Inverse Gaussian quadrature and finite normal-mixture approximation of the generalized hyperbolic distribution

TL;DR

This paper introduces a novel numerical quadrature for the generalized inverse Gaussian distribution, enabling efficient approximation of the generalized hyperbolic distribution as a finite normal mixture for accurate computation of expectations and sampling.

Contribution

It develops a new quadrature method based on Gauss-Hermite for the generalized inverse Gaussian, facilitating finite normal mixture approximation of the generalized hyperbolic distribution.

Findings

Quadrature exactly integrates polynomials of positive and negative orders.

Finite normal mixture approximation enables efficient expectation calculations.

Numerical examples demonstrate high accuracy of the proposed methods.

Abstract

In this study, a numerical quadrature for the generalized inverse Gaussian distribution is derived from the Gauss-Hermite quadrature by exploiting its relationship with the normal distribution. The proposed quadrature is not Gaussian, but it exactly integrates the polynomials of both positive and negative orders. Using the quadrature, the generalized hyperbolic distribution is efficiently approximated as a finite normal variance-mean mixture. Therefore, the expectations under the distribution, such as cumulative distribution function and European option price, are accurately computed as weighted sums of those under normal distributions. The generalized hyperbolic random variates are also sampled in a straightforward manner. The accuracy of the methods is illustrated with numerical examples.