Quasi-Monte Carlo methods in finance
In 1977 P. Boyle, University of Waterloo, proposed using Monte Carlo (MC) to evaluate options.
[1] 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.
To break this curse of dimensionality one can use the Monte Carlo (MC) method defined by where the evaluation points
Numerous LDS have been created named after their inventors, for example: Generally, the quasi-Monte Carlo (QMC) method is defined by where the
The standard terminology quasi-Monte Carlo is somewhat unfortunate since MC is a randomized method whereas QMC is purely deterministic.
In 1993, Rensburg and Torrie[6] compared QMC with MC for the numerical estimation of high-dimensional integrals which occur in computing virial coefficients for the hard-sphere fluid.
As we shall see, tests on 360-dimensional integrals arising from a collateralized mortgage obligation (CMO) lead to very different conclusions.
Traub asked a Ph.D. student, Spassimir Paskov, to compare QMC with MC for the CMO.
In 1992 Paskov built a software system called FinDer and ran extensive tests.
To the Columbia's research group's surprise and initial disbelief, Paskov reported that QMC was always superior to MC in a number of ways.
Preliminary results were presented by Paskov and Traub to a number of Wall Street firms in Fall 1993 and Spring 1994.
The firms were initially skeptical of the claim that QMC was superior to MC for pricing financial derivatives.
A January 1994 article in Scientific American by Traub and Woźniakowski[9] discussed the theoretical issues and reported that "preliminary results obtained by testing certain finance problems suggests the superiority of the deterministic methods in practice".
In Fall 1994 Paskov wrote a Columbia University Computer Science Report which appeared in slightly modified form in 1997.
This paper was followed by reports on tests by a number of researchers which also led to the conclusion the QMC is superior to MC for a variety of high-dimensional finance problems.
This includes papers by Caflisch and Morokoff (1996),[12] Joy, Boyle, Tan (1996),[13] Ninomiya and Tezuka (1996),[14] Papageorgiou and Traub (1996),[15] Ackworth, and Broadie and Glasserman (German Wikipedia) (1997).
[16] Further testing of the CMO[15] was carried out by Anargyros Papageorgiou, who developed an improved version of the FinDer software system.
This has been a very research rich area leading to powerful new concepts but a definite answer has not been obtained.
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.
If the weights decrease sufficiently rapidly the curse of dimensionality is broken even with a worst case guarantee.
On the other hand, effective dimension was proposed by Caflisch, Morokoff and Owen[21] 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).
[5] However, low effective dimension is not a necessary condition for QMC to beat MC and for high-dimensional integration to be tractable.
For example, Papageorgiou and Traub[25] reported test results on the model integration problems suggested by the physicist B. D. Keister[26] where
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[17] (QMC-GF) used only 500 points to obtain the same relative error.
In another theoretical investigation Papageorgiou[28] presented sufficient conditions for fast QMC convergence.
He presented classes of functions where even in the worst case the convergence rate of QMC is of order where
