The Ins and Outs ofBarrier Options: Part 1EMalusL DrrurreN exp Inel KAt.lt

. Euer.rusr, Denuex is head ofthe Quantitative StratcgiesGroup at Goldman, Secbs & Co.in NewYork.

Inry Kem is a vice president inthe Quantitative StrategiesGroup at Goldman, Sachs & Co.io NcwYork.

Wnrrrr 1995

arrier options are extensions of stan&rdstock options. Standard calls and putshave payoffi that depend on one marketlevel: the strike. Barrier options have

payoffs that depend on two market levels: thestrike and the barrier. Investors can use them togain exposure to (or enhance rcturns from) firturcmarket scenarios more complex than the simplebullish or bearish expectations embodied in stan-dard options. [n addition, their premiums areusudly lower than those of sandard options withthe same strike and e4pintion.

Part 1 of this article describes many of themost common barrier options available, as well aswhen to use them in place of standard options.We also describe why their sensitiviry to marketmoves often exceeds that of standard options. Thisis depicted in several cases with ables and plots,showing the theorctical vdues and sensitivities fordiferent market levels.

Part 2, which will appear in the Spring1997 issue of Deriuatiues Quarufly, expands on thispresentation. Somewhat more exotic barrieroptions are explored, and a surrmary of the math-ematicd theory behind the vduations is presented.

BARRIER OPTIONS DEFINED

Standard European-style options on stockarc characterized by their time to expiration andstrike level. At expiration, standard cdl optionspay the owner the difference between the stock

Dnnrvlrnrusouannnry 55

price and the strike level if the stock is above the

strike, and zero otherwise. Similarly' standard put

options pay the owner the difference benveen the

strike and the stock price if the stock is below the

strike, and zero otherwise. A call owner benefits

from an upward stock move; a Put o$tner benefits

from a downward stock move.

Banier options are a modified form of stan-

dard options that include both pus and calls. They

are characterized by a strike level and e banier level,

as well as by a ush rebate associated with crossing

the barrier. As with standard options, the strike

level determines the payoff at expiration. The

barrlgr options contract specifies, however, thatthe payoff depends on whether the stock price

ever crosses the barrier level during the life of theoption. In addition, if the barrier is crossed, somebarrier option contracs specify a rebate to be paid

to the optionholder.This article discusses only European-sryle

barrier options. Furthermore, in most cases weassume the rebate is zerc.

Barriers come in two types. We call a

barrier above the current stock level an up barrie6if it is ever crossed, it will be from below. We call abarrier below the current stock level t ilown baniq,if it is ever crossed, it will be from above.

Barrier options come in two types: r'roptions and out options. An in barrier option, orknock-in option, pays off only if the stock finishesin the money and if the barrier is crossed some-time before expiration. When the stock crossesthe barrier, the in barrier option is knocked in andbecomes a standard option of the same tlpe (cdl

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or put) with the same strike and expiratrthe stock never crosses the barrier, the optio.-expires worthless.

An out barrier option, or knockout option,pays off only if the stock finishes in the moneyand the barrier is never crossed before expiration.As long as the stock never crc,sses the barrier, theout barrier option remains a standard option ofthe same type (call or put) with the same strikeand expiration. If the stock crosses the barrier, theoption is knocleeil out and expires worthles.

Therefore, barrier options can be up-anil-out, up-and-in, down-anil-out, or iloum-anil-in. Thedifferent barrier options and the effect of crossingthe barrier on their payoff are described inExhibit 1.

In principle, all barrier options can haveassociated cash rebates that are paid to the option-holder when the barrier is crossed. In practice,however, only out option contracts provide cashrebates, paid to the optionholder as a sort ofconsolation prize when the option is knocked out.

WHY USE BARRIER OPTIONS?

There are three basic reasons to use barrieroprions rather than standarrd options:

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Traders value options based on optionstheory. In liquid markets, you can rnlue options

Varusrrus oF BARRTER OPTIoNS

Barrier

Option Ttp.Effect ofBarrier on Payoff

Location Crossed Not Crossed

Catl Down-and-OutDown-and-InUp-and-OutUp-and-In

Below SpotBelow SpotAbove SpotAbove Spot

WorthlesStandard CallWorthles

Standard Call

Standard CallWorthlessStandard Cdl

WorthlesPut Down-and-Out

Down-and-In

Up-and-OutUp-and-In

Belor Spot

Below Spot

Above SpotAbove Spot

WortNes

Standard Put

WorthlessStandand Put

Standard PutWorthles

Standard Put

Worthless

55 Tns INs AND Otns or Banncn Ornor.rs: Perr I WwreR1996

by calculating the expected value of their pa)'o6,averaging over all stock scenarios whose meanprice is the stock's forward price at each futuretime. According to the theory you pay for volatil-ity around the forward price.

