Pricing American Options under Stochastic Volatility and Jump Diffusion Dynamics

This paper considers the problem of pricing American options when the dynamics of the underlying are driven both by stochastic volatility following a square root process as used by Heston (1993) and by a Poisson jump process as introduced by Merton (1976). The two-factor homogeneous integro-partial differential equation for the price and early exercise surface is cast into an in-homogeneous form accord- ing to the approach introduced by Jamshidian (1992). The Fourier transform is then applied to find the solution, which generalizes in an obvious way the structure of the solution to the corresponding European option pricing problem in the case of a call option and constant interest rates obtained by Scott (1997). The price is given by an integral equation dependent upon the early exercise surface, for which a correspond- ing integral equation is obtained. An algorithm is proposed for solving the integral equation system. The method is implemented, and the resulting prices and deltas are compared with the constant volatility model. The computational efficiency of the nu- merical integration scheme is also considered by comparing with benchmark solutions obtained by a finite difference method and the method of lines applied directly to the integro-partial differential equation