Chaos Theory and Financial Markets

[Pawel]

Hello,

I have recently started reading book on Chaos Theory within Financial Markets, and it is suggeseted there that assumption of normal or lognormal distribution of returns is not a good idea, that returns have "memory" and generally that Efficient Market Theory was created only in order to prove that statistical tools based on probabililty can be used

Has anybody tried using Chaos Theory tools for modelling ? (eg using Hurst Exponent?)
What do you think about the theory ?

 

[aaronhotchner]

I don't do modeling work, but I have been interested in chaos theory for a few years. The ideas are sound, in particular the idea that markets exhibit fractal behavior (not everyone is operating with the same rules over the same time horizon like independent returns would imply). And, like you mention, returns are historically not lognormal. The actual behavior of market returns, as presented by Benoit Mandelbrot, is a Paretian distribution with infinite variance, so it has no closed form. I haven't read through all of these, but from what I did read I would recommend them:

¨Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York: W.H. Freeman & Co., 1977.​

¨Mandelbrot, Benoit B. The (Mis)Behavior of Markets. New York: Basic Books, 2004.​

¨Peters, Edgar E. Fractal Market Analysis. New York: John Wiley & Sons, 1994.​

¨Peters, Edgar E. Chaos and Order in the Capital Markets. New York: John Wiley & Sons, 1996.​

Sorry about the formatting; I pulled it from a presentation I gave in my chaos class.

 

[Jason Heale]

I've always taken issue with the efficient market hypothesis; its based off bad theoretical assumptions taken from macroeconomics (specifically, rational expectations) that were only really useful because economists wanted their linear models to arrive at unique solutions (without suffering from systematic errors).

Rantings about economics aside, while financial markets may exhibit some interesting chaotic phenomena, I think the real promising avenue of research is in complex adaptive systems, specifically agent-based modeling. There have been several papers demonstrating emergent macroeconomic behavior in multi-agent systems (a quick look up of recent publications by researchers at the Sante Fe Institute should give you a good idea and a decent reading list). The beauty of this approach is that it ties the aggregate macroeconomic behavior to microeconomic principles, something rational expectations (and the EMH) ignores.

 

[ValueSeeker]

How is this different from data mining?

 

[quotes]

Technical Analysis
Regression Data Mining
Chaos Discovery

in ascending order of complexity

 

[aaronhotchner]

Data mining knows nothing about fractal dimensions, for starters.

 

[Cary Ferguson]

I'm guessing it's a physicist who rated your comment dumb. What you mentioned is at large in econ circles. Chaos Theory was the direction economists were taking "things" in the late 1980s. I personally see the future of economics, and therefore financial markets, in the same light that physicists see particle mechanics. That is, we econ people have our "classical" models, as did the physicists. Once quantum mechanics surfaced, the physicists used statistics/data mining to predict the placement of a particle. The solution for economists should therefore be similar. The last piece for economists is behavioral economics which will allow us to use statistics and microeconomics to make macro predictions.

So take our basic models from classic econ and combine them with real moving particles/people... The outcome could have some predictive power.

 

[Piet Paulusma]

I partly agree with you Cary. But, you forget, that econ does not has its own statistics. It heavily borrows from physics. This is a crucial point. ABM are def needed. Humans are just not rational and ident. However we there is still a need for econ to derive its own statistics.
In physics you can observe a phenomena over and over again. In econ not!

 

[Cary Ferguson]

I would argue that econometrics would be econ's version of statistics. And while you're right about human behavior not being predictable, I'd say enough observation may lead to some dynamic predictive power. I especially agree with your last statement about observation... Huge prob for economists.

 

[Jamal Munshi]

I used the Rescaled Range analysis methodology described by Peters and noticed a few problems with it. When I fixed these problems the alleged fractal nature of stock returns disappeared. My paper may be downloaded from the financial economics network maintained by SSRN. Please go to the URL below and click on "Download".
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2448648

 

[tagoma]

Technical Analysis
Regression Data Mining
Chaos Discovery

in ascending order of complexity

Technical analysis?

 

[quotes]

Technical analysis?

Many build algorithm trading only on MACD.

