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Saturday, 14 December 2013

Why Modelling Power-Law Tails?

It is rather well known among investment quants that the Central Limit Theorem is based on random variables with finite second moments, such as the normal distribution. This assumption is violated when random variables exhibit power-law tail distributions as $ |x|^{−α−1} $ where $ 0 < \alpha < 2 $ and therefore having infinite variance. The sum or arithmetic mean of such random variables will converge to a stable distribution $ f(x;\alpha,0,c,0)$.

Most return data generated by financial markets is clearly not normally distributed. But then, I have never come across an asset with infinite variance. Models are abstract realities, they are always wrong in the sense that they do not exactly reproduce observable reality. The abstraction is the result of the assumptions made. Assumptions are chosen to yield a model which is relevant for a particular purpose or clarify a particular point. Assuming normally distributed (continuous or discrete) asset returns is wrong, as wrong as assuming power-law tails. But I do not see any relevant purpose or unique reason to work with power-law tails. If you do, I would be interested in learning more.

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