Efficient Weighted Lattice Rules with Applications to Finance

Abstract

Good lattice rules are an important type of quasi-Monte Carlo algorithm. They are known to have good theoretical properties, in the sense that they can achieve an error bound (or optimal error bound) that is independent of the dimension for weighted function spaces with suitably decaying weights. To use the theory of weighted function spaces for practical applications, one has to determine what weights should be used. After explaining that lattice rules based on figures of merit with classical weights (classical lattice rules) may not give good results for high-dimensional problems, we propose a "matching strategy," which chooses the weights in such a way that the typical functions of a weighted Korobov space are similar (in the sense of similar relative sensitivity indices) to those of the given function. The matching strategy relates the weights of the spaces to the sensitivity indices of the given function. We apply the Korobov construction and the component-by-component construction of lattice rules with the suitably chosen weights to the valuation of high-dimensional financial derivative securities and find that such weighted lattice rules improve dramatically on the results for the classical lattice rules; moreover, they are competitive with other types of low discrepancy sequences. We also demonstrate that lattice rules combined with variance reduction techniques and dimension reduction techniques increase the efficiency significantly. We showthat an acceptance-rejection approach to the construction of lattice rules can reduce the construction cost with almost no loss of accuracy.