Fourier space time-stepping for option pricing with Lévy models

ABSTRACT Jump-diffusion and Lévy models have been widely used to partially alleviate some of the biases inherent in the classical Black–Scholes–Merton model. Unfortunately, the resulting pricing problem requires solving a more difficult partial integro-differential equation (PIDE), and although several approaches for solving the PIDE have been suggested in the literature, none are entirely satisfactory. We present an efficient algorithm, based on transform methods, which symmetrically treats the diffusive and integral terms, is applicable to a wide class of path-dependent options (such as Bermudan, American and barrier options) and options on multiple assets, and naturally extends to regime-switching Lévy models. Furthermore, we introduce a penalty method to improve the convergence of pricing American options.