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. 1 Introduction Value-at-Risk (VAR) has become an important measure for estimating and managing portfolio risk [11, 13]. VAR is defined as a certain quantile of the change in a portfolio's value during a specified holding period. To be more specific, suppose the curre..