TheIns andOuts of BarrierOptions: Part 1

EMalusL DrrurreNexp Inel KAt.lt

arrier options are extensionsof stan&rd stock options. Standard calls and puts have payoffi that depend on one market level: the strike. Barrier options have payoffsthat dependon two market levels:the strike and the barrier. Investorscan use them to gain exposureto (or enhancercturns from) firturc marketscenarios more complex than the simple bullish or bearishexpectations embodied in stan- dard options. [n addition, their premiumsare usudly lower than those of sandard options with the samestrike and e4pintion. Part 1 of this article describesmany of the most commonbarrier optionsavailable, as well as when to use them in place of standardoptions. We alsodescribe why their sensitiviryto market movesoften exceedsthat of standardoptions. This is depicted in severalcases with ables and plots, showingthe theorcticalvdues and sensitivitiesfor diferent marketlevels. Part 2, which will appearin the Spring 1997 issueof DeriuatiuesQuarufly, expands on this presentation.Somewhat more exotic barrier optionsare explored, and a surrmaryof the math- . Euer.rusr, Denuex is head of ematicdtheory behind the vduationsis presented. the Quantitative Stratcgies Group at Goldman, Secbs& Co. BARRIEROPTIONS DEFINED in NewYork. StandardEuropean-style options on stock Inry Kem is a vice presidentin arc characterized by their time the Quantitative Strategies to and Group at Goldman,Sachs & Co. strike level. At expiration, standard cdl options io NcwYork. pay the owner the difference between the stock

Wnrrrr 1995 Dnnrvlrnrusouannnry 55 price and the strike level if the stock is above the or put) with the same strike and expiratr strike, and zero otherwise. Similarly' standard put the stock never crossesthe barrier, the optio.- options pay the owner the difference benveen the expires worthless. strike and the stock price if the stock is below the An out barrieroption, or knockoutoption, strike, and zero otherwise. A call owner benefits pays off only if the stock finishes in the money from an upward stock move; a Put o$tner benefits and the barrier is never crossedbefore expiration. from a downward stock move. As long as the stock never crc,ssesthe barrier, the Banier options are a modified form of stan- out barrier remains a standard option of dard options that include both pus and calls.They the same type (call or put) with the same strike are characterized by a strike level and e banier level, and expiration. If the stock crossesthe barrier, the as well as by a ush rebateassociated with crossing option is knocleeilout and expires worthles. the barrier. As with standard options, the strike Therefore, barrier options can be up-anil- level determines the payoff at expiration. The out, up-and-in, down-anil-out,or iloum-anil-in.The barrlgr options contract specifies,however, that different barrier options and the effect of crossing the payoff depends on whether the stock price the barrier on their payoff are described in ever crossesthe barrier level during the life of the Exhibit 1. option. In addition, if the barrier is crossed,some In principle, all barrier options can have barrier option contracs specify a rebate to be paid associatedcash rebatesthat are paid to the option- to the optionholder. holder when the barrier is crossed.In practice, This article discussesonly European-sryle however, only out option contracts provide cash barrier options. Furthermore, in most caseswe rebates,paid to the optionholder as a sort of assumethe rebateis zerc. consolationprize when the option is knocked out. Barriers come in two types. We call a barrier above the current stock level an up barrie6 WHY USE BARRIER OPTIONS? if it is ever crossed,it will be from below. We call a barrier below the current stock level t ilown baniq, There are three basic reasonsto use barrier if it is ever crossed,it will be from above. oprions rather than standarrdoptions: Barrier options come in two types: r'r options and out options. An in barrier option, or BennrcnOrnoN PavorrsMrv Monr knock-in option, pays off only if the stock finishes CrossryMercH BnlrcrsAsour rHs in the money and if the barrier is crossedsome- Ftm;nEBnrnvron oFTHE Mamrr time before expiration. When the stock crosses the barrier, the in barrier option is knockedin and Traders value options based on options becomes a standard option of the same tlpe (cdl theory. In liquid markets, you can rnlue options

Elcmrrl VarusrrusoF BARRTEROPTIoNS

Barrier Effect ofBarrier on Payoff Option Ttp. Location Crossed Not Crossed

Catl Down-and-Out Below Spot Worthles StandardCall Down-and-In Below Spot StandardCall Worthless Up-and-Out Above Spot Worthles StandardCdl Up-and-In Above Spot StandardCall Worthles Put Down-and-Out Belor Spot WortNes StandardPut Down-and-In Below Spot StandardPut Worthles Up-and-Out Above Spot Worthless StandardPut Up-and-In Above Spot StandandPut Worthless

55 Tns INsAND Otns or BanncnOrnor.rs: Perr I WwreR1996 by calculating the expected value of their pa)'o6, corresponding standard options. For example, averaging over all stock scenarioswhose mean knockout options wont pay offif the stock crosses price is the stock's at each future the knockout barrier. Therefore, they are cheaper time. According to the theory you pay for volatil- than an otherwise identical option without the ity around the forward price. knockout feature. If you think the chance of By buying a barrier option, you can elimi- knockout is smdl, you can ake advantageof the nate paying for those scenarios you think are lower premium and get the samebenefits. Or, you unlikely. Alternatively, you can enhance your can even pay more premium to receive a cash return by selling a barrier option that pays off rebateif the option is knocked out. only on scenariosyou think are improbable. Similarly, knock-in oprions have lower Suppose the forward price of the stock a premiums than corresponding standard options year from today is 105% of spor. You believe rhat with the samestrike and expintion. You may find the market is very likely ro rise, but that if it these attractively cheap ifyou feel the probability somehow drops below a supporr level of 97Yo,it of getting knocked in is high. rvill instead decline further. You can buy a down- and-out call struck at 105o/othat gets knocked out SPECIAL FEATURES if the stock, at any time, falls below 97%. In tltts OF BARRIER OPTIONS way, you avoid prying for scenariosin which the stock first declines substantially and then rises Managing the risk of an options position is again. At 2Q% , this can reduce your more complicated than managing a stock position premium by half. alone. You can dynamically hedge options by Alternatively, you can enhanceyour return going short an amount dehaof stock againsta long by selling a down-and-in cdl srruck 705yo thet options position, where delta is the theoretical ^t gets knocked in only if the market falls below hedge ratio. If delta is negarive,you go long that 97Yo.You can then pick up premium by betting amount of stock. The value of the oprion and its againsta scenarioyou feel is unlikely. delta depend upon both market level and volatili- ty. Standard call options, for example, have delta BennrsnOrnoNs May M^trcH values between zero and one, and call values HnpctNc Nsrps Monn Closnry increasewith increasing volatility. TnaN Snnrnn SreNpaRpOruons Barrier options are similar to, but more complicated than, standard options, becausethey Suppose you own a srock and have decid- sample payoft of a smaller and more specialized ed that if it ever rises by more than 5% over the set of future stock paths. You can dynamicdly next year, you will sell it, but you still want delta hedge them, just as you do with standard protection aEainst market declines of more than options, by using a theoretical model to ralue the 5%. You can buy a stan&rd struck at option and cdculate its delta. You can also use 95Yo of spot, but then you get protection even other options to create static hedges that approxi- when the market rises more then 5% and you no mately duplicate the barrier option's payo6. longer need it. lnstead, it would be better to buy The price sensitiviries of barrier options an up-and-out put with strike at g|yo, barrier at can differ markedly from those of standard l05yo, and no cash rebate.This pur will disappear options. Comparc an up-and-out cdl option with as soon as you sell the stock and have no further a standard cdl option. As the stock price moves need ofit. up, a standard call dwaln garns in ralug while an up-and-out cdl is subject to two opposing effects. pnruruus Ba,nnrsnOptrox fu the stock price moves up, the up-and-out cdl's ARE GENERALTYLowER THAN payoffbecomes potentidly larger, but the upward Tnoss oF STaNDARDOmrows move simultaneously thrcatens to extinguish the total vdue of the contract by moving it closer to Barrier options are often attractive to the out barrier. The conflict between these rwo investors becausetheir premium is lower than tendencies makes it extremely sensitive to the

