A chapter titled Very High-Dimensional Itegration and Mathemitical Finance, in Complexity and Information is a survey of our latest results.
It has long been the belief of the financial sector that high-dimensional integrals should be computed using Monte Carlo (MC) rather than Quasi-Monte Carlo (QMC). In 1992 our research group in the computer Science Department at Columbia University started testing QMC, using improved low discrepancy sequences (LDS), on a 360 dimensional CMO provided by Goldman Sachs. To our surprise QMC always beat MC.
Test results were reported in a series of papers:
Faster Valuation of Financial Derivatives, Journal of Portfolio Management, Vol. 22, No. 1, Fall 1995, 113-120 (with S. Paskov).
Beating Monte Carlo , Risk, June, 1996, 63-65 (with A. Papageorgiou).
A related paper by S. Paskov, New Methodologies for Valuing Derivatives, in Mathematics of Derivative Securities edited by S. Pliska and M. Dempster, Cambridge University Press, 1997, 545-582, is available online. This is almost identical with his original technical report, Computer Science Department, Columbia University, 1994.
There has been much interest in a theoretical explanation of why QMC is superior to MC for finance computations. A posible explanation is that QMC algorithms automatically take advantage of the non-isotropic nature of finance problems. This is made precise in When are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals, by I. Sloan and H. Wozniakowski, Journal of Complexity, March, 1998, 1-33.
For a semi-popular account of this work see B. Cipra, In Math We Trust, in 1995- 1996 What´s Happening in the Mathematical Sciences, vol. 3, American Mathematical Society, Providence, RI, 1996, 100-111.
FinDer is a software system for evaluating high-dimensional integrals.
For related material see Quasi-Monte Carlo Algorithms.
Columbia University has received a patent for an estimation method and system for complex securities using low-discrepancy deterministic sequences.