Spatial low-discrepancy sequences, spherical cone discrepancy, and applications in financial modeling

TL;DR

This paper develops a new mathematical framework for constructing low-discrepancy sequences on spheres and applies it to improve numerical integration methods, especially in financial modeling contexts like option pricing.

Contribution

It introduces a reproducing kernel Hilbert space on or rom the sphere and rom the origin, extends Stolarsky's invariance principle, and proposes new quadrature methods for high-dimensional integration.

Findings

Numerical tests show improved integration accuracy in spherical spaces.

Application to financial models demonstrates better performance than traditional methods.

Constructed point sets exhibit low discrepancy and are suitable for Quasi-Monte Carlo methods.

Abstract

In this paper we introduce a reproducing kernel Hilbert space defined on $\mathbb{R}^{d+1}$ as the tensor product of a reproducing kernel defined on the unit sphere $\mathbb{S}^{d}$ in $\mathbb{R}^{d+1}$ and a reproducing kernel defined on $[0,\infty)$. We extend Stolarsky's invariance principle to this case and prove upper and lower bounds for numerical integration in the corresponding reproducing kernel Hilbert space. The idea of separating the direction from the distance from the origin can also be applied to the construction of quadrature methods. An extension of the area-preserving Lambert transform is used to generate points on $\mathbb{S}^{d-1}$ via lifting Sobol' points in $[0,1)^{d}$ to the sphere. The $d$-th component of each Sobol' point, suitably transformed, provides the distance information so that the resulting point set is normally distributed in $\mathbb{R}^{d}$. Numerical tests provide evidence of the usefulness of constructing Quasi-Monte Carlo type methods for integration in such spaces. We also test this method on examples from financial applications (option pricing problems) and compare the results with traditional methods for numerical integration in $\mathbb{R}^{d}$.