By buying a barrier option, you can elimi-nate paying for those scenarios you think areunlikely. Alternatively, you can enhance yourreturn by selling a barrier option that pays offonly on scenarios you think are improbable.

Suppose the forward price of the stock ayear from today is 105% of spor. You believe rhatthe market is very likely ro rise, but that if itsomehow drops below a supporr level of 97Yo, itrvill instead decline further. You can buy a down-and-out call struck at 105o/o that gets knocked outif the stock, at any time, falls below 97%. In tlttsway, you avoid prying for scenarios in which thestock first declines substantially and then risesagain. At 2Q% volatility, this can reduce yourpremium by half.

Alternatively, you can enhance your returnby selling a down-and-in cdl srruck

^t 705yo thet

gets knocked in only if the market falls below97Yo. You can then pick up premium by bettingagainst a scenario you feel is unlikely.

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Suppose you own a srock and have decid-ed that if it ever rises by more than 5% over thenext year, you will sell it, but you stil l wantprotection aEainst market declines of more than5%. You can buy a stan&rd put option struck at95Yo of spot, but then you get protection evenwhen the market rises more then 5% and you nolonger need it. lnstead, it would be better to buyan up-and-out put with strike at g|yo, barrier atl05yo, and no cash rebate. This pur will disappearas soon as you sell the stock and have no furtherneed ofit.

Ba,nnrsn Optrox pnruruusARE GENERALTY LowER THANTnoss oF STaNDARD Omrows

Barrier options are often attractive toinvestors because their premium is lower than

Wnrmn1996

corresponding standard options. For example,knockout options wont pay offif the stock crossesthe knockout barrier. Therefore, they are cheaperthan an otherwise identical option without theknockout feature. If you think the chance ofknockout is smdl, you can ake advantage of thelower premium and get the same benefits. Or, youcan even pay more premium to receive a cashrebate if the option is knocked out.

Similarly, knock-in oprions have lowerpremiums than corresponding standard optionswith the same strike and expintion. You may findthese attractively cheap ifyou feel the probabilityof getting knocked in is high.

SPECIAL FEATURESOF BARRIER OPTIONS

Managing the risk of an options position ismore complicated than managing a stock positionalone. You can dynamically hedge options bygoing short an amount deha of stock against a longoptions position, where delta is the theoreticalhedge ratio. If delta is negarive, you go long thatamount of stock. The value of the oprion and itsdelta depend upon both market level and volatili-ty. Standard call options, for example, have deltavalues between zero and one, and call valuesincrease with increasing volatility.

Barrier options are similar to, but morecomplicated than, standard options, because theysample payoft of a smaller and more specializedset of future stock paths. You can dynamicdlydelta hedge them, just as you do with standardoptions, by using a theoretical model to ralue theoption and cdculate its delta. You can also useother options to create static hedges that approxi-mately duplicate the barrier option's payo6.

The price sensitiviries of barrier optionscan differ markedly from those of standardoptions. Comparc an up-and-out cdl option witha standard cdl option. As the stock price movesup, a standard call dwaln garns in ralug while anup-and-out cdl is subject to two opposing effects.fu the stock price moves up, the up-and-out cdl'spayoffbecomes potentidly larger, but the upwardmove simultaneously thrcatens to extinguish thetotal vdue of the contract by moving it closer tothe out barrier. The conflict between these rwotendencies makes it extremely sensitive to the

f)nnruamrrs ft ranrrnr.v 57

stock's movement near the barrier, and delta can

go rapidly from positive to negative rnlues.

We describe the behavior of barr ier

options in more deail as we examine each type.

Meanwhile, be awarc of two key ways in which

barrier options differ from standard optionswhen the underlying stock or index gets near

the barrier level.First, the delta of the barrier option can

differ significantly from the delta of the corre-

sponding standard option. For example, a barrier

call can have delta rnlues les than zero or greater

than one; an up-and-out deep in-the-money cdlwhose rnlue must vanish at the barrier has a nega-tive delta near the barrier because of the rapiddecline in its vdue there.

