Quasi-Monte Carlo methods in finance

We review the basic principles of quasi-Monte Carlo (QMC) methods, the randomizations that turn them into variance-reduction techniques, and the main classes of constructions underlying their implementations: lattice rules, digital nets, and permutations in different bases. QMC methods are designed to estimate integrals over the s-dimensional unit hypercube, for moderate or large (perhaps infinite) values of s. In principle, any stochastic simulation whose purpose is to estimate an integral fits this framework, but the methods work better for certain types of integrals than others (e.g., if the integrand can be well approximated by a sum of low-dimensional smooth functions). Such QMC-friendly integrals are encountered frequently in computational finance and risk analysis. We give examples and provide computational results that illustrate the efficiency improvement achieved.