, to are equivalent barriers is window barriers the within risk the no-touch knock-out, are barriers front-window If Risk Barrier expiry) Front-Window at payoff vanilla the from risk strike (e.g., risk Payoff barrier(s) the 2. from risk Touch 1. separately: (if considered hedged). higher delta moving traded before (if increases month volatility first that implied the is before for view or range the live) case, tight traded this a In hold view. will market attractive spot only specific are GBP/USD a is options have barrier barriers who Window the prices). clients touching the institutional of not to ratio spot (the of 9 Ignoring is chance in GBP%. 1 the 0.25 vanilla approximately just that the at suggests valued cheaper; this is effects, barrier significantly window smile the contract and 1.5865, the GBP% at 2.25 make spot at valued GBP/USD barriers With window 1.6000. and front 1.5500 the at month first the for barriers 26.1 EXHIBIT ttc u ti tbeeog o h ik ob fetvl fstuigasai touch static a using offset not effectively is be barriers to risks front-window the on hedge. for value option enough stable Intrinsic is risk. it barrier but the static, hedge to order in hr r w antaigrsso rn-idwbrirotosta a be can that options barrier front-window on risks trading main two are There Example ik ‘amn’ cusi h are eesdnttae lentvl,i the if Alternatively, trade. don’t levels barrier the if occurs ‘‘Payment’’ risk: B/S y .00GPcl/S u ihfotwno knock-out front-window with put call/USD GBP 1.6000 1yr GBP/USD : rn-idwdul nc-u are structure barrier knock-out double Front-window nrni value intrinsic ftebrir As barrier. the of one-touch WINDOW BARRIER AND DISCRETE BARRIER OPTIONS 475 ) AUD1.25m of touch risk in the = 2.54% × Estimating barrier risk on a front-window barrier option : AUD50m AUD/USD 1yr 0.9200 AUD call/USD put with a 3mth The trading risk from reverse American barriers is maximized at the end of Example Intrinsic value can be estimated within a pricing tool by moving the horizon this probability can be approximated byend the date. TV of a one-touch option to the barrier Payoff Risk The smile value of thevanilla option option. payoff In at addition, expiry the is probabilitybe given taken of by into the account. the option Ignoring zeta being discounting of activefront-window and the volatility at knock-out surface equivalent expiry considerations, barriers must for thisof probability a can no-touch be option approximated to by the the barrier TV end date. For front-window knock-in barriers the barrier lifeis where maximized. the The P&L samemanagement difference is challenges occur between true into touching within the barrier window the end date. barrier barrier structures; or the not main risk to the amount of touch risk contained in the barriers. front-window knock-out up barrierput at vanilla 0.9300. with The spot 9mth atthere 0.9200 0.9300 is AUD has call/USD approximately a midmarket (AUD50m window price barrier. of 2.54 AUD%. Therefore, barrier levels asconstant shown and in the expiry Exhibit can beexpiry 26.2. moved is such Alternatively, equal that to the the the number numberwindow horizon of of barrier days days can from contract. from horizon This barrier be to second endskew, approach kept date and ignores to forward forward expiry interest volatility, in rates the forward but original in most instances it will give a reasonable guide EXHIBIT 26.2 forward to the barrier end date and pricing the payoff at expiry with spot at the 476 WINDOW BARRIER AND DISCRETE BARRIER OPTIONS ifrn auiisi svtlta h ulAMtr tutr sue o pricing for management used risk is within structure bucketed correctly term are ATM to risks full vega exposures that the vega and that barriers to important vital window barriers have is knock-in barriers it from window maturities vega Because different long date. or end and barriers barrier strike) knock-out the the (from from date vega expiry short the to either vega long are options barrier Front-window ATM one only is Risks but Trading expiry final the to volatility TV? ATM calculate one to used only volatility shows 3 leg in of adjustment 0.205% ■ ■ ■ ■ 0.9300. at barrier up-and-out front-window formula. above the into substituted be can intrinsic intrinsic different average a the have so will barrier each barriers, front-window double-barrier For can adjustment: option TV barrier approximate front-window a an within into payoff combined and be barrier the from risk Adjustment smile TV The a in Risks Payoff and Barrier Combining unchanged. results drift This forward 0%. the to leaves rates interest but currency discounting payout zero the in set and forward the fix trades, trciet let eas digfotwno nc-u aressignificantly contract. barriers particularly the knock-out of are price front-window the options adding reduces barrier because clients front-window to volatility), attractive ATM long-dated than systems. ail zeta TV) Vanilla no-touch 0.9300 3mth (i.e., occurring payoff of Probability adjustment TV barrier) knock-out (since no-touch 3mth ( Value Intrinsic fteAMcrei onadsoig(hr-ae T mle oaiiyhigher volatility implied ATM (short-dated sloping downward is curve ATM the If adjustment TV barrier front-window Therefore, Example longer-dated on this like calculations from discounting of effect the remove To =+ U5mADUD1r090 U alUDptwt 3mth a with put call/USD AUD 0.9200 1yr AUD/USD AUD50m : .0AD.Ti oprswt h ie oaiiymdlTV model volatility mixed the with compares This AUD%. 0.10 + =+ .1AD hw nEhbt2..Nt o h idwbarrier window the how Note 26.3. Exhibit in shown AUD% 0.11 rbblt fpyf occurring payoff of Probability Value Intrinsic adjustment TV barrier Front-window IV .0 AUD% 0.205 ) = .4AD a e h bv calculation) above the per (as AUD% 2.54 ( IV )× oc Vadjustment TV Touch × ail zeta Vanilla = 2.54% = =+ + × .5AUD% 0.85 0.85% = 38% + 38% × WINDOW BARRIER AND DISCRETE BARRIER OPTIONS 477 Front-window barrier TV adjustment approximation in pricing tool : USD/CAD 1yr 1.1500 USD call/CAD put is 1.85 USD% premium It is also important to confirm at exactly what time the front window ceases to be The most sophisticated pricing model available should be used when pricing The initial vega exposure from this trade is shown in Exhibit 26.5. As expected, Example can be important to takemodels this do kind not of usually reasoning account into for account it. manually since pricing active on the barrier end date.not always Most the often case. the cut from the payoff is used but this is window barriers because theythe are complex forward products smile. with Forrange significant a for exposures front-window the first to barrier month, trade, atthan the if barrier was end spot envisaged date does at the ATM the stay curve option within is horizon. likely the to When be pricing lower window barrier options it (7.50% implied volatility). Adding 1mthat front-window 1.0900 and double 1.1300 knock-out reduces barriers the price to 0.50 USD% asvega shown looks in like Exhibit a 26.4. double-no-touchbarriers vega are profile no until longer the barrier activeexpiry end the date. date. vega Once will the simply be that of a vanilla option to the EXHIBIT 26.3 478 WINDOW BARRIER AND DISCRETE BARRIER OPTIONS ■ erWno are Options Barrier Rear-Window xml,7%we h pincnnvratal nc 0 ftewythrough way the of 70% knock for as, actually given never be life. can may its time option expiry stopping the one-year the when a only, month has 70% first option example, reality the barrier in for window barrier when a active expectation, if an the example, and value: For single distribution. a a of has terms it in defined is time occurring. stopping payoff the the of but probability attention the be by management weighted should risk be spreads should require two spread may The payoff risk. elements payoff both and because risk combined barrier the from spreads bid–offer 26.4 EXHIBIT oio aebtbfr h xiydt n h aresms ihrb l knock-out all be either must the barriers after the be and must date date expiry start active the barrier before are The but date. that date expiry barriers horizon the American to (two) date start double barrier or the (one) from single plus expiry called at also payoff options, barrier Rear-window ial,we sesn h tpigtm fawno are,rmme that remember barrier, window a of time stopping the assessing when Finally, the considering by calculated be can spreads bid–offer barrier Front-window rn-idwbriroto npiigtool pricing in option barrier Front-window late-starting ares aeavanilla a have barriers, WINDOW BARRIER AND DISCRETE BARRIER OPTIONS 479 the at Rear-window up-and-out barrier structure Front-window barrier vega exposure Exhibit 26.6 shows a typical rear-windowWithin up-and-out a barrier rear-window call knock-out option. barrier option: the trade does not knock out. If spot ever goes above theand knock-out expiry, barrier level the between trade thebarrier barrier expires. start start date, date Also, the trade if also spot expires. is through the barrier level Spot can go through knock-out barrier levels prior to the barrier start date and ■ rear-window single barriers mustalways always be be determined from specified the since inception their spot level. direction cannot ■ EXHIBIT 26.6 or all knock-in. The type and direction (e.g., down-and-out/down-and-in, etc.) of EXHIBIT 26.5 480 WINDOW BARRIER AND DISCRETE BARRIER OPTIONS ■ ■ characteristics: barrier gives options vanilla probability. and the barrier-knock date of the start delta of barrier The indication window date. rough the expiry very at barrier a with window expiry barrier) the one down at level; a expiry barrier for one two put the CCY1 price to and to set barrier strikes is up the probability an for barrier-knock call a (CCY1 estimating options vanilla of the way quick if One Also, knocked. expiry. knocked. because already to work have not will horizon does variation option from approach barrier this goes American TV spot, the current barrier date through European start is to barrier barrier adjustment rear-window the does TV adjustment barrier as TV American barrier adjustment from rear-window the move that simply appreciate not to important is it but ■ ■ adjustments: American TV the the and noting barrier time European each price versions, Then barrier value. smile its be ascertain to and barrier view spot sophisticated more the a to for compared allows GBP%) it barriers expressed. plus, (0.59 The prior and, option option. GBP%) month barrier (2.26 knock-out one vanilla double window range standard the 1.5000/1.7000 a cheapen the becomes significantly within option is the date, expiry expiry, spot the to before If month expires. the in option point any the at through trades 1.7000 spot (above) if or Therefore, 1.5000 1.7000. and (below) 1.5000 at month last the for knock-out oe hudb sd erwno aresuulyhv iia i–fe spread bid–offer similar a have pricing usually possible barriers Rear-window sophisticated used. most be the should front- model barriers with As rear-window option. for barrier barriers, American window standard equivalent the by generated those emr iia oteeuvln uoenbriroto n oa volatility local a and pricing. option for barrier sufficient European will be barrier equivalent may rear-window model the the on to risk similar the expiry, more the be near is barrier date rear-window start the barrier option. the barrier on If American risk equivalent the the horizon, to the similar more near be is will date start barrier the If o nc-nbrir ihso nfoto h are h otatwl be will contract the barrier the of front in spot be with will vega. long contract barrier, the knock-in barrier a the For of front in spot vega. with short barrier, knock-out a For fe h are tr ae h rdn ik n eapol ipybecome simply profile vega and risks trading the date, start barrier the After the on depends profile vega window rear the date, start barrier the to Prior being barriers rear-window the of probability the assess to useful be also may It options barrier rear-window on risks the about intuition good gives approach This no with payoff the price option, barrier rear-window a on risk smile the assess To Example B/S y .00GPcl/S u ihara-idwdouble- rear-window a with put call/USD GBP 1.6000 1yr GBP/USD : WINDOW BARRIER AND DISCRETE BARRIER OPTIONS 481 Generic window barrier structure To identify which barriers are most important, remove each barrier in turn from to expiry used (i.e., pure Black-Scholes), are two volatilities used (one to barrier Which ATM volatility curve is used to generate TV? Is only the ATM volatility EXHIBIT 26.7 Specifically, there areexposures to increased the forward smile. exposures Therefore, whenbarrier to pricing options and it risk the managing is window important ATMFor to example: assess curve, exactly which and pricing methodology additional is■ used. the probability of the barrier being live at its start date. Window barrier options have additional trading risks to American barrier options. with bespoke barriers could be constructed as per Exhibit 26.7. the structure and checkbarrier. the If there impact are on justwhole TV: one structure Big or can two TV be important changebarriers estimated barriers, in implies isolation. the by This an TV considering is adjustment important done the by on multiplying smile the the risk TV adjustment on of the the barrier important by In practice, a windowknock-in barriers barrier with different option start maycustomer and end thinks contain dates. that any For spot example, number will if a of trend sophisticated lower knock-out in or a specific way over time, a product to the equivalent Europeantrading or risk is American similar. barriers because the magnitude of their Window Barrier Risk Management Generic Window Barrier Options ■ ■ 482 WINDOW BARRIER AND DISCRETE BARRIER OPTIONS ■ iceeBrirOptions Barrier Discrete where with formulas: version these barrier using continuous adjusted the level price barrier option, the barrier discrete a price To 1997). ( Kou options and barrier Glasserman, discrete for Broadie, by developed was barriers ous be therefore TV barrier. will The knock-out option level. American barrier barrier equivalent knock-out same discrete the a for monitoring of knock. barrier will fix. continuous trade the with the calculate compared level, to barrier used methodology the exact through the is know fix to the vital once is If (i.e., it week). intervals Therefore, a regular at once usually or market, day the in a barriers fix spot Discrete a continuously. is against monitored barrier monitored being are the than that rather except fix options a barrier against monitored American like are options barrier Discrete that doubt. not, into If model the surface? by volatility generated adjustment the current TV until the the date of to end validity or the reference start brings with barrier credible the from look smile expiry forward the Does model. pricing ■ Where o r uktdGekepsrsdslyd stevg l uktdt the to bucketed within important all particularly is vega management. This risk the correctly? bucketed Is vega is displayed? or expiry, exposures option must Greek adjustment bucketed curve. TV volatility are the ATM How used, full the is of expiry effect volatility to the ATM volatility account of into structure ATM take term single full a the only is If or used? expiry), to one and date start/end neeatapoiainfrajsigdsrt aresit qiaetcontinu- equivalent into barriers discrete adjusting for approximation elegant An monitoring discrete pricing, of terms In smile the within like looks smile forward the what assess to important also is It Δ 𝜉 t steReanZt ucin( function Zeta Riemann the is stetm ewe oioigevents: monitoring between time the is are eo spot: below Barrier are bv spot: above Barrier .Bode .Gasra,S o,Mteaia Finance, Mathematical Kou, S. Glasserman, P. Broadie, M. , 𝛽 = Barrier Barrier 𝜉 2 ( 𝜋 2 1 Continuous Continuous ) … ≈ reduces o eneither me no, 0.5826 = = h rbblt fabrirknock barrier a of probability the Barrier Barrier Discrete Discrete higher … . . ). e e otniycorrection continuity A 𝛽𝜎 − hnteT fthe of TV the than 𝛽𝜎 √ √ Δ Δ t t WINDOW BARRIER AND DISCRETE BARRIER OPTIONS 483 buy spot, the spot rate may do one of two things: does buy spot, the spot rate may do one of two things: will not knock out at the fix, hence causing a loss. loss or gain. was correct not to buy spot, hence causing no loss or gain. have to sell the spot out at a lower level when it becomes clear that the barrier have to buy spot at a higher level, hence causing a loss. This is the same issue as the restricted spot market open hours case discussed in Alternatively, if the trader Suppose the trader has a stop-loss order to buy spot at a higher level if a topside Discrete barrier options also present additional risk management challenges. These formulas give good intuition about discrete barriers: A discrete barrier wider than the equivalenttrades. American barrier options, particularly on larger-sized 2. Continue higher, and hence the trader was correct to buy spot, hence causing no Chapter 25. This additional risk means that discrete barrier options are often quoted 1. Reverse back lower below the barrier level prior to the fix and the trader will discrete barrier knocks out. If spotnot goes through the barrier level but the trader does 1. Continue higher, and when the barrier officially knocks at2. the fix Reverse the back trader lower below will the barrier level prior to the fix, and hence the trader When spot goes through ancan American be barrier hedged level, the inposition resultant the unchanged. barrier spot delta However, market gap with tobarrier level a leave in-between discrete the fixes. Should barrier, delta the spot exposure trader hedge may within the go the delta gap, trading through or the not? option is equivalent to aaway from continuous spot. barrier option with the barrier placed further

PRACTICAL H

Building a Monte Carlo Option Pricer in Excel

485 he Monte Carlo pricing method is a flexible and powerful technique. Within a basic Monte Carlo pricing framework a simulation is set up that produces Trandom realized option payoffs. The simulation is then run many times and the resultant payoffs are averaged to obtain option valuations.

■ Task A: Set Up the Simulation

For each currency pair within the simulation the following market data is required:

■ Spot (S): the current exchange rate in a given currency pair

■ Interest rates (rCCY1 and rCCY2): continuously compounded risk-free interest rates in CCY1 and CCY2 of the currency pair