Second, barrier option ralues can actudlydecrease with increasing volatility. The up-and-outcall just described becomes more likely to getknocked out near the barrier as volatility increases.In options parlance, even though you are longvolatiliry or "long gamma" relative to the strikelevel, you are short volatiliry or "short gamma"relative to the barrier.

You can often get a good feel for the way abarrier option will behave as the underlying stockprice rnries by classifying the strike and barrierlevels with regad to volatiliry and then combiningtheir effects. The owner of a barrier option is longvolatility at the strike, long volatility at an in barri-er, and short volatfity at an out barrier. We makeuse of this type of analpis later.

In certain cases, like the ones listed next,the strike level of an in option relative to thebarrier is such that any non-zerio payoff at expira-tion guarantees that the barrier will be crossed.Such European-style barrier options are identicdin payoff and value to standard European-styleoptions with the corresponding strike and expira-tion. For example:

. Any up-and-in call with strike above the inbarrier is worth the same as a stan&rd cdl,because all future stock paths that lead topayoffalso lead to beingknocked in.

o For similar reasons, any down-and-in put withstrike below the in barrier is worth the same

as a stan&rd put.

Similarly, if the strike level of an out barrier

58 tic l.rs ersp orm or Blnnlrn Orrpns: Panr 1

option relative to the barrier is such that any non-zero payoffguarantees the option will be knockedout, then the option is worthles. So, an up-and-out cdl with strike above the barrier is worthless.Also, a down-and-out put with strike below thebarrier has no rnlue.

There is a simple relationship betweenEuropean-sryle in options, out options, and stan-dard options. If you own both an in option andan out option of the same type (call or put), wirhthe same e:cpiration, the same respective strike,and the same respective barrier level, you areguaranteed to receive the payoff of the corre-sponding standard option whether the barrier iscrossed or not. Therefore:

o The vdue of a down-and-in call (put) plusthe value of a down-and-out call (put) isequd to the rnlue of a corresponding stan-dard cdl (put).

o The value of an up-and-in cdl (put) plus thernlue of an up-and-out call (put) is equd ro thevalue of a corresponding san&rd cdl (put).

SOME EXAMPLES

In this section, we give examples of bothout and in European-style options with zerorebate, and compare their theoreticd values withthose of corresponding European-style standardoptions. Later, we show how their values andsensitivities vary with market levels for a widernriety of barrier options.

The down-and-out call is close in ralue tothe standard call, because it ges knocked out asthe stock moves down to levels wherc the stan-dard cdl has litde rnlue. The down-and-out put isworth much less than the sandard put, because itgets knocked out as the stock price moves downto lerrels wherc ttre san&rd put becomes deep inthe money.

The down-and-in call is worth much lessthan the standard cdl. It is knocked in only whenthe stock has made a large and unlikely down-ward move. The down-and-in put is close invdue to the standard put, because the standardput geB most of is value fiom downwand moves

in the stock price, which would also trigger the

knock-in of the down-and-in put.

Wnnen1996

Exhibits 2 and 3 show that the sum of thevalues of a down-and-in European option and adown-and-out European option equds the vdue

of a corresponding standard European option.The up-and-out cdl is worth only a frac-

tion of the standard call's value. It is knocked outfor those upward stock moves that contributemost of the value to the standard cdl. The up-

and-out put is similar in value to the standard put.

It is knocked out on upward stock moves to levelswhere the standard put is so far out of the moneythat it is almost worthless.

The up-and-in call is worth almost thesame as the standard call. [t is knocked in for thoseupward stock moves that contribute most of thevalue to the standard call. The up-and-in put isworth much less than the standard put. [t isknocked in on upward moves to levels where thestandard put is so far out of the money that it isalmost worthless.

Exhibis 4 and 5 show that the sum of thevalues of an up-and-in European option and anup-and-out European option equals the value of acorresponding stan&rd European option.

VATUES AND SENSITIVITIESOF BARRIER OPTIONS

This section presents plots of the values,

E>man2

Risk-Free Rate: 10.0% (annudly compounded)Standard European

Call Value: 7.840Down-and-Out

Call Vdue: 7.222Srandard European

Put Value: 3.749Down-and-Out

Put Vdue:

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DowN-lrrrp-IN OrnoNs

Stock Price: 100Strike: 100Barrier:Rebate:Time to E><piration: 1 year

Dividend Yield: 5.0% (annudly compounded)Volatility: 15%6 PetYeerNsk-Free Rate: 10.0% (annually compounded)Standard European

Call Value: 7.840Down-and-In

Call Value: 0.618Standard European

Put Value: 3.749Down-and-In

Put Value: 3.463

deltas, and gammas of a variety of European-stylebarrier options with zero rebate. For compari-son, we also present the same values and sensitiv-ities for the corresponding standard options.Common parameters used in all plots are shownin Exhibit 6.

