# 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.

• skew t distribution? commonly applied in garch modelling for instance. – user2763361 Nov 7 '13 at 16:55
• why would you expect stock returns to follow such a simple model? The world is far more complicated, to the point where it's impossible to exactly model returns. This is not a problem however - since you only need to model the aspect of the returns that you care about. Modelling anything is simply an interpolation scheme. Where you can interpolate situations inside the scope of the model, and anything else becomes an extrapolation. So, sure, you can probably model stock returns using a Cauchy distribution - but only where what you're looking at can be modelled using such a model. – will Dec 23 '19 at 22:09

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.

• 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 wrote a proof deriving the distribution of returns for all asset and liability classes. If there were no budget constraint, limitation on liability and liquidity had no cost, then you can prove the distribution of returns follows a Cauchy law. The budget constraint triggers skew that becomes larger and larger the higher the return. The reason is that 100% of the population would accept IBM stock at zero dollars per share, while no one would pay an infinite price. The denominator has to exist because you bought it, but the numerator does not. The probability of a trade declines as the sales price increases and returns can be thought of as the probability of a specific return, given a trade happened, times the probability a trade happened.

Nonetheless, until you get to the upper ends, the Cauchy distribution is a reasonably good fit for returns on securities that are going concerns. Firms that are going to merge or become bankrupt have different distributions. I also empirically tested this and solved the option pricing model as well.

The easiest thing is to go to my author page. https://papers.ssrn.com/sol3/cf_dev/AbsByAuth.cfm?per_id=1541471

Start with the article "The Distribution of Returns," it has everything from stocks to bonds to antiques to accounting ratios. Then go to the article on why practitioners should use Bayesian methods, then if you are interested you can look at the empirical test. Finally, that leaves you an option pricing model. It is not THE option pricing model, but a option pricing model. The article discusses realistic extensions. I am preparing two more articles for summer. One extends stochastic calculus to cover macroeconomics and finance as it does not hold under the current assumptions. The second discusses how to build a subjectively optimal portfolio. It takes little work to realize that there cannot exist an objectively optimal portfolio. A person with a mortgage and a child going to college is facing differing constrains than a pension fund.

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.

• 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

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.

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.

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.

There seems to be a consensus that the distribution of daily stock returns lies somewhere between the Gaussian Distribution and the Cauchy Distribution[1][2]. While the former will fail to encompass high volatility events, the latter typically exaggerates their occurence (tails are too fat).

I recently implemented a script for calculating european option prices using the Monte Carlo method for both of these distributions, that when compared to actual market prices resulted in underpricing in the Gaussian case and overpricing in the Cauchy case. You can check the code and plots in the link below: