Quasi-Monte Carlo Methods in Financial Engineering: An Equivalence Principle and Dimension Reduction

Quasi-Monte Carlo (QMC) methods are playing an increasingly important role in the pricing of complex financial derivatives. For models in which the prices of the underlying assets are driven by Brownian motions, the efficiency of QMC methods is known to depend crucially on the method of generating the Brownian motions. This paper focuses on the impact of various constructions. While the Brownian bridge construction often yields very good results, as Papageorgiou (J. Complexity, Vol. 18, 171-186, 2002) has pointed out, there are financial derivatives for which the Brownian bridge construction performs badly. In this paper we first extend Papageorgiou’s analysis to establish an equivalence principle: if Brownian bridge construction (or any other construction) is the preferred method of construction for some particular financial derivative, then for any other method for generating a Brownian motion, there is another financial derivative for which the latter method of construction is the preferred one. In this sense all methods of construction are equivalent and no method is consistently superior to others: it all depends on the particular financial derivative. We then show how to find a good construction for a particular class of financial derivatives, namely, weighted arithmetic Asian options (including American Asian options) based on the weighted average of the stock prices. We do this by studying a simpler problem, namely, the corresponding geometric Asian option, for which the problem of finding the best construction is analytically tractable, and then applying this construction to the weighted arithmetic Asian option. Numerical experiments confirm the success of the strategy: while in QMC all the commonly used methods (the standard method, the Brownian bridge and principal component analysis) may lose their power in some situations, the new method behaves very well in all cases. The new method can be interpreted as a practical way of reducing the effective dimension for the class of weighted Asian options. Subject classification: Finance: financial engineering, asset pricing. Simulation: quasi-Monte Carlo methods, dimension reduction, Brownian bridge, principal component analysis. Email addresses: [email protected], [email protected].