Lévy alpha-stable distribution and modelling of stock prices.

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

I asked this question 6 years ago, and in the meantime I came across this little volume:

Lévy Processes in Finance: Pricing Financial Derivatives by Wim Schoutens (2003).

Answer 6

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.

Answer 7

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.

Answer 8

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.

Answer 9

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.

Answer 10

Read: Harris, D.E. (2017) The Distribution of Returns. Journal of Mathematical Finance , 7, 769-804.

The paper derives all distributions that should be present. In practice, these are mixture distributions. The use of stable distributions came about when it was concluded that it probably was not a Cauchy distribution. Unfortunately, the methodology used to do the testing was seriously flawed in most cases. In a separate paper, it has been shown that no admissible non-Bayesian statistic exists that is also computable with regard to the bulk of all finance problems. There are deep issues in probability theory that makes such questions difficult to solve.

The simplified version of the issue goes like this. The distributions involved have two distinct issues. The first is that they lack sufficient statistics, except order statistics of course. As a result, any point statistic must lose information except Bayesian statistics where a cost function has been applied to a predictive distribution. So if you use any form of Frequentist solution, then you must lose information in order to create the measurement. This is not a problem for Bayesian methods as they always use all the information that exists in the sample and the Bayesian likelihood is always minimally sufficient.

The second issue has to do with truncation. For the distributions involved, if there were no truncation, then the median and the mode would be collocated and either could be thought of as the center of location. There is no mean, of course.

This allows the use of order statistics, it may still be the case they are not admissible for other reasons, but order statistics do not lose information other than the information loss that comes from converting things into ranks in the first place. The median is the mean rank and so you can discuss minimum variance unbiased estimators with regard to rank statistics even though the underlying data lacks a variance.

The difficulty comes from truncation. The difference between the median and the mode for US stocks from 1925-2016 is two percent per annum and the resulting difference in measurement of risk results in a four percent understatement of actual risk by using a non-Bayesian method. This can be confirmed by changing the problem from an analytic problem to a geometry problem. This bias is preserved under the transformation to logs and so log measures overstate return and understate risk.

The mode remains the center of location, but the median is now shifted as part of the left tail is missing. This is not problematic for Bayesian methods but catastrophic for Likelihoodist and Frequentist methods.

To heap insult onto injury, distribution tests are subject to the Jeffreys-Lindley paradox. Distribution tests are sharp null hypotheses. The CRSP universe has around sixty million end-of-day trades. The paradox is a theorem that states that all true sharp null hypotheses will be falsified once the sample size is large enough. Most non-Bayesian distribution tests must falsify true nulls if the sample size is large.

Answer 11

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.

Answer 12

It is very natural to use a cut-off version of stable distribution to model the return.

First of all, there is a trading curb that prevent the return blows up. In other words the heavy tail breaks outside a certain interval.

Secondly, jumps are present because there is no way to guarantee continuous trading. Jumps does not rule out a continuous distribution model. One should simply select what fits best, no matter what theory is available, after all, all models are wrong.