Thanks Robert for the simulation notebook and Stable 2:3 code.

I agree with you about trading bots forcing the intraday prices into local maxima and minima, creating a see-saw effect when it used to be more like wavelet cycles. The Stevanovich center at the university of Chicago had a series of 4 day conferences in 2017-2019 devoted to studying intraday trading. Most results confirmed massive use of trading bots and that trading at frequencies less than 1/4 a second gave no significant advantages. So much for all the hype about millisecond trading! I saw a researcher from the SEC give a presentation to this effect and then stated at the end of his talk how great millisecond trading was, even though the SEC research showed the opposite effect! It was obvious to everyone that the SEC was a toothless organization completely under the control of corporations, rather than the other way round.

So I agree that there is definitely serial dependence in the intraday prices. Perhaps another possibility would be to consider a ParameterMixtureDistribution where the parameters are taken from Fractal BrownianMotion with H>0.6 and heavy tailed distributions. The parameters would be time (for serial dependence) and the Hurst index H.

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.