EXOTIC OPTIONS I 1. List of the Most Common Exotics

EXOTIC OPTIONS I

Revised vers ion Decemb er

Exotic options are a gener ic namegiven toder ivative s ecur itie s whichhave more

complex cashow structure s than standard putsand calls The pr incipal motivation

for tradingexotic options i s thatthey p ermit a much more preci s e articulation

of views on future market b ehavior than tho s e oere d byvanilla options Like

options exotics can b e us e d as part of a

r i skmanagementstrategy or for sp eculative purposes From theinvestors p er

sp ective some exotics providehighleverage b ecaus e they can fo cus the payo struc

ture very preci s ely thi s i s the cas e of barr ier options di scus s e d b elow Exotics are

usually traded over thecounter andaremarketed tosophi sticate d corp orate in

vestors or he dge funds Exotic option de alers are generally banks or investment

houses They manage the ir r i skexp o sure by

makingtwoway marketsandattemptingtobemarketneutral as much as possi

ble and

he dging withtheir vanilla option b o ok and cash instruments

The r i skmanagement of exotics i s more delicatethan thatofstandard options

b ecaus e they are le s s liquid Thi s me ans thatthe s eller of an exotic may not b e able

to buy it backifhistheoretical he dging strategy f aile d withouthavingto pay a large

premium Therefore makinga market in exotic options require s acutetiming skills

in he dgingandthe us e of options tomanage volatility r i sk Roughly sp e aking we

can say that exotic options are tostandard options whatoptions are tothe cash

market By thi s wemean thatexotic options are very s ens itivetohigherorder

der ivative s of option pr ice s such as GammaandVega Someexotics can b e s een

essentially as b etsonthefuture b ehavior of higher order Greeks Gammaand

Vega Another imp ortant i s sue i s the notion of pin risk s ince someexotics have

di scontinuous payos they can havehuge Deltas andGammas ne ar expiration that

makethem very dicult if not imp o s s ible to Deltahe dge

Li st of the mo st common exotics

Digital options

Barr ier options

Lo okbackoptions

Averagerateoptions As ian options

Typ e s et by A ST X

M E

Options on baskets

Forwardstart options

Comp oundoptions options on options

Thi s lecture gives an intro duction tothese der ivatives We will di scus s var ious

asp ects of exotics namely i binomial tree pr icing ii clo s e dform solutions as

sumingthe sp ot pr ice follows a Geometr ic Brownian Motion iii pr ice s ens itivity

andhe dging The s econd p oint will require thatwedevelopseveral mathematical

re sultsonthe di str ibution of rstpas sage times andofthe supremumofBrownian

motion with dr ift over a given timeinterval

As idefromprovidinganintro duction tothe s e instruments thi s study i s inter

esting b ecaus e it give s us a b etter p ersp ectiveonthe r i sks as so ciate d withhe dging

der ivative pro ductsingeneral includingtheriskmanagement of p ortfolio s of stan

dard options

Digital options

A digitalorbinary optionisacontingent claim on someunderlying as s et or

commo ditythathas an allornothing payo A digital call has a the payo

if S K

F S

if S K

A digital puthas payo F S Likestandard options digitals can b e clas s ie d

as European or Amer ican style The Europ e an digital provides a payo of if

the as s et endabovethe str ike pr ice attheoptions matur itydateand zero otherwi s e

TheAmer ican digital has a payo of if theunderlying as s et re aches thevalue K

b efore or attheexpiration date T

European Digitals

Thefairvalue of thedigital call with payo can b e der ived easily under the

as sumptions of lognormal pr ice s the BlackSchole s world In f act thefairvalue

of the digital i s given by

rT rT

V S T e E f H S K g e P fS K g

T T

Themater ial for thi s lecture was taken f rom var ious re s earchpublications andmyown notes

Recommende d reading a JHull Anintroduction to chapter on exotics b MarkRubin

ste in Exotic options prepr intBerkeley Univers ity a compilation of hi s article s in RISK

Magazine andc From BlackScholes to black holesanther compilation of article s byseveral

authors f rom RISK Magazine

where r is theintere st rate as sume d constant andthe exp ectation i s taken with

re sp ect to a r i skneutral probability Here

if X

H X

if X

is the He avisidestep functionThe calculation of the last probability in i s

straightforward Since theterminal pr ice of theunderlyingassetsati se s

Z T r q T

S Se

where Z i s normal withmean zero andvar iance wehave

P fS K g e dz

N Z

Here N isthecumulative di str ibution function of thestandard normal and Z is

dened bythe equation

Z T r q T

Se K

Z ln r q T

Therefore dening

r q T

T d Z ln

we concludethatthe f air value of theEurop e an digital call i s given by

V S T e N d

Thi s formula re s emble s strongly the expression der ive d previously for for the

cashamounttobeheld in the equivalenthe dging p ortfolio for a vanil la cal lnamely

