Exotic Options 439
19.6 BARRIER OPTIONS
 
 
Barrier options are options where the payoff depends on whether the underlying asset’s pricereaches a certain level during a certain period of time. A number of different types of barrieroptions regularly trade in the over-the-counter market. They are attractive to some marketparticipants because they are less expensive than the corresponding regular options. These barrieroptions can be classified as either knock-out options or knock-in options. A knock-out option ceasesto exist when the underlying asset price reaches a certain barrier; a knock-in option comes intoexisience only when ihe underiying assei price reaches a barrier.
Equations (13.4) and (13.5) show that the values at time zero of a regular call and put option are
c= Sye N(d,) — Ke N(dy)p = Ke"™N(—dy) — Spe” N(—d,)
where7, = Sof KF 9 t F/T' oVTIn(S/K) +r = y — 0° /2)Tel - ToF | -oVT
A down-and-out call is one type of knock-out option. It is a regular cali option that ceases to exist ifthe asset price reaches a certain barrier level, H. The barrier level is below the initial asset price.The corresponding knock-in option is a dewn-and-in call. This is a regular call that comes intoexistence only if the asset price reaches the barrier level.
If H is less than or equal to the strike price, K, the value of a down-and-in call at time zero is"given by
cai = Soe (A/S) N(x) — Ke" (H/ Sp)" My — oVT )
wherer-qt a /2
A= 5
ainLH* /(SoK)1y= oT +hoV/T
Becaus¢ the value of a regular call equals the value of a down-and-in call plus the value of a down-and-optt call, the value of a down-and-out call is given by
ldo = © — Caiif H > K, then
Cao = SpN (xe? — Ke N(x, — ONT ) — Spe (H/ Sp)" N(1) + Ke (H/ Soy? Ny; — VT)
andCai = C — Caowhere_ In(So/#) In(H/So)VT +ioa VT, y= oF ro0/T440 CHAPTER 19
An up-and-out call is a regular call option that ceases to exist if the asset price reaches a barrier level,H, that is higher than the current asset price. An up-and-in call is a regular call option that comesinto existence only if the barrier is reached. When H is less than or equal to K, the value of theup-and-out call, ¢,,, is zero and the value of the up-and-in call, cy;, isc. When # is greater than K,
Cui = SoN Qe! — Ke N(x, — VT) — Spe 4A So) IN(—y) — N(—y1)]
+ Ke" H/ Soy" [N(-y + VT) — N(-y, tVT)]and
Cy SO CyPut barrier options are defined similarly to call barrier options. An up-and-out put/js a put optionthat ceases to exist when a barrier, H, that is greater than the current asset price is/reached. An up-and-in put is a put that comes into existence only if the barrier is reached. When the barrier, H7 isgreater than or equal to the strike price, K, their prices are
Bai = —Sye 2 (H/ Sy) N(—y) + Ke (A/S) 7 N(—-y + VT)andPu = P— PuWhen H is less than or equal to K,Puo = —SyN(—x) eo" + Ke N(—x, + VT) + Sye (H/Syy*N(—y1)
Ke (H/Sy)N(-y, + oVT)
25oh
Pui = P — Puo
A down-and-out put is a put option that ceases to exist when a barrier less than the current asset
price is reached. A down-and-in put is a put option that comes into existence only when the barrieris reached. When the barrier is greater than the strike price, py, — Q and py — p. When the barrier
reached. When the barrier greater than the strike price, p Vand Pg hen the barrier
is less than the strike price,
Pai = —SoN(—xy le" + Ke N(—x, + VT) + Spe (H/ So) IN(Q) — NOIKe (H/S))" (My — oVT) — Ny — oVT)I
andPdo = P — Pai
All of these valuations make the usual assumption that the probability distribution for the assetprice at a future time is lognormal. An important issue for barrier options is the frequency that theasset price, S, is observed for purposes of determining whether the barrier has been reached. Theanalytic formulas given in this section assume that S is observed continuously and sometimes thisis the case.? Often, the terms of a contract state that S is observed periodically; for example, once aday at 12 noon. Broadie, Glasserman, and Kou provide a way of adjusting the formulas we havejust given for the situation where the price of the underlying is observed discretely.4 The barrierlevel H is replaced by He* S8260VT7™ For an up-and-in or up-and-out option and by He™® 5826007 fa
3 One way to track whether a barrier has been reached from below (above) is to send a limit order to the exchange tosell (buy) the asset at the barrier price and see whether the order is filled.
4M. Broadie, P. Glasserman, and §. G. Kou, “A Continuity Correction for Discrete Barrier Options,”Mathematical Finance, 7, no. 4 (October 1997), 325-49."Exotic Options 441
for a down-and-in or down-and-out option, where m is the number of times the asset price isobserved (so that T/m is the time interval between observations).
Barrier options often have quite different properties from regular options. For example, some-times vega is negative. Consider an up-and-out call option when the asset price is close to thebarrier level. As volatility increases the probability that the barrier will be hit increases. As a result,a volatility increase causes a price decrease.
 
