style as well as European-style derivatives. However, they are difficultto apply when the payoffs depend on the past history of the state variables as well as on theircurrent values. In addition, they are liable to become computationally very time-consuming whenthree or more variables are involved.
SLICGGESTIONG END FLIDTHED DFLANDINGS RVI OV ' ce caw
vu
 
General
Clewlow, L., and C. Strickland, Implementing Derivatives Models, Wiley, Chichester, 1998.
On Tree ApproachesBoyle, P. P., “A Lattice Framework for Option Pricing with Two State Variables,” Journal of Financialand Quantitative Analysis, 23 (March 1988), 1-12.
Broadic, M., and P. Glasserman, “A Stochastic Mesh Method for Pricing High-Dimensional AmericanOptions,” Working Paper, Columbia University, 1997.
Cox, J., 8. Ross, and M. Rubinstein, “Option Pricing: A Simplified Approach,” Journal of FinancialEconomics, 7 (October 1979), 229-64.
Figlewski, S., and B. Gao, “The Adaptive Mesh Model: A New Approach to Efficient Option Pricing,”Journal of Financial Economics, 53 (1999), 313-51.
Hull, J. C., and A. White, “The Use of the Control Variate Technique in Option Pricing,” Journal ofFinancial and Quantitative Analysis, 23 (September, 1988), 237-51.
Rendleman, R., and B. Bartter, “Two State Option Pricing,” Journal of Finance, 34 (1979), 1092-1110.
On Monte Carlo Simulation
Boyle, P. P., “Options: A Monte Carlo Approach,” Journal of Financial Economics, 4 (1977), 323-38.Numerical Procedures 429
Broadie, M., and P. Glasserman, “Estimating Security Prices Using Simulation,” Management Science,42. no. 2 (February 1996), 269-85.
Broadie, M., P. Glasserman, and G. Jain, “Enhanced Monte Carlo Estimates for American OptionPrices,” Journal of Derivatives, 5 (Fall 1997), 25-44,
Brotherton-Ratcliffe, R., “Monte Carlo Motoring,” RISK, December 1994, pp. 53-58.Curran, M., “Strata Gems,” RISK, March 1994, pp. 70-71.
Moro, B., “The Full Monte,” RISK, February 1985, pp. 57-58.
Papageorgiou, A., and J. Traub, “Beating Monte Carlo,” RISK, June 1996, 65-65.
Paskov, S. H., “New Methodologies for Valuing Derivatives”, in Mathematics Derivative Securities,ed. M. A. H. Dempster and S. R. Pliska, Cambridge University Press, Cambridge, 1996.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art ofScientific Computing, 2nd edn., Cambridge University Press. Cambridge, 1992.
Sobol’, 1. M., USSR Computational Mathematics and Mathematical Physics, 7, no. 4 (1967), 86-112.
On Finite Difference Methods
Brennan, M. J., and E. 8. Schwartz, “Finite Difference Methods and Jump Processes Arising in thePricing of Contingent Claims: A Synthesis,’ Journal of Financial and Quantitative Analysis,13 (September 1978), 462-74.
Brennan, M. J., and E. $8. Schwartz, “The Valuation of American Put Options,” Journal of Finance,32 (May 1977), 449-62.
Courtadon, G., “A More Accurate Finite Difference Approximation for the Valuation of Options,”Journal of Financial and Quantitative Analysis, 17 (December 1982), 697-705.
Hull, J. C., and A. White, “Valuing Derivative Securities Using the Explicit Finite Difference Method,”Journal of Financial and Quantitative Analysis, 25 (March 1990), 87-100.
Schwartz, E. S., “The Valuation of Warrants: Implementing a New Approach,” Journal of FinancialEconomics, 4 (1977), 79-94.
Wilmott, P., Derivatives: The Theory and Practice of Financial Engineering, Wiley, Chichester, 1998.
On Analytic Approximations
Barone-Adesi, G., and R. E. Whaley, “Efficient Analytic Approximation of American Option Values,”Journal of Finance, 42 (June 1987), 301-20.
Carr, P., R. Jarrow, and R. Myneni, ‘Alternative Characterizations of American Put Options,”Mathematical Finance, 2 (1992), 87-106.
Geske, R., and H. E. Johnson, “The American Put Valued Analytically,” Journal of Finance,39 (December 1984), 1511-24.
Ho, T.-S., R. S. Stapleton, and M. G. Subrahmanyam, “The Valuation of American Options withStochastic Interest Rates: A Generalization of the Geske-Johnson Technique,”’ Journal of Finance,52, no. 2 June 1997), 827-40.
Johnson, H. E., ‘An Analytic Approximation to the American Put Price,” Journal of Financial andQuantitative Analysis, 18 (March 1983), 141-48.
MacMillan, L. W., “Analytic Approximation for the American Put Option,” Advances in Futures andOptions Research, 1 (1986), 119-39. 
 