Ke N d

The resemblance i s not accidental Theholder of a Europ e an call i s by equivalence

of thenal cashows long a contingentclaimthatdelivers oneshare if S K

or nothingif S K andshort K Europ e an digital options In other words the

standard call can b e viewed as a portfolio of two digital options onedigital with

payo cons i stingofoneshare and K digitals with payo of On theother

hand f rom theBlackSchole s formula thevalue of the call i s

qT rT

S e N d K e N d

where

r q T

d ln T

We can interpret thetwoterms in thi s formula as thevalue s of thetwo digital

payos thatmake up thestandard option

MathematicallyEurop e an digital options are even s impler topricethan stan

dard options On theother hand f rom the p oint of view of r i skmanagement the

dierence b etween vanillas anddigitals i s substantial There are twofundamental

dierence s

Digital options have mixe d convexity

Digitals have di scontinuous payos

The i s sue of mixe d convexity i s imp ortant for thehe dger b ecauseitme ans that

the r i skexp o sure i s complex as volatilitychange s Recall thatifthehe dger i s short

Gamma he i s vulnerable to large moves in theunderlyingassetwhere as if heis

long Gamma heisvulnerable tosmall moves ie totimedecay In contrast if

thehe dger i s short a standard option theriskisjustones ide d b ecaus e theposition

is short Gammaat all levels of sp ot In particular when makingamarket in digital

options agentsmay not gain a market advantage by quotingapricewithanimplie d

volatilitythat i s higher than theoneofstandard options

Let us examinethese issues bylookingatthe Greeks of the Europ e an digital

Dierentiation withrespectto S in gives

e e

dig ital

S T

and

e e

dig ital

S T

r q T

The s e s ens itivitie s b ecome large as T forS e K AsT converge s

to zero the DeltaoftheDigital option approaches the Dirac deltafunction Thi s

has two cons equence s rst f ar away f rom expiration thevalue of the digital option

is small compare d toandtheDeltas and Gammas are small As the expiration

dateapproaches he dgingthe digital b ecomes much more complicated due tothe

unbounde d Deltas andGammas The s econd cons equence i s pin riskifthe pr ice

of theunderlying as s et o scillates aroundthestrike pr ice near expiration thehe dger

will haveto buy and s ell large numb ers of share s very quickly toreplicatetheoption

At some p oint theamountofshares boughtorsold can b e so large thatthe risk

due toasmall change of thestock pr ice may excee d themaximumliabilityofthe

digital Atthi s p oint Deltahe dging b ecome s extremely r i sky

The Gammaoftheoption vani shes for

r q T T

S S T K e e

Thi s i s thevalue of sp ot for whichthe Deltaofthestandard call i s exactly

For S S T Gamma i s large andnegative Thi s means thatthehe dger i s

exposed to s ignicantriskne ar expiration if there i s a big moveinthe spot price If

SS T thehe dger i s subject totimedecay r i sk hemust rebalance hi s p o s ition

f requently in order to os et timedecayWhen S S T thehe dger i s subject to

r i sk f rom b oth large andsmall moves

The s ens itivityofthe pr ice of thedigital with re sp ect tothevolatility parameter

V S T

Vega

dig ital

e e

Vega also change s s ign at S S T In particular theselleroftheoption i s

vulnerable to an incre as e or a decre as e in market volatility accordingtowhether S

is smaller or gre ater than S T

Oneway tounderstandtheEuropean digital option in terms of standard options

i s in term of call spre ads Recall that a call spre ad i s a p o s ition whichconsistsof

being longone call with a given str ikeandshort another call with a dierent str ike

Let denotea small numb erThen theposition

Tounderstandbetter pin r i sk weshould recall thatpricesdonotchange continuously the

lognormal approximation i s just a convenientdevice for generating s imple pr icingformulas The

discrete nature of pr ice movementscanmakecontinuoustimehedgingtechnique s very r i sky when

Deltachange s rapidly as thespotmove s Thi s obs ervation applie s also to highly leverage d option p ortfolios

long calls withstrike K and

short calls with str ike K

where the calls havethe same expiration dateasthedigital has a nal payo

Min f Max S K g

Thi s function i s gre ater than H S K for all S Thi s implie s thatthevalue of the

