A spherical Monte Carlo approach for calculating value-at-risk and expected shortfall in financial risk management

Accurate and efficient calculation for expected values is challenging in finance as well as various disciplines. In general, expected values can be written as high-dimensional integrals. Monte Carlo simulation is an indispensable tool for calculating them, but it is notoriously known for its slow convergence. For spherical distributions, this paper proposes a variance reduction technique and investigates its applications in finance. By using polar transformation, the expected value is written as an integral, and the innermost integral is with respect to the radius and the outermost integral is with respect to the unit sphere. The spherical Monte Carlo estimator is the average of function values of some random points generated by lattice. We consider Value-at-Risk and expected shortfall calculation under heavy-tailed distributions and demonstrate the superiority of the proposed method via numerical studies in terms of variance, computation time, and efficiency.