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to create their own insurance by dynamically rebalancing a portfolio consisting of stocks and risk-free bonds. The weights in the portfolio show that as stock prices rise, funds are transferred from bonds to stocks, and vice versa. Compound options. An important extension of the Black-Scholes model that falls in the single underlying asset category is the compound option valuation theory developed by Geske.46 Compound options are options on options. A call on a call, for example, provides its holder with the right to buy a call on the underlying asset at some future date. Geske shows that if these options are European-style, valuation formulas can be derived. American-style call options on dividend-paying stocks. The Geske compound option model has been applied in other contexts. Roll, Geske, and Whaley developed a formula for valuing an American-style call option on a stock with known discrete dividends.47 If a stock pays a cash dividend during the call s life, it may be optimal to exercise the call early, just prior to dividend payment. An American-style call on a dividend-paying stock, therefore, can be modeled as a compound option providing its holder with the right, on the ex-dividend date, either to exercise early and collect the dividend, or to leave the position open. Chooser options. Rubinstein used the compound option framework in 1991 to value the chooser or as-you-like-it options traded in the over-the-counter market.48 The holder of a chooser option has the right to decide at some future date whether the option is a call or a put. The call and the put usually have the same exercise price and the same time remaining to expiration. Bear market warrants with a periodic reset. Gray and Whaley used the compound option framework to value yet another type of contingent claim, S&P 500 bear market warrants with a periodic reset traded on the Chicago Board Options Exchange and the New York Stock Exchange.49 The warrants are originally issued as at-the-money put options but have the distinguishing feature that if the underlying index level is above the original exercise on some prespecified future date, the exercise price of the warrant is reset at the then-prevailing index level. These warrants offer an intriguing form of portfolio insurance whose floor value adjusts automatically as the index level rises. The structure of the valuation problem is again a compound option, and Gray and Whaley s 1997 paper provides the valuation formula. Lookback options. A lookback option is another exotic that has only one underlying source of price uncertainty. Such an option s
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exercise price is determined at the end of its life. For a call, the exercise price is set equal to the lowest price that the asset reached during the life of the option; for a put, the exercise price equals the highest asset price. These buy-at-the-low and sell-atthe-high options can be valued analytically. Formulas are provided in Goldman, Sosin, and Gatto s 1979 paper.50 Barrier options. Barrier options are options that either cease to exist or come into existence when some predefined asset price barrier is hit during the option s life. A down-and-out call, for example, is a call that gets knocked out when the asset price falls to some prespecified level prior to the option s expiration. Rubinstein and Reiner s 1991 paper provides valuation equations for a large family of barrier options.51
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2.4.5.2 Approximation Methods Many valuation problems do not have explicit closed-form solutions. Probably the best-known example of this is the valuation of Americanstyle options. With American-style options, the option holder has an infinite number of exercise opportunities between the current date and the option s expiration date, making the problem challenging from a mathematical standpoint. Hundreds of different types of exotic options trade in the OTC market, and many, if not most, do not have analytical formulas. Nonetheless, they can all be valued accurately using the Black-Scholes model. If a risk-free hedge can be formed between the option and the underlying asset, the Black-Scholes model risk-neutral valuation theory can be applied, albeit using numerical methods. A number of numerical methods for valuing options are lattice based. These methods replace the Black-Scholes model assumption that asset price moves smoothly and continuously through time with an assumption that the asset price moves in discrete jumps over discrete intervals during the option s life:
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Binomial method. Perhaps the best-known lattice-based method is the binomial method, developed independently in 1979 by Cox, Ross, and Rubinstein and Rendleman and Bartter.52 In the binomial method, the asset price jumps up or down, by a fixed proportion, at each of a number of discrete time steps during the option s life. The length of each time step is determined when the user specifies the number of time steps. The greater the number of time steps, the more precise the method. The cost of the increased precision, however, is computational speed. With n time steps, 2n asset price paths over the life of the option are considered. With 20 time steps, this means more than 1 million paths. The binomial method has wide applicability. Aside from the American-style option feature, which is easily incorporated
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