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.
Comment: 37 pages, 2 figures