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Since Mandelbrot, Fama and others have performed seminal work on the topic, it has been suspected that stock price fluctuations can be more appropriately modeled using Lévy alpha-stable distrbutions other than the normal distribution law. Yet, the subject is somewhat controversial, there is a lot of literature in defense of the normal law and criticizing distributions without bounded variation. Moreover, precisely because of the the unbounded variation, the whole standard framework of quantitative analysis can not be simply copy/pasted to deal with these more "exotic" distributions.

Yet, I think there should be something to say about how to value risk of fluctuations. After all, the approaches using the variance are just shortcuts, what one really has in mind is the probability of a fluctuation of a certain size. So I was wondering if there is any literature investigating that in particular.

In other words: what is the current status of financial theories based on Lévy alpha-stable distributions? What are good review papers of the field?

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there is still alot of research effort being put into alpha-stable distributions particularly in the area of risk management. – pyCthon Apr 4 '13 at 2:17
up vote 20 down vote accepted

I recently read "Modeling financial data with stable distributions" (Nolan 2005) which gives a survey of this area and might be of interest (I believe it was contained in "Handbook of Heavy Tailed Distributions in Finance"). Another more recent reference is "Alpha-Stable Paradigm in Financial Markets" (2008).

I'm not aware of anything covering "risk of fluctuations" and this is still certainly not at the center of the field (i.e. most theory still includes some version of Gaussian or mixture of Gaussians). Would also be interested in other references.

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Nice, for now it's the best answer, but I'll wait a bit for more to come in. – Raskolnikov Feb 2 '11 at 8:41

There are several application of Lévy alpha-stable distributions to finance, especially in insurance and reinsurance. I believe that Embrechts-Kluppelberg-Mikosh's "Modelling Extremal Events for Insurance and Finance" is still an excellent reference. However, in the modeling of stock prices, this line of research is essentially inactive. The reason is that there is conclusive evidence that stock prices have finite second moments (for a survey, see Taylor's book or Cont's nice survey. This essentially rules out all stable distributions except the gaussian one. Stochastic volatility models using mixture of normals for the unconditional distributions and/or diffusion/jump processes are far more popular.

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Very nice as well! I didn't think it was already settled that Lévy-stable distributions are irrelevant to stock prices. What about the price of commodities? After all, it's closer to the kind of risks that occur in insurance and it also was the very first example Mandelbrot studied (cotton prices). – Raskolnikov Feb 8 '11 at 9:21
When I draw a log-log plot of stock prices, and measure the slope in the (apparently linear) tail region, I get values clustering around -3, thus not Levy-stable. Googling around I find this: Non-L´evy Distribution of Commodity Price Fluctuations which finds similar results for commodities. – Amos Newcombe Dec 6 '11 at 14:28
Taylors book link is broken – pyCthon Apr 4 '13 at 2:12
also stock prices and stock returns are two different things – pyCthon Apr 4 '13 at 2:13

I am still a beginner to this topic, and have been working through Cont and Tankov's textbook Financial Modelling With Jump Processes (2003), which is a fairly elementary treatment of the subject. I think a revised second edition is to come out later this year.

One interesting area of applications that has become more prominent with a recent wave of papers are those that use Bayesian methodology to evaluate stochastic volatility, for example see: Jacquier, Polson & Rossi and Szerszen among others.

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Not quite what I was looking for, but still interesting. +1 – Raskolnikov Feb 2 '11 at 8:42

I just ran across an interesting presentation comparing the effectiveness of Normal, Cauchy, and Student's-t distributions in modeling the S&P. It concludes that the normal distribution underestimates extreme movements, the Cauchy overestimates them (although a comment on the presentation points out that Mandelbrot used different parameters than the author did), and concludes that the student's-t is a fairly good fit.

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Fat tailed distributions have extreme values that follow a Frechet distribution. Try calculating VAR using method outlined in Tsay's Analysis of Financial Time Series.

If you are trying to determine a good distribution for stock price in a non academic setting use hyperbolic-secant. It is a wrong answer but it is much easier to fit then a Levy and it gives better answers then normal. Hyperbolic-Secant is bell shaped so many time series techniques "plug" right in.

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The hyperbolic-secant has finite mean and variance, what is the connection with general Lévy distributions? Your answer is not really addressing the issues in my question. It learned me a couple of new distributions though, so I'll not downvote it. – Raskolnikov Apr 10 '11 at 8:29

For starters, one can argue they provide a better fit to the distribution of asset returns than a Normal distribution simply because stable distributions allow for more degrees of freedom.

I had a discussion with a very well-known financial mathematician on the subject of using stable distributions as the return process for derivatives pricing, and his first comment regarded indeterminacy. The moments exist only in special cases. Interpreting or manipulating any point estimates or moments of the distribution that don't exist may yield results from a computational perspective, but theoretically, you may encounter problems. For example, how can you empirically estimate and interpret the volatility of a stable distribution when you know it has infinite variance?

I'm surprised John Nolan's work is not referenced here yet. I found his books and papers to be quite enlightening with respect to the applicability of stable distributions in finance.

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FinAnalytica Inc www.finanalytica.com has a multi-asset class commercial implementation including fitted classical tempered stable distributions and fitted skewed t-distributions (for lower frequency data) in its software named Cognity. You should talk to those guys.

Their backtests do all the talking...risk forecasts from left tail all the way through to the right tail are way better than Gaussian, Cauchy, more rudimentary Levy aplha-stable and of course the classical historical approaches.

They also offer GARCH models (big improvement over the ubiquitous EWMA model) to capture vol clustering and a fat tailed copula to model dependence and most importantly the joint tail dependencies which no linear model can handle.

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Hi Hank, welcome to Quant.SE! Could you please disclose your association with FinAnalytica? – Bob Jansen Mar 24 '15 at 8:05

I think, the use of stable distributions in Finance (and, probably, in Economics) is a big mistake. It is essential that the intuitive fact that the stable distributed observations possess a large number of big deviations from empirical mean is not true (see, Lev B. Klebanov, Irina Volchenkova (2015) "Heavy Tailed Distributions in Finance: Reality or Mith? Amateurs Viewpoint", arXiv:1507.07735v1, 1-17 and Lev B Klebanov (2016) "No Stable Distributions in Finance, please!", arXiv:1601.00566v2, 1-9. ) In financial data one observes many large deviations (in terms of empirical variance) from empirical mean value. It is impossible for Gaussian distributed observations. However, it is impossible for observations with stable distribution, too.

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If certain broad assumptions are correct (eg, asset prices are continuous in time, markets are efficient) then asset returns must follow a Levy process. Both the Gaussian and Stable distributions are subsets of Levy processes. The question should not be whether Gaussian or Stable is better. Neither are adequate (in fact, many Stable distributions imply infinite call prices; since call prices are finite we can safely say that these Stable distributions are not the process that generates asset returns). Instead choose a more general Levy process (eg, CGMY) and build models from there.

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