  We introduce and document a class of probability distributions, called
bilateral generalized inverse Gaussian (BGIG) distributions, that are obtained
by convolution of two generalized inverse Gaussian distributions supported by
the positive and negative semi-axis. We prove several results regarding their
analyticity, shapes and asymptotics, and we introduce the associated L\'evy
processes as well as their main properties. We study the behaviour of these
processes under change of measure, their simulations and the structure of their
sample paths, and we introduce a stock market model constructed by means of
exponential BGIG processes. Based on real market data, we show that this model
is easy to calibrate thanks notably to idiosyncratic properties of BGIG
distributions, and that it is well suited to Monte Carlo and Fourier option
pricing.
