Pricing American Options in an Infinite Activity Lévy Market: Monte Carlo and Deterministic Approaches Using a Diffusion Approximation

Abstract

Computational methods for pricing exotic options when the underlying is driven by a Lévy process are prone to numerical inaccuracy when the driving price process has infinite activity. Such inaccuracies are particularly severe for pricing of American options. In this chapter, we examine the impact of utilizing a diffusion approximation to the contribution of the small jumps in the infinite activity process. We compare the use of deterministic and stochastic (Monte Carlo) methods, and focus on designing strategies tailored to the specific difficulties of pricing American options. We demonstrate that although the implementation of Monte Carlo pricing methods for common Lévy models is reasonably straightforward, and yields estimators with relatively small bias, deterministic methods for exact pricing are equally successful but can be implemented with rather lower computational overhead. Although the generality of Monte Carlo pricing methods may still be an attraction, it seems that for models commonly used in the literature, deterministic numerical approaches are competitive alternatives.