
Hello, I am posting here the beginning of a technical treatise. Comments & corrections are welcome (there are plenty of notational typos as some were cut/paste from papers), but please have the elegance to avoid the ad hominem in this post (you can put it in another post if you wish).
https://docs.google.com/file/d/0B_31K_MP92hURjZxTkxUTFZnMVk/edit 





You used "powerlaw" twelve times, and "power law" only four. The latter is the accepted expression. 



 

jslade


Total Posts: 1217 
Joined: Feb 2007 


As a meta criticism, I would like to see this sort of reasoning applied to online techniques. If you want a specific example, the CesaBianchi/Lugosi approach to sequence prediction.
http://www.econ.upf.edu/~lugosi/longtail.pdf
Practically speaking, everyone knows that fat tails can muck up dumb econometric risk models. Maybe this is "big picture" very important, but publishing yet another technical (or non technical) paper on the subject probably won't budge folks. I dare say, examining where online techniques break down; even dumb ones like moving median filters, is a more interesting scientific question for people who care about making models that work in the field. It may also have some useful general impact: even when fitting dumb estimators on historical data, people *think* in online terms.
Another idea: demonstrating antifragility structures using a model. I'd happily move to Switzerland if I could figure out how to do so. If you're going to convince policy makers to make the world more like Switzerland, showing them which pieces are important, and which pieces can be left alone seems a useful exercise. I think it's ultimately futile, and I should simply learn more German and become a mail order bride, but the pointy heads like models. 
"Learning, n. The kind of ignorance distinguishing the studious." 


Rashomon


Total Posts: 214 
Joined: Mar 2011 


Pages 68. From my vantage, it's very common to point out that p values are misinterpreted. And that OLS doesn't converge to the right answer in all cases. These problems are well known among the cognoscenti. (So why still teach it? Because it's the closest projection, or maybe because of the history.)
• You also mention that you are going to tackle the robust methods but I didn't see it.
• It would be nice, also, to see some links to econometric papers that don't replicate.
• You'd do better, in my opinion, to quote specific papers rather than refer to "the standard general way econometrics does things". (And don't mention the charges of egomania. Fume in private; placid in public.)
• As an example of "common to nob on bad statistics": Deirdre McCloskey has assembled a catalogue of articles from top econ journals (during the 80's I think) that exhibit the errors above.
Page 16. "What are left are power laws". In what sense is that all that's left?
Page 17. s/unitary/unit/ standard deviation.
Equation esthetics. Formulae should be typeset larger and have more spaces separating characters with higher order of ops.
Thanks for opening up to feedback and good luck with the ongoing process. Also thanks for roughly 1000tupling the public interest in stats. 
"My hands are small, I know, but they're not yours, they are my own. And they're, not yours, they are my own." ~ Jewel 



ThirdEye


Total Posts: 36 
Joined: Oct 2006 


on a page 46 u say 1 Std is 68.2%, it is actually closer to 68.3% (use simply 68% or 68.3%)
you have 68.2% several places
page 15 minor spacing fix: "1 0^4sample" should be fixed to "10^4 sample"
"On the Difference Between Binaries and Vanillas with"
why are u still waisting time on finance?
Nice paper! 




In Fig 1.7 could you Nassim or anyone else explain how you get the volatility graphs? I cannot reproduce them. In the second bar graph do you take the standard deviation of 20 consecutive daily log returns and multiple the result by sqrt(252/12)? How do you get the first graph? I'm probably missing something basic. Anyway, thanks.





temnik


Total Posts: 204 
Joined: Dec 2004 


Nassim,
It seems like what you want to say to your students say boils down to these two points:
1. XIth Commandment: "Whatever you do, don't get caught" 2. Risk management and risk measurement serve to satisfy regulatory (statutory) needs  not practical needs.
This captures the aspects of the pilot of your proverbial plane being both practical and philosophical.

Mon métier et mon art, c'est vivre.




Thanks. I added about 30%, mostly about CLT and law of large numbers under fat tails.
The point is not "OLS doesn't converge to the right answer in all cases". It is that under fat tails it NEVER converges to the right answer. 





Thanks again monthly vol graph:daily dev *Sqrt[252]. Vol of vol: non overlapping std of std. But whatever you are doing, you shoudl be getting the same shape. 




