Low-Discrepancy Sequences

Abstract

Low-discrepancy (quasi-random) sampling methods offer the possibility of significantly enhancing the simulation models used in derivative valuation by using non-random random numbers to generate simulated price paths. The idea is that a set of randomly generated values for the stochastic variables in a simulation will tend to have clumps of values close to each other in some regions and bare spots elsewhere, so a large number may have to be generated in order to have good coverage everywhere. Low-discrepancy sequences are non-random sets of numbers designed to cover the space more evenly, which allows the simulation to produce accurate valuation with fewer generated price series. Several alternatives exist for producing such sets, including algorithms devised by Sobol, by Halton, and by Faure. In this article, the authors give a detailed explanation of how these procedures work and how the low-discrepancy sets are generated. They then provide a comparison test among them for several types of path-dependent options, finding that the Sobol set generally appears to do the best.