Transition Densities of Diffusion Processes

Abstract

Trying to build option models with price dynamics that better match empirical price behavior, we run into the problem that only a few returns processes, like the standard Black-Scholes lognormal diffusion, lead to closed form solutions for the transition densities. Generally these must be approximated numerically, using one of a variety of approaches. The Euler approximation is probably the most common technique of discretizing the process, but others are also in use, including the Milshtein scheme. Monte Carlo simulation is another approach. Simulation methods may make use of (pseudo-) random numbers, or deterministic quasi-random sequences that may be more efficient. Alternatives include the binomial model, Hermite expansions, and more. In this article, Jensen and Poulsen examine the comparative performance of a large number of approximation techniques in terms of accuracy and execution time. The winner in their tests, by a surprisingly large margin, is the Hermite expansion approach.