Optimized Multi-Level Monte Carlo Parametrization and Antithetic Sampling for Nested Simulations

TL;DR

This paper introduces a new MLMC parametrization and antithetic sampling method that improve the efficiency of nested Monte Carlo simulations for risk measures, especially with irregular functions, with proven theoretical guarantees and practical benefits.

Contribution

It presents a novel MLMC parametrization and demonstrates that antithetic sampling enhances efficiency, addressing limitations with irregular functions in nested Monte Carlo methods.

Findings

Improved MLMC performance in non-asymptotic settings.

Antithetic sampling enhances efficiency regardless of function regularity.

Numerical experiments confirm practical benefits in financial risk estimation.

Abstract

Estimating risk measures such as large loss probabilities and Value-at-Risk is fundamental in financial risk management and often relies on computationally intensive nested Monte Carlo methods. While Multi-Level Monte Carlo (MLMC) techniques and their weighted variants are typically more efficient, their effectiveness tends to deteriorate when dealing with irregular functions, notably indicator functions, which are intrinsic to these risk measures. We address this issue by introducing a novel MLMC parametrization that significantly improves performance in practical, non-asymptotic settings while maintaining theoretical asymptotic guarantees. We also prove that antithetic sampling of MLMC levels enhances efficiency regardless of the regularity of the underlying function. Numerical experiments motivated by the calculation of economic capital in a life insurance context confirm the practical value of our approach for estimating loss probabilities and quantiles, bridging theoretical advances and practical requirements in financial risk estimation.