19.7 BINARY OPTIONS
Binary options are options with discontinuous payoffs. A simpie exampie of a binary option is acash-or-nothing call. This pays off nothing if the asset price ends up below the strike price at time Tand pays a fixed amount, Q, if it ends up above the strike price. In a risk-neutral world, theprobability of the asset price being above the strike price at the maturity of an option is, with ourusual notation, (dy). The value of a cash-or-nothing call is therefore Qe~”? N(d>). A cash-or-nothingput is defined analogously to a cash-or-nothing cail. It pays off Q if the asset price is below the strikeprice and nothing if it is above the strike price. The value of a cash-or-nothing put is Qe’? N(—d,).
Another type of binary option is an asset-or-nothing call. This pays off nothing if the underlyingasset price ends up below the strike price and pays an amount equa! to the asset price itself if itends up above the strike price. With our usual notation, the value of an asset-or-nothing call isSpe N(d\). An asset-or-nothing put pays off nothing if the underlying asset price ends up abovethe strike price and an amount equal to the asset price if it ends up below the strike price. Thevalue of an asset-or-nothing put is Spe?7 N(—d)).
A regular European cail option is equivalent to a long position in an asset-or-nothing call and ashort position in a cash-or-nothing call where the cash payoff on the cash-or-nothing call equalsthe strike price. Similarly, a regular European put option is equivalent to a long position in a cash-or-nothing put and a short position in an asset-or-nothing put where the cash payoff on the cash-or-nothing put equals the strike price.
19.8 LOOKBACK OPTIONS
 
The payoffs from lookback options depend on the maximum or minimum asset price reachedduring the life of the option. The payoff from a European-style lookback call is the amount thatthe final asset price exceeds the minimum asset price achieved during the life of the option. Thepayoff from a European-style lookback put is the amount by which the maximum asset priceachieved during the life of the option exceeds the final asset price.
Valuation formulas have been produced for European lookbacks.° The value of a Europeanlookback“call at time zero is
. 2 ,
N(—4)) — Spine” (Man - Ge D en Ma)
a
2(r- 4)
 
