Heavy-Tailed Distributions

Heavy-tailed distributions have polynomial — not exponential — tail decay, which makes extreme outcomes vastly more likely than under a Normal. In finance this matters because real returns have heavier tails than the textbook Gaussian.

The Pareto distribution has $P(X > x) \propto x^{-\alpha}$ for some shape parameter $\alpha > 0$. It models wealth distribution, city sizes, file sizes, and many other power-law phenomena.

The Student-t$(\nu)$ distribution is bell-shaped like the Normal but has fatter tails controlled by the degrees of freedom $\nu$. It tends to a Normal as $\nu \to \infty$. Quants commonly use Student-t with $\nu$ around $4$–$6$ to model financial returns.

The Cauchy distribution is Student-t with $\nu = 1$ — so heavy that the mean and variance don't exist. Sample averages from a Cauchy never stabilize, which breaks naive simulation.

A risk model that assumes Normal returns will systematically underestimate VaR, miscalibrate option prices in the tails, and produce overconfident risk numbers. Stress tests, scenario analysis, and fat-tailed copulas exist precisely because Normal-based models miss the events that matter most.