Wnrmn1996 f)nnruamrrs ft ranrrnr.v 57 stock's movement near the barrier, and delta can option relativeto the barrier is such that any non- go rapidly from positive to negative rnlues. zero payoffguaranteesthe option will be knocked We describe the behavior of barrier out, then the option is worthles. So, an up-and- options in more deail as we examine each type. out cdl with strike above the barrier is worthless. Meanwhile, be awarc of two key ways in which Also, a down-and-out put with strike below the barrier options differ from standard options barrier hasno rnlue. when the underlying stock or index gets near There is a simple relationship between the barrier level. European-srylein options, out options, and stan- First, the delta of the barrier option can dard options. If you own both an in option and differ significantly from the delta of the corre- an out option of the same type (call or put), wirh sponding standard option. For example, a barrier the same e:cpiration, the same respective strike, call can have delta rnlues les than zero or greater and the same respective barrier level, you are than one; an up-and-out deep in-the-money cdl guaranteed to receive the payoff of the corre- whose rnlue must vanish at the barrier has a nega- sponding standard option whether the barrier is tive delta near the barrier becauseof the rapid crossedor not. Therefore: decline in its vdue there. Second, barrier option ralues can actudly o The vdue of a down-and-in call (put) plus decreasewith increasing volatility. The up-and-out the value of a down-and-out call (put) is call just described becomes more likely to get equd to the rnlue of a corresponding stan- knocked out near the barrier asvolatility increases. dard cdl (put). In options parlance, even though you are long o The value of an up-and-in cdl (put) plus the volatiliry or "long gamma" relative to the strike rnlue of an up-and-out call (put) is equd ro the level, you are short volatiliry or "short gamma" value of a correspondingsan&rd cdl (put). relative to the barrier. You can often get a good feel for the way a barrier option will behave as the underlying stock SOME EXAMPLES price rnries by classifying the strike and barrier levels with regad to volatiliry and then combining In this section, we give examples of both their effects.The owner of a barrier option is long out and in European-style options with zero volatility at the strike, long volatility at an in barri- rebate, and compare their theoreticd values with er, and short volatfity at an out barrier. We make those of corresponding European-style standard use of this type of analpis later. options. Later, we show how their values and In certain cases,like the ones listed next, sensitivities vary with market levels for a wide the strike level of an in option relative to the rnriety of barrier options. barrier is such that any non-zerio payoff at expira- The down-and-out call is close in ralue to tion guarantees that the barrier will be crossed. the standardcall, becauseit ges knocked out as Such European-style barrier options are identicd the stock moves down to levels wherc the stan- in payoff and value to standard European-style dard cdl has litde rnlue. The down-and-out put is options with the corresponding strike and expira- worth much less than the sandard put, because it tion. For example: gets knocked out as the stock price moves down to lerrelswherc ttre san&rd put becomes deep in . Any up-and-in call with strike above the in the money. barrier is worth the same as a stan&rd cdl, The down-and-in call is worth much less because all future stock paths that lead to than the standard cdl. It is knocked in only when payoffalso lead to beingknocked in. the stock has made a large and unlikely down- o For similar reasons,any down-and-in put with ward move. The down-and-in put is close in strike below the in barrier is worth the same vdue to the standard put, because the standard as a stan&rd put. put geB most of is value fiom downwand moves in the stock price, which would also trigger the Similarly, if the strike level of an out barrier knock-in of the down-and-in put.

58 tic l.rsersp orm or BlnnlrnOrrpns: Panr 1 Wnnen1996 Exhibits 2 and 3 show that the sum of the Elq{Bn3 values of a down-and-in European option and a DowN-lrrrp-INOrnoNs down-and-out European option equds the vdue of a corresponding standardEuropean option. StockPrice: 100 The up-and-out cdl is worth only a frac- Strike: 100 tion of the standardcall's value. It is knocked out Barrier: 90 for those upward stock moves that contribute Rebate: most of the value to the standard cdl. The up- Time to E>

E>man2 E:aflalr4 Dowu-aND-OurOrrrorus Up-aNn-OLrrOPTroNs

Stock Price: 100 StockPrice: 100 Strike: 100 Strike: 100 Barrier: 90 Barrier: r20 Rebate: Rebate: Time to Expiration: 1 year Time to Expiration: 1 year Dividend Yield: 5.0% (annuallycompounded) Dividend Yield: 5.0% (annuallycompounded) Volatility: l5To per yeat Volatility: 25Toper year Risk-Free Rate: 10.0%(annudly compounded) Risk-FreeRate: 10.0%(annually compounded) StandardEuropean StandardEuropean Call Value: 7.840 Call Value: 11.4U Down-and-Out Up-and-Out Call Vdue: 7.222 Cdl Value: 0.657 SrandardEuropean StandardEuropean Put Value: 3.749 Put Vdue: 7.344 Down-and-Out Up-and-Out Put Vdue: 0.286 Put Value: 6.705