In all plots, delta is the number of shares ofstock that has the same instananeous exposure :rsthe option has to infinitesimal changes in the

E:aflalr4

Risk-Free Rate: 10.0% (annually compounded)Standard European

Call Value: 11.4UUp-and-Out

Cdl Value:

Standard EuropeanPut Vdue: 7.344

Up-and-OutPut Value:

90

Dowu-aND-Our Orrrorus Up-aNn-OLrr OPTroNs

Stock Price: 100 Stock Price: 100Strike: 100 Strike: 100Barrier: Barrier:Rebate: Rebate:Time to Expiration: 1 year Time to Expiration: 1 yearDividend Yield: 5.0% (annually compounded) Dividend Yield: 5.0% (annually compounded)Volatility: l5To per yeat Volatility: 25To per year

r2090

0.657

0.286 6.705

l)mrua'mrrq ft rlnrrnrv 59

Emur5

Ur-lNp-IN Optroxs

Stock Price:

Strike:

Barrier: 720Rebate:

Time to E:eiration: 1 year

Dividend Rate: 5.0% (annually compounded)

Volatiliw: 25o/o per yezr

Risk-Free Rate: 10.0% (annudly compounded)

Standard EuropeanCdl Value: 11.434

Up-and-InCdl Value:

six months, and one month to expiration. Some ofthe feanrres we cornment on, for e:rample, negativegarnrna, may be specific to the particular barrieroption plotted, and may not apply to similaroptions with di{ferent barrier levels, di{ferentmarket parameters,'or different times to e4piration.

In the following plos we assume all barrieroptions have zero rcbate. The rebate contributionto the rnlue and sensitivity of barrier options canbe separated from the option payoff itself. Therebate associated with an up barrier has thecontribution of the binary up-and-in ull, which weexplain in Part 2 of this study (in the Spring 1997issue of Duivativa Quartulyl. The rebate associat-ed with a down barrier has the contribution ofthe binary down-anil-in prt, which will also beexplained inPert,2.

INTERPRETING THE PLOTS

Take care in interpreting the plots inExhibits 7-16. From the point of view of delta-hedging risk, the important regions of stock priceare those where delta changes rapidly, and themagnitude of gamma, positive or negative, iscorrespondingly large. For barrier options, thercare often two separate regions where gammabecomes large.

The first is exacdy at the barrier. There, asmall change in stock price causes a discontinuouschange in delta, and gamma becomes in6nite. Toi l lustrate, see Exhibi t 10, where the cal l isknocked out at a barrier et 120.Just to the left ofthis region, the gamma of the one-month up-and-out call is positive. To the right, the cdl hasbeen knocked out and gamma is zero. At thebarrier, gamma is infinite, reflecting the one-timerisk that, if the stock price is just below the barri-er and moves just above it, continuous deltahedging is impossible.

We cannot plot the infinite value ofgamma at the barrier. Therefore, for all barrieroptions, the rnlues of gamma and delta that wedisplay at the barrier are, stricdy speaking, thevalues just left and just right of the barrier. Butremember that, even though we don't display it,gamma is infinite at the barrier.

The second rcgion where gamma becomeslarge is near (but away from) the barrier, where itsmajor effect is Glt. It ir

""ry to see this region in

100100

r0.779Standard European

Put Value: 7.344

Up-and-InPut Value:

stock price. You have to short dela shares of stockto dynamicdly hedge the option. Rapid changesin delta with stock price (high gamma) make thestock dificult to hedge.

Gamma is the sensitivity of delta to changesin stock price. We define it as the ratio of thechange in delta for an infinitesimal change in stockprice to the corresponding percentage change instock price.l Therefore, gamrn:r is a measure of thedegree to which delta hedging becomes moredificult and inaccurate as the stock price changes.

Think of gamma as the risk inherent intrylng to trade a hedged options position, or as ameasure of the option's sensitivity to volatility. Anoption with greater gamma must be rehedgedmore frequendy; it takes more premium to pay forgreater rcbalancilg coss.

