Pricing and Hedging with Discontinuous Functions: Quasi-Monte Carlo Methods and Dimension Reduction

Quasi-Monte Carlo (QMC) methods are important numerical tools in the pricing and hedging of complex financial instruments. The effectiveness of QMC methods crucially depends on the discontinuity and the dimension of the problem. This paper shows how the two fundamental limitations can be overcome in some cases. We first study how path-generation methods (PGMs) affect the structure of the discontinuities and what the effect of discontinuities is on the accuracy of QMC methods. The insight is that the discontinuities can be QMC friendly (i.e., aligned with the coordinate axes) or not, depending on the PGM. The PGMs that offer the best performance in QMC methods are those that make the discontinuities QMC friendly. The structure of discontinuities can affect the accuracy of QMC methods more significantly than the effective dimension. This insight motivates us to propose a novel way of handling the discontinuities. The basic idea is to align the discontinuities with the coordinate axes by a judicious design of a method for simulating the underlying processes. Numerical experiments demonstrate that the proposed method leads to dramatic variance reduction in QMC methods for pricing options and for estimating Greeks. It also reduces the effective dimension of the problem.