An integrated quasi-Monte Carlo method for handling high dimensional problems with discontinuities in financial engineering

TL;DR

This paper introduces an integrated quasi-Monte Carlo approach combining smoothing and a novel path generation technique to effectively handle high-dimensional, discontinuous problems in financial derivative pricing, significantly improving variance reduction.

Contribution

It develops a new smoothing and path generation method that jointly address high dimensionality and discontinuities in QMC for financial models, applicable to various complex models.

Findings

Significant variance reduction in pricing exotic options.

Effective handling of discontinuities in high-dimensional settings.

Applicable to multiple financial models like Black-Scholes and Heston.

Abstract

Quasi-Monte Carlo (QMC) method is a useful numerical tool for pricing and hedging of complex financial derivatives. These problems are usually of high dimensionality and discontinuities. The two factors may significantly deteriorate the performance of the QMC method. This paper develops an integrated method that overcomes the challenges of the high dimensionality and discontinuities concurrently. For this purpose, a smoothing method is proposed to remove the discontinuities for some typical functions arising from financial engineering. To make the smoothing method applicable for more general functions, a new path generation method is designed for simulating the paths of the underlying assets such that the resulting function has the required form. The new path generation method has an additional power to reduce the effective dimension of the target function. Our proposed method caters for a large variety of model specifications, including the Black-Scholes, exponential normal inverse Gaussian L\'evy, and Heston models. Numerical experiments dealing with these models show that in the QMC setting the proposed smoothing method in combination with the new path generation method can lead to a dramatic variance reduction for pricing exotic options with discontinuous payoffs and for calculating options' Greeks. The investigation on the effective dimension and the related characteristics explains the significant enhancement of the combined procedure.