Saturday, February 23, 2008

Financial Price Change Distributions

I thought I'd try my hand at generating data that has some of the properties of real world financial data. My main focus right now is the Russell 2000 index, and so to get the ball rolling I thought I'd take a look at the distribution of daily price changes for that index.

Having read The Black Swan I was expecting that the distribution wouldn't be normal (the red line, using sigma calculated from the data). A fat tailed distribution makes a better fit, here I used the Cauchy Distribution for which there exists an analytically expressible probability density function, although Benoit Mandelbrot identified that changes in cotton prices fit a Levy skew alpha-stable distribution (of which the Gaussian and Cauchy distributions are specialisations) with alpha=1.7.

The key area of the graph is out on the extreme ends of the tails, so lets zoom in on the left hand tail.

We can see Mandelbrot's main issue with the normal distribution. The bumps to the far left in the raw data (the black line) represent highly improbable events if we are taking the data to be gaussian. Although the Cauchy distribution shown here is not a perfect fit (it is too high), it is a far far better model for those rare and extreme price movements. The normal distribution basically leads us to believe they will never happen in the lifetime of the universe, the Cauchy tells us to expect them once every few years. If you're betting your life savings on those rare events not occuring that's one hell of a difference.

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