430 CHAPTER 18QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)18.1. Which of the following can be estimated for an American option by constructing a single
18.2.
18.3.
18.4,
18.5.
—coon
18.8.
18.9.
18.10.
18.11.
18.12.
18.13.
18.14.
binomial tree: delta, gamma, vega, theta, rho?
Calculate the price of a three-month American put option on a non-dividend-paying stock whenthe stock price is $60, the strike price is $60, the risk-free interest rate is 10% per annum, and thevolatility is 45% per annum. Use a binomial tree with a time interval of one month.
Explain how the control variate technique is implemented when a tree is used to value Americanoptions.
Calculate the price of a nine-month American call option on corn futures when the currentfutures price is 198 cents, the strike price is 200 cents, the risk-free interest rate is 8% per annum,and the volatility is 30% per annum. Use a binomial tree with a time interval of three months.
Consider an option that pays off the amount by which the final stock price exceeds the averagestock price achieved during the life of the option. Can this be valued using the binomial treeapproach? Explain your answer.
“Por a dividend naving stock. the tree for the stock orice does not recombine: but the tree for thefor a diviagend paying stock, the tree tor the stock price does not recomoine; Dut the tree for the
stock price less the present value of future dividends does recombine.” Explain this statement.
. Show that the probabilities in a Cox, Ross, and Rubinstein binomial tree are negative when the
condition in footnote 7 holds.
How would you use the binomial tree approach to value an American option on a stock indexwhen the dividend yield on the index is a function of time?
Explain why the Monte Carlo simulation approach cannot easily be used for American-stylederivatives.
A nine-month American put option on a non-dividend-paying stock has a strike price of $49.The stock price is $50, the risk-free rate is 5% per annum, and the volatility is 30% per annum.Use a three-step binomial tree to calculate the option price.
Use a three-time-step tree to value a nine-month American call option on wheat futures. Thecurrent futures price is 400 cents, the strike price is 420 cents, the risk-free rate is 6%, and thevolatility is 35% per annum. Estimate the delta of the option from your tree.
A three-month American call option on a stock has a strike price of $20. The stock price is $20,the risk-free rate is 3% per annum, and the volatility is 25% per annum. A dividend of $2 isexpected in 1.5 months. Use a three-step binomial tree to calculate the option price.
A one-year American put option on a non-dividend-paying stock has an exercise price of $18.The current stock price is $20, the risk-free interest rate is 15% per annum, and the volatility ofthe stock price is 40% per annum. Use the DerivaGem software with four 3-month time stepsto estimate the value of the option. Display the tree and verify that the option prices at thefinal and penultimate nodes are correct. Use DerivaGem to value the European version of theoption. Use the control variate technique to improve your estimate of the price of theAmerican option.
A two-month American put option on a stock index has an exercise price of 480. The currentlevel of the index is 484, the risk-free interest rate is 10% per annum, the dividend yield on theindex is 3% per annum, and the volatility of the index is 25% per annum. Divide the life of theoption into four half-month periods and use the tree approach to estimate the value of theoption.Numerical Procedures 431
 
