In running Monte-Carlo simulations, I've noticed that <a href="http://en.wikipedia.org/wiki/Fat_tail">fat-tails</a> often occur, on 10,000 trade scramble runs, I've noticed several 7 sigma events are within the realm of possibility, though statistically, under a normal distribution, these would be nearly impossible. So I decided to do some research and found this great piece from wikipedia.org, which is one way of interpreting the results:<br></p><p style="font-style: italic;">One can always use <a href="http://en.wikipedia.org/wiki/Chebyshev%27s_inequality" title="Chebyshev's inequality">Chebyshev's inequality</a>:</p> <dl style="font-style: italic;"><dd>At least 50% of the values are within 1.4 standard deviations from the mean.</dd><dd>At least 75% of the values are within 2 standard deviations from the mean.</dd><dd>At least 89% of the values are within 3 standard deviations from the mean.</dd><dd>At least 94% of the values are within 4 standard deviations from the mean.</dd><dd>At least 96% of the values are within 5 standard deviations from the mean.</dd><dd>At least 97% of the values are within 6 standard deviations from the mean.</dd><dd>At least 98% of the values are within 7 standard deviations from the mean.</dd><dd>At least 1Ã¢Ëâk<sup>Ã¢Ëâ2</sup> of the values are within k standard deviations from the mean.</dd></dl><br>Without going into the statistical proofs, Chebyshev's inequality allows one to apply standard deviation to these fat-tail events, since it applies to random variables of any distribution. One can create somewhat more accurate levels of confidence using this simple principle.

Yes you have done what 99.99999% of ET members have not done. Actual research... Your findings are substantially correct. There are a lot more plus 6 (or greater) sigma events happening in the markets than would be predicted by standard distribution. Not only are there "fat tails" but the median should be slightly distorted as well depending on the data set you are testing. Now the question is what do YOU do with the result? Knowing this is true, what do you do differently with your trading? I can tell you this, knowing that it is true, I choose to model my rule set using a Linear Regression Model....I realize that I cannot calculate the odds of a specific trade accurately, but then again I don't have to... So while I may or may not be willing to discuss the specifics of my system what I will say is that once you know what the real distribution is, its possible to decide on a set of rules that capitalizes on that "little" difference at the tails. If you think about the significance of this concept, you will eventually figure out that in terms of time between "events" that little difference in the tails aint so little.

I think you mean "of any distribution with finite variance". Plenty of ET-worshipped demigods vehemently insist that the distribution of stock market returns has infinite variance. Two of them are Nassim Taleb and Benoit Mandelbrot (genuflect!). One big reason why they advocate the Pareto-Levy stable distribution as a model of stock market returns, is because this distribution has infinite variance.

This is getting complicated. How can a distribution have infinite variance? I couldn't find much on the internet

In theory there is no difference between theory and practice but in practice there is How deep are your pockets?

Every living organism has an infinite variance. Guess what, when you are dead variance drops to zero! Indeed stocks go to zero so variance is not indefinitely infinite! Variance is finite in our world

Thanks for the input, Steve. I've discussed a similar strategy to what you are implying with a fellow regarding out of the money options; the inevitable fat-tail occurence makes what seems like a lost cause profitable at times. As far as what I will do differently with my own trading, these new distribution interpretations just help me to more accurately project confidence levels. Before I had discounted large sigma events as highly improbable and thought I was in a higher confidence level than I really am... kind of like the example of Russian Roulette with a gun with 1000 chambers... I guess the number of chambers just shrunk. The idea is to apply the new interpretation with respect to consistency over a timeframe.

Deglazenbol and MGJ, thanks for the pointers. I will definitely have a look at these. I've found a nice white-paper on Pareto-Levy distribution for those who want to dig deeper: http://www.gloriamundi.org/picsresources/hfcbarhm.pdf