Fat Tails and Risk Estimate Distortions
By contrast to fat tail distributions, the normal distribution posits events that deviate from the mean by five or more standard deviations ("5-sigma event") are extremely rare, with 10- or more sigma being practically impossible. On the other hand, fat tail distributions such as the Cauchy distribution (and all other stable distributions with the exception of the normal distribution) are examples of fat tail distributions that have "infinite sigma" (more technically: "the variance does not exist").
Thus when data naturally arise from a fat tail distribution, shoehorning the normal distribution model of risk—and an estimate of the corresponding sigma based necessarily on a finite sample size—would severely understate the true risk. Many—notably Benoît Mandelbrot as well as Nassim Taleb—have noted this shortcoming of the normal distribution model and have proposed that fat tail distributions such as the stable distribution 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.
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