# Consensus on Cauchy distribution for stock prices

What is the general consensus for using a Cauchy distribution to model stock prices? I can't find much after researching online and wonder if it has been tried and discarded.

My motivation is to find a distribution for the stochastic process governing infinitesimally small stock price movements $\Delta W_t$. The standard process used is the Wiener process depending on a normal random variable $\epsilon$ i.e. $\Delta W_t = \epsilon \sqrt{t}$. This results in the problem that resulting prices are normally distributed, but it is well known that stock prices have heavier tails than that.

In fact it seems that if $\epsilon$ follows any finite variance distribution, it will result in normally distributed prices by the CLT.

I am therefore looking for a stable distribution to model stock prices and the Cauchy immediately came to mind.

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skew t distribution? commonly applied in garch modelling for instance. – user2763361 Nov 7 '13 at 16:55

Maybe this could also be a comment but I think an it is not possible to answer this question with a 'yes and here is how you do it'.

It has been tried, e.g. by me for a university research project. In this research we focused primarily on aggregation of returns and the main problem was the tractability of the resulting distributions and expressions, also when using, for example, the students $t$. Note that the idea is rather obvious and lots of people must have played with it. If it works well, we would probably know by now. I guess that's the reason our professor was immediately skeptical about this approach and I can only say he was right.

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Thanks for this. Was your conclusion was that there was no stable distribution to modelling stock prices or that the Cauchy was unsuitable? (Or something else entirely...) – SlowLearner Nov 7 '13 at 12:03
Except when $\alpha=2$ the variance of a stable distribution is infinite. This fact makes most common methods of analysis impossible. So my conclusion was all of them except the normal are unsuitable. – Bob Jansen Nov 7 '13 at 13:03
Hi Bob, thanks for the answer. I'm not so worried about whether common methods of analysis would work or not, for example I can just Monte Carlo sim, although I take your point "If it works well, we would probably know by now." I'll have a look into it anyway, can't do any harm. – rwolst Nov 8 '13 at 17:22
We found that MC doesn't really work because of the variance of the distribution you're simulating: you're pretty much guaranteed that your sample contains outliers and so even a simple statistic such as the mean will fluctuate wildly between samples (as it must). – Bob Jansen Nov 8 '13 at 18:21
Basically what I'm saying is this: don't repeat my mistakes. Playing solitaire is probably just as productive ;) – Bob Jansen Nov 8 '13 at 18:22

The consensus nowadays is that stable distributions are not a well fit, although they do possess heavy tails. In particular Cauchy has too fat tails. The reasons for this are disparate, however the first that comes to mind is that empirically longer horizons show a decrease in tail thickness, approaching normality for 1-year returns (although this has been contested e.g. by Taleb). Stable distributions by construction do not reproduce such effect; tempered-stable distributions have been introduced to adress such problem, however it's a hack that could be avoided by using other distributions in the first place. You can check the Levy family for some better alternatives.

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Those are important issues. I've tried truncated cauchy and stable as an alternative to their full versions, but I'm not familiar with tempered-stable. – John Feb 21 '14 at 18:48

I have not tried it myself but if i may be allowed to forward you to a link of a particular filter sold as an indicator called the Jurik MA. If you check the link, there is a quote where they mention 

What we mean by a random walk is a time series produced by a cumulative sum of 5000 zero-mean, Cauchy distributed random numbers.

Also this is supposed to be one of the better moving averages. So i guess this is a successful use of the Cauchy distribution. Apart from this I guess its mostly found in theory than in practice.

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I think the Cauchy distribution would result in extreme values that are much too large for a stock (asset) price returns. The stable distribution, $S(\alpha,\beta,\mu,\sigma)$ would likely fit better since it can approximate the Cauchy, normal, t etc. (with skewness) based on its parameters: characteristic exponent $\alpha$, skewness $\beta$, location $\mu$, and scale $\sigma$.

Regarding infinite variance problems and use of other distributions, the Laplace and Logistic distributions seem to fit many log-returns for various assets. Indeed, outliers are always a problem.

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When we say the stock price is fat-tailed (or Cauchy distributed) , we mean the "return" follows such distribution, which is essentially the ratio between stock price at time n+1 and time n. If you know a litter bit about Cauchy distribution, you know that it is the distribution of the ratio between two i.i.d. normal r.v..

Of course, the stock price of two consecutive days are probably not iid normal rv, so Cauchy is probably too aggressive. But the simple fact that we try to understand the stock price using the concept of "ratio" makes the fat-tail phenomena somewhat unavoidable.

To determine which specific distribution fits better, something like a likelyhood ratio test may be a good choice.

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