Variance Reduction Techniques for Value-at-Risk With Heavy-Tailed Risk Factors

The calculation of value-at-risk (VAR) for large portfolios of complex instruments is among the most demanding and widespread computational challenges facing the financial industry. Current methods for calculating VAR include comparatively fast numerical approximations - especially the linear and quadratic (delta-gamma) approximations - and more robust but more computationally demanding Monte Carlo simulation. The linear and delta-gamma methods typically rely on an assumption that the underlying market risk factors have a Gaussian distribution over the VAR horizon. But there is ample empirical evidence that market data is more accurately described by heavy-tailed distributions. Capturing heavy tails in VAR calculations has to date required highly time-consuming Monte Carlo simulation. We describe two methods for computationally efficient calculation of VAR in the presence of heavy-tailed risk factors, specifically when risk factors have a multivariate t distribution. The first method uses transform inversion to develop a fast numerical algorithm for computing the distribution of the heavy-tailed delta-gamma approximation. For greater accuracy, the second method uses the numerical approximation to guide in the design of an effective Monte Carlo variance reduction technique; the algorithm combines importance sampling and stratified sampling. This method can produce enormous speed-ups compared with standard Monte Carlo.