# 10.5  Variance Reduction

Variance reduction techniques suitable for value-at-risk were pioneered by Cárdenas et al. (1999), who invented all three of the techniques presented in this section. Their techniques can dramatically reduce the computational expense of applying a Monte Carlo transformation. Our treatment of their work clarifies some theoretical details that their practitioner-oriented presentation skirted. Glasserman, Heidelberger and Shahabuddin (2000) proposed approaches based on importance sampling and stratified sampling. In side-by-side testing, your author found these less effective than the techniques of Cárdenas et al. presented below. See also Glasserman (2003), who proposes techniques not tested by your author.

Standard variance reduction techniques employ a global remapping  to estimate the value-at-risk of a portfolio (0p, 1P). Rather than substitute  for 1P, they apply a Monte Carlo estimator directly to 1P, but use  to facilitate variance reduction. An obvious approach is to employ  as a control variate for 1P. We also may use  to implement stratified sampling.  may be a linear remapping, but best results are obtained if it is a quadratic remapping.

A shortcoming of these variance reduction techniques is the fact that they place restrictions on the distribution of 1R—typically  assuming 1R is conditionally joint normal.