■ Volatility (𝜎): the volatility of the spot log returns We start in Black-Scholes world so only a single volatility (no term structure or smile) and single interest rates (no term structure) are specified at this stage. The simulation contains multiple time steps so the time (measured in years) between steps must be defined. For daily time steps, weekends can be removed and 486 BUILDING A MONTE CARLO OPTION PRICER IN EXCEL iuaini e ppoel,teraie oaiiysol eapoiaeyequal approximately be should input: volatility volatility realized the the to properly, up set is sample. simulation new a generates and sheet the recalculates Excel in F9 Pressing where simulation: the within steps step time the hence year; the in days trading 252 are is there that assumed usually is it 252 sacek h elzdvltlt fteso ahcnb acltd fthe If calculated. be can path spot the of volatility realized simulation. the the check, of a sample one As represents sheet the on generated path spot The time between evolution spot the calculating for used is formula following The 1 sasnecek tte22dse,tm hudb xcl 1: exactly be should time step, 252nd the at check, sense a As . 𝜀 sgnrtdi xe using Excel in generated is S t + 1 = S t e ( rCCY = NORMSINV(RAND()). 2 − rCCY 1 − 2 1 𝜎 2 ) Δ t + 𝜎 √ Δ t 𝜀 BUILDING A MONTE CARLO OPTION PRICER IN EXCEL 487 , K − T S . In addition, ) T . 2 rCCY − e ( , 0) for a CCY1 put. T S − K In addition, the payoff is at maturity and it therefore needs to be present valued Try flexing market data inputs to see how the spot paths are impacted; test low It is also useful to plot spot against time to judge whether the generated path the CCY2 pips option valuedividing by at the the inception horizon spot. can be converted into CCY1 terms by into the payoff calculation area.0) Recall for a that CCY1 vanilla call option and payoffs max( are max( back to the horizon toso calculate this the option is value. present Payoff valued P&L is using naturally the in CCY2 CCY2 discount factor A vanilla option payoff atdetails and maturity copy can the now spot be and time calculated. at First, maturity set from up the appropriate the simulation payoff step volatility, high volatility, lowinterest interest rate rate differential, and differential, different high initial spot positive values. and negative looks realistic: Task B: Set Up aMonte Vanilla Carlo Option Loop Payoff and the ■ 488 BUILDING A MONTE CARLO OPTION PRICER IN EXCEL pinBs Esrsary tr t0 at RunMonteCarlo1() start Sub arrays 'Ensures 0 Base Option lines: Explicit 20 than Option fewer in written be can sheet. code the VBA onto The back them results. pushes relevant and the outputs collecting Calculates and simulation the 3. recalculating around, Loops config. 2. relevant up Loads 1. as added be should value option the and timing, count, Run outputs: input. an as added be start. to the from properly simulation and the required up be will set steps to dates, time easier expiry frequent is more different it dependence, with path options but with multiple Carlo?’’ required; options price not for Monte to are or the steps extended time within is daily framework required payoff, the vanilla steps as single time a daily for and are question, why Good maturity, at spot on h ot al scnrle yaVAsbotn htde h following: the does that subroutine VBA a by controlled is Carlo Monte The needs now rerun) is simulation the times of number the (i.e., runs of number The depends only payoff vanilla the ‘‘If be: would point this at question reasonable A update: should payoff option vanilla the recalculates, sheet the When hl (Count While 0 = Count Now = TimeStart 'Loop Double "Running" As = 1) Range("MCTimer") - VanillaPayoffs(MCRuns ReDim Range("NumberOfRuns") = MCRuns Settings 'Load Date Double As As TimeEnd Long VanillaPayoffs() Date, As Dim As Count TimeStart Long, Dim As MCRuns Dim ActiveSheet.Calculate < MCRuns) BUILDING A MONTE CARLO OPTION PRICER IN EXCEL 489 '(Recalculates the sheet:Range("Count") equivalent = to CountVanillaPayoffs(Count) pressing + = F9) 1 Range("VanillaPayoff") Count = Count + 1 Range("S_Initial"), Range("Strike"), Range("T_Maturity"),Range("rCCY1"), _ Range("rCCY2"), Range("vol")) / Range("S_Initial") .Average(VanillaPayoffs) 'Calculate Closed-form OptionRange("CFVanillaPrice") Price = OptionPrice(Range("PayoffDirection") = 1, _ 'Outputs Range("MCVanillaPrice") = Application.WorksheetFunction _ Range("MCTimer") = (TimeEnd'(Conversion - of TimeStart) Time * from 24 Days * into 60 Seconds) * 60 Wend TimeEnd = Now Due to limitations of the Application.WorksheetFunction.Average function, in For a vanilla option, the Monte Carlo Black-Scholes price can be compared If everything is set up correctly, the subroutine runs the Monte Carlo and on the Before starting the Monte Carlo, ensure that this sheet is the only one loaded earlier versions of Excel a running total variable must be used instead of an array if calculated in thecalculated VBA in subroutine the sheet and becauseUsing that pushed would the onto potentially OptionPrice slow the function downline from the sheet of Monte Practical code Carlo. rather (don’t C put than enables it this within being the with loop): one additional to a closed-form Black-Scholes price. However, the closed-form price should be sheet the run count increments upfinished, to the the timing number and of option runs. price When are the output: Monte Carlo is End Sub into the Excel, sorecalculate. the ‘‘Calculate’’ command does not also cause other sheets to 490 BUILDING A MONTE CARLO OPTION PRICER IN EXCEL ■ akC e pMlil Payoffs Multiple Up Set C: Task u RunMonteCarloMultiPayoffs() Sub payout. the as currency same the in horizon the to back valued present example, put digital European 4. call digital European 3. put vanilla European 2. call vanilla European 1. to expanded once: be at can prices simulation option Carlo four Monte output the tweaks, minor relatively only With priced. be can payoff any that is pricing Carlo Monte of feature Monte example results: an following within the runs generates of number Carlo the Changing longer. is takes possible) calculation the (if array an using of advantage The the about 65,000. information over that is runs of number the h B oeas ed ob xeddt iku h orpayoffs: four the up pick to extended be to needs also code VBA The for to, adjusted be must formula payoff the prices, digital European generate To key a but solutions closed-form have payoffs derivative simple relatively Only but accurate more get outputs Carlo Monte the increases, runs of number the As ei iiaPtaof(Cus-1 sDouble "Running" As = 1) Double Range("MCTimer") - As DigitalPutPayoffs(MCRuns 1) Double ReDim - As DigitalCallPayoffs(MCRuns 1) Double ReDim - As VanillaPutPayoffs(MCRuns 1) ReDim - VanillaCallPayoffs(MCRuns ReDim Range("NumberOfRuns") = MCRuns Settings 'Load Double As DigitalPutPayoffs() Double Dim As DigitalCallPayoffs() Double Dim As Date VanillaPutPayoffs() Double As Dim As TimeEnd Long VanillaCallPayoffs() Date, As Dim As Count TimeStart Long, Dim As MCRuns Dim = IF(FinalSpot > distribution tie ,0.Pu oeta h iia aotms be must payout digital the that note Plus 0). 1, Strike, fpyfscnb calculated. be can payoffs of BUILDING A MONTE CARLO OPTION PRICER IN EXCEL 491 MCRuns) < VanillaPutPayoffs(Count) = Range("VanillaPutPayoff") DigitalCallPayoffs(Count) = Range("DigitalCallPayoff") DigitalPutPayoffs(Count) = Range("DigitalPutPayoff") Count = Count + 1 Calculate Range("Count") = CountVanillaCallPayoffs(Count) + = 1 Range("VanillaCallPayoff") Application.WorksheetFunction.Average(DigitalPutPayoffs) Application.WorksheetFunction.Average(VanillaCallPayoffs) Application.WorksheetFunction.Average(VanillaPutPayoffs) Application.WorksheetFunction.Average(DigitalCallPayoffs) Range("MCDigitalPricePut") = _ Range("MCTimer") = (TimeEnd - TimeStart) * 24 * 60 * 60 Wend TimeEnd = Now 'Outputs Range("MCVanillaPriceCall") = _ Range("MCVanillaPricePut") = _ Range("MCDigitalPriceCall") = _ 'Loop TimeStart = Now Count = 0 While (Count The Monte Carlo now takes approximately the same time to calculate four option End Sub prices as it took to calculate one: 492 BUILDING A MONTE CARLO OPTION PRICER IN EXCEL ■ akD rcn are Options Barrier Pricing D: Task ptcnb hce gis h are ees(0 levels barrier step barrier time the each the at against through approximation, checked first trades be a can ever as spot Therefore, spot out. knocks if product options, the level, barrier knock-out American For Options Barrier American payoff. vanilla the than rather payoff barrier so using maturity framework Carlo at Monte spot same on the calculation: payoff in only adjusted priced depends an be option can barrier options barrier European European a from payoff The Options Barrier European different multiple price dates. simultaneously expiry to different reused to payoffs be option can they generated been have hntepyf utb dutdaccordingly: adjusted be must payoff the Then European the grabs code Carlo Monte the so updated be to needs VBA the Plus paths sample Once pricing: Carlo Monte of feature key another highlights This = oknock/1 no = knock): BUILDING A MONTE CARLO OPTION PRICER IN EXCEL 493 The flexibility of the Monte Carlo approach should now be starting to become Once American knock-out barrier options are being successfully priced, American This is clearly not quite right since the barrier is effectively being discretely apparent. The process of moving betweenMonte different Carlo barrier than types within is closed-form far pricing. easier within using the vanilla price, which can be effectively calculated for free. Window Barrier Options Window barriers options, cannew date be inputs implemented and adjusting within the the isKnock test framework within by the simulation: adding would be to use tighterAnother time accessible steps approach but would this be also tofor use increases the the Broadie, converting Monte Glasserman, Carlo Kou a runtime. formula Chapter continuous 26. barrier to the equivalent discreteknock-in barrier barrier options given can in be priced simply by adjusting the payoff formula or monitored once a day ratheradvanced than Monte being Carlo continuously techniques monitored. for There solving are this many problem. One simple approach 494 BUILDING A MONTE CARLO OPTION PRICER IN EXCEL ■ akE ut-se Simulation Multi-Asset E: Task eune lsbifl oeta hsfruawl ne h elso h imaginary the of realms the enter will formula this if that note briefly Plus sequence. Define of sequences generating by correlated. done be be to can This numbers random defined. normal correlations be independent framework, must Carlo paths Monte the the into between assets multiple introduce to order In | h rnfraincnb ofimdb plotting by confirmed be can transformation The simulation: the in up set be can This uncorrelated two with start case, two-asset the In 𝜌 | > Y 1. = 𝜌 X 1 + √ 1 − 𝜌 2 X 2 h e sequence new The . 𝜀 ∼ N ( 0 , 1 ) n hnajsigtesequences the adjusting then and Y X a orlto of correlation a has 1 against 𝜀 sequences, X 2 nasatrplot: scatter a in X 𝜌 1 othe to and X X 2 1 . BUILDING A MONTE CARLO OPTION PRICER IN EXCEL 495 for comparison (80% correlation shown): Y values can be used to drive the spot processes, hence making against 1 𝜀 X Multi-asset option payoffs that depend on both spot levels at maturity can The correlated Then plotting therefore be set up. Forpays example, out in a the dual common digital currencylevels: (CCY2 option in (introduced this in case) if Chapter both 30) spots are above defined spot log returns correlated, as required: 496 BUILDING A MONTE CARLO OPTION PRICER IN EXCEL ■ Extensions d h fet fpiigmdl noteoto autosa fcetya possible. to as is efficiently quants as valuations for option challenge the into the models cases, pricing both of effects In the contracts. add derivative value to evolving (PDEs) term variance additional an be time. over would spot there with model volatility stochastic a implemented. volatility rates the interest used, stochastic be or could added, rates be For interest could framework. and surface Carlo volatility Monte of basic structure this term to example, extensions possible numerous are There factorization Cholesky either decomposition. eigenvector using or sequences correlated into sequences correlated example: For defined. be must matrix correlation npatc,taigdssueete ot al rprildfeeta equations differential partial or Carlo Monte either use desks trading practice, In under example, For model. pricing a using added be would surface volatility The non- converting for multipliers into converted then is matrix correlation This an into framework two-asset a from move To N astfaeok an framework, -asset N × N CHAPTER 27

Vanilla Variations

ith only minor adjustments to the contract details, the trading risks on WEuropean vanilla options can be significantly changed. The adjustments discussed in this chapter are late-delivery, American exercise, and self-quanto payoff.

■ Late-Delivery Vanilla Options 497 Consider the options shown in Exhibit 27.1. Leg 1 is a standard European vanilla option with a 1yr expiry. Leg 2 is identical except that the delivery date is one year after the standard delivery date. For a European vanilla option with the standard delivery date, the price of a physically delivered option is the same as the price of a cash-settled option. However, for a late-delivery vanilla option the price difference can get large. In practice, Exhibit 27.1 doesn’t contain enough information to price the late- delivery option because these same contract details could represent three different derivative contracts. When pricing late-delivery options it is important to confirm exactly what payoff is required and understand what methodology the pricing model uses so any necessary additional adjustments can be made.

Late Cash Vanilla Options The late-delivery vanilla option in Exhibit 27.1 could be a late cash vanilla option. At maturity, the option is exercised against a fix. The option is then cash settled on the late delivery date (August 18, 2016). The price difference between the late cash vanilla and the standard European vanilla depends on discounting in the payout currency between the standard delivery 498 VANILLA VARIATIONS 8 06.I xrie,teoto hsclydlvr noa130 owr that forward 103.00 a into date. delivers delivery physically late the the option comparing on the matures by (August exercised, exercised date delivery If or late 2016). the expired to 18, outright is forward option prevailing the the with (103.00) maturity, strike At an spot. be into alternatively than could 27.1 Exhibit forward in on option vanilla late-delivery The Forwards on Option to due model pricing the by generated market price the discounting. rate interest make the interest will USD static effect This the rates, out. than interest pays lower option USD price the when and if higher spot be Therefore, likely spot. between will higher correlation rates a positive with USD a more the is out to there pays reference option with call considered This be payoff. must spot option which between rates, correlation interest the currency from come payout additionally will be and effect must important rates most interest The stochastic in. of effects factored USD the so positive static, small are rates to interest due 2 leg of price lower slightly rates. the interest Note 27.2. Exhibit late in vanilla. the European and regular less the worth than be cheaper positive, will be are future will currency the option payout into vanilla the further cash in paid rates settlement interest cash If fixed date. the delivery late the and date 27.1 EXHIBIT eed anyo h wppit ewe h tnaddlvr aeadtelate the and date delivery standard the between date. points delivery swap the on mainly depends h ifrnei rcn ewe hsvrainadtesadr uoenvanilla European standard the and variation this between pricing in difference The assumes that model pricing a using calculated are 27.2 Exhibit shown within mid-price Prices tool pricing the in included is discounting this of effect The enn ail pinta hsclydlvr noafradrather forward a into delivers physically that option vanilla a meaning , rcn olsoigtovnlaoto otat,oewt aedelivery late with one contracts, option vanilla two showing tool Pricing option VANILLA VARIATIONS 499 by 1.25. Note that this causes the option higher Pricing tool showing late cash vanilla option The option on forward price can be approximated using a 1yr vanilla option with delivery date and the late deliveryoption date on will be forward positive. will In be this morepayoff case expensive calculated the than CCY1 using the call the European forward vanillathe because to payoff calculated the the using late spot delivery at date maturity. will be higher than delivery date and the latecall delivery option date on will be forwardpayoff negative. will calculated In using be this the case, cheaper forwardthe the than payoff CCY1 to calculated the the using European late spot at delivery vanilla maturity. date becauseIf will the be CCY1 lower rates than are below CCY2 rates, the swap points between the standard If CCY1 rates are above CCY2 rates, the swap points between the standard a strike adjusted by thedate negative and the of late the delivery swap date.2015 For points example, and between if August the the swap 18, standardvanilla points delivery 2016 between option August with are 18, the –125, strike moved the price can be approximated using a 1yr ■ EXHIBIT 27.2 ■ 500 VANILLA VARIATIONS aeteqoe rc oe hntesai neetrt rc eeae ythe by generated price rate interest will effect static This the negative. than the more model. lower when moved pricing price rates, have interest quoted will CCY1 the points and swap make spot the option between example, out, the for correlation pays Therefore, to positive option spot. reference a higher a is with with there considered out if pays be option spot call must between CCY1 correlations which This payoff. from rates, come interest will additionally both effect be must important and rates most interest The stochastic of in. effects factored the so static are rates interest 27.3. Exhibit in European shown standard the than volatility implied different (slightly) a vanilla. at priced be to 27.3 EXHIBIT gis x u ahrta en ahstldo h tnaddlvr aea usual as date exercised delivery be standard the maturity, on settled at cash could, being than 27.1 rather But Exhibit fix. in a against option vanilla late-delivery The Options Vanilla Late-Delivery rcswti xii 73aecluae sn rcn oe htassumes that model pricing a mid-price using tool calculated pricing are the 27.3 of Exhibit 2 leg within in Prices included are points swap of effects The rcn olsoigoto nforward on option showing tool Pricing VANILLA VARIATIONS 501 at the exact only non-optimal exercise forward). However, as soon as = option is late-delivery CCY2 rates) and spot at maturity is above the strike, CCY2 rates) and spot at maturity is below the strike, > < up to the option expiry date and cut time. at any time : On August 14, 2015, USD/JPY spot is 102.75. Therefore, the option : On August 14, 2015, USD/JPY spot is 103.25. Therefore, the option American vanilla options are rarely traded in the OTC FX derivatives market, The pricing reduction from thisThis price is non-optimal calculated assuming interest exercise rates are static is so the effects shown of stochastic leg 2 of Example Example If swap points between the standard delivery date and the late delivery date are If swap points between the standard delivery date and the late delivery date are If interest rates are all statically zero, the price of the late-delivery option would The key element in this European vanilla options canoption be expiry exercised date and by cutoption the time. holder option American holder vanilla options can be exercised bywhere the the vast majority of vanilla options are traded European-style. If early exercise volatility will reduce the quoted price ofthe the chances late-delivery of vanilla non-optimal because exercise. it increases 18, 2016 is 104.00. Buying the Augustto 18, positive 2016, P&L, forward which has at not 103.00 been would realized have due lead to the exercise decision mechanism. Exhibit 27.4. interest rates must be additionally factored in. For example, higher interest rate positive (i.e., CCY1 rates not exercising a callresult option in into positive P&L the being forward missed. at the strike levelexpires. However, in the forward the points are positive future and the may forward outright to August in negative P&L. exercises into a long forward at 103.00negative for and August the 18, prevailing 2016. The forward forwardthe outright points August to are 18, August 2016 18, forward 2016 at is 103.00 102.00. leads to Buying negative P&L. the expiry decision would beinterest optimal rates (since are spot non-zero, exercise becomes non-optimal. negative (i.e., CCY1 rates exercising a call option into the forward at the strike level in the future may result compared to the option onmaturity forward. is The above late-delivery the option strike and iscould it exercised therefore is if be expired constructed spot if in at spot a at pricingdate maturity tool that is using knocks below out a below discrete the the barrier strike. strike on It and the pays expiry out a forward to thebe late the delivery same date. as the standard European vanilla option and the late cash option because (August 18, 2015), the option physicallyon delivers into the a late forward delivery at date the (August strike 18, settling 2016). American Vanilla Options ■ 502 VANILLA VARIATIONS qiaetErpa ail,wt h au ifrneicesn ihlne time longer with increasing the difference as expiry. value expensive to the as with least vanilla, at always European are equivalent vanillas American Therefore, options. vanilla assumptions. simplifying restrictive fairly closed-form on well-established rely Several they but pricing. exist within do account approximations rates into interest stochastic taken even be potentially and should curve, ATM the full dependent, path curve, are rate they interest since full Ideally, value. accurately to complicated more are Chicago the example, for American (CME). exchanges, However, Exchange on option. Mercantile products standard American same often the the are get exercising options to early vanilla spot trade as and change vanilla European position a net unwind to possible is it required, is 27.4 EXHIBIT au otfo odn h pin(anydet h owr rf)i oeta h au of value the than more is drift) forward the to due optionality (mainly remaining option the the holding from lost value h al xriefaueo nAeia ail hudb activated be should vanilla American an of feature exercise early The contain options vanilla American they hence and vanillas American pricing for exist solutions closed-form exact No rcn olsoiglt eieyvanilla delivery late showing tool Pricing . oeoptionality more hnteeuvln European equivalent the than hnthe when VANILLA VARIATIONS 503 T . ) 1 rCCY − 1) leads to a forward that is above spot. 2 rCCY ( rCCY Se 2) leads to a forward that is below spot. This > = 2 T rCCY F rCCY > 1 rCCY European option value versus early exercise value: zero interest rates )isgivenby: t F ( T Higher CCY1 interest rate cases are shown in Exhibits 27.7 and 27.8. Higher With zero interest rates and hence no forward drift or discounting, the European A higher CCY2 interest rate case is shown in Exhibit 27.6. The European option Consider the value of a 1yr European CCY1 call/CCY2 put vanilla option CCY1 interest rates ( leads to an expected callbe option lower payoff than calculated the off early the exercise lower value forward, calculated which off spot. can Higher CCY2 interest rates ( This leads to an expected callalways larger option than payoff the calculated early off exercise the valueoptimal calculated higher to off keep forward spot. the that Therefore, optionality is it and isearly hence always exercise. benefit from the forward drift rather than value is again above thethan early the exercise zero value interest at ratetime case all due spots, to plus the this forward it drift. is Recall further that over the forward to versus the equivalent earlyExhibit exercise 27.5. value. The zero interest rate case isoption shown value in is always above theto early keep exercise value. the Therefore, optionality it rather is than always early optimal exercise. EXHIBIT 27.5 504 VANILLA VARIATIONS XII 27.7 EXHIBIT 27.6 EXHIBIT uoenoto au esseryeecs au:rCCY1 value: exercise early versus value option European uoenoto au esseryeecs au:rCCY1 value: exercise early versus value option European = = 1%/rCCY2 0%/rCCY2 = = 0% 5% VANILLA VARIATIONS 505 0% = payoff is a 5%/rCCY2 = (i.e., the currency with higher interest rates). European option value versus early exercise value: rCCY1 Beyond a certain spot level it is optimal to early exercise the American vanilla. This occurs because more value is lost from the forward drift at higher interest When the payoff is a CCY1 call, the early exercise feature only has value when These examples reveal some important rules: At in-the-money spots beyond a certain level, the European option value is below When long an American vanilla,value the difference European must vanilla be price monitored. versus early exercise If it exists, the optimalthe early interest exercise rate level differential. moves closer to the strike the higher The early exercise feature of Americancall vanillas on only the has higher value yielding when currency the ■ This occurs when spot is in-the-money versus the payoff and the forward drift will ■ rate differential. CCY1 interest rates are higher thannot CCY2 a interest call rates. on When the the higherand vanilla yielding Greeks payoff currency, is in on a an staticvanilla. interest American rate world, vanilla the will pricing be identical to the equivalent European the early exercise value due toearly the exercise forward the drift. American At vanilla these option. spot levels it is optimal to ■ EXHIBIT 27.8 506 VANILLA VARIATIONS nrae am xoueo h mrcnvnlawt pthge.Byn the Beyond higher. spot with vanilla American the on exposure gamma increased vanilla the the exercising point, to exercise equivalent early 100%, optimal becomes spot. the vanilla into Beyond American spot. the higher on a delta with sharply more when up value option total the of part significant in-the-money. deep a is becomes spot 27.9) between Exhibit difference in (the optionality lines exercise the early the of Price value The risk. vanilla. European trading impacts CCY1 exercise higher 27.9. and early Exhibit in call how shown are CCY1 demonstrates profiles the case in rates vanillas interest European and American Comparing Greeks and Pricing Vanilla American money. save may deal spread (original market spreads spot option the two plus crossing unwind) avoid to plus to preferable limited ability usually with the pairs is liquidity, currency market it in spot However, optionality, option. exercise vanilla early European standard the the of trade value the on and than for rather paid been away. money has throwing optionality to optimal equivalent exercise this roughly early If used; value. not monitored, time then in not remaining is is than point over exercise away decay to value more cause XII 27.9 EXHIBIT am rfie r hw nEhbt2.1 h hre et ikpgenerates pickup delta sharper The 27.11. Exhibit in shown are profiles Gamma picks delta vanilla American The 27.10. Exhibit in shown are profiles Delta equivalent the than expensive more always is vanilla American the mentioned, As trigger external some on based is decision exercise early the when client, a For y .60ERcl/S u mrcnvnlavru uoenvnlaprice vanilla European versus vanilla American put call/USD EUR 1.3650 1yr VANILLA VARIATIONS 507 1yr 1.3650 EUR call/USD put American vanilla versus European vanilla gamma 1yr 1.3650 EUR call/USD put American vanilla versus European vanilla delta Vega profiles are shown in Exhibit 27.12. Once again, Greeks are curtailed optimal early exercise point, gammabe becomes exercised zero into spot. as it is assumed the vanilla will beyond the optimal early exercisespot point as levels, optionality vega disappears. risks At on in-the-money American vanillas move forward to tenors prior to the EXHIBIT 27.11 EXHIBIT 27.10 508 VANILLA VARIATIONS pinepr,weesfrErpa ailstevg ik r last expiry. to always are risks vega the vanillas European for whereas expiry, option 27.13 EXHIBIT 27.12 EXHIBIT pinwt ptna h pia al xriepita hw nEhbt 27.13 Exhibits in shown as point exercise early optimal 27.14. and the near spot with American option on risk trading the used. assess be must properly exposures To vega management; bucketed vanillas, risk for key is This h og n an xoue r infiatylre na mrcnvanilla American an on larger significantly are exposures vanna and volga The y .60ERcl/S u mrcnvnlavru uoenvnlavega vanilla European versus vanilla American put call/USD EUR 1.3650 1yr y .60ERcl/S u mrcnvnlavru uoenvnlavolga vanilla European versus vanilla American put call/USD EUR 1.3650 1yr VANILLA VARIATIONS 509 1yr 1.3650 EUR call/USD put American vanilla versus European vanilla vanna Finally, the bid–offer spread on American vanillas is generally slightly wider than At longer tenors, stochastic interest rates must be taken into account. If spot and At shorter tenors (e.g., under 3mth), where the early exercise optionality has The standard European vanillamaturity option it can payoff be is converted back generated to in CCY1 terms CCY2 at terms, the prevailing and spot. Self-quanto at but the interest rate differential mayvalue. flip This over effect time, is giving quantified the by early a exercise stochastic feature interest rate model. the equivalent European vanilla (particularlyrisk in management longer and tenors) monitoring due they to require. the additional interest rates move in aof correlated American manner, vanilla that options. will ConsiderCCY1 significantly the call case impact but where the CCY2 the price interestIn vanilla rates the option are basic payoff currently analysis, is higher this a than means that CCY1 the interest price rates. will be that of the European vanilla, is being used to price the optioninto in account order and to which understand are which effects being are ignored. being taken little value, it maysmile, be and sufficient then to add priceprice in difference. the a European constant vanilla volatility using American-style the versus volatility European-style EXHIBIT 27.14 American Vanilla Pricing When pricing American vanilla options, traders must understand what methodology Self-Quanto Vanilla Options ■ 510 VANILLA VARIATIONS ail C1PtOptions Put CCY1 Vanilla 27.17. Exhibit in shown as vanilla 27.16. European Exhibit in with visualized options, is of notional, strip self-quanto This the maturity. than strike same smaller the same far to notionals with strike the vanilla replicated above be European vanillas can call call call CCY1 CCY1 CCY1 to self-quanto the way a buying buying intuitive that by An is notional. self-quanto EUR1m the in about think options vanilla self-quanto and European hence: and payout vanilla regular the than larger is payout self-quanto the by divided is payout the terms, so: maturity at spot using CCY1 to CCY2 from converted be can payout The maturity: at P&L CCY2 a has payoff vanilla European the option, call CCY1 a For Options Call CCY1 Vanilla adjustment the this (usually to differently rate separately. react examined fixed payoffs be a put must they and at therefore Call CCY1 and spot. into at back than rather converted strike) payout their have options o C1ptErpa ail pin gi,tentrlvnlapyf a CCY2 has payoff vanilla natural the maturity: again, at option, P&L vanilla European put CCY1 a For hrfr,tesl-unoCY ali ogrtpievg oprdt the to compared vega topside longer is call CCY1 self-quanto the Therefore, put call/USD EUR 1.3500 long of maturity at payoff the shows 27.15 Exhibit then call, CCY1 the on maturity at in-the-money is spot If by dividing than rather case, self-quanto standard the In Notional Notional Notional CCY CCY Notional 1 1 . . max max Notional CCY ( ( CCY S K P 1 T CCY . − ( 1 − . 1 ( S S CCY K T alSelf Call T S K , K , − T : 1 0 0 K . − )= ( K )= - ) K Quanto S T + ) Notional Notional S . − + T S T > = K = ) P efQat Payout Self-Quanto CCY + efQat Payout Self-Quanto CCY CCY = 1 1 1 alVanilla Call . . Payout ( ( K S T − − S CCY K T ) S ) 1 + + T S T lsbuying plus = CCY1 to back get to CCY = > CCY Payout Payout 1 K 2 hc implies which , CCY CCY astripof 2 2 VANILLA VARIATIONS 511 1 CCY Payout = + ) T S T S − K ( . 1 CCY Notional European vanilla to self-quanto call option adjustment discrete replication European vanilla versus self-quanto call option payoff at maturity The payout can be converted from CCY2 to CCY1 using spot at maturity so: EXHIBIT 27.16 EXHIBIT 27.15 512 VANILLA VARIATIONS uoenvnlawt aestrike same with vanilla European notional. EUR1m in options vanilla self-quanto and European hence: and payout vanilla regular the than smaller is payout self-quanto the by divided is payout the terms, 27.17 EXHIBIT h tiet h aemtrt.Teeoe h efqat C1pti shorter is 27.19. Exhibit put in CCY1 shown as self-quanto vanilla the European the Therefore, to maturity. compared vega same downside the to strike the uigaCY u efqat a erpiae ybyn h C1put CCY1 the buying by replicated be can self-quanto put call CCY1 put/USD EUR a 1.3500 Buying long of maturity at payout the shows 27.18 Exhibit then put, CCY1 the on maturity at in-the-money is spot If by dividing than rather case, self-quanto standard the In Notional Notional uoenvnlavru efqat aloto eaprofile vega option call self-quanto versus vanilla European CCY CCY 1 P . CCY ( 1 . ( K 1 K u Self Put K − K : − K S T lsselling plus S - ) Quanto T + ) . + S T < = = P Self CCY Self ti fCY u ailsbelow vanillas put CCY1 of strip a - 1 unoPayout Quanto - u Vanilla Put unoPayout Quanto S S T T ogtbc oCCY1 to back get to CCY < CCY 1 K 2 hc implies which , VANILLA VARIATIONS 513 European vanilla versus self-quanto put option vega profile European vanilla versus self-quanto put option payoff at maturity EXHIBIT 27.19 EXHIBIT 27.18 514 VANILLA VARIATIONS ihe hnteeuvln ail pinbcueteei esvg ikdet the to due risk vega less is puts. there CCY1 because of option strip offsetting vanilla equivalent the than tighter the wings, within the for accounted in be optionality at should significant look which pairs be to spread. bid-offer currency taken may skew be There high must vega; in care particularly ATM However, just calls. CCY1 beyond of risk risks vega strip compounding additional wider is quoted there the because be from option should vanilla it European that equivalent suggests the replication than the call, CCY1 self-quanto a For Spread Bid–Offer Self-Quanto o efqat C1pt h elcto ugssta tsol equoted be should it that suggests replication the put, CCY1 self-quanto a For CHAPTER 28

