American Put Option Pricing for Stochastic-Volatility, Jump-Diffusion Models

The numerical treatment for the American put option pricing is discussed for a stochastic-volatility, jump- diffusion (SVJD) model with log-uniform jump amplitudes. Heston's (1993) mean reverting, square-root stochastic volatility model is used along with our uniform jump-amplitude model. However, computation is needed for nonlinear and multidimensional terms with dependence on the combined stock and volatility state space. A systematic finite difference formulation of the American put partial integro-differential complementary problem (PIDCP) is implemented using a successive over-relaxtion method projected on the maximum payoff function. Interpolation is used to construct the smooth transition to the payoff of the corresponding free boundary problem. Also, a fast, but less accurate, heuristic quadratic approximation, originally due to MacMillan (1986), is corrected and extended from pure diffusion models. The fast and simple quadratic approximation is compared with a more accurate PIDCP formulation. The simple quadratic approximation is briefly compared with market data.