The expected shortfall of quadratic portfolios with heavy-tailed risk factors

Computable expressions are derived for the Expected Shortfall of portfolios whose value is a quadratic function of a number of risk factors, as arise from a Delta-Gamma-Theta approximation. The risk factors are assumed to follow an elliptical multivariate t distribution, reflecting the heavy-tailed nature of asset returns. Both an exact expression and a uniform asymptotic expansion are presented. The former involves only a single rapidly convergent integral. The latter is essentially explicit, and numerical experiments suggest that its error is negligible compared to that incurred by the Delta-Gamma-Theta approximation.