Fat-tailed distribution

A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. In common usage, the term fat-tailed and heavy-tailed are synonymous, but different research communities favor one or the other largely for historical reasons. Fat-tailed distributions have been empirically encountered in a variety of areas: physics, earth sciences, economics and political science. The class of fat-tailed distributions includes those whose tails decay like a power law, which is a common point of reference in their use in the scientific literature. However, fat-tailed distributions also include other slowly-decaying distributions, such as the log-normal. The most extreme case of a fat tail is given by a distribution whose tail decays like a power law. That is, if the complementary cumulative distribution of a random variable X can be expressed as then the distribution is said to have a fat tail if α {displaystyle alpha } is small. For instance, if α < 3 {displaystyle alpha <3} , the variance and the skewness of the tail is mathematically undefined (a special property of the power-law distribution), and hence larger than any normal or exponential distribution. For values of α > 3 {displaystyle alpha >3} , the claim of a fat tail is more ambiguous, because in this parameter range, the variance, skewness, and kurtosis can be finite, depending on the precise value of α > 3 {displaystyle alpha >3} , and thus potentially smaller than a high-variance normal or exponential tail. This ambiguity often leads to disagreements about precisely what is or is not a fat-tailed distribution. Note: here the tilde notation ' ∼ {displaystyle sim } ' refers to the asymptotic equivalence of functions, meaning that there exists some finite value of x {displaystyle x} above which the probability distribution follows the right-hand side of the expression. In other words, asymptotically, the tail of the distribution decays like a power law. Compared to fat-tailed distributions, in the normal distribution events that deviate from the mean by five or more standard deviations ('5-sigma events') have lower probability, meaning that in the normal distribution extreme events are less likely than for fat-tailed distributions. Fat-tailed distributions such as the Cauchy distribution (and all other stable distributions with the exception of the normal distribution) have 'undefined sigma' (more technically, the variance is undefined). As a consequence, when data arise from an underlying fat-tailed distribution, shoehorning in the 'normal distribution' model of risk—and estimating sigma based (necessarily) on a finite sample size—would severely understate the true degree of predictive difficulty (and of risk). Many—notably Benoît Mandelbrot as well as Nassim Taleb—have noted this shortcoming of the normal distribution model and have proposed that fat-tailed distributions such as the stable distributions govern asset returns frequently found in finance. The Black–Scholes model of option pricing is based on a normal distribution. If the distribution is actually a fat-tailed one, then the model will under-price options that are far out of the money, since a 5- or 7-sigma event is much more likely than the normal distribution would predict.