We plot the value, delta, and gamma foreach option as the stock price rnries, for one year,

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Bannren Orutou Pen c,METERs

Strike: 100Barrier: 80 if down, 120 if upRebate:Dividend yield: 5.0% (annudly compounded)Volatility: Zff/oFer\eerNsk-free rate: 1O.0% (annually compounded)

60 ftc INs ar.rp Otm or Bsnnn OsnoNs: PARI I Wnrrrn 1996

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and with time to expiration. As expirationapproaches, the call value nears the terminalpayoff for stock prices below the strike

level of 100, increasing proportionally to the stockprice above 100.

At very low stock prices, delta approacheszero (Panel B). You have dmost no exposure tothe stock. At high stock prices delta approachesone. Owning a deep in-the-money call is like

owning the stock itself. The transition from adelte of zero to a delta of one is sharper the closeryou are to expiration.

The gamma of the sandard call is alwappositive (Panel C) and is greatest for an at-the-

money call where continuous hedging is most

dificult. The peak gets more pronounced as the

time to expiration nears.fu Panel A of Exhibit 8 shows, the value of

the down-and-out call decreases as the stock

declines, and must vanish at the barrier where a

standard cdl would still have a smdl value. The

down-and-out call is dwal's worth less than the

corresponding standard cdl, but approaches the

same ralue at very high stock prices.Below the barrier, the down-and-out call

is worthless and has zero delta (Panel B). Above

the barrier, delta is alwap positive. As the stockprice moves up from the barrier, the call value

inflates rapidly, with a delta just above the barrierthat can be larger than that ofthe correspondingstandard call.

At higher stock prices, where the effect ofthe barrier is intangible, the delta is close to that

of a standard call. Between the barrier and theregion of high stock prices, delta for this particu-

lar call with a one-year expiration actuallydecreases, as )'ou can see in Panel B.

At high stock prices the barrier is irrclerantand the gamma of the down-and-out cdl is simi-lar to that of the sandarrd call @anel C). Near thebarrier, however, the one-year cdl has a negativeganuna, reflecting the fact that the optionholder is

redly "short volatility," or short an option stmckat the barrier.

For stock prices near the barrier, anincrease in volatility would acbudly make knock-out for this particular option morc likely, and sodecrcase overall option value. Gamma is infnite atthe barrier.

As Panel A of Exhibit 9 shows, for stock

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all of our plots. In Exhibit 10, it is the rcgion withthe peak of large negative gamma around a stockprice of 112. lf the stock price oscillates in thistlpe of high-gamma region, the fuquent rcadjust-mens of delta can generate large rchedging cosr.

In Panel A of Exhibit 7, the value of thesandard cdl increases with increasing stock price

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prices below 80, the cdl is knocked in and worththe same as a sandard cdl. Its rnlue incrcases asthe stock price rises. Above 80 the cdl is not yetknocked in. As the stock price increases, thechance of getting knocked in declines, and the cdlfalls in rnlue. Is wlue is greatest at the barrier. For

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dl stock prices, it is dwap worth less than thecorrcsponding sandard cdl.

Below the barrier, the delta is that of astan&rd call and rises with stock price (Panel B).

Just aborrc the barrier the option o$/ner is effec-tively short stock (negative dela), because upward

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stock moves make knock-in less l ikely and

decrease option value. The sharp change from

positive to negative delta across the barrier makes

hedging extremely difficult. As the stock movesbelow 80, you would have to suddenly switch&om being long stock to being short.

Gamma is always positive here, as Panel Cshows, because the call owner is long volatility atboth the strike and the barrier. Gamma is infiniteat the barrier where there is a discontinuous

change in exposure.In Panel A of Exhibit 10, above the strike

of 100 the call moves in the money, and wouldincrease in value from then on were it not for

the up-and-out feature that extinguishes itsvalue at 120. This accounts for the peak in thedistribution.

For low stock prices, the exposure increasesas the stock moves up toward the strike (Panel B).Nearer the barrier, delta decreases and becomesnegative above 110, because ofthe value-quench-ing effects of the barrier.

The call owner is long gamma near thestrike and short gamma near the barrier (Panel C).The large negative garnma between 110 and 120,corresponding to the rapid change from positiveto negative delta near 110 in Panel B, makeshedging inaccurate and dificult, and makes trans-action costs large. Gamma is infinite at the barrier.