T T

digital i s le s s than thatof K K callspre ads We say thatthe call spre ads

dominate the digital Thi s ob s ervation sugge ststhatagoodhe dgingstrategy for

digital options would b e to us e callspre ads instead of Deltahe dging ie toadopt

a static he dgingstrategyIf i s large thi s will require relatively few options but

the f air value of the spre ad may b e much gre ater than thatofthe digital On the

other hand dimini shing will makethe dierence in pr ice s arbitrar ily small butthe

he dge require s manyoptions Thi s may b e dicultandcostly to execute Moreover

options with str ikes very clo s e to K may not exi st in themarket Neverthele s s the

ide a of us ing call spre ads i s a us eful one In f act he dgingwithanoption spre ad

which approximates thebinary payo but not nece s sar ily dominate s it or replicates

it exactly can help os et pin r i sk by dimini shingthemagnitudeofthejump In

other words a p ortion of the r i sk can b e divers ie d byhe dgingwithoptions and

the re s idual ie the dierence b etween the payos of the digital andtheoption

spre ad can b e hedgedinthecashmarket at a le s s er r i sk

Example An imp ortant example where digitals app e ar in nance i s in the pr ic

ingofcontingent premiumoptionsThese are der ivativesecuritie s which are

structure d as standard Europ e an options except for the f act thattheholder pays

the premium at maturity and only if the option is in the moneyAcontingent

premiumoption can b e viewe d as a p ortfolio cons i stingof

Longonestandard option withstrike K andmatur ity T

Short V binary calls with str ike K andmatur ity T

where V repre s entsthepremium to b e paid if andonlyif S K The

intr ins ic value of a contingentpremium call option i s therefore zero for S K

and S K V for S K Notice in particular thattheinvestor makes

T T

money only if S K V and will actually lo s e money if KS K V

T T

Thesituation i s analogous tothat of a p erson whichhas f ree me dical insurance

butwith with a large deductible If theoption i s structure d so thatnodown

payment i s require d then accordingtoequations and V should satisfy

rT rT

e E f Max S K g e V P f S Kg

T T

Therefore f rom and thefairdeferre d premiumshould b e

N d

r q T

V Se K

N d

More generallysuchoptions can b e structure d so thata portion of the premium

i s paid upf rontandanother i s contingentontheoption b e ing inthemoney at

matur ity All suchoptions haveembedde d Europ e an binar ie s Theidea was

implemente d in recentye ars for de s igningdebt s ecur itie s known as structure d

notes A s imple example of a structure d notewould cons i st of a notewith

coup ons indexe d to LIBOR withthefollowingcharacter i stics

Coup on payment Max LIBOR

Contingent premium if LIBOR on the coup on date

Thi s note guarantee s theholder a o or of on theintere st rate income The

structure resemble s that of a oatingrate note withanintere strate oor series

of putsonintere st rate s However theinvestor do e s not pay for theoorwhen

he buys the structure d note Instead he can andalsomust take advantage of

theintere strateoorby paying bas i s p oints if LIBOR go e s b elow

on any given coup on date Thi s der ivative s ecur ity could b e desirable toinvestors

thatbelievethatifintere st rate s go b elowthen they will b e s ignicantly below

for some p er io d of time Also the structure may b e desirable toinvestors

s eeking protection if intere st rate s dropbutwhoareunwillingtonance the

intere strate insurance upf ront

Amer ican digitals

Thepayo of theAmer ican digital option i s s imilar butnowtheholder rece ives

if and when the the underlying asset trades above K for the rst time Thi s

intro duce s additional timeoptionalitytothe problem Thefairvalue of theAmer

ican digital i s accordingtothe general pr inciple s

V S T E f e T g

where exp ectation i s taken with re sp ect toariskneutral probabilityme asure and

repre s entsthe rst hittingtime of the str ike pr ice S K

It i s worthwhile to cons ider thetwovaluation methods thatwe are f amilar with

the binomial pr icingmodel andthe lognormal approximation As weshall s ee the

former i s relatively e asy to program The lognormal approximation give s ins ight

