0
$\begingroup$

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.?

Improve this question
$\endgroup$
3
  • $\begingroup$ Are you interested in discrete time or continuous time? Or both? $\endgroup$
    – Richard Hardy
    Commented 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
    Commented Dec 13, 2021 at 9:51
  • 1
    $\begingroup$ Accepted. Sorry, I assume I was just waiting for more answers and then forgot about the thread. $\endgroup$
    – Tristan
    Commented Apr 18, 2023 at 17:25

2 Answers 2

Reset to default
2
$\begingroup$

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.)

Improve this answer
$\endgroup$
5
  • $\begingroup$ 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/… $\endgroup$
    – nbbo2
    Commented Dec 13, 2021 at 14:36
  • $\begingroup$ Are there well-known public indices you'd recommend that measure tail risk? Something similar to CBOE SKEW. $\endgroup$
    – KaiSqDist
    Commented Jun 22 at 19:39
  • 1
    $\begingroup$ @ KaiSqDist: you can calculate SKEW for any asset by its logics -- it still serves for analytical purposes, not forecast, from my viewpoint... $\endgroup$
    – JeeyCi
    Commented Jun 23 at 5:50
  • $\begingroup$ 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 $\endgroup$
    – JeeyCi
    Commented Jun 23 at 5:59
  • $\begingroup$ Thanks @JeeyCi, I will probably work on some of these computations myself. $\endgroup$
    – KaiSqDist
    Commented Jun 23 at 19:25
1
$\begingroup$

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.s. Pareto optimality (or multi-objective optimization)

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

Improve this answer
$\endgroup$
1
  • $\begingroup$ BTW, ES - seems to be the classical approach $\endgroup$
    – JeeyCi
    Commented Nov 29 at 14:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.