Comparative Study of Monte Carlo and Quasi-Monte Carlo Techniques for Enhanced Derivative Pricing

TL;DR

This paper compares Monte Carlo and quasi-Monte Carlo methods for derivative pricing, showing QMC's superior convergence and accuracy, especially in moderate dimensions, through theoretical analysis and numerical experiments.

Contribution

It provides a detailed comparison of MC and QMC techniques, highlighting the advantages of low-discrepancy sequences in derivative pricing applications.

Findings

QMC achieves faster convergence rates than MC.

QMC significantly reduces root mean square error in option pricing.

Brownian bridge RQMC improves Asian option pricing accuracy.

Abstract

This study presents a comparative analysis of Monte Carlo (MC) and quasi-Monte Carlo (QMC) methods in the context of derivative pricing, emphasizing convergence rates and the curse of dimensionality. After a concise overview of traditional Monte Carlo techniques for evaluating expectations of derivative securities, the paper introduces quasi-Monte Carlo methods, which leverage low-discrepancy sequences to achieve more uniformly distributed sample points without relying on randomness. Theoretical insights highlight that QMC methods can attain superior convergence rates of $O(1/n^{1-\epsilon})$ compared to the standard MC rate of $O(1/\sqrt{n})$, where $\epsilon>0$. Numerical experiments are conducted on two types of options: geometric basket call options and Asian call options. For the geometric basket options, a five-dimensional setting under the Black-Scholes framework is utilized, comparing the performance of Sobol' and Faure low-discrepancy sequences against standard Monte Carlo simulations. Results demonstrate a significant reduction in root mean square error for QMC methods as the number of sample points increases. Similarly, for Asian call options, incorporating a Brownian bridge construction with RQMC further enhances accuracy and convergence efficiency. The findings confirm that quasi-Monte Carlo methods offer substantial improvements over traditional Monte Carlo approaches in derivative pricing, particularly in scenarios with moderate dimensionality.