1. Introduction
Theoretical research on option valuation tends to focus on pricing the plain-vanilla European-Style derivatives.
Duffie, Pan, and Singleton (2000) showed that such options can be priced by transform methods whenever
the state vector (which includes functions of asset prices, unobserved volatilities, etc.) follows a multivariate
Gaussian-Poisson affine jump-diffusion. As a result, for a wide class of pricing problems a general solution
method has been found.
Contrastingly, no universal and analytically attractive approach is yet available for the American-style
derivatives. Still, most traded equity and FX-rate derivatives are the American-style ones. Accurate and
efficient pricing of such options is of a significant practical value.
Stochastic volatility models have been proposed in the hope of remedying the strike-price biases in option
valuation by the Black-Scholes formula. A model due to Heston (1993) has received considerable attention
in the literature. Heston’s original method has been modified and simplified by other scholars to deliver a
very efficient formula for the European-style puts. To price the American-style derivatives in the two-state-
variable setting of the model, authoritative sources (for instance, Wilmott, 2000) strongly suggest using the
finite difference (FD) schemes. FD methods, in which the partial differential equation (p.d.e.) in the value
function of a derivative security is approximated and solved for the initial option price numerically, are very
popular among practitioners and in academia. Applications of FD schemes for the Heston dynamics are
available (e.g., Winkler, 2001).
A number of efficient non-FD methods to price the American-style options have been proposed in the
context of the Black-Scholes model. A technique due to Broadie and Detemple (1996) is a smoothed binomial
scheme. The MacMillan-Barone-Adesi-Whaley approach relies on decomposing the value of an American-
style derivative into the value of a corresponding European-style option and early exercise premium. The
premium follows the fundamental p.d.e., which can be approximated by the 2nd-order ordinary differential
equation that is solved analytically. The Geske-Johnson scheme (1984) exploits the fact that an American-
style option is the limit of a sequence of “Bermudan” derivatives. The latter ones can be priced recursively
according to a simple formula.
In this paper, I adapt the Geske-Johnson method to the dynamics of the Heston model. As an empirical
test of the numerical accuracy of this approach, I consider pricing of the American-style S&P 100 index
options (OEX).
The rest of the paper is organized as follows. In Section 2, I state the assumption of the model, which are
used in Section 3, to derive the p.d.e. in the value function of a derivative security. Section 4 proceeds at a slow
pace from an analytical solution for the joint characteristic function (ch.f.) of log-price and squared volatility