Simulating Levy Processes from Their Characteristic Functions and Financial Applications

ACM Transactions on Modeling and Computer Simulation, Forthcoming

33 Pages Posted: 11 Jan 2012

See all articles by Zisheng Chen

Zisheng Chen

Department of Industrial and Enterprise Systems Engineering

Liming Feng

University of Illinois at Urbana-Champaign - Department of Industrial and Enterprise Systems Engineering

Xiong Lin

University of Illinois at Urbana-Champaign, Department of Mathematics

Date Written: July 30, 2011

Abstract

The simulation of a discrete sample path of a Levy process reduces to simulating from the distribution of a Levy increment. For a general Levy process with exponential moments, the inverse transform method proposed in Glasserman and Liu 2010 [24] is reliable and efficient. The values of the cumulative distribution function (cdf) are computed by inverting the characteristic function and tabulated on a uniform grid. The inverse of the cumulative distribution function is obtained by linear interpolation. In this paper, we apply a Hilbert transform method for the characteristic function inversion. The Hilbert transform representation for the cdf can be discretized using a simple rule highly accurately. Most importantly, the error estimates admit explicit and computable expressions, which allow us to compute the cdf to any desired accuracy. We present an explicit bound for the estimation bias in terms of the range of the grid where probabilities are tabulated, the step size of the grid, and the approximation error for the probabilities. The bound can be computed from the characteristic function directly and allows one to determine the size and fineness of the grid and numerical parameters for evaluating the Hilbert transforms for any given bias tolerance level in one dimensional problems. For multidimensional problems, we present a procedure for selecting the grid and the numerical parameters that is shown to converge theoretically and works well practically. The inverse transform method is attractive not only for Levy processes that are otherwise not easy to simulate, but also for processes with special structures that could be simulated in different ways. The method is very fast and accurate when combined with quasi-Monte Carlo schemes and variance reduction techniques. The main results we derived are not limited to Levy processes and can be applied to simulating from tabulated cumulative distribution functions in general and characteristic functions in our analytic class in particular.

Keywords: levy process, randomized quasi-Monte Carlo method, inverse transform method, Hilbert transform, analytic characteristic function, discrete Asian option, control variates

Suggested Citation

Chen, Zisheng and Feng, Liming and Lin, Xiong, Simulating Levy Processes from Their Characteristic Functions and Financial Applications (July 30, 2011). ACM Transactions on Modeling and Computer Simulation, Forthcoming, Available at SSRN: https://ssrn.com/abstract=1983134

Zisheng Chen

Department of Industrial and Enterprise Systems Engineering ( email )

IL
United States

Liming Feng (Contact Author)

University of Illinois at Urbana-Champaign - Department of Industrial and Enterprise Systems Engineering ( email )

104 S. Mathews Avenue
Urbana, IL 61801
United States

Xiong Lin

University of Illinois at Urbana-Champaign, Department of Mathematics ( email )

104 S. Mathews Avenue
Urbana, IL 61801
United States

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