Efficient Monte Carlo methods for value-at-risk

Abstract

The calculation of value-at-risk for large portfolios presents a tradeoff between speed and accuracy, with the fastest methods relying on rough approximations and the most realistic approach -- Monte Carlo simulation -- often too slow to be practical. This article describes methods that use the best features of both approaches. The methods build on the delta-gamma approximation, but they use the approximation not as a substitute for simulation but rather as an aid to it. We use the delta-gamma approximation to guide the sampling of market scenarios through a combination of importance sampling and stratied sampling. This can greatly reduce the number of scenarios required in a simulation to achieve a desired precision. We also describe an extension of the method in which "vega" terms are included in the approximation to capture changes in the level of volatility.

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