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work of MacMillan, separates the value of an American-style option into two components: the European-style option value and an early exercise premium.59 Because the Black-Scholes model formula provides the value of the European-style option, they focus on approximating the value of the early exercise premium. By imposing a subtle change on the Black-Scholes model partial differential equation, they obtain an analytical expression for the early exercise premium, which they then add to the Europeanstyle option value, thereby providing an approximation of the American-style option value. The advantages of the quadratic approximation method are speed and accuracy. 2.4.5.3 Generalizations The generalizations of the Black-Scholes option valuation theory focus on the assumed asset price dynamics. Some examine the valuation implications of modeling the local volatility rate as a deterministic function of the asset price or time or both. Others examine the valuation implications when volatility, like asset price, is stochastic. Under the assumption that the local volatility rate is a deterministic function of time or the asset price or both, the Black-Scholes model riskfree hedge mechanisms are preserved, so risk-neutral valuation remains possible. The simplest in this class of models is the case in which the local volatility rate is a deterministic function of time. For this case, Merton showed that the valuation equation for a European-style call option is the Black-Scholes model formula, where the volatility parameter is the average local volatility rate over the life of the option.60 Other models focus on the relationship between asset price and volatility and attempt to account for the empirical fact that, in at least some markets, volatility varies inversely with the level of asset price. One such model is the constant elasticity of variance model proposed by Cox and Ross.61 However, valuation can be handled straightforwardly using lattice-based or Monte Carlo simulation procedures. Derman and Kani, Dupire, and Rubinstein and Reiner recently developed a valuation framework in which the local volatility rate is a deterministic (but unspecified) function of asset price and time.62 If the specification of the volatility function is known, any of the lattice-based or simulation procedures can be applied to option valuation. Unfortunately, the structural form is not known. To circumvent this problem, these authors parameterize their model by searching for a binomial or trinomial lattice that achieves an exact cross-sectional fit of reported option prices. An exact cross-sectional fit is always possible, because there are as many degrees of freedom in defining the lattice (and, hence, the local volatilityrate function) as there are option prices. With the structure of the implied
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tree identified, it becomes possible to value other, more exotic, OTC options and to refine hedge ratio computations. The effects of stochastic volatility on option valuation are modeled by either superimposing jumps on the asset price process or allowing volatility to have its own diffusion process or both. Unfortunately, the introduction of stochastic volatility negates the Black-Scholes model riskfree hedge argument, because volatility movements cannot be hedged. An exception to this rule is provided by Merton, who adds a jump term to the usual geometric Brownian motion governing asset price dynamics.63 By assuming that the jump component of an asset s return is unsystematic, the Merton model can create a risk-free portfolio in the Black-Scholes model sense and apply risk-neutral valuation. Indeed, Merton finds analytical valuation formulas for European-style options. If the jump risk is systematic, however, the Black-Scholes model risk-free hedge cannot be formed, and option valuation will be utility-dependent. A number of authors model asset price and asset price volatility as separate, but correlated, diffusion processes. Asset price is usually assumed to follow geometric Brownian motion. The assumptions governing volatility vary. Hull and White, for example, assume that volatility follows geometric Brownian motion.64 Scott models volatility using a meanreverting process, and Wiggins uses a general Wiener process.65 Bates combines both jump and volatility diffusions in valuing foreign currency options. Except in the uninteresting case in which asset price and volatility movements are independent, these models require the estimation of risk premiums.66 The problem when volatility is stochastic is that a riskfree hedge cannot be created, because volatility is not a traded asset. But perhaps this problem is only temporary. The critical issue for all options, of course, is correct contract design.
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