Efficient risk estimation via nested multilevel quasi-Monte Carlo simulation

TL;DR

This paper introduces an efficient nested simulation method combining multilevel Monte Carlo and quasi-Monte Carlo techniques to accurately estimate large portfolio losses with reduced computational complexity.

Contribution

It presents a novel approach that integrates QMC into MLMC for risk estimation, achieving near-optimal complexity and addressing coupling issues with a smoothed MLMC variant.

Findings

QMC accelerates convergence in nested simulations.

Complexity can be reduced to nearly $O(())$ with QMC.

Smoothed MLMC mitigates indicator function coupling problems.

Abstract

We consider the problem of estimating the probability of a large loss from a financial portfolio, where the future loss is expressed as a conditional expectation. Since the conditional expectation is intractable in most cases, one may resort to nested simulation. To reduce the complexity of nested simulation, we present a method that combines multilevel Monte Carlo (MLMC) and quasi-Monte Carlo (QMC). In the outer simulation, we use Monte Carlo to generate financial scenarios. In the inner simulation, we use QMC to estimate the portfolio loss in each scenario. We prove that using QMC can accelerate the convergence rates in both the crude nested simulation and the multilevel nested simulation. Under certain conditions, the complexity of MLMC can be reduced to $O(\epsilon^{-2}(\log \epsilon)^2)$ by incorporating QMC. On the other hand, we find that MLMC encounters catastrophic coupling problem due to the existence of indicator functions. To remedy this, we propose a smoothed MLMC method which uses logistic sigmoid functions to approximate indicator functions. Numerical results show that the optimal complexity $O(\epsilon^{-2})$ is almost attained when using QMC methods in both MLMC and smoothed MLMC, even in moderate high dimensions.