American option pricing under GARCH by a Markov chain approximation

We propose a numerical method for valuing American options in general and for the GARCH option pricing model in particular. The method is based on approximating the underlying asset price process by a finite-state, time-homogeneous Markov chain. Since the Markov transition probability matrix can be derived analytically, the price of an American option can be computed by simple matrix operations. The Markov transition probability matrix is typically sparse. The use of a sparse matrix representation can substantially increase the dimension of the Markov chain to obtain better numerical results. The Markov chain method works well for the GARCH option pricing framework, and it serves as an alternative to the existing numerical methods for the valuation of American options in other pricing settings. We provide a convergence proof for the Markov chain method and analyze its numerical performance for the Black-Scholes (1973) and GARCH option pricing models. (C) 2001 Elsevier Science BN. All rights reserved.