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..

Attachments:

LNS.nb

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.

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.

Attachments:

MarketThoughtExp...nb

S23Distribution.m

Michael,

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[3], 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[3] 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.

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