  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.