 

[Ian Kaplan]

I used the Rescaled Range analysis methodology described by Peters and noticed a few problems with it. When I fixed these problems the alleged fractal nature of stock returns disappeared. My paper may be downloaded from the financial economics network maintained by SSRN. Please go to the URL below and click on "Download".
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2448648

Quickly looking at your paper I didn't see a citation for Andrew Lo's A Non-Random Walk Down Wall Street, where he discusses long range dependence and the Hurst exponent. I'll also note that the fractal techniques for calculating the Hurst exponent are more accurate than the Rescaled Range method.

 

[Vinayak Maheswaran]

Its the Universal Theory of Chaos all over again!

 

[Ian Kaplan]

I have not read the paper, but the fact that stock market returns display long range dependence is pretty solid. See Andrew Lo's "A Non-Random Walk Down Wall Street" which has a chapter on the rescaled range and dependence.

I am not a great fan of Edgar Peter's work. See http://www.amazon.com/review/R1IUC0NZGBDZ21/ref=cm_cr_pr_perm?ie=UTF8&ASIN=0471139386

I have a web page on Hurst exponent estimation which can be found here: http://www.bearcave.com/misl/misl_tech/wavelets/hurst/index.html

 

[Jamal Munshi]

i read ian kaplan's page on the hurst exponent and found it very thorough and well written. anyone with an interest in this topic should put it on his/her reading list. here is the link:
http://www.bearcave.com/misl/misl_tech/wavelets/hurst/
as for my own work, i have written a further note to the paper the link to which i had posted earlier. the note summarizes the main empirical issues with respect to estimation of the hurst component in stock returns. here is the link to my note
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2564916

ian read it and faulted it for not citing andrew lo. his comment has motivated me to write a review of the andrew lo book which i will post soon.

 

[Ian Kaplan]

I didn't mean to "fault you", but just to suggest Lo as a reference. In my opinion he has done some of the most careful work looking at the Hurst exponent and long range dependence in financial markets.

 

[vertigo]

you cannot price a derivative in a 'rigorous' sense by using chaos theory. why? because the most well known fractal process, the fractional brownian motion, is not a semi - martingale. mathematical finance, derivative pricing and stochastic calculus are contingent on using semi - martingales, if not then arbitrage is admitted. i have read mandelbrot's work and it is bizarre that for all his criticism of the stock market (it is valid criticism and one must give mandelbrot credit for his candour to publish where as others zipped their mouths), he did not realise that his model is completely useless for overcoming what he criticises.

many 'smart people' have wasted their life trying to construct stochastic integrals based on fractal processes to be used in finance, only to be embarrassed by bizarre economic statements - see Bjork's article on fractional brownian motion and the wick product.

indeed, for derivative pricing it is useless.

however i do recall reading a book, 'physicists on wall street', a long time ago, that explained of a couple of mathematicians and physicists (they may now be hedge fund managers) who applied chaos theory and monte carlo to learning the optimal strategy in blackjack and then roulette - specifically where chaos theory really came in handy is seeing the periodic behaviour of a system.

chaos theory may also be useful for spotting patterns. in this sense, it could be seen as unsupervised learning, data mining, etc, where one is extracting hidden information. the caveat here is that you must connect this analysis to the models that traders and desk office quants use to be able to explain why u are using it. otherwise u are wasting everyone's time.

 

[Ian Kaplan]

Thanks for the interesting post on brownian motion vs. fractal processes. One question: although you can't get an analytic solution from fractal processes, I wonder if fractal Monte Carlo would work.

In the book Physics on Wall Street the author has chapters on Ed Thorp, who wrote the book Beat the Dealer, which he followed with Beat the Market (which described early ways to use mispriced convertible bonds). Thorp then went on to found Princeton Newport Partners which was one of the most successful hedge funds in history. Thorp developed what was basically the Black Scholes equation before B-S-M, but he didn't publish because he was using it to make lots of money.

I think that the chaos theorists that you are referring to may be Doyne Farmer and Norman Packard (although Packard was doing cellular automata). They founded Prediction Company, which did not use any thing from chaos theory. They quickly found that chaos theory didn't have much application in the markets. Although there might be something in fractal Monte Carlo.

I've been meaning to read Calvet and Fisher's work. They attempt to apply fractal processes apparently. So while I believe that fractal mathematics doesn't lead to analytic solutions, I still wonder about simulated solutions.

We know that stocks don't have normal return distributions so Black-Scholes is problematic, which is why it has to be adjusted via "implied volatility" or extended with more complex models (e.g., Heston).