Accrual and Target Redemption Options

515 ccrual options and target redemption options are both popular within the AFX derivatives market. Accrual features and target redemption features are typically added to a forward contract or a strip of forward contracts in order to improve the transaction rate for the client.

■ Accrual Options

The key characteristic of accrual options is that the notional, rather than being static, builds up (accrues) over time. Accrual options have a fixing schedule and the rate of accrual depends on where spot fixes compared to the accrual barriers within the structure. There are two main types of accrual barrier:

1. European: If spot goes through the barrier, accrual stops with what has been accrued retained, but if spot later comes back inside the barrier, accrual restarts. 2. American Keep: If spot ever goes through the barrier, accrual stops, but what has been accrued prior to that point is retained. 516 ACCRUAL AND TARGET REDEMPTION OPTIONS where range structure the accrual within range 28.1. fixes European Exhibit typical later in A then shown accrue. is and to range) continues the notional outside the again, fixes it or fixings between the onto added is USD4k barriers, accrual maturity. at the payout between cash is spot where schedule the fixing USD1m/250 the is, agree that equally), and price. schedule, a quoting fixing to prior the client generate the with to itself schedule used are calendars holiday terms, details: % currency payout in options. digital quoted European are or options prices touch accrual like Range a expiry. is at product notional accrual simplest The Options Accrual Range uoendgtlrne xiiga ahfiigwti h ag cra,ec with each accrual, range the this within Extending fixing expiry. each at at range expiring the ranges within digital European is spot if notional further, idea full the out pays that N stettlnme ffie ntefiigschedule. fixing the in fixes of number total the is fteei utoefiiga xiy h pinbcmsaErpa iia range digital European a becomes option the expiry, at fixing one just is there If accrual: range European a within Therefore, are These (usually fixings the between split is notional the accrual, which range example confirm the Within source, fixing the specify to vital is it fixings with products For 110.00 barrier: accrual European Up 100.00 barrier: accrual European Down excluded) ECB37 holidays Source: JPY Fixing and (USD fixings daily 250 Fixings: USD1m Notional: 102.50 Spot: 1yr Tenor: USD/JPY pair: Currency contract accrual range barrier double European typical some are Following n stenme ftmsso xswti h uoenaculbrir and barriers accrual European the within fixes spot times of number the is uoenrneaccruals range European European xml:Erpa obeBrirRneAccrual Range Barrier Double European Example: ahpyu texpiry at payout Cash cra ares oi ptge usd h ag (either range the outside goes spot if so barriers, accrual = S4 e xn.Teeoe o vr xn in fixing every for Therefore, fixing. per USD4k ag accrual range a epretyrpiae yasrpof strip a by replicated perfectly be can = Notional hc asota accrued an out pays which , × N n ACCRUAL AND TARGET REDEMPTION OPTIONS 517 stop accruing and all accrued notional is kept ] cash payout late-delivered to the range accrual delivery ) 1 N ( × European range accrual structure American keep range accruals The fact that European range accruals can be replicated with a strip of European Exhibit 28.2 shows the long wing vega profile from a long European range accrued if spot stops moving; hence short vega. If spot is outsidethe the range European again range in order accrual to barriers, start spot accruing; hence needs long to vega. move into If spot is between the European range accrual barriers, more payout will be range accrual will bethe more same expensive barriers. In than turn, the thethan American American a keep no-touch keep range option range accrual in will accrual theno-touch be full with notional more notional is expensive with effectively the lost same if barriers a because barrier the level entire trades. date. Second, any well-calibrated smile pricingimpact model of will correctly the incorporate volatility the smile intoavailable the smile product. pricing Therefore, model the (often simplest local and volatility) quickest can be used. if spot ever goes through an American keep accrual barrier. Therefore, the European accrual. Recall from Chapter 18have that positive contracts TV with adjustments. long wing vega profiles usually digital ranges demonstrates two important features of Europeanaccrual range accruals. option First, pricing must takethere the is full vega ATM exposure to term every structure date into within the account fixing because schedule, not just to the expiry ■ date. As seen inis Chapter long 21, vega the in vegabetween the profile the wings barriers from (want (don’t a want spot spot longEuropean to to range European move move) accruals: and digital back this range into same profile the also range) applies■ and to short vega EXHIBIT 28.1 their [Notional 518 ACCRUAL AND TARGET REDEMPTION OPTIONS xnsotngv iia autosudrdfeetmdl n hrfr the therefore and often many can models volatility) with local similar different (often options used. model is accrual under be pricing smile this keep valuations available quickest and American similar and average. fixing simplest give an each often taking at to fixings exposures sense different some has in contract stripped the range digital European notional. equivalent is total accrual the the range in on a option spread on double-no-touch spread or bid–offer bid–offer the the than entire reason, the tighter this lose usually For or gain date. to single possible a not on is notional it because risk trading reduces positive dates a multiple have in usually will shown and as wings barriers long similar the is looks beyond adjustment. profile accrual TV vega vega range the no Again, keep with 28.3. American option, date. Exhibit double-no-touch long delivery long a accrual a of range to the profile to vega late-delivered the payouts Therefore, cash their with touches 28.2 EXHIBIT owr aota auiywt h oinldtrie yhwso moves. spot how by determined notional the with maturity at payout forward a are products accrual popular most The Forwards Accrual h tipn rcs lordcsmdlrs naculotosbecause options accrual on risk model reduces also process stripping The into date single a from risk digital or barrier stripping of process the Importantly, no- double of strip a with replicated be can accrual range keep American An uoenrneaculvg profile vega accrual range European cra forwards accrual hs rdcscontain products These . ACCRUAL AND TARGET REDEMPTION OPTIONS 519 . × = double accrual forward strike). × American keep range accrual vega profile (notional/number of fixings) for each spot fix in an area where(notional/number the of payoff fixings) is for each spot fix in an area where the payoff is × × As before, a simple smile pricing model can be used for European accrual forwards Following are some typical American keep double accrual forward contract Therefore, the product can accrue up to double the notional if spot fixes in the 2 The risk from a simple accrual forward can be decomposed into two range Long EUR range accrual in the accrual forwardShort EUR USD notional. range accrualCCY1 in notional the appropriate notional (using CCY2 notional 1 positive for the client at maturity 2 negative for the client at maturity and is often sufficient for pricing American keep accrual forwards with many fixings. details: accruals with the sameforward can accrual be barriers. replicated using: For example, a■ long EUR/USD accrual ■ accrual area at every fix. Theand different accrual hence multiples the generate value forward for ratetransacts the client for is zero moved premium. in the client’s favor, meaning that the client The notional accrues: ■ ■ EXHIBIT 28.3 There are many variations, but the most popular is a 520 ACCRUAL AND TARGET REDEMPTION OPTIONS rdr ecietersso cra owrsa ‘ail,’maigtevg and vega the meaning ‘‘vanilla,’’ as forwards accrual on risks the in describe shown Traders is This strike. the around peak vega 28.5. in a Exhibit shown with option, is knock-out This American premium. zero for 102.05 at than 28.4. rather Exhibit 98.80 at forward 1yr 2 with worse downside 28.4 EXHIBIT pedo cra owrswt uoeno mrcnke aresi fe just often bid–offer is the barriers keep Therefore spread. American fixings. vega or of the European from to strip derived with risks a forwards digital on accrual large depending on no payout spread are the there to addition, due In hedge time. over well-behaved are gamma hs rdcsaeams lasln eafo h rdn ekperspective. desk trading the from vega long an always to almost are similar products is These forward accrual double keep American this on risk vega The the making and barrier accrual keep American the with upside the capping By 1 barrier: and strike between multiple Accrual 2 strike: below multiple Accrual 106.00 barrier: forward Accrual 98.80 USD/JPY Strike: buys Client USD/JPY, sells Bank Direction: USD1m Notional: 102.05 Forward: 102.30 Spot: 1yr Tenor: USD/JPY pair: Currency xml:Aeia epDul cra Forward Accrual Double Keep American Example: mrcnke obeaculfradstructure forward accrual double keep American × cra eo h tietecin usUDJYin USD/JPY buys client the strike the below accrual × × ACCRUAL AND TARGET REDEMPTION OPTIONS 521 over time. Within a typical target target Double accrual forward with American keep barriers vega (trading desk : EUR/USD 1yr TARF with monthly fixings. Spot is at 1.2770 and the Example There are many TARF variations, often based around differences in behavior 1yr forward is 1.2800. At each1.2290 fixing, (well the below client the forward) buys provided EUR the against USD30,000 USD target at profit the has not strike been (e.g., a TARF EKI featuresdigital a range payoff). European Another knock-in variation payoffbased is on or a the a Count accumulated TARF gain, TARF but Box inpositive on features gain. which the In a the number general, of target the times is risk thatfairly not management the similar of client except these for receives their different a behavior variations at isto fixings, usually being particularly breached. when the target is close target redemption forward (TARF)whole variation, structure the expires client’s once gainstoward a the count target target. up is reached. and The the client’s losses dowhen not the count target is reached or exceeded. Plus there are variations based on the payoff The key characteristic of a target redemptionin option some is way that the on option a payoff quantity depends thatredemption counts up product, to a the clientleveraged enters forwards) with into rates a better strip than of the forwards forward (or outrights. In similar, the e.g., standard EXHIBIT 28.5 perspective) Target Redemption Options ■ 522 ACCRUAL AND TARGET REDEMPTION OPTIONS ■ ■ trade: usually the is in vega leverage ■ is total there maximum if The especially strike, perspective. various the desk at around trading trade located example the the from from levels exposures vega spot bucketed the shows 28.6 Exhibit Risk Vega has trade the occur losses that strike. client note the and below strike plus the strike, above the occur gains below client trade, this Within fixing. ■ ■ either: At buys terminates. structure client the the reached, fixing, is each profit target USD30,000 the Once reached. XII 28.6 EXHIBIT owrs hc a h aevg xoue sasrpo ail pin nthe in leveraged options of vanilla become of would strip structure strip the a a what to as 8)—exactly Chapter exposures similar (see vega notionals looks unmatched same the profile has vega which bucketed forwards, The spot: 1.1750 more is closer target to moves the risk hence vega the (and and gains sharply to has maturities. converges vega client reached), the be to where likely direction spot structure. vanilla the likely equivalent less In the is like target reduces the vega hence (and the losses reached), has be client to the where direction spot the In strike the below fixes spot EUR/USD if 1.2290 at USD against EUR600k strike the above or at fixes spot EUR/USD if 1.2290 at USD against EUR300k ftetre etr a removed. was feature target the if okn ttevg xoue tec ptlvli xii 28.6: Exhibit in level spot each at exposures vega the at Looking each at gains client’s the summing by calculated is gain positive accumulated The or agtrdmto eaexposures vega redemption Target leverage :2 × oinli transacted is notional ACCRUAL AND TARGET REDEMPTION OPTIONS 523 Target redemption versus equivalent vanilla structure vega profiles Exhibit 28.8 shows a vega chart of the TARF with different targets. With a higher Exhibit 28.7 shows the vega chart of the TARF compared to the equivalent 1.3790 spot: The structureminimal will exposure knock to implied out volatility. at the first fixing and therefore has structure will more likelyreduced knock and out there early. is Overall increased vega vega at and closer1.3280 stopping maturities. spot: time The has structurecloser will tenors is likely even knock more exaggerated. outtenor, The which soon, vega suggests is so that bucketed the the mainly stopping in vega time the on move 2mth the into trade is around two months. 1.2260 spot: Spot is closest tomaximized. the strike (1.2290) and hence optionality (vega)1.2770 is spot (current spot): The client is accumulating gains and therefore the EXHIBIT 28.7 Delta and Gamma Risk The spot ladder fromshown the in Exhibit EUR/USD 28.9. TARF from the trading desk perspective is the structure and decreases the vega, particularlywill at terminate higher more spots quickly. where the structure target, the vega on the TARF increases and the chance of knocking out decreases. ■ leveraged forward structure. The presence of the target reduces the stopping time of ■ ■ ■ 524 ACCRUAL AND TARGET REDEMPTION OPTIONS esetv h rd an oevlewt ptlwrta tlsswt spot confirms with trade loses ■ the on it exposure than delta lower the about spot Thinking this: with target. the value to more due gains higher trade the perspective 28.9 EXHIBIT 28.8 EXHIBIT ■ eli epi-h-oe o h letadtebn a omr ols u to due lose to more no the has as bank zero the reached. to and being goes client target trade the the the for on in-the-money position deep delta is the deal eventually risk higher, trading goes spot the If bank and forwards. the notional of for strip leveraged in-the-money a the more to equivalent becomes and becomes more eventually get delta will the deal until the lower, goes spot If h rd sln am o h rdn ekbcuefo h rdn desk trading the from because desk trading the for gamma long is trade The agtrdmto ptladder spot redemption Target targets different for profiles vega redemption Target ACCRUAL AND TARGET REDEMPTION OPTIONS 525 while time 30k USD target. Suppose instead spot = 300k USD × . The similarity extends to the pricing: Both products have spot Traders describe the risks on target redemption forwards as ‘‘vanilla,’’ meaning The crystallization of digital risk at the next fixing can also cause delta changes. The more frequently fixings occur within a structure, the more digital risks must that the vega andmajor gamma risk are management generally challenges well-behaved comethat from through these the time. potential fixings. digital As risks It at noted, isspread fixings the therefore are as important taken well into as account the within usual the vega-based bid–offer spreading. Target redemption options arefixing conceptually schedules, similar to and accrual inaccruals options: knock some out Both sense on have targetminimal redemptions model risk. knock This out can be on pricing confirmed models; by different pricing models the will TARF often using give various similar smile valuations. These delta gaps at fixingsTARF deal require population. careful monitoring and management within the Target Redemption Pricing out, while if spot fixesto on maturity. the Large other digital risk side is ofdetails also the generated of strike, by the it strike trade. changes will within For haveafter the the a example, contract next far if fixing, longer the the time difference remainingor in expiring strikes option at all value the move between next the to fixing option will a staying be new alive large. level fixing, it becomes known thatif the spot future fixes value above 1.2790 of at therisk the at remaining second 1.2790 fixings fixing. at will Therefore, the the be second trade lost fixing. develops digital be risk managed. These digital risksis will close be particularly to large the if strike, the so termination if level spot fixes on one side of the strike, the option will knock challenges on TARFs come fromtrade the fixings. will At knock inception, out the atclient example the the EUR/USD (1.3290 first – fixing 1.2290) iffixes at spot 1.2790 fixes at at the 1.3290knock first out or fix, at above, hence the paying second generating fixing the USD15kfixing if there profit. spot was The fixes no deal at explicit will or digital above then risk 1.2790 at again. the Before second the fixing. first However, after the first Target Redemption Fixing Risk Aside from the standard gamma and vega trading risks, the main risk management

CHAPTER 29

Asian Options

he key characteristic of Asian options is that some element of their payoff is based on an average. The most common Asian option variations have a single Taverage that is used in place of either the spot or the strike at maturity within a vanilla payoff. The average is either calculated from spot fixings that occur regularly between the horizon and expiry date as shown in Exhibit 29.1, or the fixings can be over a subsection of the period as shown in Exhibit 29.2. Fixings can be taken at different sample frequencies, for example, daily, weekly, 527 or monthly. Plus note that Asian options are always cash-settled at maturity.

■ Average Rate Options

Within an average rate option, the spot at expiry within a standard vanilla payoff is replaced with an average of fixings:

+ PayoffAverage Rate Call =(AV − K) + PayoffAverage Rate Put =(K − AV)

where AV is an average of spot fixings.