intothesensitivitie s of the pr ice withrespecttothemodel parameters as idefrom

beingusefulto b enchmark the binomial tree calculation However ir require s

intro ducingnew tools f rom ProbabilityTheory

Toimplementthe binomial approach we pro cee d as follows bydenition the

value of theAmer ican binary i s if theoption i s in the moneyTherefore with

the usual notation wehave

j j

K a if S V

n n

and

j j

V if S K b

N N

The p ortion of the binomial tree whichnee ds tobedetermined byrollback corre

sp onds tothenodes n j suchthat

S K with nN

Thevalue of the binary option atthese node s i s calculated usingthe f amiliar recur

sive relation

o n

n n

j j

j r dt

V P V V e P

n n

D U

The only dierence f rom thenumer ical p ointofviewbetween Amer ican and Euro

pean binar ie s re s ide s in thefactthatthevalue atthenodes whichareonestep away

f rom theboundary fS K g are computed using equation a for thevalue of

j j

V ie s etting V Thi s typ e of problem thatinvolves a lateral b ound

n n

ary condition i s known as a Dir ichlet problem or boundaryvalue problem in

thetheory of Partial Dierential Equations

Notice thatthe formulation allows for termstructure s of volatilityand in

tere st rates Thi s p oint i s s ignicant b ecaus e thehittingtimeofthe barr ier i s

unknown Therefore the eect on pr icingdue toatimevaryingvolatility i s non

tr ivial From thepointofviewofintere strateandvolatilityterm structure s the

main dierence b etween Europ e an andAmer ican binar ie s i s due tothefactthat

the relevantvolatility parameter for Europ e an binar ie s i s the annualize d standard

deviation of thechange of pr ice b etween nowandthematur itydate In contrast

Amer ican binar ie s are s ens itivetotheentire volatilitypath

Clo s e df rom expre s s ions for Amer ican binary options can b e obtained under the

as sumption thatthevolatilityandintere st rate s remain constant Thi s require s

intro ducingnew mathematical tools Let f repre s entthe probabilitydens ity

function of the random var iable ie

P f Tg f d

Then thevalue of theAmer ican digital in equation can b e wr itten as

In contrast Europ ean binary options can b e pr ice d in a timedep endentvolatilityenvironment

likestandard options us ing an eective meansquare volatility suchthat T dt

V S T e f d

An explicit expre s s ion for the probability di str ibution of the rstexit timecanbe

der ived from

Lemma Let Z represent a Brownian motion with drift ie

Z Z t

where Z is a Brownian motion and is a constant Let represent the rst time

t A

the path Z hits A where AThen

A T A T

p p

P f Tg N e N

T T

Wedefer the pro of of thi s Lemmafrothemoment Toapply thi s re sulttothe

cas e of a lognormal random walk of the form

t Z r q

S Se

weset

r q

and

A ln

Thereader will ver ify e as ily thatif repre s entsthersthittingtimeof S K

then

P f Tg P f Tg

Us ing equation we concludethat

r q

P f Tg N d N d

where

d ln r q T

Equation give s a clo s e df rom expre s s ion for the probability di str ibution of

the rstexit probabilityoftheset fSKgThe probabilitydens ityof can then

b e computed by dierentiatingwith re sp ect to T More preci s elywehave

e f d V S T

P f g d e

r r

e P f g r e P f g d

r rT

e P f g d e P f Tg r

Thenal expression for V S T i s f rom

T r q

r rT rT

e P f g d V S T e N d e N d r

where the probability ins idetheintegral i s given by with T replace d by

Thi s last integral can b e computed numer ically by quadrature Notice thatinthe

sp ecial cas e r the formula s implie s further The sameistrueiftheoption i s

mo die d so thattheholder collectsat time T if thepriceever touches K and

not attime In b oth cas e s thevalue of theAmer ican digital i s

r q

rT rT

V S T e N d e N d

Next we di scus s theoptions s ens itivitie s tochange s in sp ot pr ice or market

volatility

Boththe Deltaand GammaoftheAmer ican digital are monotone incre as ing for

S K and b ecomeunbounded as T inane ighborhood of S K

Therefore theoption has s ignicant pin r i sk Theworstcas e scenar io for the

he dger would b e a market rallying slowly towards thestrike level whichcollap s e s

imme diately b efore theoptions matur ityInthi s event Deltahe dging buildsupa

large sp ot p o s ition in the rally longthemarket If themarket f alls suddenlythe

he dger may incur a s ignicant lo s s defe atingthe purp o s e of Deltahe dging There

is however an imp ortant dierence withrespectto Europ e an binar ie s thehe dger

do e s not havetoworry aboutmarket whipping aroundthe str ike pr ice s ince the

option expire s after K i s hit for the rst time The exp o sure to Gammaandthe pin

