MonteCarlo 'Fat-Tails' and Chebyshev's Inequality

interesting,

i used to trade a mean reversion strategy on the SFE SPI200. I would take the difference between opening prices and the 5 day moving average and then draw a histogram of them.

Central limit Theorem applies and the output is approximately normal. I would go contrarian at 3 sigma openings and bet on a reversion to 1 sigma relative to a 5 day SMA. stops were set at 1 sigma. The strategy was quite effective, though increasing volatility in the underlying index has rendered it a tad unusable esp with 1% swings now common on the aussie index.
 
hm. it worked as well for shorts? i doubt it would on US indices.
the problem with the above graphics is that they all are smoothed.
reality shows much more ripples on the surface due to lacke of
data. usually if i recall correctly my last histograms the major
indices and fixed income markets are very dense around the mean,
thus low vola, yet have ripples at several sigma events. and
that is the dilemma: discussing the shapes as such requires
quite a lot of idealisation.
 
yeah it worked decently well enough for both shorts and longs. The one thing that was extremely important to identify when tradign that kind of strategy is whether the market is trending or ranging.

That strat would cane in a ranging market as the prices moved above/below the 5 day movign average repeatadly. on a truly rangy quarter such as in '02 qtr 1 on sfe SPI i made 20 trades and 18 of them were profitable.

But the minute u got to a trending market the large sigma opens would continue to occur and the 5day MA itself would be shifting alot day to day. there would be mean reversion eventually but what u defined as the "mean" would have moved a sh*tload and ud be screwed over.

i stopped trading the strategy due to a very bad black swan, 6% drop in teh aussie market in a day a few yrs back. lost quite a few points due to slippage.

Got burnt bad that day. i probably wouldnt recommend it to many ppl but it served its purpose because it had negative correlation with my main strategy and helped smooth the equity curve.

Probably made about 10% p.a for the 2yrs i traded it.

I used about 3yrs of backdata plotted on a histogram to create it. your right in that the shape does take a little imagination to construct. Most of teh time i just overlayed a standard normal with similar mean and standard deviation over teh Histogram. To hell with the tails.

if the 3yrs have been very biased i.e huge bull or bear market then LN results yields a better visual

Took 2 or 3 stdev as entry point. Look at the charts and guage the last major support/resistance point. If open is within those points then enter trade otherwise no entry. ADX could also be used to guage range/trend.

However i have very little hope itll ever work on american markets, the SPI tends to gap and fade making these sorts of strategies viable, but US markets dont really exhibit that kind of character.
 
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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You are halfway to understanding Nicholas Taleb’s contention that markets underprice the risk of outlier events. If you believe in a normally distributed population as outlined above, which is what undergirds all of modern portfolio theory, then it stands to reason that the markets are priced so as to be in alignment with this (far OTM options being cheap for example), right?

What’s the consequence of this? Simple: The realization that the big money in trading is made off of outlier events where most market participants are on the wrong side of a move (remember, the markets priced them as highly remote so the majority of people will go along with that thesis and never expect the 7th Std Dev to come to pass – they will hold on as it moves against them only to get stopped out by capitulation).

What’s the consequence of that? A ton of countertrend trading strategies based on this thesis, most of which fail to be profitable because they either fade moves too soon (i.e., at the 3rd std dev and it keeps going until the 7th) or they simply don’t have enough entry points to be profitable over time (Taleb himself has said that he bleeds money most years and that the long-term goal is to simply not lose – to end up flat or barely ahead).

What’s the happy medium? That’s anybody’s guess. The closest I’ve seen to it is a trend following strategy that gets you involved based on the statistically probability of the trend itself (say, a breakout from the opening range) and then just holds on until/if/when the underpriced 4,5,6,7th standard deviation events are reached.

Checkout The Logical Trader for more info.
 
Quote from CutsThrough:

What’s the consequence of that? A ton of countertrend trading strategies based on this thesis, most of which fail to be profitable because they either fade moves too soon (i.e., at the 3rd std dev and it keeps going until the 7th) or they simply don’t have enough entry points to be profitable over time (Taleb himself has said that he bleeds money most years and that the long-term goal is to simply not lose – to end up flat or barely ahead).

What’s the happy medium? That’s anybody’s guess. The closest I’ve seen to it is a trend following strategy that gets you involved based on the statistically probability of the trend itself (say, a breakout from the opening range) and then just holds on until/if/when the underpriced 4,5,6,7th standard deviation events are reached.
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Like sugar on the sell off.

Your conclusion is why, though so much of this is entertaining and seductive, there is a brutishly simple, valid approach, as described in your last sentence.

Taleb is just plain weird. Breaking even really is the goal of ... a wimp or a fool. Why not just go drive a truck and put the money in a 401k? At least you'd do more than break even...except in the event of, say, Armageddon or some other "7 sig" thing. Maybe we should all just stay in bed, then.
 
countertrend strategies are a unique balance of trying to avoid the 7sigma events but generating enough signals to make your expectancy positive even allowing for a 7 sigma event

All guys that consistently fade the trend will have teh trend fade them at least once in their trading life. Just ask Neiderhoffer.
 
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