Pricing American Options under Stochastic Volatility

This paper provides an extension of McKean’s (1965) incomplete Fourier transform method to solve the two-factor partial differential equation for the price and early exercise surface of an American call option, in the case where the volatility of the underlying evolves randomly. The Heston (1993) square-root process is used for the volatility dynamics. The price is given by an integral equation dependent upon the early exercise surface, using a free boundary approximation that is linear in volatility. By evaluating the pricing equation along the free surface boundary, we provide a corresponding integral equation for the early exercise region. An algorithm is proposed for solving the integral equation system, based upon numerical integration techniques for Volterra integral equations. The method is implemented, and the resulting prices are compared with the constant volatility model. The computational efficiency of the numerical integration scheme is also considered