boundary conditions. This hasthe advantage over delta hedging that it does not require frequent rebalancing. The staticreplication approach can be used for a wide range of derivatives. The user has a great deal offlexibility in choosing the boundary that is to be matched and the options that are to be used. Theportfolio must be unwound when any part of the boundary is reached.
SUMMARY
 
Exotic options are options with rules governing the payoff that are more complicated thanstandard options. We have discussed 13 different types of exotic option: packages, nonstandardAmerican options, forward start options, compound options, chooser options, barrier options,binary options, lookback options, shout options, Asian options, options to exchange one asset foranother, and basket options. We have discussed how these can be valued using the sameassumptions as those used to derive the Black-Scholes equations in Chapter 12. Some can bevalued analytically, but using much more complicated formulas than those for regular Europeancalls and puts, some can be handled using analytic approximations, and some can be valued usingextensions of the numerical procedures in Chapter 18. We will present more numerical proceduresfor valuing exotic options in Chapter 20.
Some exotic options are easier to hedge than the corresponding regular options; others are moredifficult. In general, Asian options are easier to hedge because the payoff becomes progressivelymore certain as we approach maturity. Barrier options can be more difficult to hedge because deltais liable to be discontinuous at the barrier. One approach to hedging an exotic option, known asstatic options replication, is to find a portfolio of regular options whose value matches the value ofthe exotic option on some boundary. The exotic option is hedged by shorting this portfolio.
SUGGESTIONS FOR FURTHER READING
 
Broadie, M., P. Glasserman, and S. G. Kou, “A Continuity Correction for Discrete Barrier Options,”Mathematical Finance, 7, no. 4 (October 1997), 325-49.
Broadie, M., P. Glasserman, and S. G. Kou, “Connecting Discrete and Continuous Path-DependentOptions,” Finance and Stochastics, 2 (1998), 1-28.
 450 CHAPTER 19
Clewlow, L., and C. Strickland, Exotic Options: The State of the Art. Thomson Business Press,London, 1997.
Curran, M., “Beyond Average Intelligence,” RISK, October 1992, pp. 60-62.
Derman, E., D. Ergener, and I. Kani, ‘Static Options Replication,” Journal of Derivatives, 2, no. 4(Summer 1995), 78-95.
aDerman, E., I. Kani, and N. Chriss, ‘Static Options Replication,” Journal of Derivatives, 1, no. 4(Summer 1994), 6-14.
Garman, M., “Recollection in Tranquility,” R/SK, March 1989, pp. 16-19.Geske, R., “The Valuation of Compound Options,” Journal of Financial Economics, 7 (1979), 63-81.
Goldman, B., 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. .
Hudson, M., “The Value of Going Out,” RISK, March 1991, 29-33.
Johnson, H., “Options on the Maximum and Minimum of Several Assets,” Journal of Financial andQuantitative Analysis, 22, no. 3 (September 1987), 277-83.
Kemna, A., and A. Vorst, “A Pricing Method for Options Based on Average Asset Values,” Journal ofBanking and Finance, 14 (March 1990), 113-29.
Levy, E., “Pricing European Average Rate Currency Options,” Journal of International Money andFinance, 11 (1992), 474-91.
Levy, E., and S. M. Turnbull, “Average Intelligence,” R/SK, February 1992, pp. 53-59.
Margrabe, W., “The Value of an Option to Exchange One Asset for Another,” Journal of Finance, 33(March 1978), 177-86.
Milevsky, M. A., and S. E. Posner, “Asian Options: The Sum of Lognormals and the ReciprocalGamma Distribution,” Journal of Financial and Quantitative Analysis, 33, no. 3 (September 1998),409-22.
Ritchken, P., L. Sankarasubramanian, and A. M. Vijh, “The Valuation of Path Dependent Contractson the Average,” Management Science, 39 (1993), 1202-13.
Ritchken, P., “On Pricing Barrier Options,” Journal of Derivatives, 3, no. 2 (Winter 1995), 19-28.Rubinstein, M., and E. Reiner, “Breaking Down the Barriers,” RISK, September 1991, 28-35.Rubinstein, M., “Double Trouble,” RISK, December 1991 / January 1992, pp. 53-56.
Rubinstein, M., “One for Another,” RISK, July/August 1991, pp. 30-32.
Rubinstein, M., “Options for the Undecided,” RISK, April 1991, pp. 70-73.
Rubinstein, M., “Pay Now, Choose Later,” RISK, February 1991, pp. 44-47.
Rubinstein, M., “Somewhere Over the Rainbow,” RISK, November 1991, pp. 63-66.
Rubinstein, M., “Two in One,” RISK, May 1991, p. 49.
Rubinstein, M., and E. Reiner, “Unscrambling the Binary Code,” R/SK, October 1991, pp. 75-83.
Stulz, R., “Options on the Minimum or Maximum of Two Assets,” Journal of Financial Economics,10 (1982), 161-85.
Turnbull, 5. M., and L. M. Wakeman, “A Quick Algorithm for Pricing European Average Options,”Journal of Financial and Quantitative Analysis, 26 (September 1991), 377-89.Exotic Options 451
 
QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)
 
19.1. Explain the difference between a forward start option and a chooser option.19.2. Describe the payoff from a portfolio consisting of a lookback call and a lookback put with the
same maturity.same maturity.
19.3. Consider a chooser option where the holder has the right to choose between a European call anda Furopean put at any time during a two-year period. The maturity dates and strike prices forthe calls and puts are the same regardless of when the choice is made. Is it ever optimal to makethe choice before the end of the two-year period? Explain your answer.
19.4. Suppose that c, and p, are the prices of a European average price call and a European averageprice put with strike K and maturity T, c. and p> are the prices of a European average strike calland European average styike put with maturity T, and ¢3; and p, are the prices of a regularEuropean call and a regular European put with strike price K and maturity 7. Show that
C1 +e, —€3 = Pi + P2— Ps
19.5. The text derives a decomposition of a particular type of chooser option into a call maturing attime T, and a pul maturing at lime 7,. Derive an alternative decomposition into a call maturingat time 7, and a put maturing at time 7).
19.6. Section 19.6 gives two formulas for a down-and-out call. The first applies to the situation wherethe barrier, H, is less than or equal to the strike price, K. The second applies to the situationwhere H > K. Show that the two formulas are the same when H = K.
19.7. Explain why a down-and-out put is worth zero when the barrier is greater than the strike price.
19.8. Suppose that the strike price of an American call option on a non-dividend-paying stock growsat rate g. Show that if g is less than the risk-free rate, r, it is never optimal to exercise the callearly.
19.9. How can the value of a forward start put option on a non-dividend-paying stock be calculated ifit is agreed that the strike price will be 10% greater than the stock price at the time the optionstarts?
19.10. Ifa stock price follows geometric Brownian motion, what process does A(t) follow where A(t) isthe arithmetic average stock price between time zero and time t?
- 19.11. Explain why delta hedging is easier for Asian options than for regular options.
19.12. Calculate the price of a one-year European option to give up 100 ounces of silver in exchange forone ounce of gold. The current prices of gold and silver are $380 and $4, respectively, the risk-freeinterest rate is 10% per annum, the volatility of each commodity price is 20%, and the correlationbetween the two prices is 0.7. Ignore storage costs.
19.13. Is a European down-and-out option on an asset worth the same as a European down-and-out____ optien on the asset’s futures price for a futures contract maturing at the same time as the option?
19.14. Answer the following questions about compound options:a. What put-—call parity relationship cxists between the price of a European call on a call and aEuropean put on a call? Show that the formulas given in the text satisfy the relationship.b. What put-call parity relationship exists between the price of a European call on a put andaEuropean put on a put? Show that the formulas given in the text satisfy the relationship.
19.15. Does a lookback call become more valuable or less valuable as we increase the frequency withwhich we observe the asset price in calculating the minimum?
 452
19.16.
19.17.
19.18.
CHAPTER 19
Does a down-and-out call become more valuable or less valuable as we increase the frequencywith which we observe the asset price in determining whether the barrier has been crossed? Whatis the answer to the same question for a down-and-in call?
Explain why a regular European call option is the sum of a down-and-out European call and adown-and-in European call. Is the same true for American call options?
than 1,000 and zero otherwise. Assume that the current level of the index is 960, the #18K-free rateis 8% per annum, the dividend yield on the index is 3% per annum, and the volatility of the
What is the value of a derivative that pays off $100 in six months if the S&P 500 rookie fee ne
. index is 20%.
19.19.
19.20.
19.21.
In a three-month down-and-out call option on silver futures, the strike price is $20 per ounce andthe barrier is $18. The current futures price is $19, the risk-free interest rate is 5%, and thevolatility of silver futures is 40% per annum. Explain how the option works and calculate jtsvalue. What is the value of a regular call option on silver futures with the same terms? What isthe value of a down-and-in call option on silver futures with the same terms?
A new European-style lookback call option on a stock index has a maturity of nine months. Thecurrent level of the index is 400, the risk-free rate is 6% per annum, the dividend yield on theindex is 4% per annum, and the volatility of the index is 20%. Use DerivaGem to value theoption. ~Estimate the value of a new six-month European-style average price call option on a non-dividend-paying stock. The initial stock price is $30, the strike price is $30, the risk-free interestrate is 5%, and the stock price volatility is 30%.
Use DerivaGem to calculate the value of:
a. A regular European call option on a non-dividend-paying stock where the stock price is$50, the strike price is $50, the risk-free rate is 5% per annum, the volatility is 30%, and thetime to maturity is one year
b. A down-and-out European call which is as in (a) with the barrier at $45c. A down-and-in Euronean call which is as in (a) with the barrier at $45
c. A down-and-in Huropean call which ts as in (a ith the barrier at »
Show that the option in (a) is worth the sum of the values of the options in (b) and (c).
ASSIGNMENT QUESTIONS
 
19.23.
19.24.
19.25,
What is the value in dollars of a derivative that pays off £10,000 in one year provided that thedollar-sterling exchange rate is greater than 1.5000 at that time? The current exchange rate is1.4800. The dollar and sterling interest rates are 4% and 8% per annum, respectively. Thevolatility of the exchange rate is 12% per annum.
Consider an up-and-out barrier call option on a non-dividend-paying stock when the stock price is50, the strike price is 50, the volatility is 30%, the risk-free rate is 5%, the time to maturity is oneyear, and the barrier at $80. Use the software to value the option and graph the relationshipbetween (a) the option price and the stock price, (b) the delta and the option price, (c) the optionprice and the time to maturity, and (d) the option price and the volatility. Provide an intuitiveexplanation for the results you get. Show that the delta, gamma, theta, and vega for an up-and-outbarrier call option can be either positive or negative.
Sample Application F in the DerivaGem Application Builder software considers the staticoptions replication example in Section 19.14. It shows the way a hedge can be constructed usingfour options (as in Section 19.14) and two ways a hedge can be constructed using 16 options.Exotic Options 453
 
19.27.
19.28,
19.29.
a. Explain the difference between the two ways a hedge can be constructed using 16 options.Explain intuitively why the second method works better.
b. Improve on the four-option hedge by changing Tmat for the third and fourth options.
c. Check how well the 16-option portfolios match the delta, gamma, and vega of the barrieroption.
Consider a down-an. it call option on a forcien rreney. The initial exchanec rate is 0.90. the
. Consider a down-and-out call option on a forcign currency. The initial exchange rate is 0.90, the
time to maturity is two years, the strike price is 1.00, the barrier is 0.80, the domestic risk-freeinterest rate is 5%, the foreign risk-free interest rate is 6%, and the volatility is 25% per annum.Use DerivaGem to develop a static option replication strategy involving five options.
Suppose that a stock index is currently 900. The dividend yield is 2%, the risk-free rate is 5%,and the volatility is 40%. Use the results in Appendix 19A to calculate the value of a one-yearaverage price call where the strike price is 900 and the index level is observed at the end of eachquarter for the purposes of the averaging. Compare this with the price calculated by DerivaGemfor a one-year average price option where the price is observed continuously. Provide an intuitiveexplanation for any differences between the prices.
Use the DerivaGem Application Builder software to compare the effectiveness of daily deltahedging for (a) the option considered in Tables 14.2 and 14.3 and (b) an average price call withthe same parameters. Use Sample Application C. For the average price option you will find itnecessary to change the calculation of the option price in cell C16, the payoffs in cells HISand H16, and the deltas (cells G46 to G186 and N46 to N186). Carry out 20 Monte Carlosimulation runs for each option by repeatedly pressing F9. On each run, record the cost ofwriting and hedging the option, the volume of trading over the whole 20 weeks, and the volumeof trading between weeks 11 and 20. Comment on the results.
In the DerivaGem Application Builder software, modify Sample Application D to test theeffectiveness of delta and gamma hedging for a call on call compound option on a 100,000 unitsof a foreign currency where the exchange rate is 0.67, the domestic risk-free rate is 5%, the foreignrisk-free rate is 6%, the volatility is 12%. The time to maturity of the first option is 20 weeks, andthe strike price of the first option is 0.015. The second option matures 40 weeks from today andhas a strike price of 0.68. Explain how you modified the cells. Comment on hedge effectiveness.454 CHAPTER 19
 
