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Most financial time series are subject to stochastic changes in volatility over time. Time-varying volatility has become an important and indisputable empirical fact. The ARCH family of models first proposed by Engle (1982) has increasingly gained prominence as the leading model for describing time-varying volatility. Its popularity is evident by referring to the comprehensive review of the earlier ARCH literature in Bollerslev et al. (1992). Stochastic asset volatility has, among others, an important implication on the valuation of derivative securities. Duan (1995) has developed an option pricing model in which the underlying asset follows a generalized ARCH (or GARCH) process of Bollerslev (1986). This model has, so far, experienced some empirical successes. 1 The purpose of this article is to devise a new numerical method for pricing European and American options, particularly in the GARCH framework. The basic idea involves the use of a finite-state, time-homogeneous Markov chain as the approximation device.2
Until very recently, Monte Carlo simulation was the only numerical method for option valuation in the GARCH framework. Unfortunately, the Monte Carlo method is inherently ill-suited for dealing with early exercise decisions required for pricing American options. Recently, three Monte Carlo simulation methods have been proposed by Tilley (1993), Barraquand and Martineau (1995) and Broadie et al. (1997) for pricing American options. These procedures are numerically feasible for the simpler pricing framework like the Black–Scholes model (1973) if the number of early exercise possibilities are restricted.
Geske and Johnson's (1984) approximation scheme is an alternative way of numerically assessing American option values in the Black–Scholes framework. In order to apply their idea in the GARCH setting, one must be able to evaluate analytically, for any intermediate time points, the maximum of the immediate exercise value and the option value if it is kept alive. This, however, poses a formidable challenge in the GARCH setting, even if the analytical formula for European options is available. Because the value of an option at any future time point (except the expiration date) is a function of both the asset price and its conditional volatility, its present value requires assessing the integral of the maximum of the live value of the option and its premature exercise value. The evaluation of this integral requires an analytical formula for the joint distribution of the asset price and its conditional volatility over an arbitrary time interval. Such an analytical formula is, unfortunately, not yet known at this moment.
A numerical method that can handle the case of American options more efficiently than Monte Carlo simulation is the binomial tree method. Although the method works well in the constant volatility case, its generalization to time-varying volatility is not a simple task. Ritchken and Trevor (1999) have proposed a lattice scheme that is efficient for option pricing in the GARCH and bivariate stochastic volatility frameworks. They accomplished the difficult task of designing a suitable lattice approximation scheme for the GARCH model by restricting the storage of conditional variance to the maximum and minimum values at every node of the discretized asset price. Since the maximum and minimum volatilities change with nodes, the partition for the conditional volatility cannot be fixed beforehand. Leisen (1997) developed a different numerical scheme specifically for the bivariate diffusion models. He constructed a two-dimensional grid for the price and volatility that is path-independent and recombining. The convergence to the bivariate diffusion model is accomplished by adjusting the transition probabilities. Leisen's (1997) method is not applicable to the GARCH model, however, because the discretized system directly converges to the bivariate diffusion model, but not the GARCH process. In contrast