Importance Sampling in the Heath-jarrow-morton Framework Importance Sampling in the Heath-jarrow-morton Framework

LIMITED DISTRIBUTION NOTICE: This report has been submitted for publication outside of IBM and will probably be copyrighted if accepted for publication. It has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and speciic requests. After outside publication, requests should be lled only by reprints or legally obtained copies of the article (e.g., payment of royalties). Copies may be requested from IBM T. J. Abstract This paper develops a variance reduction technique for pricing derivatives in high-dimensional multifactor models, with particular emphasis on term structure models formulated in the Heath-Jarrow-Morton framework. A premise of this work is that the largest gains in simulation ee-ciency come from taking advantage of the structure of both the cashhows of a security and the model in which it is priced; for this to be feasible in practice requires that the identiication and use of relevant structure be automated. We exploit model and payoo structure through a combination of importance sampling and stratiied sampling. The importance sampling applies a change of drift to the underlying factors; we select the drift by rst solving an optimization problem. We then identify a particularly eeective direction for stratiied 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. These further enhance the applicability of the method.