Jump-diffusion and Levy models have been widely used to partially alleviate some of the biases inherent in pure diffusion models. 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. This talk introduces a generic framework based on the Fourier transform for pricing and hedging of derivatives in equity, commodity, currency, and interest rate markets with jump-driven asset models. The resulting methods are efficient, applicable to a wide range of problems and naturally extend to multiple dimensions.

The Fourier transform can be applied to the pricing PIDE to obtain a linear system of ordinary differential equations that can be solved explicitly. Solving the PIDE in Fourier space allows for the integral term to be handled efficiently and avoids the asymmetrical treatment of diffusion and integral terms, common in the finite difference schemes found in the literature. For European options, prices can be obtained for a range of stock prices in one iteration of the algorithm. For exotic, path-dependent options, a time-stepping methodology is developed to handle barriers, free boundaries, and exercise policies. This talk discusses various applications of the methods and establishes their precision and convergence properties through numerical examples.

The talk is based on joint work with Kenneth R. Jackson and Sebastian Jaimungal from the University of Toronto.

Here are the slides from the talk.

Vladimir Surkov is currently a postdoctoral fellow at the Department of Applied Mathematics at the University of Western Ontario and the Fields Institute at the University of Toronto. Vladimir completed his PhD at the Department of Computer Science at the University of Toronto.