Importance Sampling and Stratification for Value-at-Risk

This paper proposes and evaluates variance reduction techniques for efficient estimation of portfolio loss probabilities using Monte Carlo simulation. Precise estimation of loss probabilities is essential to calculating value-at-risk, which is simply a percentile of the loss distribution. The methods we develop build on delta-gamma approximations to changes in portfolio value. The simplest way to use such approximations for variance reduction employs them as control variates; we show, however, that far greater variance reduction is possible if the approximations are used as a basis for importance sampling, stratified sampling, or combinations of the two. This is especially true in estimating very small loss probabilities.