EXHIBIT 29.1 Example single average fixing schedule A 528 ASIAN OPTIONS sdmntae nEhbt29.5. Exhibit in demonstrated as than cheaper is option rate average option. vanilla the European that equivalent confirms the 29.4 be Exhibit will in fixings shown spot vanilla of average the itself. spot of of volatility equivalent volatility the the the than than general, maturity lower at In payoff option. lower a put has vanilla option put rate average the hence why shows 29.3 Exhibit case. in the action is spot this realized The EUR%). (0.91 vanilla European 29.2 EXHIBIT h ieo h pinadteeoetepyf eed o uto pta maturity at spot on just not depends payoff the throughout therefore fixings and are option There the of exposure. is profile life vega vega the lower rate simply average this than of complicated story more full The option. vanilla European equivalent xii 96sosteiiilvg rfie rmteaeaert pinadthe and option rate average the from profiles vega initial the shows 29.6 Exhibit (approximately): that case the is it strikes ATM For European equivalent the to compared option rate average the of profile price The and slowly more far lower moves average the but lower sharply moves Spot equivalent the than cheaper significantly is EUR%) (0.24 option rate average The 0.8250 Strike: call put/GBP EUR Direction: EUR100m Notional: 0.8785 Forward: 0.8725 Spot: 1yr Tenor: EUR/GBP pair: Currency details: contract option rate average example are following The xml:AeaeRt pinwt 2mnhyso fixings spot monthly 12 with Option Rate Average Example: xml igeaeaefiigshdl B schedule fixing average single Example Vega AverageRate = Vega √ Vanilla 3 = 0.6 × Vega Vanilla ASIAN OPTIONS 529 EUR/GBP average rate option versus vanilla option price EUR/GBP realized spot, fixings, and the cumulative average EXHIBIT 29.4 EXHIBIT 29.3 530 ASIAN OPTIONS nta am xoue ihnteaeaert pinsoni xii 29.8. has option Exhibit rate average in profile. the price where shown the 29.4 in option Exhibit curvature in more rate seen be average also can the gamma Increased within exposures gamma initial exposures vega bucketed and the option 29.7. have on rate Exhibit notional in average dates EUR100m shown ATM to in 1yr option exposures EUR/GBP vanilla vega a European have a example, For to have occur. options options fixings rate Asian which average way, another causes Put dependence takes. path it path the on but 29.5 EXHIBIT erae xii 99sosvg rfie fe i otl xnswti the within fixings monthly six after profiles option. vega rate shows average 29.9 Exhibit decrease. stm assadtefiig e,teepsrso h vrg aeoption rate average the on exposures the set, fixings the and passes time As increased the to leads horizon the toward optionality moving of process This U/B vrg aeoto npiigtool pricing in option rate average EUR/GBP ahdependence path This . ASIAN OPTIONS 531 EUR/GBP average rate option versus equivalent vanilla bucketed vega profile EUR/GBP average rate option versus equivalent vanilla vega profile The gamma on the vanilla option increases closer to expiry as expected but the The vega has reduced on both options but it has reduced far more on the average options. gamma on the average rate decreases over its life as shown in Exhibit 29.10. rate option. Whenvega pricing exposure average leads ratevanilla to options, option. smaller as A TV a local adjustments sense volatility than model check, is the the often zeta reduced sufficient of for the pricing average equivalent rate EXHIBIT 29.7 EXHIBIT 29.6 532 ASIAN OPTIONS XII 29.8 EXHIBIT otl fixings monthly 29.9 EXHIBIT U/B vrg aeoto esseuvln ail am profile gamma vanilla equivalent versus option rate average EUR/GBP U/B vrg aeoto esseuvln ail eapol fe six after profile vega vanilla equivalent versus option rate average EUR/GBP ASIAN OPTIONS 533 EUR/GBP average rate option versus equivalent vanilla gamma profile after six Another feature of Exhibits 29.9 and 29.10 is the average rate vega and gamma Average rate options are easier to risk manage than the equivalent European Toward the end of the fixings, average rate options have virtually no gamma risk Before any fixings have occurred, the average rate will be most sensitive to peak exposures moving to loweroptionality spot occurs levels. when The the reason expected for future this value is of that the maximum underlying (in this case, at expiry by, for example, setting daily fixings for the lastvanilla week options due of to the reduced option. risk atTherefore, expiry an and generally average reduced rate Greek option exposures. isspread often than quoted the with equivalent a vanilla tighter option. premium bid–offer because so much ofitself, the there will average be a has deltathrough already jump the at been the strike. determined. spot The level Atdivided at size which the by of the final this resulting the average fixing delta number goes fixings. jump of For is this fixings: reason, the very average average rate small rate options are if option often the notional used to average reduce rate strike pin option risk has daily changes in spot because alltime future passes fixings and are fixings impacted occur, by spotthe spot moves payoff moves. will and However, have therefore as a exposures smaller reduce. and smaller impact on EXHIBIT 29.10 monthly fixings 534 ASIAN OPTIONS asadi sipratta rdr prcaeta hsdfeec xss The exists. difference this observations: of that number the appreciate by traders divided that in important average relevant is arithmetic particularly it is and effect ways This half. in tenors. longer drift at pairs the currency cut high-interest-rate-differential would averaging the 29.11. purely Exhibit derived in average shown an as with drift, along the option, from AUD/USD 1yr a spot on drift lower the Consider to pulled are be exposures must peak spot 0.8250), rate (also average the strike levels. the an the for have to hence Therefore, equal fixings 0.8250. 0.8140; be completed of around to spot six maturity prevailing the at a average months, with expected six compared After 0.8360 strike. of the average at is average) the 29.11 EXHIBIT npatc,aeae ihnAinotoscnb osrce ntodifferent two in constructed be can linear, options were Asian within drift averages forward practice, In the If drift. forward the curtails important. averaging also The is options rate average within drift forward Understanding U/S vrg aeoto owr drift forward option rate average AUD/USD stesmls omo vrg:tesmo h observations the of sum the average: of form simplest the is Average Arithmetic = ∑ i n = n 1 S i ASIAN OPTIONS 535 + + ) ) K K − − + + ) ) T Inverse S AV Arithmetic i 1 1 S − − 1 n T = i S AV ∑ Average within the standard vanilla payoff is n 1 Average ( =( =( ( . = . 1 1 is spot at maturity. strike CCY T CCY Inverse S Average Strike Put Average Strike Call Average Notional Notional uses the arithmetic average of the inverse observations: = = Payoff 2 2 Payoff CCY CCY Payoff Payoff is an average of fixings and Example: Average Strike Option with 12 monthly spot fixings harmonic average AV Forward: 0.8785 Notional: EUR100m Direction: EUR put/GBP call Currency pair: EUR/GBP Tenor: 1yr Spot: 0.8725 The following are example average strike option contract details: These variations basically come down to either calculating the average of the The where Within an averagereplaced strike with option, an average the of fixings: fixings in standardof terms the and fixings thendifferences and inverting but then such the calculating flexibility averageclients the is that or average. exactly required match in These taking their order FX may the flows. to seem inverse provide like precise irrelevant payoffs for For a harmonic average payoff,a the CCY1 reciprocal notional of option this payoff. inverse For average a CCY1 is call used harmonic within average rate option: The arithmetic average payoff isCCY1 naturally call quoted arithmetic in average CCY2 rate option: per CCY1 terms. For a Average Strike Options ■ 536 ASIAN OPTIONS tielvli nw,tegma(n l te re xoue)o h average the of exposures) Greek the other when all fixing, last (and the gamma After the time. known, over is up level builds strike gradually and equivalent occurred, the have than spot higher as profile 29.12. Exhibit vega spot in the to shown of closer as strike wings option strike the vanilla the the European keeps occur, keeps turn fixings averaging in As the This spot. although moves. with known, moves be strike to (expected) starts the because flat is ■ ■ ■ This path. ■ option: forward put the strike of average average an the within to because set occurs strike with EUR%) (2.85 option vanilla XII 29.12 EXHIBIT sso oe ihr h xetdsrk ee oe ihradteeoethe option. therefore vanilla equivalent and the higher to relative moves decreases level payoff call strike CCY1 expected strike average the higher, decreases. option moves call spot the As from payoff expected the lower, moves spot As decreases. option put the from payoff expected the the higher, moves therefore option. spot and vanilla As equivalent lower the to moves relative level decreases payoff strike put CCY1 expected strike the average lower, moves spot As h am rfiefo h vrg tieoto trsa eobfr n fixings any before zero at starts option strike average the from profile gamma The option strike average an from profile vega the occurred, have fixings any Before option: call strike average an within Whereas European equivalent the than cheaper is EUR%) (1.85 option average the Again, U/B vrg tieoto eapol sfiig occur fixings as profile vega option strike average EUR/GBP ASIAN OPTIONS 537 + ) 2 AV − 1 AV + 2) =( AV − 1 AV 1) daily for the last three months and ‘‘strike’’ fixings AV Double Average Rate Payoff Example: Double Average Rate Option 2 are averages of fixings. This is shown in Exhibit 29.13. Note Example double average fixing schedule AV 1and 2) daily for the first three months AV AV ( Within this trade, the ‘‘strike’’ is determined first, then ‘‘spot.’’ Exhibit 29.14 Tenor: 1yr Spot: 0.8725 Forward: 0.8785 Notional: EUR100m Payoff: EUR call/GBP put, i.e., ( Description: ‘‘Spot’’ fixings ( Currency pair: EUR/GBP The following are example double average rate option contract details: In general, average strike options will be quoted with a bid–offer spread roughly EXHIBIT 29.13 shows the bucketed vega exposures from the trade. where that these fixing periods may overlap. Within a double average rate option, bothEuropean the vanilla strike payoff and are the spot replaced within with the averages standard of fixings: equal to the equivalent Europeanaverage strike vanilla. options Like are average usually subdued rate comparedand options, to a the smile local equivalent volatility risks vanilla model option on is often sufficient for pricing. strike option will be exactly equal tothe the regular fact vanilla to that the expiry the date.pre-hedging However, of strike the is strike risk not impossible. exactly known until the final fixing makes precise Double Average Rate Options ■ 538 ASIAN OPTIONS ti adt eeaie u Vajsmnso obeaeaertsaeoften are single rates the than average more double even on Greeks the adjustments subdues variations. TV average averaging double but the generalize, because to small hard is strike. it the hence from possible spot, as because to far vega as close average long spot is average the trade moves the strike volatility fixed, implied the has increasing strike keeps the Once that optionality. move, the maximizing doesn’t spot if because 29.14 EXHIBIT omn osbepout a ecetdwti h obeaeaeframework average double the within created be can products possible many So short, is vega trade, the of months three first the during fixing is strike the While obeaeaert pinbcee eaprofile vega bucketed option rate average Double CHAPTER 30

Multi-Asset Options

hen the payoff from an FX derivatives contract is based on more than one Wcurrency pair it is described as a multi-asset option. Having multiple currency pairs within an option structure adds extra dimensions and can significantly increase risk management complexity.

