Quasi-Monte Carlo Methods in Finance - Isotropic Integrals

Isotropic Integrals

QMC can also be superior to MC and to other methods for isotropic problems, that is, problems where all variables are equally important. For example, Papageorgiou and Traub reported test results on the model integration problems suggested by the physicist B. D. Keister

where denotes the Euclidean norm and . Keister reports that using a standard numerical method some 220,000 points were needed to obtain a relative error on the order of . A QMC calculation using the generalized Faure low discrepancy sequence (QMC-GF) used only 500 points to obtain the same relative error. The same integral was tested for a range of values of up to . Its error was

, where is the number of evaluations of . This may be compared with the MC method whose error was proportional to .

These are empirical results. In a theoretical investigation Papageorgiou proved that the convergence rate of QMC for a class of -dimensional isotropic integrals which includes the integral defined above is of the order

This is with a worst case guarantee compared to the expected convergence rate of of Monte Carlo and shows the superiority of QMC for this type of integral.

In another theoretical investigation Papageorgiou presented sufficient conditions for fast QMC convergence. The conditions apply to isotropic and non-isotropic problems and, in particular, to a number of problems in computational finance. He presented classes of functions where even in the worst case the convergence rate of QMC is of order

where is a constant that depends on the class of functions.

But this is only a sufficient condition and leaves open the major question we pose in the next section.