l)mrua'mrrq ft rlnrrnrv 59 Emur5 six months, and one month to expiration. Some of Ur-lNp-IN Optroxs the feanrreswe cornment on, for e:rample, negative garnrna, may be specific to the particular barrier Stock Price: 100 option plotted, and may not apply to similar Strike: 100 options with di{ferent barrier levels, di{ferent Barrier: 720 market parameters,'or different times to e4piration. Rebate: In the following plos we assumeall barrier Time to E:eiration: 1 year options have zero rcbate. The rebate contribution to the rnlue and sensitivity Dividend Rate: 5.0% (annuallycompounded) of barrier options can be separatedfrom the option payoff itself. The Volatiliw: 25o/oper yezr rebate associatedwith an up barrier has the Risk-Free Rate: 10.0%(annudly compounded) contribution of the binaryup-and-in ull, which we StandardEuropean 2 Cdl Value: 11.434 explain in Part of this study (in the Spring 1997 issueof Duivativa The Up-and-In Quartulyl. rebate associat- Cdl Value: r0.779 ed with a down barrier has the contribution of StandardEuropean the binary down-anil-in prt, which will also be Put Value: 7.344 explainedinPert,2. Up-and-In Put Value: INTERPRETING THE PLOTS stock price. You have to short dela sharesof stock Take care in interpreting the plots in to dynamicdly hedge the option. Rapid changes Exhibits 7-16. From the point of view of delta- in delta with stock price (high gamma) make the hedging risk, the important regions of stock price stock dificult to hedge. are those where delta changesrapidly, and the Gamma is the sensitivity of delta to changes magnitude of gamma, positive or negative, is in stock price. We define it as the ratio of the correspondingly large. For barrier options, therc changein delta for an infinitesimal changein stock are often two separate regions where gamma price to the corresponding percentage change in becomeslarge. stock price.l Therefore, gamrn:ris a measureof the The first is exacdy at the barrier. There, a degree to which delta hedging becomes more small changein stock price causesa discontinuous dificult and inaccurate as the stock price changes. change in delta, and gamma becomes in6nite. To Think of gamma as the risk inherent in illustrate, see Exhibit 10, where the call is trylng to trade a hedged options position, or as a knocked out at a barrier et 120.Just to the left of measureof the option's sensitivity to volatility. An this region, the gamma of the one-month up- option with greater gamma must be rehedged and-out call is positive. To the right, the cdl has more frequendy; it takes more premium to pay for been knocked out and gamma is zero. At the greater rcbalancilg coss. barrier, gamma is infinite, reflecting the one-time We plot the value, delta, and gamma for risk that, if the stock price is just below the barri- each option as the stock price rnries, for one year, er and movesjust above it, continuous delta hedging is impossible. We cannot plot the infinite value of E:amn6 gamma at the barrier. Therefore, for all barrier Bannren Orutou Pen c,METERs options, the rnlues of gamma and delta that we display at the barrier are, stricdy speaking, the Strike: 100 valuesjust left and just right of the barrier. But Barrier: 80 if down, 120 if up remember that, even though we don't display it, Rebate: gamma is infinite at the barrier. Dividend yield: 5.0% (annudly compounded) The second rcgion where gamma becomes Volatility: Zff/oFer\eer large is near (but away from) the barrier, where its Nsk-free rate: 1O.0%(annually compounded) major effect is Glt. It ir to see this region in ""ry

