Simulating Lévy Processes from Their Characteristic Functions and Financial Applications

Abstract

The simulation of a discrete sample path of a Lévy process reduces to simulating from the distribution of a Lévy increment. For a general Lévy process with exponential moments, the inverse transform method proposed in Glasserman and Liu [2010] 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 article, 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 Lévy 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 Lévy processes and can be applied to simulating from tabulated cumulative distribution functions in general and characteristic functions in our analytic class in particular.

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