Generalized Hyperbolic Distributions

Asset returns are famously not normal: they have fatter tails than a bell curve predicts, and often some skew — bigger or more frequent moves in one direction than the other. Rather than picking one specific fat-tailed shape and hoping it fits, the generalized hyperbolic (GH) distribution is built to contain many of the common choices as special cases of a single five-parameter formula, controlling location, scale, skewness, tail heaviness, and one extra shape parameter that interpolates between families.

Set its parameters one way and GH reduces to the Student's t distribution (heavy, symmetric tails). Set them another way and it becomes the normal-inverse Gaussian (heavy tails, allows skew). Push the shape parameter to an extreme and it approaches the ordinary normal distribution. This is the appeal for a risk model: instead of testing several distribution families against the data and picking a winner, you can fit one GH distribution and let the data itself decide, through the fitted parameters, which "corner" of the family it lives closest to.

The cost is that GH's likelihood function involves a Bessel function and five parameters to estimate simultaneously, which is numerically harder and slower to fit than a plain normal or even a Student's t, and with a modest sample the extra flexibility can just as easily overfit noise as capture real skew.