Value-at-risk with heavy-tailed risk factors

This paper develops methods for computationally efficient calculation of value-at-risk (VAR) in the presence of heavy-tailed risk factors. The methods model market risk factors through a multivariate t-distribution, which has both heavy tails and empirical support. Our key mathematical result is a transform analysis of a quadratic form in multivariate t random variables. Using this result, we develop two computational methods. The first uses Fourier transform inversion to develop a heavy-tailed delta-gamma approximation; this method is extremely fast, but like any delta-gamma method is only as accurate as the quadratic approximation. For greater accuracy, we therefore develop an efficient Monte Carlo method; this method uses our heavy-tailed delta-gamma approximation as a basis for variance reduction. Specifically, we use the numerical approximation to design a combination of importance sampling and stratified sampling of market scenarios that can produce enormous speed-ups compared with standard Monte Carlo.