# What are the most common methods to model fat tails in the changes of asset prices?

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.?

• Are you interested in discrete time or continuous time? Or both? Commented Dec 13, 2021 at 8:46
• 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. Commented Dec 13, 2021 at 9:51
• Accepted. Sorry, I assume I was just waiting for more answers and then forgot about the thread. Commented Apr 18, 2023 at 17:25

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.)

• If you want a reference for "GARCH Student t" it is Bollerslev (1987) in Review of Economics and Statistics, 69(3), p. 542-547. jstor.org/stable/1925546 as discussed by Richard Hardy here stats.stackexchange.com/questions/500445/… Commented Dec 13, 2021 at 14:36
• Are there well-known public indices you'd recommend that measure tail risk? Something similar to CBOE SKEW. Commented Jun 22 at 19:39
• @ KaiSqDist: you can calculate SKEW for any asset by its logics -- it still serves for analytical purposes, not forecast, from my viewpoint... Commented Jun 23 at 5:50
• in general, predicting Volatiliy is not very usefull because is non-system-risk vs. delta-risk as system-risk that can be submitted to hedging... though sometimes appear new instruments like e.g. S&P 500 Dispersion Index (DSPX℠) and asserts - such as Volatility-Quoted FX Options Commented Jun 23 at 5:59
• Thanks @JeeyCi, I will probably work on some of these computations myself. Commented Jun 23 at 19:25

see Extreme Value Theory for Risk Estimation coded to find VaR & C.I. for extreme value risk estimates of financial time series - example for stocks is given by link... and look through Regression Estimator for the Tail Index as non-parametric approach... and others at 19.Strategies for Modeling and Predicting Heavy-Tailed Data

Generalized Pareto distribution is a very important distribution in the extreme value investigation

there is a set of Estimation procedures, such as the maximum likelihood (ML), the method of moments (MOM) and the probability weighted moments (PWM) method

can see here implementation in R or scikit-extremes-package in Python or phat-tails-package or example of Hill estimator use... Though, data-driven tail index finding through "minimizing the asymptotic mse do not perform well in finite samples"... So, algorithmization can vary for fitting Pareto distribution with real data, and model-free approches perhaps can benefit compared with pareto modeling or change distribution estimation with some sorts of indexes to make analytical aims easier to achieve algorithmically.

In general, there are 4 Ways to Quantify Fat Tails

• Power Law Tail Index. ...
• Kurtosis (i.e. non-Gaussianity) ...
• Log-normal's σ ...
• Taleb's κ

Tail Comparison & survival_probability_plots can see here for BTC as part of FergM's work - seems to be interesting approach for analysing purposes of tails comparison!

p.p.s. In general there is vast number of fat-tailed distributions & you should always make assumption about your distribution to consider either for Outliers or for Fat-tails in your analysis, as e.g. here- Identifying multiple outliers in heavy-tailed distributions with an application to market crashes

-- remember the nature (either discrete or continuous) of your distribution & choose the appropriate one for fitting depending on aims of your analysis...

p.p.p.s. extrapolation over the unit of the return period