■ Multi-Asset Trading Risks 539 Standard Greek exposures are not sufficient for managing multi-asset risk. Single values for delta, gamma, and vega make little sense when the exposures in a given currency pair depend not just on changes in that currency pair but on all the currency pairs within the structure. It is therefore vital for a trader to understand the methodology used to calculate exposures on multi-asset products within their pricing and risk management systems. If standard Greek exposures are used to risk manage multi-asset options, they certainly rely on strict assumptions about how the underlying assets move together. This can lead to big risk management shocks when the assumptions break down. Correlations between spot log returns are key parameters within the multi-asset framework. When pricing the two-currency-pair case, for intuition it is often useful to consider how the option payoff is impacted by perfectly positively correlated spots, perfectly negatively correlated spots, and spots with zero correlation. For multi-asset options with more than two currency pairs, correlations are often viewed within a matrix. For example, the following three-asset correlation matrix shows a 25% correlation between asset 1 and asset 2: ⎡100% 𝟐𝟓% 75% ⎤ ⎢ 𝟐𝟓 ⎥ ⎢ % 100% 30% ⎥ ⎣ 75% 30% 100%⎦ 540 MULTI-ASSET OPTIONS eemn pta auiyi ifrn urnypismyb ape tdifferent at sampled be may pairs currency different in maturity and at 1% spot by determine rise generated? rates is USD P&L what if 1%, example, by For to drop used position. all are volatilities the P&Ls implied in resultant risks the and major ways multi-asset the different trading identify in for flexed tool is data management Market risk positions. important more relatively a becomes to continue quants which in a area power. an model. of brain is notion considerable volatility this their the but apply mixed plus explored, developed, the been been has to smile have correlation equivalent models correlation no stochastic is and Local there Certainly calculations. models. within used pricing be must correlation CCY2 the and of pair negative one market the in the pair, CCY1 If other as cases. the currency both in common in the CCY2 has or ordering CCY1 pair is convention currency common the that such pairs. pairs major between to correlation to possible exposure Alternatively, as pairs. therefore viewed currency is be relevant can all it risk across cross-volatility terms 16, vega Chapter in risk from multi-asset view framework triangle pairs volatility currency ATM major in surfaces volatility Rebonato’s Riccardo in exists issues 2004). these Edition, on material best book The about. worry to correlated: correlated, not 100% are are 3 3 and and 1 2 assets assets and correlated, but 100% easier meaning are ‘‘valid,’’ 2 often and be 1 is assets always example, process must flexing matrix this correlation semi-definite However, The impacted. done. than is said price option the how ares(e hpe 5 osmlf ut-se ikmanagement. risk multi-asset simplify to 25) Chapter (see barriers this where cases in charged be should arise. spread may bid–offer issue Additional day. the of times ihnrs aaeeti scmo oflxtecreainmti n observe and matrix correlation the flex to common is it management risk Within rcn oesfrmliastotosaenta eletbihda single-asset as well-established as not are options multi-asset for models Pricing currency two between quoted are correlations that 16 Chapter from also Recall tradable are there FX in cross-stocks, of notion no is there where Equities, Unlike correlation terminal versus correlation instantaneous around issues also are There ial,taesotnatvl rt-f ik(e hpe 5 n s shadow use and 15) Chapter (see risk write-off actively often traders Finally, to used fixes the that is options multi-asset within factor complicating Another analysis scenario sophisticated, less generally are models pricing multi-asset Since oaiiyadCreain h efc egradteFox the and Hedger Perfect The Correlation: and Volatility nmteaia em.Ti rvnscreainmtie hr,for where, matrices correlation prevents This terms. mathematical in ⎣ ⎢ ⎢ ⎡ 100 100 100 % % % 100 100 0 % % % and 100 100 rs urnypis ealn the Recalling pairs. currency cross 0 % % % ⎦ ⎥ ⎥ ⎤ Jh ie os 2nd Sons, & Wiley (John positive MULTI-ASSET OPTIONS 541 . basket sum to .Vega i 𝜔 in currency t . Weights i is the spot at time replication spreading i i i , , , t t 0 S S S confidence interval spreading i 𝜔 1 n = i ∑ is generally lower for short-dated options. = price 𝜕 basket correlation , t 𝜕 S but the flexibility exists to offer different variations. n 1 is the inception spot in currency pair i , 0 S currency pairs in the basket, n , and therefore i move. At maturity, the basket spot is compared to a basket strike and a cash The following formula can be used to calculate the basket spot. Each currency A combination of these approaches can also be used, with replication spreading The second approach can be thought of as where there are pair 1 and are almost always set to pair within the basketThis must may be mean, for quoted example, such flipping AUD/USD that into CCY1 USD/AUD is terms: the common currency. Basket options contain a number of currencythem. pairs Spot with a moves common in currency the between basketspot pairs are normalized and averagedsettlement is to generated create based a on a basket vanilla or basket digital payoff. used in major currencycurrency pairs. pairs and confidence-interval spreading applied in cross- exposures are calculated inATM volatility all bid–offer major spread in and each pairspread. and cross then This summed currency to approach get pairs, a is total multiplied bid–offer appealingspreads by because but the it it uses will realany usually discount implied overestimate for volatility offsetting bid–offer bid–offer risk. spread because it does not give However, it does notthe take product into and account will theothers. generate liquidity It must too of also the much be understood currency spreadthan that long-dated pairs in short-dated correlations correlations within some and are cases much lessTherefore, and stable correlation not must be enough flexed in more on contracts with shorter maturities. options. The first approach can beCorrelations thought of are as flexedassessing up the and variability down ofdetermined. to historical As generate correlation a a an rough1yr), rule appropriate change of correlation correlation in thumb, flexes flex option in around is and G10 value. 10% convenient pairs to By method, for 15% plus medium it are tenors gives often (1mth a appropriate. to good This intuitive is feel for a the neat risk in the trade. Two standard approaches exist for calculating bid–offer spreads on multi-asset Basket Options Multi Asset Bid–Offer Spreading ■ ■ 542 MULTI-ASSET OPTIONS oe rs-oaiiy aktotosaeln eai h ao urnypairs currency major the in vega long are options Basket static. cross-volatility. be would lower spot basket the directions; opposite where and the basket equal of two-pair in stylized average move a spots Consider an the reduces. as volatility basket volatile lower the move as 100%, pairs between than be correlations the would as spots However, spot volatilities. component spot basket all component If the basket. correlated, the in 100% spot pairs were currency basket the depends the between spots and component correlations the the drift) to on compared forward spot basket component the of the volatility The in of volatility volatility. average seen spot some options basket (i.e., forward the Asian the reduces the basket from with volatility. the spot returns As component within the spot. realized to averaging compared basket shows the the 30.1 cash 29, plus the Exhibit Chapter and rates calculated. specified spot is is notional individual expiry the which at in USD payout currency common has basket than cheaper are they because case. clients standard by the preferred often are payoffs basket Inverted maturity: at payoff the flipping And spot: basket the calculate to used formula the is: maturity at payoff call basket A 100%). (e.g., percentage a as quoted often Brazil, index. of currencies financial the a contains constructing which China. to BRIC, and India, is similar Russia, basket is FX basket well-known most a The constructing of process The uuaiesra eutn rmvg iki l ao n rs urnypairs currency cross and major offset. risks all vega in the since risk basket, the vega within from the resulting in spread vega cumulative short hence pairs, currency major the pairs. cross-currency between correlation long and aktotosaeuulyqoe ihatgtrbdofrsra hnthe than spread bid–offer tighter a with quoted usually are options Basket implies pairs currency major between correlation higher 16, Chapter in noted As basket the on dependent is basket the of value the option, vanilla regular a Like Example flipping by basket ‘‘self-quanto’’ or ‘‘inverted’’ an construct to possible also is It are strikes basket therefore and 1 at starts spot basket the framework, this Within y U/B/U aktcl essUDwt 0%srk.The strike. 100% with USD versus call basket EUR/GBP/AUD 1yr : Payoff Payoff aktCall Basket Call Basket = = Notional Notional S t , basket CommonCCY CommonCCY = ∑ i = n 1 . . 𝜔 max max i S S 0 t ( ( , , i i S K T T , , basket basket − − K S T T , , basket basket , , 0 0 ) ) MULTI-ASSET OPTIONS 543 between correlation GBP/USD is below 1.6000 at expiry. At inception, and . If the basket spot is near the basket digital level at expiry, this can Realized component spot and basket spot returns One variation in which the risk management becomes more challenging is a Baskets sound complex but in most cases the averaging effect significantly reduces EUR/USD is above 1.3000 if EUR/USD spot is atand 1.3000 there and was GBP/USD spot noaround is significant 50%. at drift The 1.6000 or price (the of digital discounting,EUR/USD levels) the and each GBP/USD dual individual spots. digital digital then depends will on cost the Dual digital options arecurrency combinations pairs of but two to the Europeanare same digital expiry in-the-money options date. at in The dual separate expiry. digital For pays out example, if a both digitals dual digital might payout USD1m if basket digital be difficult to manage because all spots withinpossible the to basket accurately impact hedge the the payoff, digital so risk it with is a not standard European digital contract. their trading risks. Avolatility copula surface into (see account Chapter andto this 12) generally the approach captures the volatility can majorityproduct be surface. of and exposures used they Basket are to options attractive because takeindividual are the vanilla the basket commonly options. is usually traded cheaper than as buying the an investment EXHIBIT 30.1 Dual Digital Options ■ 544 MULTI-ASSET OPTIONS up o8%adteda iia Vjmst 4.Ti ikpi au rm25% from value in pickup This 64%. to TV jumps digital TV digital individual dual each the currency, and jump common spot 80% the correlated to in highly jumps event a external If 25%. an is an to TV has due digital option occurs dual digital the each and 50% where EUR/GBP) of product TV digital (i.e., individual dual a cross Consider product. the the to within exposure long a has option digital volatility. dual this volatility: that cross and correlation between higher link the again Recall exposure. be shown will is TV digital digital dual dual the and spots, correlation 30.2. the Exhibit between in relationship between The correlation 50% 25%. 0% at at is valued valued triggering there be lower) GBP/USD If will (and higher payout. higher digital EUR/USD EUR/USD the of dual chance requires half the a is that manner, there because since correlated negatively payout perfectly would to a There product 0%. the at valued for and be chance will no digital dual be the manner, correlated positively XII 30.2 EXHIBIT uldgtlotoscnb ifiutt ikmng eas fteimplicit the of because manage risk to difficult be can options digital Dual correlation short significant a has digital dual this how shows 30.2 Exhibit perfectly a in move spots GBP/USD and EUR/USD if world, stylized a In B/S oe.Atraiey fERUDadGPUDsosmv in move spots GBP/USD and EUR/USD if Alternatively, lower. GBP/USD = rs-oaiiylower cross-volatility uldgtlT esscorrelation versus TV digital Dual hsi eosrtdi xii 03 hc confirms which 30.3, Exhibit in demonstrated is This . correlation leverage MULTI-ASSET OPTIONS 545 Dual digital TV versus cross volatility If one digital is far out-of-the-money and the second is at-the-money, the primary If one digital is far in-the-money and the second is at-the-money, the primary currency pair moves in-the-money. Again, trading riskcurrency is mainly pair concentrated rather in one thantogether. being dependent on how the currency pairs are moving this case, trading risk isdependent mainly on concentrated how in the one currency currency pairs pair are rather moving than together. being trading risk transferspayout onto it the is out-of-the-money most digital. important Intuitively, that to get the the spot within the out-of-the-money digital Therefore, the main exposures withinhow in-the-money the each product digital change is. depending on relatively trading risk transfers ontoto a the standard at-the-money digital digital option.maturity, hence and Intuitively, it one the is less digital risks important is than will the likely digital be to payout which be similar is in-the-money in at the balance. In to 64% is difficultfrom to this hedge issue using increases for vanilla e.g. options. a The triple risk digital contract. management challenge Trading Risks The dual digital product pays out if both digitals are in-the-money at maturity. EXHIBIT 30.3 546 MULTI-ASSET OPTIONS xouei ocreain u htcreainepsr sbigrpeetdas represented being is exposure correlation main the that contract vega. but digital EUR/USD correlation, dual to this Within is price. exposure GBP/USD higher versus a EUR/USD therefore lower In and in triangle. correlation results volatility volatility valid EUR/USD a is lower maintain that case, to this updated one is cross- the correlation EUR/GBP for instead the and impact except not volatility constant does volatility kept EUR/USD Therefore, are flexed. system being the within volatilities implied payout. a of chance volatile less the a increase of having not simply chance does Intuitively, the spot why? increases EUR/USD but value, that option lower, the goes hence and volatility payout implied EUR/USD if 30.4. 1.6000, Exhibit at in shown level as digital lower GBP/USD moves the profile as vega Then, have EUR/USD reversal. will the risk digital lower, a EUR/USD moves GBP/USD to the similar the payout, profile certain If vega a familiar level. with a digital high, GBP/USD very starts variable level a digital vega with the EUR/USD consider example, in digital dual exposure GBP/USD and EUR/USD the to back Going Risk Vega their (i.e., together moving are pairs currency correlation). two the in spots how on concentrated XII 30.4 EXHIBIT hsepsr eut rmasmtosmd ihntevg aclto.All calculation. vega the within made assumptions from results exposure This at-the-money-spot, level digital GBP/USD the With moment: a for this Ponder mainly is risk the value, same the roughly have digitals individual both If uldgtlERUDvg o ifrn B/S iia levels digital GBP/USD different for vega EUR/USD digital Dual MULTI-ASSET OPTIONS 547 Dual digital GBP/USD vega profile These examples show why, when risk managing multi-asset options, it is vital to Correlation also impacts the GBP/USD vega exposure in the same way. Again, but cross-volatility moving) Short EUR/USD versus GBP/USD correlation exposure Flat EUR/USD vega (with correlation and thebut other cross-volatility major volatility moving) held constant Flat GBP/USD vega (with correlation and the other major volatility held constant Short GBP/USD vegacorrelation (with moving) all other impliedLong EUR/GBP volatilities cross vega held (withcorrelation all constant moving) other but implied volatilities held constant but Short EUR/USD vegacorrelation (with moving) all other implied volatilities held constant but EXHIBIT 30.5 ■ But using a different calculation methodologyrisk it as: would be possible to view the same ■ ■ ■ ■ understand what assumptions areexample, used the exposures to calculated calculate are: Greek exposures.■ Within this for an at-the-money-spot EUR/USD digitalat the current GBP/USD spot. This vega is becomes shown negative in Exhibit 30.5. 548 MULTI-ASSET OPTIONS ■ eto n os-fOptions Worst-of and Best-of hr h pta auiyi urnypair currency in maturity at spot the where is worst-performing it if or exercised best- be CCY1 only the will be only it can option date, vanilla currency expiry normal common a the in-the-money. like the On and and exercised pair. puts is the option or in calls CCY2 be between currency or can common options. a Payoffs rainbow be must pairs, traded there and commonly the date most expiry same the the has far the payoff by out Each (pays are worst-of and options payoff) payoff) maximum the minimum payoffs. out vanilla ranked (pays of best-of function derivatives, some FX is In option rainbow the of payoff the maturity, options Rainbow using pair exposures. currency vanna major each or in chart risk spot smile versus the vega assess a to easier it makes also It trade. ■ ■ prices: option Price Price h olwn rcn dniisgv nutv onso eto n worst-of and best-of on bounds intuitive give identities pricing following The cannot: worst-of the where reduced be can payout best-of The pair currency in CCY2 is currency common the pair If currency in CCY1 is currency common the If in options for symbols, In the on risks the view to way intuitive more a arguably is formulation second The Payoff Payoff Best Worst - of - of Worst Best ≥ ≤ - n igevnlaoto price option vanilla single any of - Price of Payoff = = Payoff = max max Best min aepyfsgnrtdars utpecrec ar.At pairs. currency multiple across generated payoffs have Worst - Best of ( ( ( - - max S max of of T , = = 1 n S ( T − ( urnypairs: currency , max min 1 S K S T T 1 , ( ( 1 S , , 1 payout S payoff T − T S − , 1 , T 1 K , 2 K S 1 T 1 − 1 , 1 , , 2 , , 0 payoff 0 i K payout ) ) sdenoted is 2 , , , max … max 2 2 , , , ( … ( 0 i i … : : ) S S payoff payoff T S , T , 2 T , S , 2 S payoff , T i − payoff T . − , 2 , 2 i i K K = = 2 2 n , ) , n 0 S S ) 0 T T S ) , , K i i ) T − − , i K K , , … … ) ) MULTI-ASSET OPTIONS 549 sum of the two vanilla option prices = of - Worst any single vanilla option price sum of all vanilla option prices Price ≤ ≤ + of - of of - - : Best-of EUR put/USD call and GBP put/USD call option. In this example, Worst Best Best In general, it is important not to over-hedge best-of and worst-of options. The When pricing a worst-of contract it can be instructive to remove each currency Plus, if only two currency pairs are included within the structure: Worst-of options are often a lot cheaper than the equivalent vanilla options. Price Price Price the major pairs are EUR/USD andoption GBP/USD and is the cross-pair long is vega EUR/GBP.volatility This in plus the the best-of major hasdirections. pairs Therefore more the since best-of value is option if long buyers vega theexposure in implies two always the long cross-pair major like since cross-vega. spots short increasing correlation move in opposite keep changing over their life. Initial hedgesor will almost rebalanced. certainly need to be unwound Vega Risk Example 1 often be moved deep in-the-money at veryof little being cost the because it lowest has payout. aeffectively Moving low removes probability the that strike pair such fromcorrelation that the risk. spot worst-of is and deep hence in-the-money reduces potential future dynamics of these products mean that their risk profiles change dramatically and pair in turn anddiscount compare within the the price new is TVs.are coming deep This from in-the-money. approach and This technique it shows canworst-of is a also particularly position: be trader applicable When used where if a when payouts the risk particular managing spot a is short in-the-money versus the strike, it can in one major currencyhow pair. certain it The is split which option betweenthe (if rainbow the any) option will two be have exercised. types similar Ifvanilla values, of the options correlation vanilla risk has risk options a depends is within value maximized. on than significantly If larger the one (for rest, of best-of) the it orrisk becomes smaller reduces, (for clearer and worst-of) which vega risk vanilla in the that trade currency pair will increases. become, correlation strong directional spot views over multiple currency pairs. Trading Risks At different times during the lifecould of be a to best-of or correlation worst-of between option, currency the pairs main exposure or it could be to implied volatility ■ This makes them attractive to institutional investors (particularly hedge funds) with ■ ■ 550 MULTI-ASSET OPTIONS ssoni xii 08 gi,hge orlto mle ihrwrto price worst-of higher a case implies direction correlation payoff higher same Again, the 30.8. to Exhibit similar in shown is as relationship correlation put/ versus EUR TV (e.g., The different is currency common option. second-order this zero implies relationship linear price. best-of lower worst-of higher a a and to leads price correlation higher profiles; correlation versus TV the put/ shows EUR (i.e., pairs both in same that the risks currency different common the the on is direction call/put options not. same does worst-of the it in and or out best-of pays option of the if aspects occur key the of One Risk Direction Payout should change. and will multidimensional delta are such. options AUD/USD as viewed worst-of the be and therefore moves, best-of EUR/USD in as exposures Delta example, this In pairs. for spots correct the being how of being chance purposes. little of management have chance risk numbers little single-delta has hence and This move and strain. actually move, can the not all delta does takes AUD/USD EUR/USD EUR/AUD itself, the hence by AUD/USD, moves and AUD/USD assuming EUR/USD For calculated isolation. in caution. be in worst-of with moves pair a treated currency for be major to example, each that need assumes options often worst-of calculation and The best-of on exposures Delta calculation? Risk the Delta within moves what and constant held is What vega: calculate to cross-pair used the in vega spots short major cross-vega. is increasing short two worst-of implies like the the correlation if buyers Therefore long value since direction. option more same again, has the since, worst-of in pairs the move major time this the However, in volatility. vega long is option h aofdrcino h omncrec sdfeet h eainhpis is relationship it The case; different. direction is payoff derivative currency second same common the the from on different direction payoff the price. best-of lower a and xii 09sosterltosi ewe rs-oaiiyadT when TV and cross-volatility between relationship the shows 30.9 Exhibit the on direction payoff the where case two-pair simplified The a TV. consider and Now cross-volatility between relationship the shows 30.7 Exhibit is currency common the on payout the where case two-pair simplified a Consider currency other in delta the impacts pair currency one in move a practice, In methodology the understand to taken be must care case, digital dual the in As 2 Example os-fERptUDcl n B u/S aloto.This option. call put/USD GBP and call put/USD EUR Worst-of : 𝜕 𝜕 crossvol price ie,cosvla xoue hslast nrae risk increased to leads This exposure. cross-volga) (i.e., S call USD n U put/ AUD and 𝜕 𝜕 crossvol S call USD price cosvla xouewithin exposure (cross-volga) olne linear longer no n U call/USD AUD and S call USD .Ehbt30.6 Exhibit ). mliga implying , put ). MULTI-ASSET OPTIONS 551 TV versus cross-volatility profiles for best-of and worst-of options: Same payoff TV versus correlation profiles for best-of and worst-of options: Same payoff direction on common currency EXHIBIT 30.7 EXHIBIT 30.6 direction on common currency 552 MULTI-ASSET OPTIONS XII 30.9 EXHIBIT currency common on direction 30.8 EXHIBIT aofdrcino omncurrency common on direction payoff Vvru orlto rfie o eto n os-fotos ifrn payoff Different options: worst-of and best-of for profiles correlation versus TV Vvru rs-oaiiypolsfrbs-fadwrto pin:Different options: worst-of and best-of for profiles cross-volatility versus TV MULTI-ASSET OPTIONS 553 . ) K ( ) 0 , K options to distinguish them from − QF T S ( max . GBP Notional third currency quanto : 1yr EUR/USD EUR call/USD put 1.3500 option that pays out at is called a quanto-factor and is usually equal to the strike QF Example These strategies give intuition as to why worst-of options are cheap and best-of For worst-of options, buy the least expensive single vanilla initially and switch it In general, this cross-volga exposure causes worst-of options with different payoff where are sometimes called self-quanto options (see Chapter 27). maturity in GBP. To calculateis the typically payoff, achieved a by calculating, GBP% for payoff example, must be calculated. This Within quanto options,rather the than option CCY1 or payoff CCY2 is within denominated the in currency pair. a Therefore, third these currency, products at each switch). At expiry, this strategy will have the worst vanilla payoff asoptions required. are expensive. Itwhen vanilla also values shows are similar why becauseswitching more the hedge. switches would risks be are required within more the difficult to manage worst-of options. For best-of options, buyand the switch most expensive it single into vanilla(costing another initially money currency pair at when eachpayoff the switch). as most At required. expensive expiry, vanilla this changes strategy will have theinto best another currency vanilla pair when the least expensive vanilla changes (earning money double-no-touch options: Flex the cross-volatilityfind up the and average down TV a to fixed put amount an and approximate value on the second-order exposure. Switching Hedge Finally, it is interesting to consider a ‘‘switching hedge’’ strategy for best-of or direction on the common currency to tradepayoff more direction over on TV the than common trades currency. withpricing the If models same no are sophisticated available stochastic to correlation using a value simple the method that cross-volga is exposure, analogous to it the can old-school ‘‘alpha’’ method be for estimated pricing management challenges plus the different payoff directionsusually on the cause common the currency option toleverage have that will a be low difficult valuation to and hedge using hence vanilla the options. trade has increased Quanto Options ■ 554 MULTI-ASSET OPTIONS rrs aaeetise nesterntoasaevr ag.Tecreainrisk correlation The large. very are notionals pricing their significant unless any issues cause management rarely risk they quanto or position, Once trading model. the volatility be into local also booked quanto should are a options surface example, volatility for the using, wings account or into skew taken significant but with pricing, pairs the currency within adjustments in main the are discounting and adjustment forward ■ ■ ■ that: such shifts therefore 𝜌 where becomes: standard the Specifically, currency. quanto the equation: in differential stochastic discounting applying and drift the third- to due perspective, vanillas management regular than risk complex element. a more third-currency slightly From are currency. options specific quanto currency a in payout their nqat pin ssalrltv otersso uldgtlo os-foptions. worst-of or digital dual on risks the to relative small is options quanto on stecreainbtenCY/C2adCY/C2 h unoforward quanto The CCY3/CCY2. and CCY1/CCY2 between correlation the is eaieylwradtepieo unocl pini oe ntequanto the moves in forward quanto lower the correlated, is negatively USD. are in option GBP/USD is and EUR/USD call option If call quanto vanilla European a regular of the than price (GBP) currency the and moves forward lower quanto the relatively pricing correlated, positively are only GBP/USD and the EUR/USD If volatility, zero come discounting. will has currency option quanto European GBP/USD regular the the from and or option quanto zero the between is difference correlation the If eaieyhge n h rc fteqat aloto shge ntequanto the USD. in in higher is option is call option vanilla call European regular quanto the the than of (GBP) price currency the and higher relatively ti osbefrayoto aoft eqate noatidcrec.The currency. third a into quantoed be to payoff option any for possible is It option: call GBP into quanto EUR/USD example the pricing when Therefore, the adjusting by priced are options quanto framework, Black-Scholes a Within require clients where products investment as used often most are options Quanto 𝜎 1 stevltlt fCCY1/CCY2, of volatility the is dS S t t =( dS S rCCY t t =( 2 rCCY F − quanto rCCY 2 = − 1 F rCCY − . e 𝜎 − 2 𝜌.𝜎 𝜌.𝜎 1 stevltlt fCY/C2 and CCY3/CCY2, of volatility the is ) 1 1 dt .𝜎 .𝜎 2 + 2 ) 𝜎 dt dW + t 𝜎 1 dW t CHAPTER 31

Miscellaneous Options

xotic FX derivative option types not yet examined include ‘‘volatility’’ products Elike volatility swaps, variance swaps, and forward volatility agreements, plus forward start options, and compound options. 555

■ Volatility and Variance Swaps

In FX derivatives:

■ A volatility (vol) swap is a forward contract on the realized volatility of spot over an agreed period.

