Variance reduction techniques for estimating value-at-risk

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 effective variance reduction techniques; specifically importance sampling and stratified 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 stratified sampling can result in large variance reductions when estimating the probability of large portfolio losses.