Fast Simulation of Levy Processes

Abstract

We present a robust method for simulating an increment of a Levy process, based on decomposing the jump part of the process into the sum of its positive and negative jump components. The characteristic exponent of a spectrally one-sided Levy process has excellent analytic properties, which we exploit to design a fast and accurate algorithm for calculating the cumulative distribution function of an increment of such a process. That algorithm is based on the parabolic inverse Fourier transform method introduced by S. Boyarchenko and S. Levendorskii, while the method of simulating a random variable using the values of its cumulative distribution function goes back to the work of P. Glasserman and Z. Liu.

C code based on the simulation algorithm described in this article is available on the author's website for Levy processes of class KoBoL (a.k.a. the CGMY model). It can be used as a building block of any Monte-Carlo program for pricing derivative securities under a KoBoL process. Our method typically performs faster than the CGMY simulation method introduced by Madan and Yor by a factor of 10--100, and sometimes even higher, depending on the type of the option.

Keywords: Monte-Carlo simulation, levy processes, KoBoL process, CGMY model, parabolic inverse Fourier transform, path-dependent option pricing

JEL Classification: C63, G13