@NassimNTaleb" "monthly vol graph:daily dev *Sqrt[252]. "
Thanks. That is daily st dev, annualized, AFAIK. Where does the "monthly" fit? Is that maybe a "rolling annualized st dev calculated over a month's worth of daily returns"?
If this is the case, how many data points of daily returns do you use for calculating the standard deviation at each point? I used 20 and I get close to your graph but there are differences.
Regardless, a "monthly vol graph" does not quite correspond to a "graph of the rolling annualized volatility over a period of a month", AFAIK. But if they match, it is OK but for now they do not exactly. I used data from Yahoo for S&P 500.





 


Hi Nassim,
One thing I would like to see you address is how skewness and kurtosis affect diversification. In the gaussian world, diversification is the great free lunch and you get something like this:
But in the nongaussian world, life is not as good as in the gaussian fairy tale. And let's not even talk about correlation.
I have not seen this specific point addressed in the literature, and I think it would fit nicely with the rest of the material you presented, in the spirit of "things that you've been taught that ain't so" 
Inflatable trader 



Baltazar


Total Posts: 1775 
Joined: Jul 2004 


There is the start of an answer to that in "Bouchaud and Potters" Theory of financial risk and derivative pricing.
" for the sum of independant random variable, all the cumulant simply adds (..) Normalized cumulant thus decay with N (number of asset) for n (order of the cumulant) >2. The higher the cumulant the faster the decay \lambda^N_n >N^{1n/2} Kurtosis, defined as the fourth order normalized cumulant thus decreases as 1/N"
They note however further that for case of power law (instead of gaussian) the cumulant diverge and the sum of power law still behaves as a power law (page 2223). They stress that the central limit theorem doesn't say anything about the tails of the portfolio, just about the "center".
Further in the book they discuss portfolio theory using power law distribution while minimizing the VaR. 
Qui fait le malin tombe dans le ravin 



Thanks a million Nero Tulip. I think that my new Chapter 2 has the basic material: I look at the Mean deviation of the sum of random variables and show how slowly the Law of Large Number operates, even if sigma finite. So will replicate your graph for diffferent alpha tails. 





That will be an interesting graph, looking forward to seeing it.

Inflatable trader 



I just did it numerically with a few points. Very time consuming: take the characteristic function^N, then invert numerically the fourier transform. Quite shocking (unexpectedly), but amazed at how slowly diversification operates. Will post soon as soon as I get to run the whole thing. 




Cheng


Total Posts: 2869 
Joined: Feb 2005 


I just did it numerically with a few points.
That is precisely the way a mathematical proof works. Especially when we talk about concepts like . 
"Sun's going down / Moon's rising high / We'll pay you Bodom beach terror"




@Nassim: you didn't expect that diversification would work slower with fat tails? 
Inflatable trader 



Baltazar


Total Posts: 1775 
Joined: Jul 2004 


Maybe this reference will be helpful http://finance.martinsewell.com/stylizedfacts/distribution/MantegnaStanley1994.pdf
They consider the convergence of a sum of truncated levy toward gaussian and show that for some parameters choice you need a very large (10^4) sum to get proper convergence.

Qui fait le malin tombe dans le ravin 



merci balthazar, my chapter 2 has results for all distributions. Rate of convergence by CLT for all, pulling a Lambert function product log. It is very slow. 





Nero, of course, but seeing is believing. And we can show that an alpha close to 1 needs 400 times the sample of an alpha of 300 (for the LLN). 



Baltazar


Total Posts: 1775 
Joined: Jul 2004 


Nassim: happy to be helpful. I would reiterate the proposal to look into Bouchaud and Potter. A bit further in the text they explain convergences toward gaussian or Levi and take the example of sum of exponential distributed rv. They explain that the convergence toward gaussian distribution is mostly valid toward x=0 and even give formula to link the number of rv added and the range of x for which the gaussian approximation will be valid. There is also have a worked out example for powerlaw distribution and truncated levi distributions. I must admit I am not familiar with Lambert function, I will educate myself... 
Qui fait le malin tombe dans le ravin 




@Nassim: sorry, I got confused because you wrote: "Quite shocking (unexpectedly)"
Anyways, why did you pick alphas of 1 and 300?

Inflatable trader 



Hi I know the bouchaud and potters very well. They see the crossover at Sqrt[n Log[n]] for the tails of a powerlaw to escape CLD (no longer Gaussian). I generalized it to all tails and got a different result. Chapter 2. Thanks.




