The Use of Low Discrepancy Points in Valuing Complex Financial Instruments

Abstract

Modern finance has evolved the use of very complex financial instruments. Stock and interest rate options fit this description. Another example of such an instrument would be a mortgage pool involving many tranches and providing relationships between the tranches so that the payoff on one tranche depends upon the amounts paid upon other tranches over the whole history of the pool. Since the valuation of this last instrument would involve a separate probability distribution for each period over the whole period of the pool, the calculation could involve 360 separate probability distributions over the whole period. It would require then, a multiple integration over all these periods, all 360 of them. Such calculations are generally not possible on an exact basis so that numerical integration must be used. In such an environment only Monte Carlo methods are practical. Certain selected sequences of values called “low discrepancy points” are theoretically more efficient in this kind of calculation than the random numbers usually generated for Monte Carlo calculations. This paper discusses the theoretical basis for such a claim (Niederreiter covers much of the material in a more rigorous fashion.), the calculation of such points, and illustrations of the results of using such methods on several real problems.