Theoretical Explanations
The results reported so far in this article are empirical. A number of possible theoretical explanations have been advanced. This has been a very research rich area leading to powerful new concepts but a definite answer has not been obtained.
A possible explanation of why QMC is good for finance is the following. Consider a tranche of the CMO mentioned earlier. The integral gives expected future cash flows from a basket of 30 year mortgages at 360 monthly intervals. Because of the discounted value of money variables representing future times are increasingly less important. In a seminal paper I. Sloan and H. Woźniakowski introduced the idea of weighted spaces. In these spaces the dependence on the successive variables can be moderated by weights. If the weights decrease sufficiently rapidly the curse of dimensionality is broken even with a worst case guarantee. This paper led to a great amount of work on the tractability of integration and other problems. A problem is tractable when its complexity is of order and is independent of the dimension.
On the other hand, effective dimension was proposed by Caflisch, Morokoff and Owen as an indicator of the difficulty of high dimensional integration. The purpose was to explain the remarkable success of quasi-Monte Carlo (QMC) in approximating the very high dimensional integrals in finance. They argued that the integrands are of low effective dimension and that is why QMC is much faster than Monte Carlo (MC). The impact of the arguments of Caflisch et al. was great. A number of papers deal with the relationship between the error of QMC and the effective dimension .
It is known that QMC fails for certain functions that have high effective dimension. However, low effective dimension is not a necessary condition for QMC to beat MC and for high dimensional integration to be tractable. In 2005, Tezuka exhibited a class of functions of variables, all with maximum effective dimension equal to . For these functions QMC is very fast since its convergence rate is of order, where is the number of function evaluations.
Read more about this topic: Quasi-Monte Carlo Methods In Finance
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