r i sk are s impler than for the Europ e an counterpart

TheVega of theAmer ican digital option is positiveatallvalue s of sp ot Thi s

follows f rom the convexity with respect tothe sp ot pr ice Thus the r i skexp o sure

due to an incorrect e stimateofthevolatility i s only onesided Just likewith

standard options the s eller fe ars an incre as e in volatilityandthe buyer a decre as e

in volatility

Ne ar the barr ier S K we can make a straightforward analys i s of the s en

sitivityoftheDeltaoftheoption tovolatilitywhatsometraders call collo quially

DDeltaDVol In f act we knowthat V S T i s nondecre as ingasa function of

thevolatilityandthat V K T Thi s implie s thatthe dierence quotient

V K T V K T

K T

decre as e s as incre as e s Hence var ie s invers ely to in a ne ighb orhood of

K Therefore incre as ingthevolatility parameter with re sp ect to saytheimplie d

volatilityofvanilla options trade d in themarket will provideprotection against slip

page when thespotisaway f rom K improvethe exp o sure topinriskbydecre as ing

the Deltaatthe barr ier

Barr ier options

Barr ier options are a gener ic namegiven toder ivative s ecur itie s withpayos

which are contingentonthe sp ot pr ice re aching a given level or barr ier over the

lifetimeoftheoption The mo st common typ e s or barr ier options are

Kno ckoutoptions The s e are contingentclaimsthat expire automatically

when the spot price touches one or more pre determine d barr iers

Recall thatthe Europ ean digital has twos ided volatilityrisk

Of cours e s ince increas ingthevolatility increas e s thepremium the s eller will havetocharge

more if hewishes to followthi s augmented volatility strategyHemust therefore charge abovethe

market volatilityorelsesetaside someofhisother funds tonance the strategy

Kno ckin optionsThese contingent claims are activated when the spot price

touches one or more pre determine d barr iers

The mo st common barr ier options are structure d as standard European putsand

calls withone kno ckin or kno ckout barr ier For instance

a downandout call with str ike K barr ier H andmatur ity T i s an option to

buy theunderlyingassetforK attime T provided thatthe sp ot pr ice never

goes belowH between nowandthematur itydate

An upandoutcallwith str ike K barr ier H andmatur ity T i s an option to

buy theunderlyingassetforK attime T provided thatthe sp ot pr ice never

go e s aboveH between nowandthematur itydate

A downandin call with str ike K barr ier H andmatur ity T i s an option to

buy theunderlying as s et for K attime T provided thatthe spot price goes

b elowH between nowandthematur itydate

An upandin call with str ike K barr ier H andmatur ity T i s an option to buy

theunderlying asset for K attime T provided thatthe spot price goes above

H between nowandthematur itydate

Similar denitions apply toputs Barr ier options are e sp ecially us e d in fore ign

exchange der ivatives markets The London Financial Times of Novemb er

rep orted that exotic options now constitutearound of the currency option

bus iness Barr ier options which are relatively s imple var iations on the European

putand call enjoyagreatpopular ity

A rst ob s ervation regarding barr ier options i s that are muchcheaper than stan

dard options Theoptionalityfeature can b e targete d more preci s ely byintro ducing

a barr ier Thi s i s illustrated in thefollowingexample described tomeby a trader

Example A large multinational corp oration bas e d in Europe must convert its

US bus ine s s revenue into DEM p er io dicallyGiven theweakness of theDollar

with re sp ect totheDeutschemarkinthe past years anddropoftheDollar atthe

b eginning of the company fe ars a decre as e in revenue s in DEM terms Its

tre asury department could haveanticipated theproblem by purchas ingstandard

options but did not do thi s Ideallythe companywould liketohaveanatthe

money DEM callDollar put withsixmonths toexpiration If thespotexchange

rate i s DEMUSD thevalue of a dollar putwithstrike expir ingin

days i s p er dollar notional On a million notional thecostof

thi s option i s therefore approximately On thother hand suppose

thatthe company purchases nowa downanoutdollar put or equivalentlyan

upandoutMark call with a kno ckout barr ier at DEMUSD Thevalue of

thi s option i s inste ad p er dollar notional or tothenearest

We will der ivebelow a pr icing formula for kno ckoutoptions Thus

G BowleyNewBreed of exotics thrives LFTNov Supplementonder ivatives

Weusedavolatility of a US dep os it rateofanda German dep os it rateof

Theresultwas rounded totheneare st

the kno ckoutoption with a barr ier i s nearly times cheaper than thevanilla

Therefore if the tre asurer b elieves thatthedollar will not drop b elow over

thenext s ix months the kno ckoutoption provides a cheaper alter nativewiththe

same terminal payo

Theoption de scr ib e d in theabove example which kno cks outwhen theoption i s

inthemoney i s often calle d a revers e kno ckoutThe dierence b etween inthe

money andoutofthemoney barr iers i s s ignicant b ecaus e the former have di scon

