The representation of American options prices under stochastic volatility and jump-diffusion dynamics

This paper considers the problem of pricing American options when the dynamics of the underlying are driven by both stochastic volatility following a square-root process as used by Heston [<italic>Rev. Financial Stud.</italic>, 1993, <bold>6</bold>, 327--343], and by a Poisson jump process as introduced by Merton [<italic>J. Financial Econ.</italic>, 1976, <bold>3</bold>, 125--144]. Probability arguments are invoked to find a representation of the solution in terms of expectations over the joint distribution of the underlying process. A combination of Fourier transform in the log stock price and Laplace transform in the volatility is then applied to find the transition probability density function of the underlying process. It turns out that the price is given by an integral dependent upon the early exercise surface, for which a corresponding integral equation is obtained. The solution generalizes in an intuitive way the structure of the solution to the corresponding European option pricing problem obtained by Scott [<italic>Math. Finance</italic>, 1997, <bold>7</bold>(4), 413--426], but here in the case of a call option and constant interest rates.