The practice of Delta–Gamma Value at Risk: Implementing the quadratic portfolio model

This paper intends to critically evaluate state-of-the-art methodologies for calculating the value-at-risk (VaR) of non-linear portfolios from the point of view of computational accuracy and efficiency. We focus on the quadratic portfolio model, also known as “Delta–Gamma”, and, as a working assumption, we model risk factor returns as multi-normal random variables. We present the main approaches to Delta–Gamma VaR weighing their merits and accuracy from an implementation-oriented standpoint. One of our main conclusions is that the Delta–Gamma-Normal VaR may be less accurate than even Delta VaR. On the other hand, we show that methods that essentially take into account the non-linearity (hence gammas and third or higher moments) of the portfolio values may present significant advantages over full Monte Carlo revaluations. The role of non-diagonal terms in the Gamma matrix as well as the sensitivity to correlation is considered both for accuracy and computational effort. We also qualitatively examine the robustness of Delta–Gamma methodologies by considering a highly non-quadratic portfolio value function.