Low-discrepancy sequences | Monte Carlo methods in finance
Quasi-Monte Carlo methods in finance
High-dimensional integrals in hundreds or thousands of variables occur commonly in finance. These integrals have to be computed numerically to within a threshold . If the integral is of dimension then in the worst case, where one has a guarantee of error at most , the computational complexity is typically of order . That is, the problem suffers the curse of dimensionality. In 1977 P. Boyle, University of Waterloo, proposed using Monte Carlo (MC) to evaluate options. Starting in early 1992, J. F. Traub, Columbia University, and a graduate student at the time, S. Paskov, used quasi-Monte Carlo (QMC) to price a Collateralized mortgage obligation with parameters specified by Goldman Sachs. Even though it was believed by the world's leading experts that QMC should not be used for high-dimensional integration, Paskov and Traub found that QMC beat MC by one to three orders of magnitude and also enjoyed other desirable attributes. Their results were first published in 1995. Today QMC is widely used in the financial sector to value financial derivatives; see list of books below. QMC is not a panacea for all high-dimensional integrals. A number of explanations have been proposed for why QMC is so good for financial derivatives. This continues to be a very fruitful research area. (Wikipedia).