Spe" N(a,) — Spe”
5 See B. Goldman, H. Sosin, and M. A. Gatto, “Path-Dependent Options: Buy at the Low, Sell at the High,”Journal of Finance, 34 (December 1979), 1111-27., M. Garman, “Recollection in Tranquility,” R/SK, March 1989,pp. 16-19.442 CHAPTER 19
where
a, = Sol Smin) += + 0°/2)To/T ,
Qn = ay oVT
ay = USo/ Smin) + (=r +94 07/2)T; oVT
y= 29 = 12) 1S 5/ Sin)Gg
and Syiq is the minimum asset price achieved to date. (If the lookback has just been originated,then Sinin = So.) .The value of a European lookback put is
2
2Se (Mlb) — pel MBs) + Sue 5 Mb) — Sye* NU)
q) 2(r - 4)wherepy = tMSmnax/ So) + (r+. + o/2T= oVTby =b, —oVTjpg wx I Smax/ Sa) + (7 = 9 = 0° /2)T3= oVTr= gq — 0 /2)\a(Smax/ So)Y= oo
and Smax is the maximum asset price achieved to date. (If the lookback has just been originated,then Snax = So-)
Example 19.1 Consider a newly issued lookback put on a non-dividend-paying stock where thestock price is 50, the stock price volatility is 40% per annum, the risk-free rate is 10% per annum, andthe time to maturity is three months. In this case, Sax = 50, Sp = 50, r=0.1, g =0, o = 0.4, andT = 0.25. From the formulas just given, b; = —0.025, b, = —0.225, b; = 0.025, and ¥; = 0, so thatthe value of the lookback put is 7.79. A newly issued lookback call on the same stock is worth 8.04.
A lookback call is a way that the holder can buy the underlying asset at the lowest price achievedduring the life of the option. Similarly, a lookback put is a way that the holder can sell theunderlying asset at the highest price achieved during the life of the option. The underlying asset ina lookback option is often a commodity. As with barrier options, the value of a lookback option isliable to be sensitive to the frequency that the asset price is observed for the purposes of computingthe maximum or minimum. The formulas above assume that the asset price is observed con-tinuously. Broadie, Glasserman, and Kou provide a way of adjusting the formulas we have justgiven for the situation where the asset price is observed discretely.°
® M. Broadie, P. Glasserman, and S. G. Kou, “Connecting Discrete and Continuous Path-Dependent Options,”Finance and Stochastics, 2 (1998), 1-28.Exotic Options 443
19.9 SHOUT OPTIONS
 
 
A shout option is a European option where the holder can “shout” to the writer at one time duringits life. At the end of the life of the option, the option holder receives either the usual payoff from aEuropean option or the intrinsic value at the time of the shout, whichever is greater. Suppose thestrike price is $50 and the holder of a cail shouts when the price of the underlying asset is $60. Ifthe final asset price is less than $60 the holder receives a payoff of $10. If it is greater than $60, theholder receives the excess of the assel price over $50.
A shout option has some of the same features as a lookback option, but is considerably lessexpensive. It can be valued by noting that if the option is shouted at a time 1 when the asset price isS, the payoff from the option is
max(0, Sp ~ S,) +(S,- K)
where, as usual, K is the strike price and S7 is the asset price at time T. The value at time rt if theoption is shouted is therefore the present value of S, — K plus the value of a European option withstrike price S,. The latter can be calculated using Black-Scholes formulas.
We value a shout option by constructing a binomial or trinomial tree for the underlying asset inthe usual way. As we roll back through the tree, we calculate at each node the value of the option ifwe shout and the value if we do not shout. The option’s price at the node is the greater of the two.The procedure for valuing a shout option is therefore very similar to the procedure for valuing aregular American option.
19.10 ASIAN OPTIONS
 
 
Asian options are options where the payoff depends on the average price of the underlying assetduring at least some part of the life of the option. The payoff from an average price call ismax(0, Syye — K) and that from an average price put is max(0, K — Saye), where Saye is the averagevalue of the underlying asset calculated over a predetermined averaging period. Average priceoptions are less expensive than regular options and are arguably more appropriate than regularoptions for meeting some of the needs of corporate treasurers. Suppose that a U.S. corporatetreasurer expects to receive a cash flow of 100 million Australian dollars spread evenly over the nextyear from the company’s Australian subsidiary . The treasurer is likely to be interested in an optionthat guarantees that the average exchange rate realized during the year is above some level. Anaverage price put option can achieve this more effectively than regular put options.
Another type of Asian option is an average strike option. An average strike call pays offmax(0, Sp ~— Saye) and an average strike put pays off max(0, S,,.— Sr). Average strike optionscan guarantee that the average price paid for an asset in frequent trading over a period of time isnol greater than the final price. Alternatively, il can guarantee that the average price received for anssi ent trading over a period of time is not less than the final price.
f the underlying asset price, S, is assumed to be lognormally distributed and S,,, is a geometricaverage of the S’s, analytic formulas are available for valuing European average price options.’This is because the geometric average of a set of lognormally distributed variables is alsolognormal. Consider a newly issued option that will provide a payoff at time T based on the
7 See A. Kemna and A. Vorst, “A Pricing Method for Options Based on Average Asset Values,” Journal of Bankingand Finance, 14 (March 1990), 113-29.444 CHAPTER 19
 
