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

Xiaoqun Wang () and Ian H. Sloan ()
Additional contact information
Xiaoqun Wang: Department of Mathematical Sciences, Tsinghua University, 100084 Beijing, China, and School of Mathematics, University of New South Wales, Sydney 2052, Australia
Ian H. Sloan: School of Mathematics, University of New South Wales, Sydney 2052, Australia, and Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong

Operations Research, 2011, vol. 59, issue 1, 80-95

Abstract: 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 performance of QMC methods is known to depend crucially on the construction of Brownian motions. This paper focuses on the impact of various constructions. Although the Brownian bridge (BB) construction often yields very good results, as Papageorgiou pointed out, there are financial derivatives for which the BB construction performs badly [Papageorgiou, A. 2002. The Brownian bridge does not offer a consistent advantage in quasi-Monte Carlo integration. J. Complexity 18 (1) 171--186]. In this paper we first extend Papageorgiou's analysis to establish an equivalence principle: if the BB construction (or any other construction) is the preferred construction for a particular financial derivative, then for any other construction, there is another financial derivative for which the latter 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. In practice, our strategy is to find a good construction for an “easy” problem and then apply it to more complicated problems related to the easy one. This strategy is applied to the arithmetic Asian options (including Bermudan Asian options) based on the weighted average of the stock prices. We do this by studying a simpler problem, namely, the geometric Asian option, for which the best construction is easily available, and applying it to the arithmetic Asian option. Numerical experiments confirm the success of this strategy: whereas in QMC all the common methods (the standard method, BB, and principal component analysis) may lose their power in some situations, the new method behaves very well in all cases. Further large variance reduction can be achieved in combination with a control variate. The new method can be interpreted as a practical way of reducing the effective dimension for some class of functions.

Keywords: finance; financial engineering; asset pricing; simulation; quasi-Monte Carlo methods; dimension reduction; Brownian bridge; principal component analysis (search for similar items in EconPapers)
Date: 2011
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (13)

Downloads: (external link)
http://dx.doi.org/10.1287/opre.1100.0853 (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:inm:oropre:v:59:y:2011:i:1:p:80-95

Access Statistics for this article

More articles in Operations Research from INFORMS Contact information at EDIRC.
Bibliographic data for series maintained by Chris Asher ().