The Normal Inverse Gaussian Distribution

Real asset returns are both fatter-tailed and more asymmetric than a normal distribution allows, extreme days happen more often than a bell curve predicts, and crashes tend to be sharper than rallies. The normal inverse Gaussian (NIG) distribution was built to capture exactly this: it's constructed by mixing normal distributions together, where the variance of each normal is itself drawn randomly from an inverse Gaussian distribution. That mixing is what produces fat tails, a return sample that happened to draw a high-variance normal that day looks like an outlier relative to the average, without needing a separate "regime" or jump model.

The NIG has four parameters, each controlling one visible feature of the distribution: a location parameter shifting the whole distribution left or right, a tail-heaviness parameter controlling how fat the tails are, a skewness parameter allowing crashes and rallies to have different likelihoods, and a scale parameter setting overall spread. This four-knob flexibility is why NIG fits daily equity or FX return histograms noticeably better than a normal distribution while still being tractable enough for closed-form option pricing formulas, unlike some other fat-tailed alternatives that require simulation.