Application of Quasi Monte Carlo and Global Sensitivity Analysis to Option Pricing and Greeks

TL;DR

This paper demonstrates that Quasi Monte Carlo methods, combined with global sensitivity analysis, significantly improve the efficiency and stability of option pricing and Greeks computation in high-dimensional financial models.

Contribution

The study compares QMC and MC methods, showing QMC's superior performance and analyzing how GSA explains variance reduction and effective dimension in financial simulations.

Findings

QMC outperforms MC in convergence speed and stability.

Finite differences with QMC can match adjoint methods for small Greeks.

GSA reveals reduced effective dimension in QMC simulations.

Abstract

Quasi Monte Carlo (QMC) and Global Sensitivity Analysis (GSA) techniques are applied for pricing and hedging representative financial instruments of increasing complexity. We compare standard Monte Carlo (MC) vs QMC results using Sobol' low discrepancy sequences, different sampling strategies, and various analyses of performance. We find that QMC outperforms MC in most cases, including the highest-dimensional simulations, showing faster and more stable convergence. Regarding greeks computation, we compare standard approaches, based on finite differences (FD) approximations, with adjoint methods (AAD) providing evidences that, when the number of greeks is small, the FD approach combined with QMC can lead to the same accuracy as AAD, thanks to increased convergence rate and stability, thus saving a lot of implementation effort while keeping low computational cost. Using GSA, we are able to fully explain our findings in terms of reduced effective dimension of QMC simulation, allowed in most cases, but not always, by Brownian Bridge discretization or PCA construction. We conclude that, beyond pricing, QMC is a very effcient technique also for computing risk measures, greeks in particular, as it allows to reduce the computational effort of high dimensional Monte Carlo simulations typical of modern risk management.