geometric average calculated between time zero and time T. In a risk-neutral world, it can beshown that the probability distribution of the geometric average of a asset price over a certainperiod is the same as that of the asset price at the end of the period if the asset’s expected growthrate is set equal to }(r — g — o° /6) (rather than r — g) and its volatility is set equal to o//3 (ratherthan o). The geometric average price option can therefore be treated like a regular option withthevolatility set equal to a/./3 and the dividend yield equal to es
1 o 1 oro 5 rT 4a-— =3 rats]
When, as is nearly always the case, Asian options are defined in terms of arithmetic averages, exactanalytic pricing formulas are not available. This is because the distribution of the arithmeticaverage of a set of lognormal distributions does not have analytically tractable properties.However, the distribution is approximately lognormal and this leads to a good analytic approxi-mation for valuing average price options. We calculate the first two moments of the probabilitydistribution of the arithmetic average in a risk-neutral worid exactly and then assume thisdistribution is the lognormal.®
Consider a newly issued Asian option that provides a payoff at time T based on the arithmeticaverage between time zero and time T. The first moment, M, and the second moment, Mb, of theaverage in a risk-neutral world can be shown to be
 
(r—q)Fe -1M, =-———_S,(r=gytandme 2el2r-a+0" IT 52 283 1 ear2 qt PW 2940? C-OE\WAr-gtor r—-qte
If we assume that the average asset price is lognormal, we can regard an option on the average aslike an option on a futures contract and use equations (13.17) and (13.18) with
oS~~.X=
Fo= M,and1 M.o=-In (7) (19.2)T \M?
Example 19.2 Consider a newly issued average price call option on a non-dividend-paying stockwhere the stock price is 50, the strike price is 50, the stock price volatility is 40% per annum, the risk-free rate is 10% per annum, and the time to maturity is one year. In this case, Sy = 50, K = 50,r=0.1,g=0,0 = 0.4, and T = |. If the average is a geometric average, we can value the option as aregular option with the volatility equal to 0.4/3, or 23.09%, and dividend yield equal to(0.14 0.47 /6)/2, or 6.33%. The value of the option is 5.13. If the average is an arithmetic average,we first calculate M, = 52.59 and M> = 2,922.76. When we assume the average is lognormal, theoption has the same value as an option on a futures contract. From equations (19.1) and (19.2),Fy = 52.59 and a = 23.54%. DerivaGem gives the value of the option as 5.62.
The formulas just given for M,; and My, assume that the average is calculated from continuous
8 See S. M. Turnbull and L. M. Wakeman, “A Quick Algorithm for Pricing European Average Options,” Journal ofFinancial and Quantitative Analysis, 26 (September 1991), 377-89.Exotic Options 445
observations on the asset price. Appendix 19A shows how M, and M, can be obtained when theaverage is calculated from observations on the asset price at discrete points in time.
We can modify the analysis to accommodate the situation where the option is not newly issuedand some prices used to determine the average have already been observed. Suppose that theaveraging period is composed of a period of length t; over which prices have already been observedand a future period of length t, (the remaining life of the option). Suppose that the average assetprice during the first time period is S. The payoff from an average price call is
nay St t Savel2ma t) + ft
x, 0)}
where Saye is the average asset price during the remaining part of the averaging period. This is thesame as
 
— K*i+h MaAX(Save , 0)
where
th tt tis
ate Ksty fy
K*=
When K* > 0, the option can be valued in the same way as a newly issued Asian option providedthat we change the strike price from K to K”* and multiply the result by t:/(t; + 12). When K* <0the option is certain to be exercised and can be valued as a forward contract. The value is
 
