  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.
