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

TL;DR

This paper develops and analyzes a randomized quasi-Monte Carlo method for estimating the sensitivity of Conditional Value at Risk (CVaR), demonstrating improved accuracy over traditional Monte Carlo methods especially in lower dimensions.

Contribution

It introduces a strongly consistent RQMC-based estimator for CVaR sensitivity and derives its error rate, showing theoretical and numerical advantages over Monte Carlo methods.

Findings

RQMC estimator is strongly consistent under mild conditions.

RQMC outperforms Monte Carlo in all tested cases.

Error rate of RQMC decreases with lower dimensions, aligning with theoretical predictions.

Abstract

Conditional value at risk (CVaR) is a popular measure for quantifying portfolio risk. Sensitivity analysis of CVaR is very useful 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. We first prove that the RQMC-based estimator is strongly consistent under very mild conditions. Under some technical conditions, RQMC that uses $d$-dimensional points in CVaR sensitivity estimation yields a mean error rate of $O(n^{-1/2-1/(4d-2)+\epsilon})$ for arbitrarily small $\epsilon>0$. The numerical results show that the RQMC method performs better than the Monte Carlo method for all cases. The gain of plain RQMC deteriorates as the dimension $d$ increases, as predicted by the established theoretical error rate.