; th (Mye arty Kte")1
19.11 OPTIONS TO EXCHANGE ONE ASSET FOR ANOTHER
 
 
Ostions to exchanse one asset for ancther fsaometimes referred to a vohawes ontions\) ariseUpiions to excnange one asset ior another (sometimes reierrca to 45 excnaige options; arise in
various contexts. An option to buy yen with Australian dollars is, from the point of view of a US.investor, an option to exchange one foreign currency asset for another foreign currency asset.A stock tender offer is an option to exchange shares in one stock for shares in another stock.
Consider a European option to give up an asset worth Uy at time T and receive in return anasset worth V;. The payoff from the option is
max{V, — U,, 0)
A formula for valuing this option was first produced by Margrabe.? Suppose that the asset prices Uand V both follow geometric Brownian motion with volatilities oy and oy. Suppose further thatthe instantaneous correlation between U and V is p, and the yields provided by U and V are qyand gy, respectively. The vaiue of the option at time zero is
Voe 4? N(dy) — Une 7 N(d2) (19.3)
® See W. Margrabe, “The Value of an Option to Exchange One Asset for Another,” Journal of Finance, 33 (March1978), 177-86.446 CHAPTER 19
wheredf, =! Wo/Uo) + au — av + 6° /2)T\ G/Tdy =d,-6VT
6 = of, + oF — 2poyoy
and Uy and Vp are the values of U and V at times zero.
This result will be proved in Section 21.6. It is interesting to note that equation (19.3) isindependent of the risk-free rate r. This is because, as r increases, the growth rate of both assetprices in a risk-neutral world increases, but this is exactly offset by an increase in the discount rate.The variable 6 is the volatility of V/U. Comparisons with equation (13.4) show that the optionprice is the same as the price of Uy European call options on an asset worth V/U when the strikeprice is 1.0, the risk-free interest rate is gy, and the dividend yield on the asset is gy. MarkRubinstein shows that the American version of this option can be characterized similarly forvaluation purposes.'° It can be regarded as Uy American options to buy an asset worth V/U for 1.0when the risk-free interest rate is gy and the dividend yield on the asset is gy. The option cantherefore be valued as described in Chapter 18 using a binomial tree.
It is worth noting that an option to obtain the better or worse of two assets can be regarded asaposition in one of the assets combined with an option to exchange it for the other asset:
min(Uy, Vr) = Vr — max(V; — Uz, 0)max(lp, Vp) = Up + max(Vp — U;, 0)
19.12 BASKET OPTIONS
 
Options involving two or more risky assets are sometimes referred to as rainbow options. Oneexample is the bond futures contract traded on the CBOT described in Chapter 5. The party withthe short position is allowed to choose between a large number of different bonds when makingdelivery. Another example is a LIBOR-contingent FX option. This is a foreign currency optionwhose payoff occurs only if a prespecified interest rate is within a certain range at maturity.
Possibly the most popular rainbow option is a basket option. This is an option where the payoffis dependent on the value of a portfolio (or basket) of assets. The assets are usually eitherindividual stocks or stock indices or currencies. A European basket option can be valued withMonte Carlo simulation, by assuming that the assets follow correlated geometric Brownian motionprocesses. A much faster approach is to calculate the first two moments of the basket at thematurity of the option in a risk-neutral world and then assume that value of the basket islognormally distributed at that time. The option can then be regarded as an option on a futurescontract with the parameters shown in equations (19.1) and (19.2). Appendix 19A shows how themoments of the value of the basket at a future time can be calculated from the volatilities of, andcorrelations between, the assets.
'© See M. Rubinstein, “One for Another,” RISK, July/August 1991, 30-32Exotic Options 447
19.13. HEDGING ISSUES
 
 
Before trading an exotic option, it is important {or a financial institution to assess not only how itshould be priced but also the difficulties that are likely to be experienced in hedging it. The generalapproach described in Chapter 14 involving the creation of a delta-neutral portfolio and themonitoring of gamma, vega, and other Grock Icttcrs can be used.
Exotic options sometimes prove to be easier to hedge using the underlying asset than thecorresponding plain vanilla option. Consider for example an average price option where theaveraging period is the whole life of the option. As time passes, we observe more of the assetprices that will be used in calculating the final average. This means that our uncertainty about thepayoff decreases with the passage of time. As a result, the option becomes progressively easier tohedge. In the final few days, the delta of the option always approaches zero because pricemovements during this time have very little impact on the payoff.
Barrier options can, in certain circumstances, be significantly more difficult to hedge thanregular options. Consider a down-and-out call option on a currency when the exchange rate is0.0005 above the barrier. If the barrier is hit, the option is worth nothing. If the barrier is not hit,the option may prove to be quite valuable. The delta of the option is discontinuous at the barrierand hedging using conventional techniques is difficult. The approach in the following section isoften more appropriate.
19.14 STATIC OPTIONS REPLICATION
 
