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-tailed Distribution
Famous quotes containing the words fat, tails, risk and/or estimate:
“One year
They sent a million here:
Here men were drunk like water, burnt like wood.
The fat of good
And evil, the breast’s star of hope
Were rendered into soap.”
—Randall Jarrell (1914–1965)
“I think mice
Are rather nice.
Their tails are long,
Their faces small,
They haven’t any
Chins at all.”
—Rose Fyleman (1877–1957)
“I saw the man my friend ... wants pardoned, Thomas Flinton. He is a bright, good-looking fellow.... Of his innocence all are confident. The governor strikes me as a man seeking popularity, who lacks the independence and manhood to do right at the risk of losing popularity. Afraid of what will be said. He is prejudiced against the Irish and Democrats.”
—Rutherford Birchard Hayes (1822–1893)
“The man possessed of a dollar, feels himself to be not merely one hundred cents richer, but also one hundred cents better, than the man who is penniless; so on through all the gradations of earthly possessions—the estimate of our own moral and political importance swelling always in a ratio exactly proportionate to the growth of our purse.”
—Frances Wright (1795–1852)