Levy distribution

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The Levy distribution is a continuous probability distribution which is unbounded below and above. It has four parameters\[\mu\], a location (mean); \(0<\alpha \leq 2\), the peakedness; \(-1<\beta<1\), the skewness; \(\nu>0\), a spread. The probability density function does not have a simple closed form. Instead it must be written in terms of its characteristic function. If \(P(x)\) is the probability density function then the moment generating function is given by

\(M(z)=\int_{-\infty}^{\infty}e^{izx}P(x)\;dx\),

where \(i=\sqrt{-1}\). For the Levy distribution

\(\ln(M(z))=i\mu z-\nu^{\alpha}|z|^{\alpha}\left(1-i\beta\mbox{sgn}(z)\tan(\pi a/2)\right),\;\;\;\mbox{for }\alpha\ne 1\)

or

\(\ln(M(z))=i\mu z-\nu |z|\left(1+\frac{2i\beta}{\pi}\mbox{sgn}(z)\ln(|z|)\right),\;\;\;\mbox{for }\alpha=1\).

The normal distribution is a special case of this with \(\alpha=2\) and \(\beta=0\), and with the parameter \(\nu\) being one half of the variance.

The Levy distribution, or Pareto Levy distribution, is increasingly popular in finance because it matches data well, and has suitable fat tails. The tail of the distribution decays like \(|x|^{-1-\alpha}\).

Mean = \(\mu\) and Variance = \(\mbox{infinite, unless }\alpha=2,\mbox{ when it is }2\nu\).

Stable distribution

The distribution also has the important theoretical property of being a stable distribution in that the sum of independent random numbers drawn from the Levy distribution will itself be Levy. This is a useful property for the distribution of returns. If you add up \(n\) independent numbers from the Levy distribution with the above parameters then you will get a number from another Levy distribution with the same \(\alpha\) and \(\beta\) but with mean of \(n^{1/\alpha}\mu\) and spread \(n^{1/\alpha}\nu\).

References

  • Spiegel, MR, Schiller, JJ, Srinivasan, RA 2000 Schaum's Outline of Probability and Statistics. McGraw-Hill