18.15.
18.16.
18.17.
18.18.
18.19.
18.20.
18.21.
18.22.18.23.
18.24.
How can the control variate approach improve the estimate of the delta of an American optionwhen the tree approach is used?
Suppose that Monte Carlo simulation is being used to evaluate a European call option on a non-dividend-paying stock when the volatility is stochastic. How could the control variate andantithetic variable technique be used to improve numerical efficiency? Explain why it is necessaryto calculate six values of the option in each simulation trial when both thc control variate andthe antithetic variable technique are used.
Explain how equations (18.25) to (18.28) change when the implicit finite difference method isbeing used to evaluate an American call option on a currency.
An American put option on a non-dividend-paying stock has four months to maturity. Theexercise price is $21, the stock price is $20, the risk-free rate of interest is 10% per annum, andthe volatility is 30% per annum. Use the explicit version of the finite difference approach to valuethe option. Use stock price intervais of $4 and time intervais of one month.
The spot price of copper is $0.60 per pound. Suppose that the futures prices (dollars per pound)are as follows:
3 months 0.59
6 months 0.579 months 0.54
12 months 0.50
The volatility of the price of copper is 40% per annum and the risk-free rate is 6% per annum.Use a binomial tree to value an American call option on copper with an exercise price of $0.60and a time to maturity of one year. Divide the life of the option into four 3-month periods forthe purposes of constructing the tree. (Hint: As explained in Section 13.7, the futures price of avariable is its expected future price in a risk-neutral world.)
Use the binomial tree in Problem 18.19 to value a security that pays off x” in one year where x isthe price of copper.
When do the boundary conditions for § = 0 and 5 > oo affect the estimates of derivative pricesin the explicit finite difference method?
How can finite difference methods be used when there are known dividends?
A company has issued a three-year convertible bond that has a face value of $25 and can beexchanged for two of the company’s shares at any time. The company can call the issue forcingconversion when the share price is greater than or equal to $18. Assuming that the company willforce conversion at the earliest opportunity, what are the boundary conditions for the price ofthe convertible? Describe how you would use finite difference methods to value the convertibleassuming constant interest rates. Assume there is no risk of the company defaulting.
Provide formulas that can be used for obtaining three random samples from standard normaldistributions when the correlation between sample i and sample j is 9; ;.
ASSIGNMENT QUESTIONS
 
 
. An American put option to sell a Swiss franc for dollars has a strike price of $0.80 and a time to
maturity of one year. The volatility of the Swiss franc is 10%, the dollar interest rate is 6%, theSwiss franc interest rate is 3%, and the current exchange rate is 0.81. Use a three-time-step treeto value the option. Estimate the delta of the option from your tree.432
CHAPTER 18
 
