# Poisson Distributions, from James Sogi

December 31, 2008 |

David Stirzaker's Probability and Random Variables, a Beginner's Guide discusses Poisson distributions  which can be used to determine the probability of some rare independent event in a specific period of time. At higher n it approximates the normal distribution. Applied to the market the issue can be stated, during the time that I am long, what is the probability of a crash in the magnitude of 10-10-08 or over 100 points high to low or greater than 7 standard deviation moves. Of 13 such occurrences since 94, 9 were this last October and November. The prior occurrence was in 2000. The clustering reflects that the events are not truly independent negating one of the assumptions of the Poisson model. The clustering seems to be the more important survival factor rather than the probability distribution model. Thus once these outliers appear, the probability of another occurring might be better modeled by a cluster model rather than a distribution curve. It was shown by Rama Cont of the Ecole Polytechnique in 2005 that volatility clusters. We've argued about the effect of these outliers on the distribution curves, but the two regime analysis might be safer. There are other solutions obviously.

Looking at a search for cluster probability models;

2. A cluster-based probability model has been found to perform extremely well at capturing the complex structures in natural textures (e.g., better than Markov random field models).

Having said this, the vacation trade is markedly dampened, and I wonder if the volatility is wearing off as time passes. Some argue for another volatility event in early or mid 2009 but that would not really fit a cluster model of volatility. There seems to be a nice wall of worry to climb and some nice symmetrical drops and gaps in September trade for the elephants to mirror.

Anecdotally, at the store the other day, I heard a lady loudly commenting on how many private jets were lined up on the runway. I'm heading down to the beach after close and will do the jet count for this year. The indicator didn't work too well last year. I'll include my friend's private jet that comes in on the third. Which brings to mind Monty Python… "I'm not dead yet!"

## Bruno Ombreux adds:

There is one thing with the Poisson distribution. It converges to Gaussian but not uniformly. Tails converge more slowly than the body. One needs quite a few observations in the tails to reach normality, but there are not a lot, else this wouldn't be tails.

This implies we can't even be sure that these events are outliers. Fat tails may not be fat. They may look so only because we don't have enough observations.

Add to this regime changes, volatility clustering etc… The only solution I can think of is not to use too much leverage.

## Adam Kretschmann adds:

Sharp Sports Betting by Stanford Wong has a good chapter on Poisson distributions for those less mathematically inclined.

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