APPENDIX 19A
Calculation of the First Two Moments of Arithmetic Averagesand Baskets
Consider first the problem of calculating the first two moments of the value of a basket of assets ata future time, 7, in a risk-neutral world. The price of each asset in the basket is assumed to belognormal. Define:n: The number of assetsS;: The value of the ith asset at time TF;: The forward price! of the ith asset for a contract maturing at time To;: The volatility of the ith asset between time zero and time To;;: Correlation between returns from the ith and jth assetP: Value of basket at time TM,: First moment of P in a risk-neutral worldM2: Second moment of P in a risk-neutral worldBecause P = 9°_, Si, E(S;) = F,, M, = E(P), and M) = E(P’), where E denotes the expected valuein a risk-neutral world, it follows thatntM = )O%i=lAlso,
P?= > SS;i=1 j=l
From the properties of lognormal distributions,E(S;S;) = FF chia?
A am= SS nent1 j=)
Hence,
We now move on to the related problem of calculating the first two moments of the arithmeticaverage price of an asset in a risk-neutral world when the average is calculated from discreteobservations. Suppose that the asset price is observed at times 7; (1 < i < m). We redefine variablesas follows:
S;: The value of the asset at time 7;
F;: The forward price of the asset for a contract maturing at time 7;
o;: The implied volatility for an option on the asset with maturity 7;
6;;: Correlation between return on asset up to time 7; and the return on the asset up to time T;
2 Strictly speaking, F, should be the futures price rather than the forward price. In practice, analysts usually assumeno difference between the two when calculating moments.Exotic Options 455
P: Value of the arithmetic averageM,: First moment of P in a risk-neutral worldM,: Second moment of P in a risk-neutral world
In this case,
In this case,E(S)S8)) = FE, Fyes v7
= oT,iT
E(S)S,) = FiFye%™
m=, (Re 42 KFT2=s3(d Fe +230 RF
i=] i<j
It can be shown that, when i < j,
 
so that
andCHAPTER 20
MORE ON MODELS ANDNUMERICAL PROCEDURES
Up to now the models we have used have been based on the original Black-Scholes assumptionsand the numerical procedures we have used have been relatively straightforward. In this chapter weintroduce a number of new models and explain how the numerical procedures can be adapted tocope with particular situations. .
When the exotic option pricing formulas of Chapter 19 are used, there is no simple way of ©calculating the volatility that should be input from the volatility smile applicable to plain vanillaoptions. Sometimes the correct volatility is counterintuitive. As a result it is difficult for traders toincorporate a volatility smile into their pricing of exotic options. The first part of this chapterpresents a number of aiternatives to the geometric Brownian motion modei that are designed toovercome this problem. The parameters of the models we will consider can be chosen so that theyare approximately consistent with the volatility smile observed in the market.
The second part of the chapter extends our discussion of numerical procedures. We explain howsome types of path-dependent derivatives can be valued using trees. We discuss the special problemsassociated with valuing barrier options numerically and how these problems can be handled. Weoutline alternative ways of constructing trees for two correlated variables. Finally, we show howMonte Carlo simulation can be used to value derivatives when there are early exercise opportunities.
As in Chapter 19, our results are presented for derivatives dependent on an asset providing ayield at rate g. For an option on a stock index g should be set equal to the dividend yield on theindex, for an option on a currency gq should be set equal to the foreign risk-free rate, and for anoption on a futures contract g should be set equal to the domestic risk-free rate.
20.1 THE CEV MODEL
 