18.26. A one-year American call option on silver futures has an exercise price of $9.00. The current
18.27.
18.29.
—_wtaoO
futures price is $8.50, the risk-free rate of interest is 12% per annum, and the volatility of thefutures price is 25% per annum. Use the DerivaGem software with four 3-month time steps toestimate the value of the option. Display the tree and verify that the option prices at the final andpenultimate nodes are correct. Use DerivaGem to value the European version of the option. Usethe control variate technique to improve your estimate of the price of the American option.
A six-month American call option on a stock is expected to pay dividends of $1 per share at theend of the second month and the fifth month. The current stock price is $30, the exercise price is$34, the risk-free interest rate is 10% per annum, and the volatility of the part of the stock pricethat will not be used to pay the dividends is 30% per annum. Use the DerivaGem software withthe life of the option divided into six time steps to estimate the value of the option. Compareyour answer with that given by Black’s approximation (see Section 12.12).
. The current value of the British pound is $1.60 and the volatility of the pound—dollar exchange
rate is 15% per annum. An American call option has an exercise price of $1.62 and a time tomaturity of one year. The risk-free rates of interest in the United States and the United Kingdomare 6% per annum and 9% per annum, respectively. Use the explicit finite difference method tovalue the option. Consider exchange rates at intervals of 0.20 between 0.80 and 2.40 and timeintervals of 3 months.
Answer the following questions concerned with the alternative procedures for constructing treesin Section 18.5.
a. Show that the binomial model in Section 18.5 is exactly consistent with the mean andvariance of the change in the logarithm of the stock price in time é¢.
b. Show that the trinomial model in Section 18.5 is consistent with the mean and variance ofthe change in the logarithm of the stock price in time 6t when terms of order (é¢) andhigher are ignored.
c. Construct an alternative to the trinomial model in Section 18.5 so that the probabilities are1/6, 2/3, and 1/6 on the upper, middle, and lower branches emanating from each node.Assume that the branching is from § to Su, Sm, or Sd, with m? = ud. Match the mean andvariance of the change in the logarithm of the stock price exactly.
The DerivaGem Application Buider functions enable you to investigate how the prices ofoptions calculated from a binomial tree converge to the correct value as the number of time stepsincreases (see Figure 18.4 and Sample Application A in DerivaGem). Consider a put option on astock index where the index level is 900, the strike price is 900, the risk-free rate is 5%, thedividend yield is 2%, and the time to maturity is 2 yearsa. Produce results similar to Sample Application A on convergence for the situation where theoption is European and the volatility of the index is 20%.b. Produce results similar to Sample Application A on convergence for the situation where theoption is American and the volatility of the index is 20%.c. Produce a chart showing the pricing of the American option when the volatility is 20% asafunction of the number of time steps when the control variate technique is used.d. Suppose that the price of the American option in the market is 85.0. Produce a chartshowing the implied volatility estimate as a function of the number of time steps.Numerical Procedures 433
APPENDIX 18A
Analytic Approximation to American Option Prices of MacMillan andof Barone-Adesi and Whaley
Consider an option on a stock providing a dividend yield equal to g. We will denote the differencebetween the American and European option price by v. Because both the American and theEuropean option prices satisfy the Black-Scholes differential equation, v also does so. Hence,
dv av 19252 ap _ap PY DSagt 20 S gerFor convenience, we define
t=T-t A(t)=l-e", aa, 8B
We also write, without loss of generality,v = h(t) gS, A)With appropriate substitutions and variable changes, this gives
xsr 4 ps
0g @a agaa (1 — how = 0
ash?
The approximation involves assuming that the final term on the left-hand side is zero, so that
2.$74 + psbA y= 0 (18A.1)The ignored term is generally fairly small. When 7 is large, 1 — A is close to zero; when 7 is small,ag/dh is close to zero.
The American call and put prices at time ¢ will be denoted by C(S, ) and P(S, t), where S is thestock price, and the corresponding European call and put prices will be denoted by c(S, 1)and p(S, t). Equation (18A.1) can be solved using standard techniques. After boundary conditionshave been applied, it is found that
CS,1) = (os t) + Ax(S/S*)? if S < S*| S-K if S> S*
The variable S* is the critical price of the stock above which the option should be exercised. It isestimated by solving the equation
S*-Kac(S*, n+ [l-e@ON(d, sy]2
iteratively. For a put option, the valuation formula is
P(S. A) +A,(S/S")" if S> s*
P(S,O=(5) [eee ifs<s™434 CHAPTER 18
The variable §** is the critical price of the stock below which the option should be exercised. It isestimated by solving the equation
+K-S*= p(s, t)- [1 _ et? ON (- dy (s*))] —
iteratively. The other variables that have been used here are as follows:
n=4(-0--fe- 0 +=)n= (-B- D+ ev)
A, = -(—)p — eT N(-d(S**))]1
Ay = Ce — eT FO N(d(5*))]
2In($/K) + —¢ +07 /2\T — 1)
d(S)=1(5) CRAs Pointed out in Chapter 13, options on stock indices, currencies, and futures contracts are
analogous aptiong aro constant dividend vield ence the adratic
analogous to options on a stock providing a constant diviaena yicia. Hlence the quaaratic
approximation approach can easily be applied to all of these types of options.CHAPTER 19
EXOTIC OPTIONS |
 
