5.9  Variance Reduction

We have considered two techniques of numerical integration: quadrature and the Monte Carlo method. Quadrature is deterministic. The Monte Carlo method is random. We may reduce the standard error of a Monte Carlo estimator by making it more deterministic. Variance reduction techniques are one way of doing this. These encompass a variety of methods that make an estimator more deterministic by incorporating additional information about a problem. The resulting estimators can have dramatically reduced standard errors while remain immune to the curse of dimensionality.

There are various approaches to variance reduction. In this section we will describe two that have proven effective with value-at-risk analyses: control variates and stratified sampling. We describe them generally here. Based on that discussion, in Chapter 10, we will show how to apply them to value-at-risk analyses.