Importance sampling in the heath-jarrow-morton framework

Abstract

This article develops a variance-reduction technique for pricing derivatives by simulation in highdimensional multifactor models. A premise of this work is that the greatest gains in simulation efficiency come from taking advantage of the structure of both the cash flows of a security and the model in which it is priced. For this to be feasible in practice requires automating the identification and use of relevant structure. We exploit model and payoff structure through a combination of importance sampling and stratified sampling. The importance sampling applies a change of drift to the underlying factors; we select the drift by first solving an optimization problem. We then identify a particularly effective direction for stratified sampling (which may be thought of as an approximate numerical integration) by solving an eigenvector problem. Examples illustrate that the combination of the methods can produce enormous variance reduction even in high-dimensional multifactor models. The method introduces some computational overhead in solving the optimization and eigenvector problems; to address this, we propose and evaluate approximate solution procedures, which enhance the applicability of the method.