The derivatives we have covered in the first 18 chapters of this book are what are termed plainvanilla products. They have standard well-defined properties and trade actively. Their prices orimplied volatilities are quoted by exchanges or by brokers on a regular basis. One of the excitingaspects of the over-the-counter derivatives market is the number of nonstandard (or exotic)products that have been created by financial engineers. Although they are usually a relativelysmall part of its portfolio, these exotic products are important to an investment bank because theyare generally much more profitable than plain vanilla products. :
Exotic products come about for a number of reasons. Sometimes they meet a genuine hedgingneed in the market; sometimes there are tax, accounting, legal, or regulatory reasons why corporatetreasurers find exotic products attractive; sometimes the products are designed to reflect a corporatetreasurer’s view on. potential future movements in particular market variables; occasionally anexotic product is designed by an investment bank to appear more attractive than it is to an unwarycorporate treasurer.
In this chapter we describe different types of exotic options and discuss their valuation. We use acategorization of exotic options similar to that in an excellent series of articles written by EricReiner and Mark Rubinstein for RISK magazine in 1991 and 1992. We assume that the assetprovides a yield at rate g. As discussed in Chapter 13, for an option on a stock index we set g equalto the dividend yield on the index, for an option on a currency we set q equal to the foreign risk-freerate, and for an option on a futures contract we set g equal to the domestic risk-free rate. Most ofthe options discussed in this chapter can he valued using the DerivaGem software
19.1 PAGKAGES
A package i$ a portfolio consisting of standard European calls, standard European puts, forwardcontracts, cash, and the underlying asset itself. We discussed a number of different types of packagesin Chapter 9: bull spreads, bear spreads, butterfly spreads, calendar spreads, straddles, strangles, etc.
Often a package is structured by traders so that it has zero cost initially. An example is a rangeforward contract.' A short-range forward contract consists of a long position in a put with a lowstrike price, K,, and a short position in a call with a high strike price, K>. It guarantees that theunderlying asset can be sold for a price between K, and K, at the maturity of the options. Along-range forward contract consists of a short position in a put with the low strike price, Ky.
' Other names used for a range forward contract are zero-cost collar, flexible forward, cylinder option, option fence,min-max, and forward band.
435436 CHAPTER 19
 
(a) (b)
Figure 19.1 Payoffs from (a) a short position and (b) a long position in a range forward contract
and a long position in a call with the high strike price, Ky. It guarantees that the underlying assetcan be purchased for a price between K, and K, at the maturity of the options. The price of thecall equals the price of the put when the contract is initiated. Figure 19.1 shows the payoff fromshort- and long-range forward contracts. As K, and K, are moved closer to each other, the pricethat will be received or paid for the asset at maturity becomes more certain. In the limit whenK, = K», the range forward contract becomes a regular forward contract.
It is worth noting that any derivative can be converted into a zero-cost product by deferringpayment until maturity. Consider a European call option. If c is the cost of the option whenpayment is made at time zero, then A =ce’” is the cost when payment is made at time T, thematurity of the option. The payoff is then max(S; — K, 0) — A, or max(S; — K — A, —A). Whenthe strike price, K, equals the forward price, other names for a deferred payment option are breakforward, Boston option, forward with optional exit, and cancelable forward.
19.2 NONSTANDARD AMERICAN OPTIONS
 
In a standard American option, exercise can take place at any time during the life of the optionand the exercise price is always the same. In practice, the American options that are traded in theover-the-counter market sometimes have nonstandard features. For example:
1. Early exercise may be restricted to certain dates. This instrument is called a Bermudan option.
2. Early exercise may be allowed during only part of the life of the option. For example, theremay be an initial “lock out” period with no early exercise.
3. The strike price may change during the life of the option.
The warrants issued by corporations on their own stock often have some of these features, Forexample, in a seven-year warrant, exercise might be possible on particular dates during years 3 to 7,with the strike price being $30 during years 3 and 4, $32 during the next two years, and $33 duringthe final year.
Nonstandard American options can usually be valued using a binomial tree. At each node, thetest (if any) for early exercise is adjusted to reflect the terms of the option.~ Exotic Options 437
 