tinuous payos atexpiration Thus Deltahe dgingrevers e kno ckin andknockout

options may le ad to s ignicant pin r i sk s imilar totheone encountere d in digitals

In contrast options withoutofthemoney barr iers do not s eem tobevery intere st

ing f rom a he dging p ersp ective Wewilltherefore di scus s pr imar ily barr ier options

which kno ckinoroutwhen theoption i s inthemoney

Kno ckin and kno ckoutoptions are related bythesimple formula

KI KO Vanilla

Thi s formula i s s elfevident theholder of a p ortfolio cons i stingofone kno ckin call

andone kno ckout call with same str ike barr ier andmatur ity will eectively hold

a call atmatur ity regardle s s of whether the barr ier was cro s s e d or not Wecan

therefore re duce the que stion pr icing barr ier options tothepricing of kno ckouts

We notethat in some cas e s the structure of barr ier options i s more complicated

We notetwo cas e s thatwere mentioned tousby profe s s ional traders

Double kno ckin or double kno ckout options whichhavetwo barr iers

kno ckoutoptions withrebateTheholder rece ive s a consolation pr ize in

the form of a cash rebateonthepremium paid if theoption kno cks out

Pr icing barr ier options

Barr ier options are pr ice d bysolvinga boundaryvalue problem s imilar totheone

for Amer ican digitals In the cas e of an upandout call thevalue of thi s der ivative

s ecur ityisdetermine d recurs ively bysolvingthe problem

h i

n n

j j

j r dt j

V e P V P V if S H a

n n

U D

n n

where P and P are r i skneutral probabilitie s

U D

j j

H b if S V

n n

and

Thi s option cons i sts of a regular kno ckoutoption withanattached Amer ican digital option

