Wiener-Hopf Factorization for the Normal Inverse Gaussian Process

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.