19.3 FORWARD START OPTIONS
 
 
Forward start options are options that will start at some time in the future. They are sometimesused in employee incentive schemes, The terms of the options usually specify that they will be atthe money at the time they start.
Consider a forward start at-the-money European call option that will start at time 7, and matureat time T,. Suppose that the asset price is Sp at time zero and S, at time 7;. To value the option, wenote from the European option pricing formulas in Chapters 12 and 13 that the value of an at-the-money call option is proportional to the asset price. The value of the forward start option attime T, is therefore cS,/So, where c is the value at time zero of an at-the-money option that lastsfor T, — T,. Using risk-neutral valuation, the value of the forward start option at time zero is
where E denotes the expected value in a risk-neutral world. Because c and Sp are known andE(S,) = Spe?" it follows that the value of the forward start option is ce~7”!, For a non-dividend-paying stock, g = 0 and the value of the forward start option is exactly the same as thevalue of a regular at-the-money option with the same life as the forward start option.
19.4 COMPOUND OPTIONS
 
 
Compound options are options on options. There are four main types of compound option: acall on a call, a put on a call, a call on a put, and a put on a put. Compound options have twostrike prices and two exercise dates. Consider, for example, a call on a call. On the first exercisedate, T;, the holder of the compound option is entitled to pay the first strike price, K,, andreceive a call option. The call option gives the holder the right to buy the underlying asset for thesecond strike price, K>, on the second exercise date, T,. The compound option will be exercisedon the first exercise date only if the value of the option on that date is greater than the first strikeprice.
When the usual geometric Brownian motion assumption is made, European-style compoundoptions can be valued analytically in terms of integrals of the bivariate normal distribution.” Withour usual notation, the value at time zero of a European call option on a call option is
Spe 4? M(a,. bys VT, /T;) — Kp? M(ag, by; /T)/Tz) — 6 K N(a)wherein(So/S") + (rg + /2)NiQos a =a, —-0VT,o/T,In(S/K2) +r —q +.0°/2)TpoVT;
2 See R. Geske, “The Valuation of Compound Options,” Journal of Financial Economics, 7 (1979), 63-81;M. Rubinstein, “Double Trouble,” R/JSK, December 1991 / January 1992, pp. 53-56.
b, by = b — oT)438 CHAPTER 19
The function M is the cumulative bivariate normal distribution function as defined inAppendix 12B. The variable S* is the asset price at time 7, for which the option price attime T, equals Ky. If the actual asset price is above S* at time 7;, the first option will be exercised;if it is not above $*, the option expires worthless.
With similar notation, the value of a European put on a call is
Kye” M(—ay, by; —/T,/T,) — Sge 4? M(—a,. by; —/T,/T2) + eK N(—as)12, B25 al Ti / Tn) — So (44.845 i/ 42) 4 iN{-ay)}The value of a European cail on a put is /
e? M(—ay, —by; JT; /Tz) — Spe 1? M(—ay, —by; VT) /T2) — eK, N(fan)
The value of a European put on a put is
Spe 4”? M(ay, —by; — JT) /T2) — Kre ? M(ar, —by; —/T,/T,) + 6" K Naa)
A procedure for computing M is provided in Appendix 12C.
19.56 CHOOSER OPTIONS
 
A chooser option (sometimes referred to as an as you like it option) has the feature that, after aspecified period of time, the holder can choose whether the option is a call or a put. Suppose thatthe time when the choice is made is T,. The value of the chooser option at this time is
max(c, p)
where c is the value of the cail underlying the option and p is the value of the put underlying theoption.
If the options underlying the chooser option are both European and have the same strike price,put-call parity can be used to provide a valuation formula. Suppose that S$, is the asset price attime T,, K is the strike price, 75 is the maturity of the options, and r is the risk-free interest rate.Put-cail parity implies that
max(c, p) = max(c, c+ Ke“@-™ — §, e427)
=cet+e 4h) max(0, Ket ORT) _ S,)
This shows that the chooser option is a package consisting of the following:
1. A call option with strike price K and maturity T>
2. eo ARN) 1)
put options with strike price Ke’-%™?-™ and maturity T;
As such, it can readily be valued.
More complex chooser options can be defined where the call and the put do not have the samestrike price and time to maturity. They are then not packages and have features that are somewhatsimilar to compound options.