Approximation of Multiple Integrals over Hyperboloids with Application to a Quadratic Portfolio with Options

TL;DR

The paper addresses VaR computation for a quadratic portfolio of options under elliptic joint log-returns, which leads to evaluating multi-dimensional integrals over hyperboloids. It introduces a transformation that converts hyperboloid regions into a product of radial and spherical integrals, then applies numerical schemes based on Genz–Monahan and Sheil–O'Muircheartaigh to compute these integrals. Two numerical CAC 40-based examples illustrate the method for a Delta-hedged portfolio ($\\Delta=0$) and a mixed portfolio with both calls and puts, reporting $R$ and $V$ values from $G(R)=\\alpha$. The work extends VaR analysis beyond Normal portfolios, providing a practical tool for risk management of nonlinear option portfolios under elliptic risk factors.

Abstract

We consider an application involving a financial quadratic portfolio of options, when the joint underlying log-returns changes with multivariate elliptic distribution. This motivates the needs for methods for the approximation of multiple integrals over hyperboloids. A transformation is used to reduce the hyperboloid integrals to a product of two radial integrals and two spherical surface integrals. Numerical approximation methods for the transformed integrals are constructed. The application of these methods is demonstrated using some financial applications examples.