Dimension Reduction Techniques in Quasi-Monte Carlo Methods for Option Pricing
INFORMS Journal on Computing
Enhancing Quasi-Monte Carlo Methods by Exploiting Additive Approximation for Problems in Finance
SIAM Journal on Scientific Computing
Constructing adapted lattice rules using problem-dependent criteria
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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The valuation of many financial derivatives leads to high-dimensional integrals. The constructions of robust or universal good lattice rules for financial applications are both important and challenging. An important common feature of the integrands in computational finance is that they can often be well approximated by a sum of low-dimensional functions, i.e., functions that depend on only a small number of variables (usually just two variables). For numerical integration of such functions the quality of the low-order (i.e., low-dimensional) projections of the node set is crucial. In this paper we propose methods to construct good lattice points with “optimal” low-order projections. The quality of a point set is measured by a new measure called elementary order-$\ell$ discrepancy, which measures the quality of all order-$\ell$ projections and is more informative than usual measures. Two constructions, namely the Korobov and the component-by-component constructions, are studied such that the low-order projections are optimized. Numerical experiments demonstrate that even in high dimensions it is possible to construct new good lattice points with order-$2$ projections that are better than those of the Sobol’ points and random points and with higher-order projections that are no worse (while the Sobol’ points lost the advantage over random points in order-$2$ projections on the average). The new lattice rules have the potential to improve upon the accuracy for favorable functions, while doing no harm for unfavorable ones. Their applications for pricing path-dependent options and American options (based on the least-square Monte Carlo method) are studied and their high efficiency is demonstrated. A nice surprise revealed is the robustness property of such lattice rules: the good projection property and the suitability for a large range of problems. The potential possibility and limitations of good lattice points in achieving good quality of moderate- and high-order projections is investigated. The reason why classical lattice rules may not be efficient for high-dimensional finance problems is also discussed.