Importance Sampling for Jump–Diffusions Via Cross-Entropy

Abstract

This paper develops efficient importance sampling schemes for a class of jump–diffusion processes that are commonly used for modeling stock prices. For such financial models, related option pricing problems are often difficult, especially when the option under study is out-of-the-money and there are multiple underlying assets. Even though analytical pricing formulas do exist in a few very simple cases, often analysts must resort to numerical methods or Monte Carlo simulation. We demonstrate that efficient and easy-to-implement importance sampling schemes can be constructed via the method of cross-entropy combined with the expectation–maximization algorithm, when the alternative sampling distributions are chosen from the family of exponentially tilted distributions or their mixtures. Theoretical justification is given by characterizing the limiting behavior of the cross-entropy algorithm under appropriate scaling. Numerical experiments on vanilla options, path-dependent options and rainbow options are also performed to illustrate the use of this technology.