The constant elasticity of variance (CEV) model assumes that the risk-neutral process for a stockprice, S, isdS = (r—q)Sdt+ oS°* dz
where r is the risk-free rate, g is the dividend yield, dz is a Wiener process, o is a volatilityparameter, and aq is a positive constant.! The stock price has volatility oS*!, When a = }, the
' See J. C. Cox and S. A. Ross, “The Valuation of Options for Alternative Stochastic Processes,” Journal ofFinancial Economics, 3 (March 1976), 145-66.
456More on Models and Numerical Procedures 457
CEV model is the geometric Brownian motion model we have been using up to now. When a < 1,the volatility increases as the stock price decreases. This creates a probability distribution similar tothat observed for equities with a heavy left tail‘and a jess heavy right tail? (see Figure 15.4). Whena > |, the volatility increases as the stock price increases, giving a probability distribution with aheavy right tail and a less heavy left tail. This corresponds to a volatility smile where the implied
volatility ig an increasine function of the strike price. This type of volatility smule is sometimes
olallity is an increasing tunchon of sirike ice. i NS Olahiity smile is sometimes
observed for options on futures (see Assignment 13.40.The valuation formulas for European call and put options under the CEV model is
c= Sel — "(a,b + 2,0] — Ke 7c, ba)
p=Ke[l-x(c,b,a| —- Se" (a,b +2,0when 0 <a < 1, andc= Soe" [1 — 7(c, —b, a) — Ke" (a, 2-5, 0)p= Ke" [1 — 7 (@, 2-6, 0] — Sye 7(c, -b, a)when w > 1, whereKu-@) It (Se “OT yrd-2)
; b= ; =(l-ay’o°T l-a (l-«a)’o°T
 
 
and x’(z, v, &) is the cumulative probability that a variable with a noncentral x” distribution withnoncentrality parameter v and k degrees of freedom is less than z. An efficient procedure forcomputing x°(z, v, &) is provided by Ding.?
20.2 THE JUMP DIFFUSION MODEL
 
 
 
Merton has suggested.a mode! where the asset price has jumps superimposed upon a geometricBrownian motion.’ Define:
uw: Expected return from asset net of the dividend yield, g2. Average number of jumps per yeark: Average jump size measured as a percentage of the asset priceThe percentage jump size is assumed to be drawn from a probability distribution in the model.
The probability of a jump in time é¢ is 46t. The average growth rate in the asset price from thejumps is therefore 4k. The process for the asset price is
=(u-Akjdt+oadz+dp
? The reason is as follows. As the stock price decreases, the volatility increases making even lower stock price morelikely; when the stock price increases, the volatility decreases making higher stock prices less likely.
3 See C. G. Ding, “Algorithm AS275: Computing the Non-central x? Distribution Function,” Applied Statistics,41 (1992), 478-82.
4 See R. C. Merton, “Option Pricing When Underlying Stock Returns Are Discontinuous.” Journal of FinancialEconomics, 3 (March 1976), 125-44.458 CHAPTER 20
 
where dz is a Wiener process, dp is the Poisson process generating the jumps, and o is the volatilityof the geometric Brownian motion. The processes dz and dp are assumed to be independent.The key assumption made by Merton is that the jump component of the asset’s return representsnonsystematic risk (i.e., risk not priced in the economy).° This means that a Black-Scholes type ofportfolio, which eliminates the uncertainty arising from the geometric Brownian motion (but notthe jumps) must earn the riskless rate.An important particular case of Merton’s model is where the logarithm of the size of thepercentage jump is normal. Assume that the standard deviation of the normal distribution is s.Merton shows that a European option price can then be written© LAT yanEur,n=0
where 4’ = 4(1 +k). The variable f, is the Black-Scholes option price when the dividend yield is q,
the variance rate is2
nso +E
and the risk-free rate isnyr-Ak+ TT
where y = In(1 +4).
The model gives rise to heavier left and heavier right tails than Black-Scholes and is consistentwith the implied volatilities observed for currency options (see Section 15.2 and Problem 20.20).
20.3. STOCHASTIC VOLATILITY MODELS
 