60 ftc INsar.rp Otm or Bsnnn OsnoNs:PARI I Wnrrrn 1996 EnuBmT and with time to expiration. As expiration Srurlpmp Cetr: Srrurs 100 approaches, the call value nears the terminal payoff for stock prices below the strike the PlNsr. A. Veruu level of 100, increasing proportionally to stock price above 100. At very low stock prices, delta approaches zero (Panel B). You have dmost no exposure to I ra]'DarDldon - the stock. At high stock prices delta approaches --. airEnl||D'dftlon -- tm({|b.|Dk bn one. Owning a deep in-the-money call is like fe owning the stock itself. The transition from a E delte of zero to a delta of one is sharper the closer c you are to expiration. The gamma of the sandard call is alwap positive (Panel C) and is greatest for an at-the- to tro money call where continuous hedging is most $dFb. dificult. The peak gets more pronounced as the to expiration nears. PINSL B. DstrA time fu Panel A of Exhibit 8 shows, the value of the down-and-out call decreasesas the stock 3 declines, and must vanish at the barrier where a a - ltil'lrlhrcl standard cdl would still have a smdl value. The c --. tirEd|rbarDidotl -- tnErirbrsricl down-and-out call is dwal's worth less than the l3 corresponding standard cdl, but approaches the 5 E: sameralue at very high stock prices. the barrier, the down-and-out call g Below is worthless and has zero delta (Panel B). Above 3 the barrier, delta is alwap positive. As the stock 16 ilo price moves up from the barrier, the call value .!*tlo. inflates rapidly, with a delta just above the barrier PaNer C. Gluul that can be larger than that ofthe corresponding standard call. At higher stock prices, where the effect of the barrier is intangible, the delta is close to that ,l - trurb.dcl 3 -- tiF,ilrD'arltrbl t of a standard call. Between the barrier and the -- lt! rd|lrdrd.n I t I region of high stock prices, delta for this particu- t expiration actually Ir I lar call with a one-year I 8s decreases,as )'ou can see in Panel B. At high stock prices the barrier is irrclerant and the gamma of the down-and-out cdl is simi- lar to that of the sandarrd call @anel C). Near the r to rr0 r20 tI barrier, however, the one-year cdl has a negative d9tb ganuna, reflecting the fact that the optionholder is redly "short volatility," or short an option stmck all of our plots. In Exhibit 10, it is the rcgion with at the barrier. the peak of large negativegamma around a stock For stock prices near the barrier, an price of 112. lf the stock price oscillatesin this increase in volatility would acbudly make knock- tlpe of high-gammaregion, the fuquent rcadjust- out for this particular option morc likely, and so mens of delta can generatelarge rchedging cosr. decrcaseoverall option value. Gamma is infnite at In PanelA of Exhibit 7, the value of the the barrier. sandard cdl increaseswith increasingstock price As Panel A of Exhibit 9 shows, for stock

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t 3 1,' I g t, 1 r! t €! I t I , t E5 E3 t., ts Is q i------ll I o q c ro ilo ro tto rdDrta d.ftr prices below 80, the cdl is knocked in and worth dl stock prices, it is dwap worth less than the the same as a sandard cdl. Its rnlue incrcasesas corrcspondingsandard cdl. the stock price rises. Above 80 the cdl is not yet Below the barrier, the delta is that of a knocked in. As the stock price increases, the stan&rd call and riseswith stock price (PanelB). chance of getting knocked in declines, and the cdl Just aborrcthe barrier the option o$/ner is effec- falls in rnlue. Is wlue is greatest at the barrier. For tively short stock (negativedela), becauseupward

62 nra ns elo Oursor BennnnOnom: Prrr 1 Wnnrn1996 stock moves make knock-in less likely and Exruilr 10 decreaseoption value. The sharp change from Ur-aNp-Our Calu positive to negative delta acrossthe barrier makes Srntrn 10O BennIER120 hedging extremely difficult. As the stock moves below 80, you would have to suddenly switch PaNnrA. Verus &om being long stock to being short. Gamma is always positive here, as Panel C a\ shows, becausethe call owner is long volatility at ,r t I both the strike and the barrier. Gamma is infinite I I t I at the barrier where there is a discontinuous t I I I changein exposure. f. , I , I In Panel A of Exhibit 10, above the strike t I , I I of 100 the call moves in the money, and would I

increase in value from then on were it not for i'rl the up-and-out feature that extinguishes its Iroiloteio value at 120. This accounts for the peak in the d!b distribution. For low stock prices, the exposureincreases Pr,xn B. DErra as the stock moves up toward the strike (PanelB). Nearer the barrier, delta decreasesand becomes negativeabove 110, becauseofthe value-quench- I ing effectsof the barrier. I q I The call owner is long gamma near the strike and short gamma near the barrier (PanelC). l" I Iq t tl I The large negative garnmabetween 110 and 120, I t corresponding to the rapid change from positive q t I to negative delta near 110 in Panel B, makes ? hedging inaccurate and dificult, and makes trans- Itoilou0tI action costslarge. Gamma is infinite at the barrier. Hrlr As Panel A of Exhibit 11 shows, the up- and-in call will knock in only if the stock crosses PnNrr C. G.l,uua the barrier before expiration. In that case,the option immediately becomes a deep in-the- q c money call. That is why the value of the one- I q I month call increasesso sharply as the stock c approaches120. tq t F9 t . I Analogously, as Panel B shows, at 120 the !o t T; I delta of the one-month cdl spikes far above the c5 t .: t delta of a standard call. Above 120, the call has I t I R t knocked in and has the same positive delta as a 9 deep in-the-money standardcall. ,0 T F to lr0 1n t! The owner of the cdl is long considerable volatility at the 120 barrier, whose crossing -4tr suddenly brings into existence a valuable deep in- the-money option (Panel C). The call close to Note that a short-term deep in-the-money put is e4piration has a gamma about three times larger at worth more than a long-term one, because the 110 than that of the corresponding sandard call at present vdue of the strike received is worth more. the strike of 100. Gamma is infinite at the barrier. As Panel B shows, at lower stock prices the In Panel A of Exhibit 12, the value of the put owner is effectively short the stock, with a standardput increaseswith decreasingstock price. delta value of -1. At high stock prices the delta

