Handling Discontinuities in Financial Engineering: Good Path Simulation and Smoothing

Abstract

Discontinuities are common in the pricing and hedging of complex financial derivatives. Quasi-Monte Carlo (QMC) methods for high-dimensional finance problems with discontinuities can be inefficient because of the lack of good smoothness and high dimensionality. Interestingly, path simulation method (PSM) may affect both factors, implying its significance in QMC methods. What defines a “good” PSM for problems with discontinuities? The ability to align the discontinuities with the coordinate axes is a desirable property for a PSM. We show that for an arbitrary PSM, there exists a class of options with discontinuous payoff functions such that the transformed functions have only axis-parallel discontinuities, for which good QMC performance can be expected. In this sense, any PSM can be “good” in QMC methods for a specific class of problems. We analyze the structure of discontinuities for digital options using the new approach and show the superiority and the uniqueness (up to a permutation) of the standard construction. We develop a two-step procedure for pricing and hedging derivatives with discontinuous payoff functions. The first step is to design a good PSM that has the ability to align the discontinuities with the coordinate axes and the second step is to further exploit this nice property to remove the discontinuities completely. We prove that the new estimate is unbiased and has smaller variance. Numerical experiments demonstrate that the two-step procedure is very effective in QMC methods for pricing options and estimating Greeks, leading to a dramatic variance reduction. Both the path simulation step and the smoothing step are crucial and beneficial for QMC methods, with the contribution from each step varying depending on the severity of discontinuity.