Avoiding Zero Probability Events When Computing Value at Risk Contributions: A Malliavin Calculus Approach

Abstract

This paper is concerned with the process of risk allocation for a generic multivariate model when the risk measure is chosen as the Value-at-Risk (VaR). Making use of Malliavin calculus, we recast the traditional Euler contributions from an expectation conditional to an event of zero probability to a ratio of conditional expectations, where both the numerator and the denominator's conditioning events have positive probability. For several different models we show empirically that the estimator using this novel representation has no perceivable bias and variance smaller than a standard estimator used in practice.