■ A variance (var) swap is a forward contract on the realized variance of spot over an agreed period. Most commonly, daily spot fixings are used to determine realized volatility and variance. Mathematically, for daily spot fixings, realized variance (V) is calculated using this formula: ∑N 252 2 V = × ai N i=1 ( ) Si where ai are spot log returns, that is, ai = ln , Si is spot fixing number i,and Si−1 in total there are N log returns. 556 MISCELLANEOUS OPTIONS ainesa aofcnb efcl elctdb otatta asota out date. expiry pays same that the contract to the a options rates), by interest replicated perfectly deterministic contract be plus can jumps payoff Black- a swap spot in variance no because, product (specifically, natural world more the Scholes theoretically are entitled swaps 1999 variance how from Zou and Kamal, Derman, ‘‘ Demeterfi, mean. by they paper what Sachs is Goldman this ‘‘price’’, swap payment. var upfront or zero swap vol for the quoted about is institutional talk strike traders for variance When quoted or are volatility volatility swaps two-way the variance a to and clients, equal volatility swaps approximately the volatility is from When payoff away payoff. swap percent swap variance 1 the is maturity, volatility at realized strike if that ensures formula This squared: point volatility per amount cash amount: vega a where is return mean the If required. is sometimes it clients whether becomes: check However, calculation to the result. important added, is the it impact and significantly it request not will this usually eo h oaiiysrk h ainesa oe esta h oaiiyswap. swap volatility variance the the on than quoted rate less the tenor, loses and swap pair currency whereas variance given swap, a the volatility in the Therefore, strike than volatility more as gains the above volatility, swap i.e., below variance realized volatility, realized the in in strike linear convex volatility is is the payoff payoff swap swap variance volatility the The while expected, strike. 12% a and vega oeTa o vrWne oKo bu oaiiySwaps Volatility about Know to Wanted Ever You Than More oaiiysa n ainesa aof r hw nEhbt3. o USD10k for 31.1 Exhibit in shown are payoffs swap variance and swap Volatility a is swaps var and swaps vol on document research quantitative classic The formula: this with vega to linked usually is notional variance The a is notional the and terms volatility-squared in quoted is strike the swap, var a In effectively is notional the and terms volatility in quoted is strike the swap, vol a In and formula variance realized the in included not usually is return mean The a stema ftelgreturns. log the of mean the is tepr,wihi uncnb elctduigaprfloo vanilla of portfolio a using replicated be can turn in which expiry, at Payoff Payoff volswap varswap V Notional = = = 252 N Notional var Notional × = ∑ i = N Notional 1 vol ( 2 var a . . i K 2 . ( vol √ ( − V vol V a − ) − K ’ h ouethighlights document The .’’ vol K vol 2 ) ) log MISCELLANEOUS OPTIONS 557 Volatility swap and variance swap payoffs Prior to any fixings, as spot moves, the value of the vol swap remains constant. Variance swaps are a popular product in the equity derivatives market but they Vol swaps and var swaps are attractive products for clients because they give a throughout its life, as fixingsThis occur, makes more is the fixed payoff and lessand less sensitive hence is to exposures still changes generally to reduce in be over market determined. time. data (like Asian options), This is shown in Exhibit 31.2. volumes. Volatility Swap Greeks In terms of risk management, the important thing about the volatility swap is that pure payoff based onvanilla realized options volatility do not or give variance.or pure time As exposures passes has because their as exposures been change. spot observed, moves, ATM forwards move, are relatively less popular in FX. Inin FX, the both interbank volatility broker and market variance but swaps in are volumes traded which are a fraction of vanilla option will be higher than thevol equivalent swap rate versus quoted var on swapthis the strike ‘‘convexity volatility adjustment’’ difference can swap. can be Tracking be valued used the by to modeling volatility price itself. volatility swaps, or EXHIBIT 31.1 558 MISCELLANEOUS OPTIONS up aetegmaepsr adrt rd hntegmao standard a value. peak on delta the gamma These to the back fixing. jumps than previous gamma trade fixing, the each to from at fixing harder Plus, far option. each exposure is vanilla at gamma spot jump the if delta make a large jumps causes particularly This be reset. will exposures that the occurs, fixing. fixing previous each the As at is gamma peak the that such moves it that except 31.5. Exhibit in shown as fixing delta The lower. spot with 31.4. shorter Exhibit and in shown higher is spot profile with longer moves then fixing, strike. the from moves spot further the log the 31.3. as Exhibit increases in value shown swap is volatility This increases. the return hence and increases volatility realized 31.2 EXHIBIT n h easasucagda h xn aebtmvslwri h wings. the in point lower reference moves a but rate has fixing trade the the at occurs, unchanged fixing stays vega first the the and When values. spot all across eoeayfiig aebe ae h eao o wpi oiieconstant positive a is swap vol a of vega the made, been have fixings any Before practice. in work exposures these how on moment a for dwell to important fixing is each It at unchanged approximately remains profile gamma the of previous shape the The at gamma peak the with gamma long is product the way, another Put the at zero at starts that position delta a implies 31.3 Exhibit in profile rises value value The option; ATM overnight an long being to similar is payoff this that Note lower, or higher either fixing, each from away moves then spot as fixes, spot After oaiiysa au ro oayfixings any to prior value swap Volatility MISCELLANEOUS OPTIONS 559 Volatility swap delta after fixing Volatility swap value after fixing EXHIBIT 31.4 EXHIBIT 31.3 560 MISCELLANEOUS OPTIONS wyfo xdsrk.I o wp ti si h ‘tie’kesrstigback resetting keeps ‘‘strike’’ the if as is moves it spot fixing. swap, each as at vol reduce ATM a volatility the permanently In why to strike. not reason at fixed one does is a is vega peak from This clients; vega away fixing. this with previous but popular the profile, at are the peak swaps of the than peak level the lower to a resets vega fixes, swap volatility wings 31.7. the Exhibit in in lower fixing. shown is is each vega This after and fixing. wings tightens each the distribution after in the volatility, higher implied is lower vega At and widens distribution set the volatility, are fixings all because symmetric is 31.6 Exhibit 100.00. in to shown profile volatility vega the realized that fixing, reduces. previous vega ‘‘optionality,’’ this the from from away further lower wings, or the In higher increases. either moves spot If 31.5 EXHIBIT eestevg oiina ahfiig lsvltlt wp aemnmlinterest minimal effects. have second-order minor swaps some volatility and Plus discounting from fixing. apart each exposures, at rate position vega the resets oaiiysashv otaal an rvlaepsrssneteposition the since exposures volga or vanna tradable no have swaps Volatility the When practice. in evolves vega how understand to important is it Again, implied higher At volatility: implied of level the Note by impacted occur. also fixings is vega more swap Vol as time over rate linear a at reduces vega peak The straddle: ATM overnight an to similar optionality of type a contain swaps Volatility oaiiysa am fe fixing after gamma swap Volatility MISCELLANEOUS OPTIONS 561 Volatility swap vega at different levels of implied volatility Volatility swap vega over time EXHIBIT 31.7 EXHIBIT 31.6 562 MISCELLANEOUS OPTIONS wyrglryoe iea xnsocr hsi hw nEhbt31.10. Exhibit in shown is This occur. fixings as time over regularly away ■ ■ as constant ■ approximately stays results: important also are that These exposure passes. gamma time constant a implies This shown is fixing, This each increases. spot from value all away swap 31.8. over moves Exhibit variance in static spot hence and as and increases Then, constant variance occurring. is realized fixings value any swap to prior variance values swaps, volatility with As Greeks Swap Variance 31.8 EXHIBIT am’ r‘srkls vega.’’ ‘‘strikeless 31.11. or gamma’’ Exhibit in shown is This spots. all over exposure volga long h o otatgnrtsacntn am xoueoe l spots. all over exposure gamma constant a generates contract log The spots. all over gamma constant contract’’. have ‘‘log swaps called Variance so a with replicated be can swaps Variance osatgmaadvg rfie vrso r oeie ald‘‘strikeless called sometimes are spot over profiles vega and gamma Constant constant decays a implies and which volatility, spots implied all in linear over is swap constant variance a is on vega swap The variance a on vega way, similar a In 31.9. Exhibit in shown as spot with linearly moves delta swap, variance a Within ainesa au fe fixing after value swap Variance MISCELLANEOUS OPTIONS 563 Variance swap vega over time Variance swap delta after fixing EXHIBIT 31.10 EXHIBIT 31.9 564 MISCELLANEOUS OPTIONS ihCY aotwl ewrhmr hnteeuvln otatwt CCY1 with contract equivalent the than more worth be swap variance lower, will pair or spot payout volatility With currency a CCY2 downside. so implies a CCY1, with the than strikes to in valuable further downside more example, relatively moving on becomes of For CCY2 volatility chance surface. higher increased an reversal, volatility has risk the spot downside when payout in large occur which skew a differences confirm large with valuation to a important large is is Particularly it there requires. so client different the quite currency be can payouts CCY2 the is swaps effect. variance is this it by swaps so lower vol fixings pulled more for rate therefore quoted used and the often USD see the to most than common calendar days is formulas holiday holiday the fewer The in has chapter. which quoted The the WMR, as calculation. of 252, the start is within the trades fixings swap at interbank of volatility within exactly number the used almost the However, multiplier tenor. by standard is same impacted price the additionally swap to is volatility rate volatility ATM the the world, to Black-Scholes equivalent theoretical a In Pricing Swap Volatility 31.11 EXHIBIT ar esspiigte nmngdeegn aktcrec ar.I liquid In pairs. currency market emerging managed in them pricing versus pairs payout. hr sabgdfeec ewe rcn o wp nlqi 1 currency G10 liquid in swaps vol pricing between difference big a is There and swaps volatility quoting when of aware be to point important Another ainesa eaa ifrn eeso mle volatility implied of levels different at vega swap Variance aotcurrency payout h utdvltlt tie o C1versus CCY1 for strikes volatility quoted The . MISCELLANEOUS OPTIONS 565 , not the realized forward spot volatility at maturity. The log contract replication sets up vanilla log contract By trading strikes with these wide gaps between them (10 figures), the accuracy In practice, wing vanilla options are often used to hedge a volatility swap vega It is also important to understand the spot dynamic in the currency pair. This Volatility swaps and variance swaps need special attention in emerging market In managed emerging market currency pairs it is important to remember that of the replication reduces towardgamma expiry. exposure A over perfect all replicationpossible providing spots strikes between constant would zero and require infinity. vanilla (No, that options isn’t to actually possible.) be traded at all Variance swaps can be perfectly replicateddate. with Specifically, vanillas the to replication the variance involvespays swap transacting expiry out a strip a ofnotionals vanilla such options that that they arenotionals are inversely shown proportional in to Exhibit the 31.12. strike squared. Example range. However, overincreases time if this spot moves hedge closeway must to through one be the of strikes, rebalanced the the hedge hedge as will strikes. no gamma Plus, longer if exposure work. spot jumpsVariance a Swap long Pricing currency pairs because the vegaexchange stays with rate spot jumps as it andswings moves. for If implied these the volatility products emerging than market spikes, vanilla options this with can equivalent vega. causeprofile. far Specifically, larger 10 delta P&L strangles usually hedge vega over a wide enough spot volatility using daily fixings and comparing it to the ATM volatility. allows a traderintervene to in assess the the spot market?will expected this Is spot impact the realized distribution: fixing volatility? What an Does else average might the or cause a spot central to point-in-time bank jump? fix? How track the differential between theirover models time and and the quote interbank reflecting broker this market adjustment. prices the volatility swap pays outvolatility, on so the forward realized volatilitypricing must a be vol stripped swap. out This effect of can the be ATM assessed volatility by when calculating historic realized spot This counterintuitive result occursthe because same both variance models andof since (if the local well-calibrated) volatility volatility have has willgood a be indication lower for higher. vol-of-vol short-dated A the volatilitystochastic well-calibrated swaps expectation interest while mixed rates at should vol longer additionally model tenors be the often quantified. effect In gives of practice, a traders often G10 currency pairs,Local pricing volatility models gives can anstochastic be approximate volatility upper used gives bound an to for approximate generate the lower reference quoted bound vol for points: the strike quoted and vol strike. 566 MISCELLANEOUS OPTIONS ■ owr oaiiyAgreements Volatility Forward owr oaiiyareet FA)aeaohrcmol rddvolatility dates. expiry two traded between volatility commonly implied forward another a as are quoted are (FVAs) They product. agreements volatility Forward contracts. swap variance on moves. to an P&Ls where daily big added pairs generate maximum could currency sometimes de-pegging pegged or are rate in exchange barriers applicable restrictions particularly knock-out reason are e.g., considerations this These contract, For swap enough. no variance contract far this the will swap practically variance moves in hedge no short spot a swaps Therefore, the from squared. if protection volatility way, volatility provide will than on long hedge based dangerous vanilla a is attainable more payoff jumps the are spot because swaps regard if Variance although work. swap, longer variance a above from be to the swaps than variance higher for price usually market value. is model the theoretical strikes causes the the turn these in in leverage for which the offer offer, to are model market due traders options practice, the low-premium in Therefore, of 15, product. amounts Chapter large in sell mentioned to As a reluctant midmarket. with at reality traded options meets wing be low-premium theory of can where point, amounts the large is this assumes match this method At smile replication since replication. The volatility downside bump. the the the within to rise of particularly notionals wings options, the the delta low whether very consider for to market vital is it strikes, xiydt.Ms omnytedtsaeseie ntrso akttenors. market of terms in final specified the are to fix) dates spot option the the the commonly and from source (calculated Most agreed strike date. an fixed from a expiry taken with is ATM fixing an spot becomes a then date, expiry first the At rnatn ti fvnlaotoswl prxmtl eg h eaprofile vega the hedge approximately will options vanilla of strip a Transacting delta low very buying involves replication the since view, of point pricing a From 4.01,0 510,204 591,716 694,444 826,446 1,000,000 1,234,568 1,562,500 2,040,816 19,600 2,777,778 16,900 14,400 12,100 10,000 8,100 6,400 4,900 140.00 3,600 130.00 120.00 110.00 100.00 90.00 80.00 70.00 60.00 ( Strike XII 31.12 EXHIBIT K Strike ) o otatVnlaReplication Vanilla Contract Log 2 oinl(CCY1) Notional MISCELLANEOUS OPTIONS 567 is 2 𝜎 ,and 1 t . 2 t increases the forward ATM implied volatility ) 2 𝜎 ( increases the forward implied 1 forward t ) . 1 2 1 𝜎 1 ( 𝜎 t expiry dates within the contract. For − − 2 2 t t . both 2 2 𝜎 √ = 12 𝜎 FVA dates structure is the forward ATM implied volatility between the first and second expiry is the ATM implied volatility to the first expiry date at time 12 1 𝜎 𝜎 This overlaps with trader intuition about how attractive an FVA contract is to Usually, the ATM curve rises at longer tenors due to interest rate volatility but An important point is that the vega exposure is to The vega exposure from an FVA is flat over different spots until the contract Forward vol agreements are closely linked to forward implied volatility, a Lower ATM volatility at thevolatility. first expiry date Higher ATM volatilityimplied at volatility. the second expiry date buy or sell. If thebe ATM high curve and is the upward FVA sloping, contract the will forward often implied be volatility more will attractive to sell. there is no exposure toonly interest forward rates (except implied discounting) vol onvolatility exposure. an should Therefore, FVA be until on stripped it long-dated out fixes, curve FVAs, of (if such interest the a pricing. rate thing existed) Put should another be way, much flatter an than FVA the equivalent volatility ATM curve. to the final expiry date. Looking at the forward■ implied volatility calculation: ■ fixes, at which point it becomesis a standard shown ATM in vanilla Exhibit to 31.14. the final expiry date. This and there are therefore vega exposuresa to long FVA position, the vega exposure will be short to the first expiry date and long where dates, the ATM implied volatility to the second expiry date at time into a 3mth ATM in six months’ time. variance-based calculation given in Chapter 11: EXHIBIT 31.13 For example, Exhibit 31.13 shows a ‘‘three-month-in-six-month’’ FVA that turns 568 MISCELLANEOUS OPTIONS ■ owr tr Options Start Forward ntoepr ae nteftr.Aan hs ae a eseie ntrsof terms in specified be can dates these Again, future. the in dates expiry two on called (also options start Forward rebalanced. be to need may the and to reduce over will again, strangles accuracy Although delta hedge exposure. the matching 25 volatility time, gives outright some hedge and the sell exposures that and such volatility configured forward date be some should expiry buy Notionals to first date. be expiry the would final FVA of to long shape a strangles the for to delta hedge exposure better 10 significant A a unhedged. be left will curve there ATM hedged, the is choice date expiry model final good the to a not model. therefore the within is deterministic volatility is volatility Local used, (local) model because random. pricing the is within that itself preferable volatility is it Therefore, surface. volatility the 31.14 EXHIBIT uoenbrir rAeia are pin.Srksado areso these on barriers and/or The Strikes date. options. expiry final barrier digital, the American European to be or also options can barrier, they of but European range vanillas European wider commonly a most are into options settle 31.15. can Exhibit in option shown start is dates the of diagram A tenors. market or dates actual ttefis xiydt,rte hnstln noa T iea V,teforward the FVA, an like ATM an into settling than rather date, expiry first the At position vega net the only If FVA. an for hedges vanilla possible consider Finally, within volatility implied forward the on mainly depends FVA an of pricing The V eapol hnea rtfixing first at change profile vega FVA cliquets r ieFA nta hi tutr relies structure their that in FVAs like are ) MISCELLANEOUS OPTIONS 569 95%. × constant constant + × spot fix on first expiry date spot fix on first expiry date = = Forward start vega profile Forward start option dates structure : EUR/USD forward start EUR Put/USD Call vanilla from January 23, The forward start vega exposure is similar to the FVA in that it remains flat until Example EXHIBIT 31.16 flat to having a large vanna/skewin exposure Exhibit from 31.16. the long downside vanilla as shown the moment spot fixes ona the first standard expiry FX date,However, at derivatives note which that product point the the vega (e.g., may optionEUR/USD not forward becomes a be start at example, vanilla as the spot peak option fixes, when the it or vega fixes. position In a jumps the from barrier case being of option). the 1. Strike or barrier2. level Strike or barrier level 2015, into June 23, 2015, with strike set at the ECB fixing EXHIBIT 31.15 options are set withstrike reference or to barrier a levels can spot generated fixing in taken one of on two the ways: first expiry date. The 570 MISCELLANEOUS OPTIONS ■ opudOptions Compound eg o nucrancs o.Cmonsaegnrlyfrceprta the than cheaper far generally are Compounds flow. is cash option vanilla uncertain underlying an the for if appropriate hedge is a This trade. to decision the deferring ■ ■ ■ their of terms in ■ described therefore be are can options directions: option put Compound vanilla and put. underlying call the a turn, or In call date. a expiry either final the to option vanilla vanilla specific a sell compound to the called right (sometimes the price pre-agreed or a strike). at buy date expiry to final right the to the option either has option compound be can shown dates is dates These these date. of diagram expiry A 31.17. tenors. final Exhibit market a in or dates and actual date of terms decision in a specified dates: expiry two have an is product option compound The start forward the on opinion an enables model price. the Viewing option by dynamics. generated smile forward smile calibrated different forward models very the pricing produce Different surface vanilla consideration. same key the a to is smile forward the the rates, spot in to role equal future strike important the a on more for depend a volatility will implied pricing plays of example, now expectation start forward smile EUR/USD volatility the In the pricing. but date expiry final XII 31.17 EXHIBIT alo put a on Call call a on Call u naput a on Put call a on Put let rd opudotosbcuete oki h rc ftevnlawhile vanilla the of price the in lock they because options compound trade Clients the sell) to right (the put a or buy) to right (the call a either are Compounds the of owner the date), decision the called (sometimes date expiry first the At interest and volatility ATM of structure term full the as well the as and Therefore, date expiry first the both to exposures vega are there FVA, an Like h ih osl ail C1ptoption put CCY1 vanilla a sell to right the : h ih osl ail C1cl option call CCY1 vanilla a sell to right the : h ih obyavnlaCY u option put CCY1 vanilla a buy to right the : h ih obyavnlaCY aloption call CCY1 vanilla a buy to right the : opudoto ae structure dates option Compound pino noption an on option hrfr opudoptions compound Therefore . × 95%. MISCELLANEOUS OPTIONS 571 . moves in spot space Compound option vega profile : USD/JPY 3mth compound call option on a 3mth 100.00 strike USD Exhibit 31.18 shows the vega profilesThe from relative vega the exposures between above the USD/JPY two dates compound depend on the probability of Note that the decision on whether to exercise or expire the compound is based Compound owners have long vega exposures to both the decision date and the Example EXHIBIT 31.18 compound exercise. For example, if the compound strike on a compound call is 0%, exercised—the center of the optionality. Theits vega peak to at the the final strike expiryof as date the will usual compound have but being that exercised exposure on the will decision be date. weighted byoption. the probability on spot, interest rates, andThe implied consequence volatility of this market is data that for the the exercise final point expiry date. final expiry date becausedecision there date is is optionality around at the spot both level dates. at The which peak the option vega will on either the be expired or Put/JPY Call option at 0.30compound USD%. option decides Therefore, whether to in exercise three thethe compound months optionality. compound the Exercising involves owner paying of 0.30 the date. USD% The compound for should the be exercised 3mth ifexpiry option the date to market is price the higher of final the than vanilla 0.30 expiry to USD%, the otherwise final the compound should be expired. equivalent outright vanilla option, althoughoption the also premium must for be the paid underlying at vanilla the decision date. 572 MISCELLANEOUS OPTIONS stergtt u rtergtt elaprfloo uuly ail options. vanilla (usually) of portfolio a sell to right the or buy possible to right as the elements is these of pricing. many when as account into options, taken start be should forward Like compound rates. the interest because more date be expiry always final must the vanilla to final optionality. additional strike option the contains premium plus vanilla the compound the As the date. than expiry will of expensive final profile price the vega total to The vanilla the zero). a rises, of at that (bought be exercised simply be therefore always should compound the ial,amr eea eso facmon san is compound a of version general more a Finally, forward and smile volatility forward the on depends pricing option Compound pino strategy on option ,which FURTHER READING

hese are the books that I get the most from, in terms of either entertainment or Tinformation, ordered in increasing quantitative content. Reminiscences of a Stock Operator by Edwin Lefevre (John Wiley & Sons, Abridged Edition, 2004) A classic story about the ups and downs of trading written in the early twentieth 573 century. Amazingly, most of the wisdom still holds true today. There’s a hardcover version available that features lovely period artwork. Where Are the Customer’s Yachts? by Fred Schwed Jr. (John Wiley & Sons, 1st Edition, 2006) A cynical description of how Wall Street operates based around the 1929 Crash. When so much material written and presented about finance and financial markets is completely humorless, this is a cracking read. The parts on options are particularly interesting, written as they are before the development of the Black-Scholes model. It is fascinating to consider how the derivatives market developed even without a consistent way of pricing volatility. Volatility Trading by Euan Sinclair (John Wiley & Sons, 2008) Explains how buyside (i.e., hedge funds) trade volatility with a nice balance of practical and technical material. The approach is centered on picking individual trades, managing them through to maturity, and tracking their P&L rather than risk managing a portfolio of options. For a derivatives trader at a bank it is interesting to see how hedge funds assess value and manage their risk; many of the techniques are applicable to market-makers, too. 574 FURTHER READING rte yaqatwoevltlt ml oeshv enaotdtruhu the throughout adopted been have models smile volatility whose quant a by Written through hand the by you Leads start. to It place the is this Black-Scholes, underpinning reference. trader a quant-centric as every desk that their achievement on have fantastic will A payoffs. of variety massive cover to updated often products. is financial it new Plus content. worked mathematical good pitched clarity, with nicely Written and classes. examples, asset all across products financial of basics but application. perspective real-world quantitative the a from of topics sight of losing range without massive a Covers industry. the in hard be packed is can book It the derivatives material. verve. with and this familiar wisdom are from the of you of full directly once but Much comes joiners, trading. book new for of this going thrill in and used experience style the analysis with details technical combines nuty rbbymr sflfrqat hntaes u ie h opeiyof complexity accessible. the still given is but it traders, material than the quants for useful more Probably industry. acls atnae,adfial Black-Scholes. finally and Martingales, Calculus, o ¯ nielbo o nesadn o h oaiiysraei oee npractice. in modeled is surface volatility the how understanding for book ideal An Guide framework Practitioner’s A mathematical Surface: the Volatility The understand to want you If book: a of gem A Calculus Financial a covering formulas options containing book a cover: the on says it what Does Formulas Pricing Option to Guide Complete The the learning for book good a is This traders. junior and students for favorite A Derivatives Other and Futures, Options, figures key the of one from finance quantitative to introduction comprehensive A Finance Quantitative on Wilmott Paul successfully It published. was it when time its of ahead years book: stunning A Hedging Dynamic 1 1 Press, 2 Professional, 2005) 2006) Edition, st dto,2006) Edition, st dto,1996) Edition, yNsi ae Jh ie os 1 Sons, & Wiley (John Taleb Nassim by nd yMri atradAde ene(abig University (Cambridge Rennie Andrew and Baxter Martin by dto,2007) Edition, yPu imt Jh ie os 2 Sons, & Wiley (John Wilmott Paul by yJh .Hl Petc al 6 Hall, (Prentice Hull C. John by yJmGtea Jh ie Sons, & Wiley (John Gatheral Jim by yEpnGadrHu (McGraw-Hill Haug Gaarder Espen by st dto,1996) Edition, th Edition, nd ABOUT THE COMPANION WEBSITE

his book is accompanied by a companion website at www.wiley.com/go /FXtraderschool (wiley15). TReaders will find the following completed practicals, which correspond to the practicals in this book:

■ Practical A: Building a Trading Simulator in Excel 575 ■ Practical B: Building a Numerical Integration Option Pricer in Excel

■ Practical C: Building a Black-Scholes Option Pricer in Excel

■ Practical D: Generating Tenor Dates in Excel

■ Practical E: Constructing an ATM Curve in Excel

■ Practical F: Constructing a Volatility Smile in Excel

■ Practical G: Generating a Probability Density Function from Option Prices in Excel