Hedging an option position involves replicating the opposite position. The procedures described inChapter 14 involve what is sometimes referred to as dynamic options replication. They require theposition in the hedging assets to be rebalanced frequently and can be quite expensive because ofthe transaction costs involved.
An alternative approach that can sometimes be used io hedge a position in exotic options isstatic options replication.'' This involves searching for a portfolio of actively traded options thatapproximately replicate the option position. Shorting this position provides the hedge. The basicprinciple underlying static options replication is as follows. If two portfolios are worth the same onacertain boundary, they are also worth the same at all interior points of the boundary.
Consider as an example a nine-month up-and-out call option on a non-dividend-paying stockwhere the stock price is 50, the strike price is 50, the barrier is 60, the risk-free interest rate is 10%per annum, and the volatility is 30% per annum. Suppose that f(5S, ¢) is the value of the option attime t fora stock price of S. We can use any boundary in (S, ¢) space for the purposes of producingthe replicating portfolio. A convenient one to choose is shown in Figure 19.2. It is defined byS= 60 and ¢ = 0.75. The values of the up-and-out option on the boundary are given by
F(S, 0.75) = max(S — 50, 0) when S < 60F(60, t) = 0 when 0<t < 0.75
There are many ways that we can approximately match these boundary values using regular
1 See E. Derman, D. Ergener, and I. Kani, “Static Options Replication,” Journal of Derivatives, 2. no. 4 (Summer1995), 78-95.448 CHAPTER 19
AS
60
50 -
 
0.25 0.50 0.75
Figure 19.2 Boundary points used for the static options replication example
options. The natural instrument to match the first boundary is a regular nine-month European calloption with a strike price of 50. The first instrument introduced into the replicating portfolio is
therefore ely to be one unit of this option. (We refer to this option as option A.) One way of then
kely to be one unit of this opt e refer 1s option as option A.) One way of thenproceeding is as follows. We divide the life of the option into a number of time steps and chooseoptions that satisfy the second boundary condition at the beginning of each time step.
Suppose that we choose time steps of three months. The next instrument we choose should leadto the second boundary being matched at t = 0.5. In other words, it should lead to the value of thecomplete replicating portfolio being zero when t = 0.5 and S = 60. The option shouid have theproperty that it has zero value on the first boundary since this has already been matched. Onepossibility is a regular nine-month European call option with a strike price of 60. (We will refer tothis as option B.) Black-Scholes formulas show that this is worth 4.33 at the six-month point whenS = 60. They also show that the position in option A is worth 11.54 at this point. The position werequire in option B, therefore, is —11.54/4.33 = —2.66.
We next move on to matching the second boundary condition at t = 0.25. The option usedshould have the property that it has zero value on all boundaries that have been matched thus far.One possibility is a regular six-month European call option with a strike price of 60. (We refer tothis as option C.) This is worth 4.33 at the three-month point when S = 60. Our position inoptions A and B is worth —4.21 at this point. The position we require in option C, therefore, is4.21/4.33 = 0.97.
Finally, we match the second boundary condition at t = 0. For this we use a regular three-monthEuropean option with a strike price of 60. (We refer to this as option D.) Calculations similar tothose above show that the required position in option D is 0.28.
The portfolio chosen is summarized in Table 19.1. It is worth 0.73 initially (i.e., at time zerowhen the stock price is 50). This compares with 0.31 given by the analytic formula for the up-and-out call earlier in this chapter. The replicating portfolio is not exactly the same as the up-and-outoption because it matches the latter at only three points on the second boundary. If we use thesame scheme, but match at 18 points on the second boundary (using options that mature every half
  Exotic Options 449
Table 19.1 The portfolio of European call options used toreplicate an up-and-out option
ees
 
Strike Maturity Initial
Option price (years) Position valueA 50 0.75 1.00 +6.99B 60 0.75 —2.66 ~8.21Cc 60 0.50 0.97 +1.78D 60 0.25 0.28 +0.17
 
month), the value of the replicating portfolio reduces to 0.38. If 100 points are matched, the valuereduces further to 0.32.
To hedge a derivative, we short the portfolio that replicates its