Variance Reduction Techniques for Estimating Value - at - RiskPaul

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 describes, analyzes and evaluates an algorithm for estimating portfolio loss probabilities using Monte Carlo simulation. Obtaining accurate estimates of such loss probabilities is essential to calculating value-at-risk, which is a quantile of the loss distribution. The method employs a quadratic (\delta-gamma") approximation to the change in portfolio value to guide the selection of eeective variance reduction techniques; speciically importance sampling and stratiied sampling. If the approximation is exact, then the importance sampling is shown to be asymptotically optimal. Numerical results indicate that an appropriate combination of importance sampling and stratiied sampling can result in large variance reductions when estimating the probability of large portfolio losses.