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approacheszero. strike of 100, the one-month put moves rapidly The gamma of the sandard European-sryle into the money. Ticvnrd the barrier of 80 its wlue put is identicd to the gamma of the correspond- declinesto zeto. The peak occuts at about 90. ing standard call (Panel C). This follows from Just above the barrier, clelta is large and put-cdl parity. positive - the put owner benefits grcady from As Panel A of Exhibir 13 shows, below the upward price moves away from the extinguishing

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€ ,ta o' a I o t --. t F I gE Ecl to t .{ EI f d a t, ? t, t, s to fio tto t$ ostotrotars 6d(p&. .bat!.ta barrier (Panel B). For higher stock prices, the The put owner is long volatility (positive effect of rhe barrier diminishes, and delta gamma) at the srike of 100 and shorr volatility approaches the negative values of a standard put. (large negativegamma) near the barrier of 80 (panel The rapid change from positive to negative dela C). In addition, gamma is infinite at the barrier. makes hedging inaccurate. For large stock prices ln Panel A of Exhibit 14, the down-and-in delta approacheszero. one-month put knocls in deep in the money at

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80. Approaching the stock price of 80 from above, standard put becauseof the rapid decline in its its value therefore stePsuP sharply. Below 80, it is value as the chance of knock-in decrcaseswith knocked in and its value is that of a standatd pur increasing stock price- Dela is everywhete nega- In Panel B, below 80 the delta is that of a tive, and approacheszero for large stock prices of the money. standardput, close to -1. Just above 80, the delta where the put is far out of the one-month put droPs far below that of a The put owner benefis &om any volatility

66 Tru Irrrser.rp Ocm or BennsnOrrroNs: P*r 1 Wnrrm196 that moves the stock into the money or near the SI,JMMARY knock-in barrier. Therefore, gamma is large and positive near 80 (Panel C). In addition, gmma is Barrier options on stocl$ create opportuni- infinite at the barrier. ties for investors that are not arailable with san- In Panel A of Exhibit 15, the value must dard puts and calls, frequently with lower \nnish at the 120 barrier, where the standardput premiums than standard options with similar is worth litde aq'uny. Thereforc, the dependence strike. The payo6, being a function of both the on stock price is similar to that of a sandanil put. strike and the barrier, can be complicated. The The barrier has little effect in this case,and the pricing characteristics,captured by delta and option is no riskier than a standardput. gamrna, can change dramaticdly through time, Below the 120 barrier @anelB), the delta and exhibit discontinuities. is similar to that of a standardput and is dways Options vduation models are theory and, negative. Above the barrier, the put is extin- like all models, are more limited than the real guished and the delta is zerc. world they attempt to represent. Provided the Gamma is similar to that of a standald put potential users of barrier options are aware of below the barrier, and zero above (Panel C). At these limitations, the additiond power and flexi- the barrier, gamma is infinite. bility of such options can be important contribu- Panel A of Exhibit 16 shows that, as the tors to an overall investrnent strategT. stock rises to the 120 barrier, the put value increaseswith its chance of getting knocked in. ENDNOTES Above 120, the put is worth the same as a standand put with strike 100, and its vdue declines as the This article is basedon "The Ins and Ous of Barri- stock p:ice increases.Therc is a sharppeak at 120. er Options," a Goldman SachsQuantitative Strategies The sharp break in delta at 120 makesthis ResearchNotes paper,June 1993. put difrcult to hedge dynamically (Panel B). As The authon are gntcfirl to Ken Iseya and Yoshi the stock moves above 120, you would have to Kobashi, who cncouraged them to write this article aftcr suddenly switch from shorting stock to going long many stimulating conversatioruabout barrier options. They also thank Deniz Ergencr, who helped and buying it. obtain the resuls dccribcd here, and Alcx Bergier and Barbara Dunn, who In Panel C, gamma is everywhere positine, patiendy read the manuscript and made many hclpful and and is very large exacdy at the 120 barrier, where clri$ing sugcstiors. sensitivity to volatility is great. Gamma is infinite 'Mathematically, T = (5/100) aLlas, where S is at the barrier. thc stockprice.

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