Intermediate rank lattice rules and applications to finance

This paper discusses the intermediate rank lattice rule for the general case when the number of quadrature points is ntm, where m is a composite integer, t is the rank of the rule, n is an integer such that (n,m)=1. By using the technique of averaging the quantity Pα, we obtained a general expression for the average of Pα over a subset of Zs, derived an upper bound and the asymptotic rate for intermediate rank lattice rule. The results recover the cases of the conventional good lattice rule and the maximal rank rule. Computer search results show that P2's by the intermediate rank lattice rules are smaller than those by good lattice rule, while searching intermediate rank lattice points is much faster than that of good lattice points for very close numbers of quadrature points. Numerical tests for application to an option pricing problem show that the intermediate rank lattice rules are not worse than the conventional good lattice rule on average. All the lattice rules show superiority over the Sobol' sequence, which beats the pseudo-random point sets.