Variance reduction techniques for estimating value-at-risk

This paper describes, analyzes and evaluates an algorithm for estimating po rtfolio 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 qu adratic (''delta-gamma'') approximation to the change in portfolio value to guide the selection of effective variance reduction techniques; specifical ly importance sampling and stratified sampling. If the approximation is exa ct, then the importance sampling is shown to be asymptotically optimal. Num erical results indicate that an appropriate combination of importance sampl ing and stratified sampling can result in large variance reductions when es timating the probability of large portfolio losses.