 
When the instantaneous volatility is a known function of time, the risk-neutral process followed bythe stock price isdS = (r — q)S dt + o()Sdz (20.1)
The Black-Scholes formulas are then correct providing the variance rate is set equal to the averagevariance rate during the life of the option (see Problem 20.6). The variance rate is the square of thevolatility. Suppose that during a one-year period the volatility of a stock will be 20% during thefirst six months and 30% during the second six months. The average variance rate is
0.5 x 0.20? +. 0.5 x 0.307 = 0.065
It is correct to use Black-Scholes with a variance rate of 0.065. This corresponds to a volatility of¥0.065 = 0.255, or 25.5%. We implicitly used this result in Section 17.6 when discussing how theGARCH(I, 1) model can be used to calculate volatility term structures.
Equation (20.1) assumes that the instantaneous volatility of an asset is perfectly predictable. Inpractice volatility varies stochastically. This has led some researchers to develop more complexmodels where there are two stochastic variables: the stock price and its volatility.
* This assumption is important because it turns out that we cannot apply risk-neutral valuation to situations where thesize of the jump is systematic. For a discussion of this point, see E. Naik and M. Lee, “General Equilibrium Pricing ofOptions on the Market Portfolios with Discontinuous Returns,” Review of Financial Studies, 3 (1990), 493-521.More on Models and Numerical Procedures 459
Hull and White consider the following stochastic volatility model for the risk-neutral behavior ofa price:
dSparade VV dzs (20.2)dV =a(V, — Vydt +€V" dzy (20.3)
where a, V,, &, and a are constants, and dzs and dzy are Wiener processes. The variable V in thismodel is the asset’s variance rate. The variance rate has a drift that pulls it back to a level V; at rate a.Hull and White compare the price given by this model with the price given by the Black-Scholesmodel when the variance rate in Black-Scholes is put equal to the expected average variance rateduring the life of the option.
Hull and White show that, when volatility is stochastic but uncorrelated with the asset price, theprice of a European option is the Black-Scholes price integrated over the probability distributionof the average variance rate during the life of the option.® Thus a European call price is
oO =| c(V)g(V) dV0where V is the average value of the variance rate, c is the Black-Scholes price expressed as afunction of V, and g is the probability density function of V in a risk-neutral world. This result canbe used to show that Black-Scholes overprices options that are at the money or close to the money,and underprices options that are deep in or deep out of the money. The model is consistent withthe pattern of implied volatilities observed for currency options (see Section 15.2) .
The case where the asset price and volatility are correlated is more complicated. Option pricescan be obtained using Monte Carlo simulation. In the particular case where a = 0.5, Hull andWhite provide a series expansion and Heston gives an analytic result.’ The pattern of impliedvolatilities obtained when the volatility is negatively correlated with the asset price is similar to thatobserved for equities® (see Section 15.3).
Chapter 17 discusses exponentially weighted moving average (EWMA) and GARCH(I, |)models. These are alterative approaches to characterizing a stochastic volatility model. Duanshows that it is possible to use GARCH(1, 1) as the basis for an internally consistent optionpricing model.° (See Problem 17.14 for the equivalence of GARCH(, 1) and stochastic volatilitymodels.)
For options that last less than a year, the pricing impact of a stochastic volatility is fairly small inabsolute terms (although in percentage terms it can be quite large for deep-out-of-the-moneyoptions). It becomes progressively larger as the life of the option increases.
The impact of a stochastic volatility on the performance of delta hedging is generally quite large.
This can be tested by carrying out simulations of delta hedging, such as those in Section 14.4,6 Sec J.C