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I was wondering what the most common, or most popular, ways - in both academia, and industry - there were to model the fat tails of volatility in asset prices changes.

I am presuming a basic Brownian motion random walk, is not what is used, because it will not replicate fat tails. Is that correct? Or am I wrong, and in most cases, a basic Brownian motion is "good enough"? What are more advanced methods that are used, whether it be in terms of stochastic calculus, statistical methods, etc.?

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  • $\begingroup$ Are you interested in discrete time or continuous time? Or both? $\endgroup$
    – Richard Hardy
    Dec 13, 2021 at 8:46
  • $\begingroup$ I would be interested in either, whatever methods are in common use. I do not know enough about the industry to know what is currently in popular use, or in which circumstances you might prefer one over the other. $\endgroup$
    – Tristan
    Dec 13, 2021 at 9:51
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    $\begingroup$ Accepted. Sorry, I assume I was just waiting for more answers and then forgot about the thread. $\endgroup$
    – Tristan
    Apr 18 at 17:25

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For the case of discrete time, consider a GARCH model with standardized innovations that follow a Student-$t$ or another (somewhat) heavy-tailed distribution. The dependent variable will have a tail heavier than that due to the GARCH model. (The model generates heavier tails than present in the distribution assumed for the standardized innovations.)

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