Representation and Numerical Approximation of American Option Prices under Heston Stochastic Volatility Dynamics

In this paper we consider the evaluation of American call options on dividend paying stocks in the case where the underlying asset price evolves according to Heston’s (1993) stochastic volatility model. We solve the Kolmogorov partial differential equation associated with the driving stochastic processes using a combination of Fourier and Laplace transforms and so obtain the joint transition probability density function for the underlying processes. We then use Duhamel’s principle to obtain the expression for the American option price, which depends upon the unknown early exercise surface. By evaluating the pricing equation along the free surface boundary, we obtain the corresponding integral equation for the early exercise surface. An algorithm is proposed for solving the integral equation system, based upon numerical integration techniques for Volterra integral equations. The method is used to explore the impact of stochastic volatility on the price and free boundary of American call options.