Wiener-Hopf Factorization for the Normal Inverse Gaussian Process

TL;DR

This paper derives explicit Wiener-Hopf factor representations for the NIG process, showing their connection to generalized gamma convolutions, and develops Padé approximation methods for practical computation with applications in finance and risk.

Contribution

It provides new explicit representations of Wiener-Hopf factors for the NIG process and introduces two Padé-based approximation methods for their efficient computation.

Findings

Wiener-Hopf factors are MGFs of GGCs for certain parameters.

Padé approximations can accurately estimate Wiener-Hopf factors.

Applications demonstrated in ruin probabilities and perpetual option pricing.

Abstract

We derive the L\'evy-Khintchine representation of the Wiener-Hopf factors for the Normal Inverse Gaussian (NIG) process as well as a representation which is similar to the moment generating function (MGF) of a generalized gamma convolution (GGC). We show, via this representation, that for some parameters the Wiener-Hopf factors are, in fact, the MGFs of GGCs. Further, we develop two seperate methods of approximating the Wiener-Hopf factors, both based on Pad\'e approximations of their Taylor series expansions; we show how the latter may be calculated exactly to any order. The first approximation yields the MGF of a finite gamma convolution, the second that of a finite mixture of exponentials. Both provide excellent approximations as we demonstrate with numerical experiments and by considering applications to the ultimate ruin problem and to the pricing of perpetual options.