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I have read the Wikipedia article on volatility, and Nassim N. Taleb's Incerto, and found two statements attributed to Mandelbrot's views, which appear to be in contradiction.

  1. Taleb (who was mentored by Mandelbrot) writes that the standard deviations of the returns of numerous instruments are infinite. He also mentions that the Lévy alpha-stable distributions are better descriptors of returns, and their variance is infinite when the stability parameter $\alpha<2$.
  2. In the Wikipedia page, it says

Some use the Lévy stability exponent $α$ to extrapolate natural processes: $$\sigma_T = T^{1/\alpha} \sigma. $$ If $α = 2$ the Wiener process scaling relation is obtained, but some people believe $α < 2$ for financial activities such as stocks, indexes and so on. This was discovered by Benoît Mandelbrot, who looked at cotton prices and found that they followed a Lévy alpha-stable distribution with $α = 1.7$. (See New Scientist, 19 April 1997.)

My question is: In item #2, if $\alpha<2$, doesn't it mean that $\sigma$ doesn't exist (in light of item #1)? How did they arrive instead at a different scaling exponent?

Any help is greatly appreciated.

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I don't think the claim that "Lévy alpha-stable distributions are better descriptors of returns" is universally accepted.

While Mandelbrot (and others before him) has correctly identified non-normality of returns in financial time series, he wasn't really equipped at the time (1963) to pursue its real nature. Appropriate models appeared only much later, with ARCH/GARCH (1982) and then stochastic volatility.

Empirically, returns do seem to have first and second moment, and possibly more. For example R. Cont in his review "Empirical properties of asset returns" says about heavy tails:

the (unconditional) distribution of returns seems to display a power-law or Pareto-like tail, with a tail index which is finite, higher than two and less than five for most data sets studied. In particular this excludes stable laws with infinite variance and the normal distribution. However the precise form of the tails is difficult to determine.

Also, it's worth noting that while unconditional returns are non-normal, returns scaled by volatility are much much closer to N(0,1), which shouldn't happen with stable distributions. See eg. "The Distribution of Stock Return Volatility" by Andersen, Bollerslev, Diebold, Ebens (there is also a paper "Exchange Rate Returns Standardized by Realized Volatility are (Nearly) Gaussian" by similar set of authors):

the daily DJIA returns ... have fatter tails than the normal and, for the majority of the stocks, are also skewed to the right. Quite remarkably, however, ... all of the thirty standardized return series ... are approximately unconditionally normally distributed. In particular, the median value of the sample kurtosis is reduced from 5.416 for the raw returns to only 3.129 for the standardized returns.

There is a huge body of research on the distribution of returns, out of which I've only quoted two, but I think few people focus on stable distributions nowadays. Perhaps the field has moved from modelling returns to modelling volatility and correlations/copulas.

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