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