Calculating risk measures and pricing financial derivatives using spherical Monte Carlo simulation

Abstract

In finance, accurate and efficient calculation for risk measures and derivatives' prices remains critical and challenging both in theory and practice. Many of these values are related to the expectation of a performance function under a complex random system (or mathematically, an integral). Although Monte Carlo simulation is indispensable for high-dimensional integration, it is formidable due to its slow convergence. To overcome this problem, this paper proposes a spherical estimator as a randomized integration rule, and provides a simple yet general importance sampling method for the spherical distribution. The importance sampling can be combined to the spherical estimator to gain further variance reduction. Illustrations are given in two types of nancial applications: (1) the Value-at-Risk and Expected Shortfall for the quadratic portfolio under heavy-tailed risk factors and (2) exotic options prices under GARCH models. Numerical experiments con rm that the superiority of our proposed simulation scheme in terms of variance reduction and computation efficiency.