Aug 1, 2011

Stocks are not Wiener Processes.

The controversy over the distribution that best fits the change of stock prices is apparently fairly recent. In turns out that stocks are actually Levy processes. I was stumbled across this fact while trying to fit the data onto the normal distribution and failing. Having read somewhere that the differential in stock prices is normal (see Black Sholes model), I assumed it would be at least a reasonable fit. After failing, I programmatically tried a ton of random distributions till a suitable fit was found.

Here is a histogram of the daily closing differences for General Electric. On top of it is superimposed a fit of the histogram with first the Normal distribution and then the Cauchy distribution. The fit was found using maximum likelihood estimation.

One of these is a better fit. Which do you think?

It blows my mind how terrible of a fit the Normal distribution is - why did it take Mandelbrot and Nassim Taleb to bring this fact up? In fact, running a simple test for normality on the data shows an incredibly small chance of it being normally distributed.

Of course this is not proof that the differential is Cauchy distributed. For that however you have to simply look at the properties of each distribution. For example the sum of two Cauchy random variables is another Cauchy random variable with parameters equal to the sum of the two previous parameters. Define beforehand the properties of stocks and you can derive the behavior of the distribution which should match it.

Using the normal distribution is fine if you are making some kind of approximation. However whenever an approximation is made, you have to ask how good it will be and under what conditions it fails. It looks like this hasn´t been seriously tried until recently.