Sensitivity estimation of conditional value at risk using randomized quasi-Monte Carlo

Conditional value at risk (CVaR) is a popular measure for quantifying portfolio risk. Sensitivity analysis of CVaR is common in risk management and gradient-based optimization algorithms. In this paper, we study the infinitesimal perturbation analysis estimator for CVaR sensitivity using randomized quasi-Monte Carlo (RQMC) simulation. RQMC has proved valuable in financial option pricing with a better rate of convergence compared to Monte Carlo sampling, but theoretical guarantees for this new application of RQMC shall be studied. To this end, we first prove that the RQMC-based estimator is strongly consistent under very mild conditions. Under some technical conditions, RQMC yields a mean error rate of O ( n − 1 / 2 − 1 / ( 4 d − 2 ) + ϵ ) for arbitrarily small ϵ > 0 , where d represents the dimension of RQMC points and n is the sample size. Some typical applications of CVaR sensitivity estimation are conducted to both show how the theoretical results can be applied, as well as to provide numerical results documenting the superiority of the RQMC estimator.