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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.
Thanks for the comment, This paper has intrigued me since I heard Mikosch present it at the Extreme Value Theory Conference in Colorado in 2009. I can visualize it working in the domain of attraction of a stable distribution, but the proof gets complicated when you get to lighter power tails and how they should be scaled to have the limit work, and I am not sure that I fully understand it.
If you take a lighter power tail, say a StudentTDistribution, with summation the power tail behavior when alpha = 3 moves further and further out on the distribution, until it becomes unlikely to ever happen, I have not been able to simulate the evolution to a Fréchet type of extreme value distribution with various scalings of a StudentTDistribution random walk. The good fit of the Log[hi/lo] measure to the MaxStable/distribution is interesting and hard to explain when you look at higher frequency price data. The strong serial dependence of the measure should be a warning that this is not an independent random variable.
The measure tells us something about the intraday trading, and it can be applied to individual stocks with the same results, but how it evolves is a mystery to me.
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].
Michael, I fooled around with a Stable 2:3 order statistic distribution for a while about 9 years ago, which has an alpha parameter range [0, 4], If you think it has any merit I can work more on it, see attached. But I have kind of moved on. My current thinking is that the whole idea of market returns arising from some sort of random summation process is flawed. I suspect that trading behavior and now deterministic algorithms drive prices toward local minima and maxima, which generate a distribution of returns that have power tail appearance when you try to describe them as a distribution, but they have also have serial dependence. The whole process I think has a recursive fractal behavior. I am collecting minute intraday data now. I have about a year's worth and may be able to post an example soon.
The attached script doesn't have much documentation, but I can give you more if you are interested in this distribution.
An even simpler volatility measure would be:
Just in case that anybody would be afraid of Log[ ] ;-) .
That's neat, and it could have been published in newspapers with daily high and low stock prices since the 1920s, if anybody had thought of it.
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.
Michael, the reason I was at the Extreme Value Analysis conference where I met Tom Mikosch was to present a lognormally scaled stable distribution, The mathematics of the combination is that the stable tail eventually shows up and dominates as the distribution tails get longer. So if you have enough data, you should eventually be able to find a power tail in the stable domain of attraction. I have attached part of a notebook that was written in version 7; none of the code will work, It was written before Mathematica had stable distributions and used a MathLink to a program by John Nolan. The notebook with the whole presentation was too big to upload so I cut to the section with the distribution, but you can see the pictures and the math is all there.
You probably have the connections to get data from some place like WRDS and put some millisecond data for something like the SPY ETF in the Wolfram Data Repository. I would love to play with accurate data say from 2020 to present. I am exploring some data I have been scraping from the NASDAQ website which appears to be tick by tick data probably traded on their system; the heaviest tail I can find is from 2.6 to 3.1..