h i

j j j

V Max S K if S K c

N N N

In the cas e of an upandin call theboundary conditions b and c are

replace d by

j j j

V V if S H a

n n n

repre s entsthevalue of a vanilla call atthenoden j and where V

j j

V if S K b

N N

Thevalidityofthe s e equations follows f rom i theterms of the barr ier options

whichdeterminetheir value atthe barr ier andatmatur ityandiitheab s ence

of arbitrage whichimplie s a Thevalue s of barr ierputsaredetermined by

makingobvious mo dications

Next we cons ider thepricingassumingthat S i s a lognormal random walk with

constant q and r constant geometr ic Brownian Motion As in the cas e of the

Amer ican binary option we will nee d someauxiliary re sultsonthepropertie s of

Brownian motion with dr ift

Lemma Let Z t represent a Brownian motion with drift Then if A

and B arepositive numbers with B A

BB dB A and Z P Max Z

t T

T B

dB dB e e

Weshall provethi s Lemmalater

Toapply thi s re sult let denotethersttimethatthe lognormal walk

hit the level S H Then thevalue of a downandoutputwith str ikepriceK

kno ckoutat H HKandmatur ity T sati se s

P S T K H e E f Max K S Tg

KO T

E Max K S Min S H e

T t

t T

Thi s last expre s s ion can b e rewr itten as

e E K S H S K Min S H

T T t

t T

e E f K S H S Kg

T T

e E K S H S K Min S H

T T t

t T

e E f K S S Kg

T T

e E f K S S Hg

T T

Ke P H S K Min S H

T t

t T

e E S H S K Min S H

T T t

t T

Notice thatthe rst term corre sp onds tothevalue of a standard European putwith

str ike K The s econdterm can b e calculated easily usingthesame re asoningasin

theder ivation of theBlackSchole s formula To calculatethetworemainingterms

we will us e theresult of LemmaIntro ducingtheparameters

A ln

and

r q

wehave us ing Lemma

P H S K Min S H P A Z A Min Z A

T t H K H

t T t T

P A Z A Max Z A

K H H

t T

A B

H dB

B T

e e

Here weusethefactthat Z t and Z t havethe same probability

t t

di str ibution Similarly

E S H S K Min S H

T T t

t T

S E e A Z A Max Z A

K H H

t T

A B

H dB

B T B

S e e e

Calculating explicitly thetwointegrals in and andusingthe BlackSchole s

formula to calculatethe rst twoterms in we arr iveatthenal re sult

P S T K H

rT K qT K

Ke N d Se N d

rT H qT H

Ke N d Se N d

r q

f N d N d g Ke

r q

Se fN d N d g

where

r q T ln d

d ln r q T

ln d r q T

ln r q T d

d ln r q T

T ln r q d

and

d ln r q T

The formula for an upandout call i s obtained imme diately bya change of nu

meraire an upandout call on the r i sky as s et with str ike K i s nothingbuta

downandoutput on cash withtheunderlyingassetviewe d as theunit of account

The pr icing formulas for upandoutputs downandout calls are obtaine d us ing

very s imilar technique s Weleavethem as an exerci s e for theintere ste d re ader

Finallythe f air value s of kno ckin options can b e obtained usingthe par ity

relation For instance us ing and wendthatthevalue of a down

andin putis

qT H rT H

Se N d P S T K H Ke N d

r q

Ke f N d N d g

r q

fN d N d g Se

Toendthi s s ection we presentsomenumer ical value s for a particular barr ier

option thi s example was mentioned earlier

Example Arevers ekno ckoutdollar put DEM call

USD intere st rate

DEM intere st rate

volatili ty

Str ike DEMUSD

Kno ckoutat DEMUSD

days tomatur ity

Sp ot

Val

days tomatur ity

Sp ot

Val

days tomatur ity

Sp ot

Val

days tomatur ity

Sp ot

Val

He dging barr ier options

The r i skmanagement of barr ier options should takeinto cons ideration themixed

Gamma exp o sure of the s e instrumentsforrevers ekno ckoutsand kno ckins as well

as the pin r i sk atthe barr ier

The r i skexp o sure of a reverse knockoutputoption can b e understood intuitively

as follows f rom equation theholder of thi s option i s

Longa standard put with str ike K

short an Amer ican digital option with barr ier at H which pays oneputwith

str ike K up on hittingthe barr ier in other words a kno ckin put

Ignor ingthe dierence b etween a kno ckin putandanAmer ican digital option

with payo H K atthe barr ier i s not a bad approximation near the expiration

date Wethen s ee immediately thattheoption has mixe d Gamma exp o sure f ar

away f rom the barr ier thestandard put dominates andtheholder of the kno ckout

i s longGammaVega where as near S H the digital dominates andtheholder

is short GammaVega

The s eller thatwishes tohe dge f ace s the mirrorimage position short Vega and

Gammanear the str ike or inthemoney and long GammaVega clo s er tothe barr ier

However at the barrier the Gamma r i sk i s complex if thespotpriceisjustbelow

the barr ier thehedger must adjust hi s Deltainorder toearntimedecay hi s

liabilityisthatofastandard putiftheoptions f ails toknockout The Delta

incre as e s withoutbounds near the barr ier On theother hand if theoption does

not kno ckout the large Deltaposition may b e detr imental in cas e of a large market

b ecaus e thi s would le ad to a lo s s in thspotmarket

Example Cons ider theoption de scr ib e d in the previous s ection as sumingthat

an agentsold theoption with days toexpiration when thespotpricewas

at p er dollar notional If s even days b efore expiration thespot

trades at DEMUSD andthe agent Deltahe dge s accordingtotheabove

table s hi s sp ot p o s ition in USD would b e long p er dollar notional A

drop of in theexchange rateinoneday will re sult in a lo s s of p er

dollar notional Tomakethi s more concrete as sumethatthe notional amountis

dollars The premiumcollecte d for theoption was The

sp ot p o s ition on theother handisawhopping The loss in the

dollar p o s ition if themarket move s suddenly down by through the barr ier

would b e Tohaveabetter ide a of thelikelihood thatthi s happ ens

notethatthi s repre s ents a moveofspotinonedayAt an annual volatility

of thi s would b e a threestandard deviation moveinoneday Theevent

has lowprobabilitybut i s not imp o s s ible Would you r i sk a lo s s of million

