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Intermediate rank lattice rules and applications to finance. (English) Zbl 1161.65003

In scientific computation the Monte Carlo (MC) simulation method is the main tool that deals with high-dimensional problems. The quasi-Monte Carlo (QMC) method uses quasi-random or low-discrepancy point sets to reduce the rate of convergence of the obtained errors.
The author discusses the intermediate rank lattice rule for the general case when the number of quadrature points is \(n^{t}.m,\) where \(n,m\) are integers such that \((n,m)=1\) and \(t\) is the rank of the rule.
In Section 2 the idea of the lattice rule for the calculus of approximations of high-dimensional integrals of one-periodic in each argument integrands is developed. Some known results of the expression of the lattice rule are reminded. Notions as a good lattice rule, a maximum rank rule and intermediate rank lattice rules are discussed. The functional class \(S_{s}^{\alpha}(C)\) is considered. The lattice rule error is bounded in terms of the quantity \(P_{\alpha}({\mathbf g},N).\) The quantity \(M_{\alpha}(N)\) which is the mean of \(P_{\alpha}({\mathbf g},N)\) over the generating vector \({\mathbf g}\) is introduced. Theorem 2 gives an expression of \(M_{\alpha}(N)\) in the terms of the Riemann zeta function and the order \( {\mathcal O}\left({(\log N)^{\alpha s} \over N^{\alpha}}\right)\) of \((M_{\beta(N)}(N))^{\alpha /\beta(N)},\) for a special value of the parameter \(\beta(N)\) is obtained.
In Section 3 intermediate rank lattice rules are considered. The existence of good rank \(t\) rules is established. The mean \(M_{\alpha,t}^{(n)}(m)\) of \(P_{\alpha}(Q_{t})\) over a subset of \(\mathbb{Z}^{s}\) is introduced. Theorem 3 is the main theoretical result of the paper and gives an expression of \(M_{\alpha,t}^{(n)}(m).\) Under the conditions of Theorem 3 in Corollary 1 an upper bound of \(M_{\alpha,t}^{(n)}(m)\) is given and the order \({\mathcal O}\left( {\log \log m \over m}\right)\) as \(m \to \infty\) of \(M_{\alpha,t}^{(n)}(m)\) is obtained. Theorem 4 states the existence of good intermediate rank \(t\) lattice rules and the order \({\mathcal O}\left( { (\log m)^{\alpha s} \over n^{\alpha} }\right)\) as \(m \to \infty\) of \((M_{\beta(m),t}^{(n)}(m))^{\alpha / \beta(m)},\) for a special value of the parameter \(\beta(N)\) is obtained.
In Section 4 comparisons of some computer search results about Korbov type lattice rules and intermediate rank lattice rules are made.
In Section 5 an example of applications of lattice rules to option pricing is developed. Both MC and QMC methods are applied to finance area-option pricing and the efficiencies of these methods are compared. For the QMC methods Sobol’ sequences and both rank-1 and intermediate rank lattice points are used.

MSC:

65C05 Monte Carlo methods
11K38 Irregularities of distribution, discrepancy
11K45 Pseudo-random numbers; Monte Carlo methods
91G60 Numerical methods (including Monte Carlo methods)

Software:

QSIMVN
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