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.
Read more about this topic: Fat Tails
Famous quotes containing the words fat, tails, risk and/or estimate:
“An artist should remain true. Otherwise his talent, like his stomach, grows fat and stuffy.”
—Norman Reilly Raine (1895–1971)
“All along the backwater,
Through the rushes tall,
Ducks are a-dabbling,
Up tails all!”
—Kenneth Grahame (1859–1932)
“The Englishman’s strong point is his vigorous insularity; that of the American his power of adaptation. Each of these attitudes has its perils. The Englishman stands firmly on his feet, but he who merely does this never advances. The American’s disposition is to step forward even at the risk of a fall.”
—Thomas Wentworth Higginson (1823–1911)
“I will frankly declare, that after passing a few weeks in this valley of the Marquesas, I formed a higher estimate of human nature than I had ever before entertained. But alas! since then I have been one of the crew of a man-of-war, and the pent-up wickedness of five hundred men has nearly overturned all my previous theories.”
—Herman Melville (1819–1891)