given the o dds Moreover let us mention the imp ortant p ointofliquidityA

s ellingorder of ne arly million dollars as theexchangerategoesthrough the

barr ier may caus e a further dropinthedollar as there will b e few buyers and

many s ellers Thi s will have dire cons equence s for thehe dger

Pro ofs of Lemmas and

Thi s s ection contains sketche s of theproofsofthetwo Lemmas us e d toder ive

clo s e dform solutions for barr ier options andAmer ican digitals

A cons equence of theinvar iance of Brownian Motion under reec

tions

Lemma Let Z denote standardBrownian motion on the interval T Then

for al l A and B A we have

P max Z A Z B B dB e dB

T T

t T

Oneshould also takeinto accountthattheannualize d volatilitymay very well undere stimate

thedaily moveofthe exchange rate

Tondout more aboutthe r i skmanagementofexotic options s ee N Taleb Dynamic

Hedging manuscr ipt in preparation where in particular the liquidity i ssue in the trading

of barr ier options i s di scuss e d in greatdepth

Pro of Cons ider a s imple random walk dened by

X X dt n N

n n

p p

where dt repre s entsa small p o s itivenumb er The probabilitie s for dt and dt

are as sumed tobe Set

A dt

where X repre s entsthe integer part of X Therefore A repre s entsthe large st

integer multiple of dt whichis A

As sumethat a given path or re alization of therandom walk i s suchthat X

A for some n N andthat X A for mnWe ob s ervethatthepath

which coincide s withthi s re alization of the random walk for m n andwhichis

reectedabout the line X A for m n o ccurs withthesameprobabilityas

the or iginal onenamely Therefore we concludethat for BA

P max X A X B P max X A X A B

j N j N

P f X A B g

Thi s last equalityholds b ecaus e A B A

Let T Ndt By the Central Limit Theorem the joint di str ibution of the

approaches thatoraBrownian Motion as dt N random var iable s X

Therefore if we replace formally max X by max Z X by Z and

j t N T

j N t T

A by Awe conclude f rom thatequation holds The Lemmaisproved

Notice thatwehaveestabli shed theanalogue of Lemmainthecase

The cas e

To prove Lemmawewillnee d thefollowingresultaboutBrownian Motion

withdrift

For a r igorous pro of of the formal passage tothe limit s ee for instance

Billingsley ConvergenceofProbability Measures Wiley

Theorem Cameron Martin Let F z z z bea continuous func

tion Then

E F Z Z Z

t t t

n o

Z t

t n

E F Z Z Z e

t t t

Thi s re sultstates thatthe exp ectation of a function of Brownian motion with dr ift

i s equal tothe exp ectation of thesamefunction of regular Brownian multiplie d by

an exp onential f actor namely

Z T

Pro of of theTheorem Denethe incrementsoftheBrownian path

Z Y Z

t t

j j

Z Z t t

t t j j

j j

Also s et

G y y y F y y y y y y

n n

Us ingtheexplicit form of theGaus s ian di str ibution andthe f act thatthe increments

Y are indep endentrandom var iable s withmean t t andvar iance t t

j j j j j

we obtain

E F Z Z Z

t t t

E f G Y Y Y g

dy dy dy y t t

n j j j

G y y y exp

t t t t t t

j j n n

n R

dy dy dy

j n

G y y y exp

t t t t t t

j j n n

where

y t

j n

G y y y G y y y e

n n

Makinga change of var iable s wendthatthelastintegral in i s equal to

n o

Z t

t n

F Z Z Z e E

t t t

whichiswhatwewanted toshow Thi s concludes the pro of of theTheorem

We are now re ady for the

Pro of of Lemma Us inganapproximation argumentwhichwe omit it can b e

shown thattheabovetheorem can b e also applie d tothe functional of thepath

F Z Max Z

t T

Thi s i s a continuous function of thepaththat can b e approximate d in a suitable

s ens e bycontinuous functions of n var iable s as in the previous theorem

ApplyingtheTheorem tothi s functional we concludethatifC Athen

C P Max Z A Z

t T

T Z

E Max Z A Z C e

t T

B T

E Max Z A Z B e dB

t T

A B

B T

e e

See Billingsley ibid

where we us e d Lemmatoder ivethe last equalityEquation follows immedi

ately by dierentiatingbothside s of

Finallyweprove Lemma on the di str ibution of the rstpas sage time for Brown

ian motion withdrift

Pro of of Lemma Us ingthenotation of Lemmas andwendthat

A P f Tg P Max Z

t T

P Max Z A Z B dB

t T

Z Z

A A

B dB B g dB P Max Z A Z P fZ

T T

t T

Z Z

A A

B dB dB

T T

p p

e e

T T

The conclus ion of Lemmafollows byevaluatingthi s last expre s s ion in terms of

thecumulative normal di str ibution