MonteCarlo 'Fat-Tails' and Chebyshev's Inequality

there's a nice chapter in Carmona's book "Statistical Analysis of Financial Data in S-Plus" how to fit these distributions to your data using S-PLUS/R. He seems to prefer a kernel smoother for values around the mean, and to model the decay at the tails parametrically.

I'm wondering if people actually use these kinds of models for trading? Anyone?
 
Are you using daily returns in your dataset?

I once heard it mentioned if you analyze other sets of returns (2 day, monthly, etc..) the fat tails disappear. Would that make sense?
 
Can you give us some information how you actually run your simulation?

Quote from tireg:

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 1−k<sup>−2</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.
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Even when using monthly data, October 1987 was still a 4 sigma event.

Quote from kevinmr:

Are you using daily returns in your dataset?

I once heard it mentioned if you analyze other sets of returns (2 day, monthly, etc..) the fat tails disappear. Would that make sense?
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Quote from MGJ:

Just be sure you realize that when you say "Stock market returns do not have infinite variance," you are also saying "Mandelbrot is wrong and I am right."

Here's an early Mandelbrot paper on the topic: http://www.garfield.library.upenn.edu/classics1982/A1982NR91700001.pdf

And here are some quantitative finance people ("quants") talking it over: http://www.wilmott.com//messageview.cfm?catid=3&threadid=30632 Contributor "N" says variance is finite but the group convinces her otherwise.
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I wasn't even thinking about contesting Mandelbrot. It's hard to do that if I don't understand what he is talking about. To me the variance must be finite. Happily I am not the only one :).
On the same page Kurtosis seems to give an explanation, but I'll have to think about it. That definition of infinite, i.e. not integrable, variance, is a bit obscure.
But thanks
 
Quote from kevinmr:

Are you using daily returns in your dataset?

I once heard it mentioned if you analyze other sets of returns (2 day, monthly, etc..) the fat tails disappear. Would that make sense?
More...

No.

To answer your question, I was using returns on a system that generates daily signals. Daily returns would imply that I took the average return and divide it by # of days held, which is an efficiency measure, which is not what I am referring to in the observation.

If I were to group the returns, instead of 'x% per trade' to 'x% per month' the buckets, in essence, would become larger due to larger variance of returns in a given month. Yet this could also provide fat tails... in the instances of MFE and MAE - if I had a horribly down month, or an incredible month. Say hypothetically I typically average 3% a month, but one month I have a huge drawdown of 20% and another I have a large gain of 120%. While these may be due to some concentrated number of trades, the fat tails nonetheless still manifest themselves.
 
i'm no expert but to me the theory of infinite variance is necessary simply because there are no other answers. beyond an undefined boundary returns become progessively random which in theory means that infinite variance exists, where total randomness is reached.

but the real world is not the theoretical world. in theory you can take a trading system that works on a $50 stock and make it work just as well on a 50c stock by simply moving the decimal point. But if you try to actually trade that system on a 50c stock you will get a vastly different result (and a vastly different distribution). so infinate variance is a theory that gives us a control to test variance against. that may seem to give us comfort until you ponder the fact that we are attempting to measure the stability of randomness.
 
excuse my ignorance here, but "fat-tails" are in essence distributions that do not fall to near-zero at their extremes?

im very interested but this is a little over my head... if you could answer or point me to an answer somewhere i would be grateful - thanks
 
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