Fat-tailed distribution

fat tailfat tailsfatter tailsfat tail distributionfat-tailFat-tail distributedFat-tailedfat-tailed modelsfattenedfive-sigma event
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.wikipedia
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Heavy-tailed distribution

heavy tailsheavy-tailedheavy tail
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.
There are three important subclasses of heavy-tailed distributions: the fat-tailed distributions, the long-tailed distributions and the subexponential distributions.

Log-normal distribution

lognormallog-normallognormal distribution
However, fat-tailed distributions also include other slowly-decaying distributions, such as the log-normal.
In finance, in particular the Black–Scholes model, changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest, not like simple interest, and so are multiplicative). However, some mathematicians such as Benoît Mandelbrot have argued that log-Lévy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crashes. Indeed, stock price distributions typically exhibit a fat tail.; the fat tailed distribution of changes during stock market crashes invalidate the assumptions of the central limit theorem.

Power law

power-lawscaling lawpower laws
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.
Fat-tailed distribution

Cauchy distribution

LorentzianCauchyLorentzian profile
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).
However, because of the fat tails of the Cauchy distribution, the efficiency of the estimator decreases if more than 24% of the sample is used.

Kurtosis

excess kurtosisleptokurticplatykurtic
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 terms of shape, a leptokurtic distribution has fatter tails.

Seven states of randomness

Seven states of randomness
The importance of seven states of randomness classification for mathematical finance is that methods such as Markowitz mean variance portfolio and Black–Scholes model may be invalidated as the tails of the distribution of returns are fattened: the former relies on finite standard deviation (volatility) and stability of correlation, while the latter is constructed upon Brownian motion.

Black swan theory

black swanblack swansblack swan event
Black swan theory
These concerns often are highly relevant in financial markets, where major players sometimes assume normal distributions when using value at risk models, although market returns typically have fat tail distributions.

The Fat Tail

In The Fat Tail: The Power of Political Knowledge for Strategic Investing, political scientists Ian Bremmer and Preston Keat propose to apply the fat tail concept to geopolitics.
As William Safire notes in his recent etymology of the term, a fat tail occurs when there is an unexpectedly thick end or “tail” toward the edges of a distribution curve, indicating an irregularly high likelihood of catastrophic events.

Probability distribution

distributioncontinuous probability distributiondiscrete probability 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.

Skewness

skewedskewskewed 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.

Normal distribution

normally distributednormalGaussian
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). 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.

Exponential distribution

exponentialexponentially distributedexponentially
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.

Cumulative distribution function

distribution functionCDFcumulative probability distribution function
That is, if the complementary cumulative distribution of a random variable X can be expressed as

Random variable

random variablesrandom variationrandom
That is, if the complementary cumulative distribution of a random variable X can be expressed as

Big O notation

Obig-O notationΘ
Note: here the tilde notation "\sim" refers to the asymptotic equivalence of functions, meaning that there exists some finite value of x above which the probability distribution follows the right-hand side of the expression.

Mean

mean valuepopulation meanaverage
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.

Standard deviation

standard deviationssample standard deviationsigma
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.

Variance

sample variancepopulation variancevariability
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).

Benoit Mandelbrot

MandelbrotBenoît MandelbrotBenoit Mandelbrot’s mathematics
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.

Nassim Nicholas Taleb

Nassim TalebTaleb, Nassim NAntifragility
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.

Stable distribution

stable distributionsStablealpha stable 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). 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.

Finance

financialfinancesfiscal
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.

Black–Scholes model

Black–ScholesBlack-ScholesBlack–Scholes formula
The Black–Scholes model of option pricing is based on a normal distribution.

Option (finance)

optionsoptionstock options
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.