Hi Robert

An even simpler volatility measure would be:

2*(high-low)/(high+low)

Just in case that anybody would be afraid of Log[ ] ;-) .

Thanks Robert for your insightful reply. I agree that the proof given in Mikosch's paper is very involved and has a nested aspect to it in which each result depends upon another subresult until the required conclusion is reached. My intuitive feel of the result is that volatility is a process that must satisfy the requirement of representing the variability of market processes, which can be represented by stable distributions, hence it must have a max Stable or extreme value distribution like Frechet. It leaves hope that intraday distributions might be expressed in terms of a generalized Stable Distribution where the stability parameter has a wider range of values than [0,2].

A very interesting result! The original paper is even more general in that it considers instances where the jump sizes occur in a separable Banach space. It would be interesting to see if the option valuations under this version of stochastic volatility remain consistent according to the book on this topic by Alan Lewis, who also wrote Mathematica code for his book. I guess it depends upon the extent to which truncated Lognormal distributions behave like heavy tailed random walk distributions in the limit.