■ Practical H: Building a Monte Carlo Option Pricer in Excel

INDEX

A regular: Accrual barriers, 515 knock-in call options, 390, Accrual forwards, 395, 518–521 443–444 Accrual options, 395, 515–521 knock-out call options, 390, 439–443 accrual forwards, 518–521 577 range accrual, 516–518 pricing, 445 Adapted Greeks, 275–285 reverse: delta, 276–279 knock-in, 390, 445, 450 knock-out, 390, 445–450 gamma, 279–281 pricing, 450–452 risk management with, 282–285 selling, 468–471 vega, 221, 281–282 strike-out options, 454–460 Agreeing broker market data, 311–312 transatlantic, 392–394 American barrier options, 388–394, American digitals, 388 424, 439–460 American keep accrual discrete, 394 barrier, 515 double knock-out and knock-in, American keep range accruals, 390, 391, 452–453 517–518 in-the-money, 392 American vanilla options, 394, knock-in/knock-out replication, 501–510 453–454 call and put, 14 knock-out–knock-in, 394 and Greeks, 506–509 Monte Carlo option pricer, pricing, 509 492–493 Arithmetic average, 534 578 INDEX T uv,103–106 curve, ATM spreads: calendar ATM ATM, Ask, 527–538 396, options, Asian nmre ntuetanalysis, instrument market in 171–191 of, construction 132 vanilla, 132 exposures, trading 132 making, price options, rate average double 535–537 options, strike average 527–535 options, rate average see eka ainepatterns, variance weekday 171–172 variance, 172–178, interpolation, using 178–179, model, a using 180–191 short-dates, for options, same-day pricing Friday, a on ATM overnight cut Tokyo vs. cut York New 187 patterns, variance intraday a over patterns volatility implied market derivatives FX and 193–204 Excel, in 188–189 holidays, and events 172–179 curve, ATM core 199–204 weights, adding 344–346 189–190 193–197 197–199 190–191 184–185 185–187 pricing, 182 week, 182–184 pricing, 537–538 see Offer(s) At-the-money T oaiiyadcorrelation and volatility ATM position: ATM 272 gamma, ATM are et a tuhoptions), (touch gap delta Barrier 39–40. desks, trading Bank B 468 Axe, 535–537 options, strike Average 527–535 396, options, rate Average (ATM). At-the-money 313–319 triangles, volatility ATM rdn,152–155 trading, 138 defined, 154 in, changes weighted 272 271, down, roll and theta 354–356 slope, 346 seasonality, tutr,41–44 structure, 40–41 types, client 122 straddles, zero-delta 113–114 spreads, volatility 205 smile, volatility and 209–211 exposures, vega 118–120 conventions, market 103–105 of, volatilities implied 171–172 volatility, implied forward 319–320 vega, dephased positions cross-currency 313–319 triangles, volatility ATM upr ucin,43 functions, support 42–43 desks, sales 43–44 trading, internal rmwr,313–321 framework, 429–430 market broker Interbank curve 321 management, e also See e also See ATM INDEX 579 Interbank broker see Best-of options see 311–312 57–62 calculating option values, 65–66 91–101 equation (SDE), 57–62 market 25d vs. 10d, 229–230 drivers of, 228 price making, 128 stochastic differential equation, terminal spot distributions in generate first-order Greeks, 98–100 plot exposures, 100–101 set up simple option pricer, 91–96 set up VBA pricing function, 96–98 drift in, 58–61 solving, 62–65 uncertainty in, 61–62 multi-asset options, 541 one-touch options, 434 self-quanto vanilla options, 514 vanilla, 113–116 Black-Scholes formula, 66–67 solving Black-Scholes SDE, 62–65 Broker market data, agreeing, Bucketed vega exposures, 154 Butterfly (fly), 126–128, 225–230 Black-Scholes option pricer (Excel), Black-Scholes stochastic differential Bleed, delta, 266, 268–269 BO options, Breakeven calculation, 323–325 Broker fly, 127, 225 Broker market, Binary options, 388 Black-Scholes delta, 276 Black-Scholes formula, 18, 66–67 Black-Scholes framework, 57–68 See also See also 476–478 European barrier options American barrier options risk management, 481–482 front-window, 395, 473–478 generic, 481 Monte Carlo option pricer, 493 rear-window, 395 European digital options, 405 exotic FX derivatives, 462–463 front-window barrier options, in inverted market, 21 language of, 23 leaving orders, 22 midmarket, 117–118 European barrier options, 418 trading risks, 549–553 in choice markets, 21 defined, 19 switching hedge, 553 late-starting, 478 shadow, 466–467 transatlantic, 392–394 window, 395, 473–482 continuous, 388, 424 discrete, 394, 482 European, 388–389. in-the-money, 392 American, 389–394. Bid–offer spread, 22–23 Bet spread, 410 Bid(s), 19–22 Basket digitals, 543 Basket options, 396, 541–543 Bending barriers, 466 Best-of (BO) options, 396, 548–553 Basis point, 120, 164 Barrier options, 388–394 580 INDEX lsn u ik 24 risk, out Closing 98 approach, Closed-form 568 Cliquets, market), (derivatives types Client reversals), (risk price Choice 21 markets, Choice 304 spread), (in Choice CCY, settlement: Cash Black-Scholes Excel in price, Cash 388 options, Cash-or-nothing 272 theta, balance Cash 352–354 trades, Carry spreads: Call/put 14 Call, C 50 interest, Buying Buying: ( (fly), Butterfly otnosbrir,38 424 388, barriers, Continuous 541 spreading, interval Confidence 248 intervals, Confidence 570–572 396, options, Compound Collar, fvnlaF eiaie,18 derivatives, FX vanilla of 399 options, digital European of 132–134 vanilla, 134 making, price 20 of, methods 21 markets, faster in 205 106, smile, volatility and 213–215 exposures, vega 128 exposures, trading 228–229 trading, 40–41 130 129, 94 pricer, option see see Currencies ikrvra (RR) reversal Risk Continued ) oeAMcreconstruction, curve ATM Core 224 approach, Copula Conventions: interest compounded Continuous urnybok,42 blocks, Currency 3 (CCY), Currencies 224–225 reversal, risk Cross 7 pairs, Cross-currency 397 342, swaps, Correlation 313. Correlation, 40–41 clients, Corporate sn neplto,172–178 interpolation, using 178–179 model, a using 264 delta, quoting 306 market, self-quanto: 219 of, strength relative CCY1 variations options one-touch 4 G10, replication: option digital European premium, CCY2 vs. CCY1 and delta 321 in, exposures managing 319–320 vega, dephased 313–319 triangles, volatility ATM correlation, implied trading rate, interest vs. spot realized C1ptotos 510–513 options, put CCY1 510 options, call CCY1 402 CCY2, 402–403 CCY1, 172–179 160 rate, s C2pyu,434–435 payout, CCY2 vs. 265–267 341–342 336–339 framework correlation and volatility e also See ATM INDEX 581 foreign See also options, 388 537–538 exchange (FX) derivatives options, 550 trading risks, 545–546 vega risk, 546–548 bid–offer spread, 405 European digital range, 401–411 Greeks, 405–409 pricing, 403–405 replication, 400–403 European digital spread, 410 principal protected, 466 structured, 465–466 American, 388 basket, 543 dual, 397, 543–548 European, 388, 399–411 European digital range, 401–411 American barriers, 469 touch options, 429–430 best-of and worst-of target redemption options, 524 Direct market, 39 Discrete barrier options, 394, 482 DNT (double-no-touch) Double average rate options, Dephased vega, 319–320 Deposits (depos), 160 Depth (market), 20 Derivatives, 11. Digital bets, 388 Digital options: Delta exchange, 463 Delta gap: Delta hedging, 118, 144–145 Delta neutral, 83 Delta risk: Multi-asset see Cross-currency See also 310–311 98–101 options pairs vanilla FX derivatives, 83–87 vanilla price making, 118 vanilla trading low delta options, forward delta vs., 263–264 low delta vanilla options, 310–311 quoting conventions, 264 sticky, 276 touch options, 424–425, 429–430 adapted, 276–279 bleed, 266, 268–269 CCY1 vs. CCY2 premium, 265–267 in Excel Black-Scholes option pricer, in describing vanilla FX options, 15 New York vs. Tokyo, 185–187 multiple, payoff in, names of, 8–9 pegged, 306–307 quoting, 8 in describing vanilla FX options, 14 G10, 8, 9 major, 7 market conventions, 118–120 ATM volatility triangles, 313–319 cross, 7. Delta bleed, 266, 268–269 Daily variance, 173, 175 DCDs (dual currency deposits), 41 Decay, 143 Delivery dates, 17, 162–163 Delta: Cut(s): D Currency pairs, 3, 8 582 INDEX uoenbrirotos 388–389, options, barrier European 515 barrier, accrual European countries: (EM) market Emerging EKO, EKI, 200 time, Economic 473 barriers, ending Early E 543–548 397, options, digital Dual 41 (DCDs), deposits currency Dual 58–61 (SDE), Drift 388 options, (DNT) Double-no-touch knock-in and knock-out Double uoendgtlotos 388, options, digital European urnypi ae,9 names, pair currency 8 mechanisms, flows currency 546–548 risk, vega 545–546 risks, trading 452–453 barrier, American 391 390, American, res 419–422 Greeks, 418 spread, bid–offer uoendgtlrne 401–411 range, digital European 405 spread, bid–offer replication: 422 pricing, 492 pricer, option Carlo Monte 416–417 413, knock-out, 418–420 413, knock-in, 417–418 value, intrinsic see uoenkokot 416–417 knock-out, European 418–420 knock-in, European 420–422 risk, vega 422 risk, pin and gamma 399–411 413–422 options: see uoenknock-in European uoenknock-out European uoenkoki EI,39 413, 389, (EKI), knock-in European 410 spread, digital European 401–411 range, digital European Excel: Weighted (Exponentially EWMA Events: options: vanilla European 516–517 accruals, range European 413, 389, (EKO), knock-out European elcto,400–403 replication, 403–405 pricing, 405–409 Greeks, osrcigAMcre 193–204 curve, ATM constructing risks trading changing for variations 14 put, and call lc-coe pinpie,91–101 pricer, option Black-Scholes 188 defined, construction, curve ATM and C2 402 CCY2, 402–403 CCY1, 406–408 risk, vega 408–409 risk, pin and gamma e pVApiigfunction, pricing VBA up set pricer, option simple up set 100–101 exposures, plot Greeks, first-order generate 509–514 payoff, self-quanto 497–503 late-delivery, 501–510 exercise, American 96–98 91–96 98–100 331–333 volatility, Average) Moving 188–189 497–514 on, 416–417 418–420 INDEX 583 368–372 384–385 377–380 from delta, 235 basket options, 396 best-of and worst-of options, 396 compound options, 396 correlation swaps, 397 dual digital options, 397 European digital options, 388 first-generation exotics, 387–395 example of, 359–360 path dependence, 373 stopping time, 370–371 volatility smile pricing, 360–367 VVV (vega/volga/vanna) pricing, interest rate models, 375 jump diffusion models, 383–384 local volatility models, 380–382 mixed volatility models, 382–383 smile models, 375 stochastic interest rate models, stochastic volatility models, accrual options, 395 American vanilla options, 394 Asian options, 396 barrier options, 388–394 use Black-Scholes to get strike pricing models, 375–385 product classification, 387–397 vanilla call options, 12 vanilla put options, 13 defined, 11, 355 pricing, 357–373 Exchange rate, 3 Exercise: Exotic FX derivatives, 355–356 233–234 price-taking functionality, 32–35 233–240 238–240 234–235 with VBA functions, 236–238 option price, 72–75 69–72 functionality, 35–36 27–31 485–496 Monte Carlo loop, 487–490 69–75 functions from option prices, 253–259 set up Malz smile model, set up two-way price and investigate strike placement, plot implied volatility vs. delta, plot implied volatility vs. strike set up terminal spot distribution, testing, 75 extensions, 36–37 introduce price-making set up ticking market price, extensions, 496 multi-asset simulation, 494–496 pricing barrier options, 492–493 set up multiple payoffs, 490–492 set up simulation, 485–487 set up vanilla option payoff and set up option payoff and calculate volatility smile construction, trading simulator, 27–37 numerical integration option pricer, generating tenor dates, 165–168 Monte Carlo option pricer, generating probability density 584 INDEX xoetal egtdMoving Weighted Exponentially date: Expiry/expiry 370 life, Expected ( derivatives, FX Exotic xesos nEcltaigsimulator, trading Excel in Extensions, ail u pin,1,17 13, options, put vanilla 162–163 derivatives, FX vanilla 17 options, call vanilla 351 analysis, strike stick 185–187 cuts, Tokyo vs. York New 151 dates, ‘‘bad’’ and ‘‘good’’ 15 options, FX vanilla describing in 461–471 trading, tutrdF egn strategies, hedging FX structured 466–467 barriers, shadow barriers, American selling and 461–462 management, risk 467–468 risk, recycling 465–466 products, investment 463 market, broker interbank 462–463 spread, bid–offer 395 options, barrier window 397 swaps, variance and volatility 388 options, touch exotics, third-generation 395 options, redemption target exotics, second-generation 394–395 options, quanto agreements, volatility forward 396 options, start forward 36–37 331–333 volatility, (EWMA) Average 463–465 468–471 396–397 395–396 397 Continued ) iigrs tre redemption (target risk Fixing 8 rate), (FX Fix 387–395 exotics, First-generation 370 time, exit First 98 approach, difference Finite 22 orders, Fill-or-kill 4 moves), (spot Figure 244–248 distributions, Fat-tailed F oeg xhne(X akt 3–9 market, (FX) exchange Foreign derivatives (FX) exchange Foreign derivatives, (FX) exchange Foreign Fly, eiaie aktlanguage, market derivatives 3 pairs, currency 3 currencies, 7 pairs, currency cross 7–8 of, aspects 293–294 speed, transaction 294 sizes, transaction 39–55 of, structure 294–296 liquidity, and construction curve ATM 11–18 options, put and call vanilla 18 volumes, trading see rdn nenhptp,44 tips, internship trading 41–44 structure, desk trading 44–46 tips, trader junior 46–47 market, direct interbank market, broker interbank 40–41 types, client pin) 525 options), 156–157 47–55 182–184 pricing, derivatives) (FX market. 11–18 Butterfly e also See aktanalysis Market INDEX 585 Swaps See also Foreign exchange see 506–509 150–152 options), 524 market P&L distributions from, 273–274 trading, 139–143 P&L distributions from, 273–274 trading, 145–146 delta hedging, 144–145 long gamma, 139–143 short gamma, 145–146 and American vanilla options, European barrier options, 420–422 gamma/strike profile, 151–152 long: positive, 79 ‘‘renting,’’ 184 short: smile gamma effect, 272–273 touch options, 428–429 trading, 138–146 in trading short-date position, vanilla FX derivatives, 88–89 adapted, 279–281 ATM, 272 European barrier options, 422 European digital options, 408–409 Garman and Kohlhagen formula, 67 Gartman’s rules of trading, 300 ‘‘Given’’ bids, 23 Greeks: Gamma adaption effect, 279 Gamma risk (target redemption FX market, FX swap contracts, 4. G Gamma, 270–271 Swap points See also Foreign exchange see Forward volatility see derivatives 473–478 derivatives), 160 agreements vol), 171–172 568–570 470–471 341, 397, 566–568 11, 288 risk, 474–475 trading risks, 476–478 payoff risk, 475–476 size of, 7 spot rate (spot), 3 swap points (forward points), 3 long positions, 5 main product areas, 11 major currency pairs, 7 names of currency pairs, 8–9 short positions, 6 exchange rate, 3 forwards (forward outrights), 3 FX swap contracts, 4 FVAs, FX derivatives, Future cash, interest rate risk and,Future 288 valuing (vanilla FX Forward volatility agreements (FVAs), Front-window barrier options, 395, Forward roll, theta, 271, 272 Forward start options (cliquets), 396, Forward volatility, spot volatility vs., Forward delta, spot delta vs., 263–264 Forward extra strategy, 464 Forward implied volatility (forward Forward points, 4. Forwards (forward outrights), 3, 586 INDEX oiotlsras 132 spreads, Horizontal put and call (vanilla date Horizon construction curve ATM Holidays, and, spread bid–offer period, Holding 23 bids, ‘‘Hit’’ volatility, spot Historic 41 funds, Hedge Hedges: H ( Greeks: mle kw 220 skew, Implied 341–342 trading, correlation, Implied I eahde 463 hedge, vega 553 hedge, switching exotics, for strategies structured 463 hedge, rho 118 87, transactions, hedged delta 557–561 swaps, volatility 562–564 swaps, variance 83–90 derivatives, FX vanilla pricer, option Black-Scholes Excel in 405–409 options, digital European ea 89–90 vega, 88–89 gamma, 83–87 delta, 406–408 risk, vega 408–409 risk, pin and gamma 420–422 risk, vega 422 risk, in and gamma pin) 17 options), 189 and, 22 volatility spot 463–465 98–100 Continued ) see Realized mle volatility: Implied nebn rkrmre,39–40, market, broker Interbank 41 clients, Institutional 388 options, touch Instant 435 options, one-touch Instant 130 differential, volatility Implied mle s elzdmre analysis, market realized vs. implied 171–172 ATM, forward 103–105 contracts, at-the-money utn mle oaiiyi,294 in, volatility implied quoting 50 49, pricing, 47 firms, main 463 derivatives, FX exotic 127 contracts, fly broker data, market broker agreeing 108 smiles, volatility 339–341 trading, 294 quoting, 182 week, a over patterns 113 midmarket, 328 tenors, longer at elzdso s neetrate interest vs. spot realized volatility spot realized Moving Weighted Exponentially rdn mle volatility, implied trading correlation, implied trading convexity, volatility realized forward realized vs. spot realized aclto,328–331 calculation, 331–333 volatility, Average 325–343 311–312 47–55 339–341 341–342 342–343 333–336 volatility, 336–339 calculations, INDEX 587 Knock-out–knock-in barrier see 418–420 options, 394 383–384 options regular barrier, 443–444 reverse barrier, 445, 450 double, 452–453 replication, 453–454 front-window, 473–474 rear-window, 479–480 regular barrier, 439–443 reverse barrier, 445–450 late cash, 497–498 option on forwards, 498–500 American barrier options: European, 389, 413, 418–420 European barrier options, 413, American: European, 413, 416–417 American barrier options: European, 389, 413, 416–417 Late cash vanilla options, 497–498 Late delivery, 161, 501 Late-delivery vanilla options, 497–503 Late-starting barriers, 478 Leaving orders, 22 Knock-out–knock-in (KIKO) barrier Knock-out options: L J Jargon, 156–157 Jump diffusion pricing models, Junior trader tips, 44–46 K KIKO, Knock-in options: In-the-money construction and, 187 384–385 43–44 correlation vs., 336–339 see defined, 417 European barrier options, 417–418 front-window barrier options, 474 barrier options, 392 vanilla call and put options, 17 long ATM straddle, 291–292 long vanilla call options, 288–290 long vanilla put options, 290–291 future cash and forwards, 288 and risk reversals, 219–220 zero, 160 transaction process in, 50–54 continuous compounded, 160 correlation between spot and, 434 relative power of brokers, 50 structure, 47–49 trader/broker relationship, 49–50 Inventory management, 26 Inverted market, 21 Investment products, exotic, 465–466 ITM, Intrinsic value, 79 In-the-money (ITM): Intraday variance patterns, ATM curve Interest rate volatility, 433 Internal trading, among trading desks, Interest rates markets, 306 Interest rate pricing models, 375, Interest rate risk (rho), 287–292 Interest rate carry, 456 Interest rate correlation, realized spot Interbank direct market, 46–47 Interest rate(s), 4 588 INDEX Markets: 232 formula, smile volatility Malz 232–234 model, smile Malz 7 pairs, currency Major M 310–311 options, delta Low 143 position, strike Long positions: Long 305–306 derivatives, FX Long-dated 556 contract, Log models, pricing volatility Local requests), price (vanilla trading Live 22 Liquidity, 174–177 variance, Linear 21 orders, Limit 23 offers, ‘‘Lifted’’ 311 131, forward, Leveraged 447 Leverage, 242 distributions, Leptokurtotic Legs,4,304 aktaayi F derivatives), (FX analysis Market ar rds 352–354 trades, carry calculation, breakeven 21 inverted, 306 rates, interest 21 choice, risk: rate interest 5 defined, 306 market, 294–296 market, derivatives FX in ail u pin,290–291 options, put vanilla 288–290 options, call vanilla 291–292 straddle, ATM long 323–325 323–354 380–382 118 aktlqiiy 306 liquidity, Market analysis, instrument Market instruments: Market 306 conventions, Market mle s elzdanalysis, realized vs. implied oaiiysie 4,351 346, smile, volatility 346–350 analysis, value 351–352 positioning, market 350–351 historical, 344–346 curve, ATM 209–216 exposures, vega smile, volatility defining 118–120 making, price vanilla 118–120 at-the-money, analysis, instrument market elzdso s elzdforward realized vs. spot realized rate interest vs. spot realized volatility spot realized Moving Weighted Exponentially oaiiysie 4,351 346, smile, volatility 346–350 analysis, value 351–352 positioning, market 350–351 historical, 344–346 curve, ATM volatility, implied trading correlation, implied trading convexity, volatility realized oaiiy 333–336 volatility, 336–339 correlations, 328–331 calculation, 331–333 volatility, Average 325–343 344–352 205–208 344–352 339–341 341–342 342–343 INDEX 589 400–401 424, 438 (Excel), 69–75 option price, 72–75 69–72 contracts, 8 trading risks, 545–546 vega risk, 546–548 in inverted market, 21 language of, 23 leaving orders, 22 midmarket, 118 bid–offer spread, 434 pricing, 430–434 butterfly, 128 in describing vanilla FX options, 15 European digital replication, vanilla call and put options, 12 set up option payoff and calculate set up terminal spot distribution, testing, 75 bid–offer spread, 22–23 in choice markets, 21 defined, 19 dual digital options, 543–548 quanto options, 553–554 trading risks, 539–540 One-touch (OT) options, 388, 424 Notional, 4 No-touch (NT) options, 388, Numerical integration option pricer O Offer(s), 19–22 N Non-Deliverable