Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Quasi-random sequences and their discrepancies
SIAM Journal on Scientific Computing
Quasi-Monte Carlo methods in numerical finance
Management Science
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
The effective dimension and quasi-Monte Carlo integration
Journal of Complexity
Why Are High-Dimensional Finance Problems Often of Low Effective Dimension?
SIAM Journal on Scientific Computing
Complexity and effective dimension of discrete Lévy areas
Journal of Complexity
Constructing Robust Good Lattice Rules for Computational Finance
SIAM Journal on Scientific Computing
Low discrepancy sequences in high dimensions: How well are their projections distributed?
Journal of Computational and Applied Mathematics
New Brownian bridge construction in quasi-Monte Carlo methods for computational finance
Journal of Complexity
Dimension Reduction Techniques in Quasi-Monte Carlo Methods for Option Pricing
INFORMS Journal on Computing
Smoothness and dimension reduction in Quasi-Monte Carlo methods
Mathematical and Computer Modelling: An International Journal
Fast orthogonal transforms and generation of Brownian paths
Journal of Complexity
Enhancing Quasi-Monte Carlo Methods by Exploiting Additive Approximation for Problems in Finance
SIAM Journal on Scientific Computing
Fast orthogonal transforms for pricing derivatives with quasi-Monte Carlo
Proceedings of the Winter Simulation Conference
American option pricing with randomized quasi-Monte Carlo simulations
Proceedings of the Winter Simulation Conference
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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. Complexity18(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.