Value-at-Risk and Expected Shortfall for Quadratic portfolio of securities with mixture of elliptic Distributed Risk Factors

Abstract

Generally, in the financial literature, the notion of quadratic VaR is implicitly confused with the Delta-Gamma VaR, because more authors dealt with portfolios that contains derivatives instruments. In this paper, we postpone to estimate the Value-at-Risk of a quadratic portfolio of securities (i.e equities) without the Delta and Gamma greeks, when the joint log-returns changes with multivariate elliptic distribution. We have reduced the estimation of the quadratic VaR of such portfolio to a resolution of one dimensional integral equation. To illustrate our method, we give special attention to the mixture of normal and mixture of t-student distribution. For given VaR, when joint Risk Factors changes with elliptic distribution, we show how to estimate an Expected Shortfall .