As Panel A of Exhibit 11 shows, the up-and-in call will knock in only if the stock crossesthe barrier before expiration. In that case, theoption immediately becomes a deep in-the-money call. That is why the value of the one-month call increases so sharply as the stockapproaches 120.. Analogously, as Panel B shows, at 120 thedelta of the one-month cdl spikes far above thedelta of a standard call. Above 120, the call hasknocked in and has the same positive delta as adeep in-the-money standard call.

The owner of the cdl is long considerablevolatility at the 120 barrier, whose crossingsuddenly brings into existence a valuable deep in-the-money option (Panel C). The call close toe4piration has a gamma about three times larger at110 than that of the corresponding sandard call atthe strike of 100. Gamma is infinite at the barrier.

In Panel A of Exhibit 12, the value of thestandard put increases with decreasing stock price.

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Note that a short-term deep in-the-money put isworth more than a long-term one, because thepresent vdue of the strike received is worth more.

As Panel B shows, at lower stock prices theput owner is effectively short the stock, with adelta value of -1. At high stock prices the delta

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strike of 100, the one-month put moves rapidlyinto the money. Ticvnrd the barrier of 80 its wluedeclines to zeto. The peak occuts at about 90.

Just above the barrier, clelta is large andpositive - the put owner benefits grcady fromupward price moves away from the extinguishing

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As Panel A of Exhibir 13 shows, below the

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barrier (Panel B). For higher stock prices, theeffect of rhe barr ier diminishes, and deltaapproaches the negative values of a standard put.The rapid change from positive to negative delamakes hedging inaccurate. For large stock pricesdelta approaches zero.

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The put owner is long volatility (positivegamma) at the srike of 100 and shorr volatility(large negative gamma) near the barrier of 80 (panelC). In addition, gamma is infinite at the barrier.

ln Panel A of Exhibit 14, the down-and-inone-month put knocls in deep in the money at

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In Panel B, below 80 the delta is that of a

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that moves the stock into the money or near theknock-in barrier. Therefore, gamma is large andpositive near 80 (Panel C). In addition, gmma isinfinite at the barrier.

In Panel A of Exhibit 15, the value must\nnish at the 120 barrier, where the standard putis worth litde aq'uny. Thereforc, the dependenceon stock price is similar to that of a sandanil put.The barrier has little effect in this case, and theoption is no riskier than a standard put.

Below the 120 barrier @anel B), the deltais similar to that of a standard put and is dwaysnegative. Above the barrier, the put is extin-guished and the delta is zerc.

Gamma is similar to that of a standald putbelow the barrier, and zero above (Panel C). Atthe barrier, gamma is infinite.

Panel A of Exhibit 16 shows that, as thestock r ises to the 120 barr ier, the put valueincreases with its chance of getting knocked in.Above 120, the put is worth the same as a standandput with strike 100, and its vdue declines as thestock p:ice increases. Therc is a sharp peak at 120.

The sharp break in delta at 120 makes thisput difrcult to hedge dynamically (Panel B). Asthe stock moves above 120, you would have tosuddenly switch from shorting stock to going longand buying it.

In Panel C, gamma is everywhere positine,and is very large exacdy at the 120 barrier, wheresensitivity to volatility is great. Gamma is infiniteat the barrier.

SI,JMMARY

Barrier options on stocl$ create opportuni-ties for investors that are not arailable with san-dard puts and calls, frequently with lowerpremiums than standard options with similarstrike. The payo6, being a function of both thestrike and the barrier, can be complicated. Thepricing characteristics, captured by delta andgamrna, can change dramaticdly through time,and exhibit discontinuities.

Options vduation models are theory and,like all models, are more limited than the realworld they attempt to represent. Provided thepotential users of barrier options are aware ofthese limitations, the additiond power and flexi-bility of such options can be important contribu-tors to an overall investrnent strategT.

ENDNOTES

This article is based on "The Ins and Ous of Barri-er Options," a Goldman Sachs Quantitative StrategiesResearch Notes paper,June 1993.

The authon are gntcfirl to Ken Iseya and YoshiKobashi, who cncouraged them to write this article aftcrmany stimulating conversatioru about barrier options. Theyalso thank Deniz Ergencr, who helped obtain the resulsdccribcd here, and Alcx Bergier and Barbara Dunn, whopatiendy read the manuscript and made many hclpful andclri$ing sugcstiors.

'Mathematically, T = (5/100) aLlas, where S isthc stock price.

Wnrmn196 Drnrvnrnres Quenrrnrv 57