Forward (NDF) Non-optimal exercise, 501 Tenor See also Monte Carlo loop, 487–490 Weighted, 331–333 548–553 382–383 485–496 165–168 calculation, 162–163 trading risks, 549–553 switching hedge, 553 expiry dates and delivery dates spot dates calculation, 161–162 bid–offer spread, 541 basket options, 541–543 best-of and worst-of options, multi-asset simulation, 494–496 pricing barrier options, 492–493 set up multiple payoffs, 490–492 set up simulation, 485–487 set up vanilla option payoff and extensions, 496 ATM contracts, 103–104 generating dates in Excel, vanilla FX derivatives, 161–163 Moving Average, Exponentially Multi-asset options, 539–554 Mixed volatility pricing models, Monte Carlo option pricer (Excel), Merton model, 383–384 Middle office, 42 Midmarket bid, 117–118 Midmarket offers, 117 ‘‘Mine!,’’ 23 Mark-to-market P&L, 155 Maturity, 3, 15. Market tenor: Market making, 23–26 Market positioning, 306, 351–352 Market sentiment, 117 590 INDEX pinpies(Excel): pricers Option price: Option premium: Option numerical Excel in payoff, Option 303–304 orders, Option 572 strategy, on Option 498–500 forward, on Option 79 contracts, of Optionality options, O/N ( options, (OT) One-touch ot al pinpricer, option Carlo Monte 91–101 pricer, option Black-Scholes option integration numerical in density probability generating 180–181 variance, and 12 calls, vanilla 265 delta, and FX vanilla for conversions 434–436 variations, rcn are pin,492–493 options, barrier pricing 494–496 simulation, multi-asset 496 extensions, function, pricing VBA up set pricer, option simple up set 100–101 exposures, plot Greeks, first-order generate pay-at- vs. pay-at-maturity payout, CCY2 vs. CCY1 485–496 96–98 91–96 98–100 72–75 pricer, 253–259 from, functions 163–164 derivatives, 72–75 pricer, option integration 435–436 touch, 434–435 see vrih options Overnight Continued ) a-ttuhone-touch Pay-at-touch 436 options, one-touch Pay-at-maturity SDE, in options, Path-dependent FX exotic pricing in dependence, Path 151 options, exercised Partially 106 shift, ATM Parallel 23 offers, ‘‘Paid’’ P 39, market, (OTC) Over-the-counter options: (O/N) Overnight 17 (OTM), Out-of-the-money options, OT 39, market, (over-the-counter) OTC 20 book, Order values: Option ueia nerto pinpricer, option integration numerical ail rdn,296–299 trading, vanilla 162 dates, delivery and expiry 184–185 Friday, a on ATM 77–82 derivatives, FX vanilla in distributions spot terminal e ptria ptdistribution, spot terminal up set calculate and payoff option up set and payoff option vanilla up set 485–487 simulation, up set 490–492 payoffs, multiple up set etn,75 testing, 69–72 72–75 price, option 69–75 487–490 loop, Carlo Monte pin,436 options, 59–60 373 derivatives, 294 294 65–66 calculating, see n-oc options One-touch INDEX 591 Option pricers (Excel) See also 182–184 368–372 103–116 trading simulator, 32–35 445 450–452 example of, 359–360 path dependence, 373 stopping time, 370–371 volatility smile pricing, 360–367 VVV (vega/volga/vanna) pricing, maintaining volatility surfaces, price making, 116–120 jump diffusion, 383–384 local volatility, 380–382 mixed volatility, 382–383 smile, 375 stochastic interest rate, 384–385 stochastic volatility, 377–380 European barrier options, 422 European digital options, 403–405 exotic FX derivatives, 357–373 one-touch options, 430–434 same-day options, 190–191 target redemption options, 525 vanilla FX derivatives, 103–120 variance swaps, 565–566 volatility swaps, 564–565 interest rate, 375 American regular barrier options, American reverse barrier options, American vanilla options, 509 and ATM curve construction, Principal protected deposits, 466 Pricing models, 375–385 Price takers, 19–20 Price-taking functionality, in Excel Pricing. Profit and loss Probability density functions trading simulator, 35–36 118 159, 160 derivatives, 18 options), 475–476 worst-of options), 550–553 see market conventions, 118–120 overview, 116–118 transacting delta hedged or live, see straddles, 122 strangles, 125–126 success in, 26 vanilla FX derivatives, 116–120 ATM calendar spreads, 132 butterfly, 128 call/put spreads, 134 risk reversal, 129–130 seagull, 135 vanilla trading, 310 European barrier options, 422 European digital options, 408–409 touch options, 428–429 Price-making functionality, in Excel Premium firm orders, 303–304 Present valuing (vanilla FX derivatives), Price makers, 19–20 Price making: Pips (points), 4 P&L, Positions, quoting, 5 Positive spreads, 307–308 Pin risk: pdfs, Pearson’s coefficient, 336 Pegged currency pairs, 306–307 Physical delivery, of vanilla FX Payoff risk (front-window barrier Payout direction risk (best-of and 592 INDEX Ranges: 548 options, Rainbow R 553–554 394–395, options, Quanto 42 analysts), (quantitative Quants Q 86–87 parity, Put–call 189 holidays, Public 7 6, (P&L), loss and Profit (pdfs), functions density Probability ag cra pin,516–518 options, accrual Range uoen 516–517 European, 517–518 range, keep American 401–411 digital, European 388 as, options DNT 553–554 395, currency, third 509–514 394, self-quanto, 15–16 options, put and call vanilla position, short-date trading in 155–156 trading, 5 quoting, short or long from distributions parameterization smile volatility and in prices option from generated 244–248 distributions, fat-tailed integration numerical Excel in 248 intervals, confidence C1ptotos 510–513 options, put CCY1 510 options, call CCY1 514 spread, bid–offer 148–149 273–273 gamma, 249–252 limitations, 253–259 Excel, 70–72 pricer, option 241–252 elzd(itrc ptvolatility: spot (historic) Realized 220 skew, Realized Replication: (American): options barrier Regular 41 banks, Regional 231 230, Rega, 467–468 risk, Recycling 388 Rebates, 395, options, barrier Rear-window 41 money, Real 342–343 convexity, volatility Realized nmre nlss 325–343 analysis, market in 328–331 326, calculating, uoendgtlotos 400–403 options, digital European barrier American 445 pricing, 439–443 options, call knock-out 390 knock-in, and knock-out 443–444 options, call knock-in time, frequency/sample sample and elzdso s elzdforward realized vs. spot realized rate interest vs. spot realized Moving Weighted Exponentially C2 402 CCY2, 402–403 CCY1, volatility, implied trading correlation, implied trading convexity, volatility realized oaiiy 333–336 volatility, 336–339 correlations, 331–333 volatility, Average 453–454 knock-in/knock-outs, 478–481 327–328 339–341 341–342 342–343 INDEX 593 156–157 146–152 476–478 549–550 420–422 406–408 522–524 467–468 549–553 trading ATM position, 152–155 trading gamma, 138–146 trading P&L, 155–156 trading short-date position, multi-asset options, 539–540 best-of and worst-of options, dual digital options, 546–548 European barrier options, European digital options, target redemption options, touch options, 425–428 FX derivatives market language, touch options, 428–429 vanilla trading, 310 best-of and worst-of options, dual digital options, 545–546 European vanilla options, 497 front-window barrier options, volatility smile, 230–231 vega risk: warehousing, 24 writing off risk, 308–310 with adapted Greeks, 282–285 exotic FX derivatives, 461–462 success in, 26 vanilla FX derivatives, 137–157 recycling risk, exotic FX derivatives, trading risks: Risk management: Interest rate risk 408–409 options, 475–476 worst-of options, 550–553 options, 525 474–475 options, 524 439 European digital options, future cash and forwards, 288 long ATM straddle, 291–292 long vanilla call options, 288–290 long vanilla put options, 290–291 European barrier options, 422 best-of and worst-of options, 550 target redemption options, 524 see barrier, 416–420 digital, 402–403 pin risk: payoff risk, front-window barrier payout direction risk, best-of and gamma risk, target redemption interest rate risk, 287–292 fixing risk, target redemption front-window barrier options, closing out, 24 cross risk reversal, 224–225 delta risk: knock-out, 445–450 knock-out and knock-in, 390 pricing, 450–452 knock-in, 445, 450 European options: Risk: Rho, Rho hedge, 463 Replication spreading, 541 Retail clients, 41 Reverse barrier options (American), 594 INDEX efqat ail pin,509–514 options, vanilla Self-quanto 394 options, Self-quanto 231 230, Sega, 395–396 exotics, Second-generation 134–135 Seagull, SDE, 190–191 pricing, options, Same-day desks trading of interaction desks, Sales 232 model, SABR S 103 prices), (of Run RR, bid–offer preference, Risk/reward 222 multipliers, reversal Risk 217–225 128–131, (RR), reversal Risk ( management: Risk eln neet 50 interest, Selling Selling: ehd f 20 of, methods 21 markets, faster in 468–471 barriers, American 510–513 options, put CCY1 510 options, call CCY1 514 spread, bid–offer 205 106, smile, volatility and 211–213 exposures, vega 130–131 exposures, trading 220–222 trading, 129–130 making, price 219–220 of, drivers 224–225 cross, 222–224 10d, vs. 25d 308–310 off, writing 481–482 options, barrier window see see ifrnilequation differential 42–43 and, 22–23 and, spread ikreversal Risk lc-coe stochastic Black-Scholes Continued ) hr-aepositions: Short-date 466–467 barriers, Shadow ln,156–157 Slang, Skew: 5 dollar-cad,’’ ten ‘‘Short 6 positions, Short T uv osrcin 180–191 construction, curve ATM oaiiysie 0,12 1,346. 217, 112, 106, smile, volatility 220 realized, 361–364 the, pricing 242–243 tilt, pdf and 220 implied, 346–348 value, analyzing in 302–303 trading, vanilla 146–152 trading, 138 ladder, spot 302–303 wings, short vs. ATM long 138 defined, e okctv.Tkocut Tokyo vs. cut York New 187 patterns, variance intraday over patterns volatility implied pricing, market derivatives FX 188–189 holidays, and events ht,149–150 theta, 150–151 strikes, 148–149 balance, P&L 151–152 profile, gamma/strike 150 gamma, patterns, variance weekday options, same-day pricing Friday, a on ATM overnight ek 182 week, a 182–184 e also See 189–190 190–191 184–185 184–187 pricing, ikreversal Risk INDEX 595 Risk reversal (RR) see derivatives, 370–371 291–292 384–385 377–380 positive, 307–308 quoting, 304–305 one-touch options, 434 self-quanto vanilla options, 514 vanilla, 113–116 price making, 134 vanilla, 132–134 long ATM, interest rate risk, price making, 122 trading exposures, 124 vanilla, 121–124 replication spreading, 541 vanilla trading: vertical, 134 call/put, 134 risk reversals, 130 call/put: confidence interval spreading, 541 defined, 304 horizontal, 132 positive, 307–308 Straddles: Spread contracts, Spread price: Standardized language, 8–9 Stick strike analysis, 351 Sticky delta, 276 Sticky strike, 276 Stochastic interest rate pricing models, Stochastic local volatility models, 382 Stochastic Volatility Inspired (SVI), 232 Stochastic volatility pricing models, Stop-loss orders, 22, 429 Stopping time, in pricing exotic FX Delta see 477–478 470–471 market, 293–294 multi-asset options, 541 price making, 132 trading exposures, 132 vanilla, 132 European barrier options, 418 European digital options, 405 exotic FX derivatives, 462–463 front-window barrier options, bet, 410 bid–offer, 22–23 ATM calendar: limited open hours, 469–470 speed of transactions, 3 jump diffusion, 383–384 local volatility, 380–382 mixed volatility, 382–383 stochastic volatility, 377–380 Spot volatility, forward volatility vs., Spread(s): Spot ladder, 138 Spot market: Spot rate (spot), 3–6, 11 Spot dates, 3, 161–162 Spot delta, Spot dynamic, 469 Spot firm orders, 303 Spot jumps, 245, 246 Smoothing barriers, 466 Sovereigns, 41 Speed of transactions, in FX derivatives Spot (spot rate), 3–6, 11 Smile volatility roll, theta, 271 Smile gamma effect, 272–273 Smile position, 272 Smile pricing models, 375 596 INDEX wp,4 Swaps, Volatility (Stochastic SVI 42 Structurers, strategies, hedging FX Structured 465–466 deposit, Structured 117 sentiment, market Structural 147 146, topography, Strike placement: Strike 454–460 392, options, Strike-out 392 options, Strike-in 225 fly, Strike Strike(s): 125–126 Strangles, ( Straddles: oaiiy 4,37 555–557 397, 340, volatility, 555–557 397, variance, 11 defined, 397 342, correlation, 122–124 straddles, zero-delta 221 trading, reversal risk and 12 options, put and call vanilla position, short-date trading in 276 sticky, 351 analysis, strike stick 151–152 profile, gamma/strike 15 options, FX vanilla describing in 126 exposures, trading 125–126 making, price zero-delta: rcn,564–565 pricing, 557–561 Greeks, 565–566 pricing, 562–564 Greeks, 122–124 placement, strike 122 contracts, ATM nprd,232 Inspired), 463–465 150–152 Continued ) agtrdmto owr (TARF), forward redemption Target 153 month, Target 429 22, orders, Take-profit T 86–87 forwards, Synthetic worst-of and (best-of hedge Switching 335 4, 3, points), (forward points Swap ikn aktpie nEcltrading Excel in price, market Ticking 396–397 exotics, Third-generation 395, options, quanto currency Third 270–271 Theta, 357–358 (TV), Value Theoretical 156–157 Terminology, distributions: spot Terminal 3. forwards, on Tenor, 117 sentiment, market Temporary 395, options, redemption Target oaiiysierl,271 roll, smile volatility position, short-date trading in 271–272 roll, forward 143 defined, 272 balance, cash 272 271, roll, curve ATM integration numerical Excel in 65–66 values, option calculating in 522–524 risk, vega 525 pricing, 525 risk, fixing 524 risk, gamma and delta 395 553 options), iuao,27–31 simulator, 553–554 149–150 69–72 pricer, option tenor 521–525 e also See Market INDEX 597 549–553 476–478 functionality, 35–36 price-taking functionality, 32–35 274–276 zeta, 285–287 adapted Greeks, 275–285 delta, 263–269 gamma and theta, 270–274 interest rate risk, 287–292 vega and weighted vega, quoting, 5 size of, 19, 294 speed of, 293–294 best-of and worst-of options, dual digital options, 545–546 European vanilla options, 497 front-window barrier options, multi-asset options, 539–540 extensions, 36–37 introduce price-making set up ticking market price, 27–31 set up two-way price and risk reversal, 130–131 straddles, 124 strangles, 126 vanilla FX derivatives, 263–292 Transatlantic barrier options, 392–394 TV (Theoretical Value), 357–358 TV adjustment, 358 Two-way price, 20, 32–35, 113 Trading internship tips, 44 Trading risks: Trading short-date position, 150 Trading simulator (Excel), 27–37 Trading volumes, 18 Transactions: 434–435 435–436 and, 153 bid and offer language, 23 bid–offer spread, 22–23 leaving orders, 22 CCY1 vs. CCY2 payout, pay-at-maturity vs. pay-at-touch, bid–offer spread, 434 pricing, 430–434 ATM calendar spreads, 132 butterfly, 128 FX derivatives market, 18 market making, 23–26 price making, 26 risk management, 26 bids and offers, 19–22 vega risk, 425–428 one-touch variations, 434–436 barrier delta gap, 429–430 delta risk, 424–425 gamma and pin risk, 428–429 no-touch options, 438 one-touch options: economic, 200 stopping, 370–371 Trading desk structure, 41–44 Trading exposures: Traders, 41–42 Trading, 19–26 Tradable rates, 104 Trade queries, 146, 147 Time decay, 143 Time value, 79 Time zones, expiry and delivery dates Touch options, 388, 423–438 Time: 598 INDEX ail Xderivatives: FX Vanilla 346–350 analysis, Value Value: V 110–113 surfaces, volatility Updating 61–62 SDE), (in Uncertainty U rcn,103–120 pricing, 160 159, valuing, present settlement cash vs. delivery physical 77–82 value, option conversions, premium option 161–163 calculations, tenor market Greeks: 160 valuing, future 11–18 options, put and call 77–82 derivatives, FX vanilla 79 time, 357–358 Theoretical, in distributions spot terminal 79 intrinsic, ea 89–90 vega, 88–89 gamma, 83–87 delta, 288–291 on, risk rate interest 14 European, describe, to required details 14 American, options, barrier front-window options, barrier European 417 defined, f 18 of, 163–164 14–15 65–66 calculating, 474 417–418 ikmngmn,137–157 management, risk rdn xoue,263–292 exposures, trading 293–312 trading, 121–135 structures, rdn hr-aeposition, short-date trading 155–156 P&L, trading 138–146 gamma, trading 152–155 position, ATM trading language, market derivatives FX 116–120 making, price surfaces, volatility maintaining dpe res 275–285 Greeks, adapted 308–310 risk, off writing 301–302 positioning, vega 302–303 trading, short-date 304–305 spreads, quoting 307–308 spreads, positive 310 risk, pin 306–307 pairs, currency pegged 296–299 options, overnight 310–311 options, delta low derivatives, FX long-dated 300 trading, of rules Gartman’s 303–304 orders, option client data, market broker agreeing 125–126 strangles, 121–124 straddle, 134–135 seagull, 128–131 reversal, risk 131 forward, leveraged 132–134 spreads, call/put 126–128 butterfly, 132 spreads, calendar ATM 156–157 103–116 305–306 311–312 146–152 INDEX 599 ATM volatility and See also volatility, 333–336 342–343 549–550 368–372 correlation framework; Implied volatility Average, 331–333 328–331 368–372 at-the-money, 209–211 butterfly, 213–215 risk reversal, 211–213 realized volatility convexity, spot vs. forward, 470–471 vanilla call options, 12 zero, 58 best-of and worst-of options, dual digital options, 546–548 European barrier options, 420–422 European digital options, 406–408 target redemption options, 522–524 touch options, 425–428 and bid–offer spread, 22 Exponentially Weighted Moving and liquidity, 295 realized spot volatility calculation, realized spot vs. realized forward market instruments, 209–216 vanilla FX derivatives, 89–90 VVV (vega/volga/vanna) pricing, weighted, 154, 274–276 Volatility cones, 344 Vega risk: Vega/volga/vanna (VVV) pricing, Vertical spreads, 134 Volatility. Vega hedge, 463 Vega positioning, 301–302 98–101 368–372 189–190 497–514 CCY1 put options, 510–513 and Greeks, 506–509 pricing, 509 late cash, 497–498 option on forwards, 498–500 bid–offer spread, 514 CCY1 call options, 510 delta, 263–269 gamma and theta, 270–274 interest rate risk, 287–292 vega and weighted vega, 274–276 zeta, 285–287 defined, 209 dephased, 319–320 in Excel Black-Scholes option pricer, Greeks, 562–564 pricing, 565–566 adapted, 221, 281–282 bucketed vega exposures, 154 defined, 171 linear, 174–177 and option premium, 180–181 weekday variance patterns, 189–190 defined, 209 VVV (vega/volga/vanna) pricing, and ATM construction, 171–172, self-quanto, 509–514 late-delivery, 497–503 American, 501–510 Variance (var) swaps, 397, 555–557 Vega, 274 Variance: Vanna, 210, 211, 213–215, 230 Vanilla FX derivatives variations, 600 INDEX oaiiysmile: Volatility 245 volatility, of Volatility 304 orders, firm Volatility ig,16 1,348–349, 112, 106, wings, 106–110 derivatives, FX vanilla 351 analysis, strike stick 361–364. 346, 112, 106, skew, 205, contracts, reversal risk 230–231 management, risk limitations, parameterization exposures, vega instrument market defining, instruments market 346, analysis, instrument market in pricing, derivatives FX exotic in 106 defined, 232 methods, construction 225–230 205, contracts, butterfly 205 contracts, at-the-money rdn,220–222 trading, 211–213 exposures, 219–220 of, drivers 224–225 cross, 222–224 10d, vs. 25d 211–213 reversal, risk 213–215 butterfly, 209–211 at-the-money, 228–229 trading, 213–215 exposures, 228 of, drivers 229–230 10d, vs. 25d 364–367. also 217–225 249–252 209–216 205–208 351 360–367 ikreversal Risk e also See utry(fly) Butterfly See oaiiysiecntuto (Excel), construction smile Volatility eka ainepten,189–190 patterns, variance Weekday 24 risk, Warehousing W pricing, (vega/volga/vanna) VVV 231 230, 216, 215, 210–213, Volga, 313–319 ATM, triangles, Volatility 397, 340, swaps, (vol) Volatility 169 surfaces, Volatility s lc-coe ogtsrk from strike get to Black-Scholes use 233–234 model, smile Malz up set with strike vs. volatility implied plot delta, vs. volatility implied plot placement, strike investigate V vg/og/an)pricing, (vega/volga/vanna) VVV 209 defined, 564–565 pricing, 557–561 Greeks, 169. smile, volatility derivatives, FX vanilla pricing in 169. curve, ATM oaiiysie 106–110 smile, volatility 110–113 updating, 113–116 spreads, bid–offer 103–106 curve, ATM B ucin,236–238 functions, VBA 234–235 238–240 233–240 368–372 368–372 555–557 smile Volatility 103–116 curve 235 delta, e also See e also See ATM INDEX 601 548–553 ATM contracts, 122 strike placement, 122–124 switching hedge, 553 trading risks, 549–553 -score, 350 ‘‘Yours!,’’ 23 Z Zero-delta straddles: Zero interest rates, 160 Zero-premium collar, 129 Zero premium transactions, 464 Zero volatility, 58 Zeta, 285–287 Z Worst-case selling rate, 13 Worst-of (WO) options, 396, Write-off book, 309 Writer, 12 Writing off risk, 308–310 Y Worst-of options See also see Butterfly (fly) 473–482 348–349. 199–204 payoff risk, 475–476 risk, 474–475 trading risks, 476–478 and pdfs, 241–244 pricing, 364–367 risk management, 481–482 in analyzing value, 349–350 generic, 481 Monte Carlo option pricer, 493 rear-window, 395, 478–481 front-window, 395, 473–478 WO options, Worst-case purchasing rate, 12 Wings (volatility smile), 106, 112, Weighted vega, 154 Window barrier options, 395, Weights, added to ATM curve, Weighted ATM shift, 106, 107 WILEY END USER LICENSE AGREEMENT Go to www.wiley.com/go/eula to access Wiley’s ebook EULA.