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they imply. For example, an investment strategy may have an expected return, after one year, that is five times its standard deviation. Assuming a normal distribution, the likelihood of its failure (negative return) is less than one in a million; in practice, it may be higher. Normal distributions
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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 understate the true degree of predictive difficulty (and of risk). Many—notably
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and characterized by a narrower and larger maximum, and by a fatter tail than in the normal distribution case. On the other hand, this distribution has only one fat tail associated with an increase in sales due to promotion of the new records that enter the charts.
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are sometimes synonymous; fat-tailed is sometimes also defined as a subset of heavy-tailed. Different research communities favor one or the other largely for historical reasons, and may have differences in the precise definition of either.
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compared to
Brownian motion (below). Central events are more common and rare events more extreme in the Cauchy distribution than in Brownian motion. A single event may comprise 99% of total variation, hence the "undefined
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provides for such a distribution. However, traumatic "real-world" events (such as an oil shock, a large corporate bankruptcy, or an abrupt change in a political situation) are usually not mathematically
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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. For
602:("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
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Fat tails in market return distributions also have some behavioral origins (investor excessive optimism or pessimism leading to large market moves) and are therefore studied in
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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
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frequently found (e.g. "20% of customers account for 80% of the revenue") is a manifestation of a fat tail distribution underlying the data.
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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
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that emerge in finance generally do so because the factors influencing an asset's value or price are mathematically "well-behaved", and the
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Steven R. Dunbar, Limitations of the Black-Scholes Model, Stochastic
Processes and Advanced Mathematical Finance 2009
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for various location and scale parameters. Cauchy distributions are examples of fat-tailed distributions.
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http://www.math.unl.edu/~sdunbar1/MathematicalFinance/Lessons/BlackScholes/Limitations/limitations.xml
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Compared to fat-tailed distributions, in the normal distribution, events that deviate from the
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Buda, A. (2012). "Does pop music exist? Hierarchical structure in phonographic markets".
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The most extreme case of a fat tail is given by a distribution whose tail decays like a
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moment is infinite, so for every power law distribution, some moments are undefined.
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Fat-tailed distributions have been empirically encountered in a variety of areas:
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Regular variation subexponentiality and their applications in probability theory
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Fractals and
Scaling in Finance: Discontinuity, Concentration, Risk
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The 80/20 principle : the secret of achieving more with less
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Quantitative
Finance and Risk Management: A Physicist's Approach
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then the distribution is said to have a fat tail if
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863:. Springer.
793:. Springer.
741:leptokurtic
626:as well as
580:Lévy flight
564:Lévy flight
1046:Categories
889:(4): 394.
775:References
731:or in the
722:80-20 rule
571:variance".
183:log-normal
112:April 2010
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718:marketing
534:∼
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262:∼
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179:power law
1033:Archived
912:Archived
748:See also
616:variance
155:kurtosis
151:skewness
18:Fat tail
663:finance
647:options
636:finance
582:from a
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