Benoit B. Mandelbrot: How Fractals Can Explain What's Wrong with Wall Street | Hacker News Reader

Benoit B. Mandelbrot: How Fractals Can Explain What's Wrong with Wall Street

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24 pointsprakash12 comments

As Mandelbrot stated :

These techniques do not come closer to forecasting a price drop or rise on a specific day on the basis of past records. But they provide estimates of the probability of what the market might do and allow one to prepare for inevitable sea changes.

In my opinion multifractals can explain the Dow and Nasdaq fluctuations, but it is not enough to predict them. That the market has fractal structure as Mandelbrot noticed is probably very good assumption, but it depends on so many factors that the dimentionality of the fractal will be enormous, and one of the dimencions would be time. Also every stock history, every index, or whatever other value used is just a function of that fractal structure with respect to current and past time. Knowing the values of these functions can give a lot information about the structure of the market to estimate what would be their future values but this is not sure to happen. However some trends could be estimated but their time scale cannot be known for sure ( we have multifractal in which the time can be streched ). So we can say we expect that there will be crisis between 3 and 6 months from now on but we may never the exact that until it hits us.

Very similar is the situation with earthquake predictions, but there we have I believe much more knowledge about the underlying fractal structure (the Earth, continents, continental drift, internal Earth energy etc.) We can say there would be big earthquake in Cali every several decades but short term predictions are hard to make.

I will be happy to hear your opinion about fractals and stock market.

People seem to be in the habit of mistaking "description" with "explanation" or predictive power. I have no doubt that you can describe short term stock market fluctuations in many different ways (fractals and multi-fractals being one of them) that still doesn't get you anywhere close to explaining or predicting those fluctuations. There is good reason for this, of course. Mainly, that short term behavior is mostly psychological. If investors are happy (confident, bull, whatever you want to call it) prices go up, if they are unhappy (scared, bearish, etc) prices go down. To this short time scale behavior you have to add the long time scale component that makes the average market value increase at roughly 10% a year, and the "invisible hand" effect that makes individual stock prices converge (eventually) to the correct value.

IMHO, It's the combination of these three effects that make (short term) stock market fluctuations practically unpredictable.

Here's an hilarious description of this "market sentiment"...

      “You have to remember two things about the markets.
      One is that they are made up of very sharp and 
      sophisticated people, these are the greatest brains. 
      And the second thing you have to remember, is that 
      the financial markets, to use the common phrase, are 
      driven by sentiment.

      What does that mean?

      What does that mean? Well, things, lets say, are just
      going along as normal in the market. And then, 
      suddenly, out ot the blue, one of these very sharp 
      and sophisticated people says “MY GOD, SOMETHING 
      AWFUL IS GOING TO HAPPEN! WE LOST EVERYTHING! OH MY 
      GOD, WHAT ARE WE GOING TO DO, WHAT ARE WE GOING TO 
      DO?!? SHALL I JUMP OUT OF THE WINDOW? LET’S ALL JUMP 
      OUT OF THE WINDOW! SELL, SELL, SELL! Precisely. And 
      then, a few days later, this same, sophisticated 
      person says ‘you know, I think things are going 
      rather well’ and everybody else says ‘I agree with 
      you, I think we’re rich.’ [...] And that’s what we 
      call market sentiment.”

http://www.dailymotion.com/video/k2rLNuMggOKSJWHSpY

very nice video. Surely, psychology factors play quite a part in the way market works. And also having several variables determined by such factors in chain you can expect to have big fluctuations - exactly the ones that the Gaussian("normal distribution") portfolios fail to explain.

the last part of the video explains an example of such chain- and the effect- the subprime crisis.

Predictions are irrelevant in risk management. You don't want to know what will happen, you want to know the worst thing that could possibly happen. What you need is a theory of probability for price movements. Financial theory uses the normal distribution, which underestimates the likelihood of big crashes.

Mandelbrot says fractals give a better picture of market behaviour. They can be used to generate scenarios of price movements which give you different values for your portfolio. This is the probability distribution of your portfolio. It tells you how much debt you can take if you want to be able to pay it back 99.99% of the time.

He makes an interesting analogy with the likelihood of a storm. Ships are built to survive waves of a certain height, and the highest probable wave is also calculated with a probability theory. The theory of waves had to be revised because a lot more ships were sinking than was thought to be likely. Someone came up with a better theory, and now safer ships can be built. In the same way, fractals could help build better portfolios.

=========================

First, a bit of context

=========================

This article was written shortly after the spectacular collapse of Long Term Capital Management in late 1998. At the core of LTCM were Robert Merton and Myron Scholes, who shared the 1997 Nobel Prize in Economics for the Black-Scholes model. The Black-Scholes model was inspired by some calculus developed for rockets. Rockets have to constantly recalculate their trajectories, constantly reassuming certain assumptions about the path ahead. It didn't take much to see how similar math could be very useful for trading: it's simply a matter of choosing the right space and the right projectile (eg, imagine the space of all stocks and your portfolio as the projectile).

The Black-Scholes model is at the core of what Mandelbrot calls "portfolio theory". That is what collapsed in late 1998, what would prompt Mandelbrot to write an article or motivate some editors to ask him to write an article. Keep in mind, the one of the claims the fractals guys make is that fractals help explain turbulence, which is very related to, again, rockets. Interestingly, from 1947 to 1949, Mandelbrot studied aeronautics at Cal Tech (which gets huge USG and DoD money for aeronautics). Shortly after returning to France, his focus appears to have sharpened on what would become the seeds of his fractal theory. He started publishing papers in applied fields, like fluid dynamics and economics. So, just based on history, without thinking about the content of the theories themselves, I think it's reasonable to assume that Mandelbrot would have some professional interest and awareness of the Black-Scholes model.

Now, considering that the Black-Scholes model is quite competitive with fractal theory in the big-money arenas of fluid dynamics and economics, and considering the bells of Stockholm have not yet tolled for Mandelbrot, it is little wonder he would be keenly interested in discussing the merits of his theory in the context of the competing theory's recent and spectacular failure.

Scholes et al claimed to provide predictive power with their model at work in LTCM's computers. It is worth noting, in the current recession, the Black-Scholes model remains at the heart of the false confidence all those "Quants" on Wall Street gave their bosses, which is what led to their bosses feeling comfortable with 30:1 leverage positions.

I couldn't agree more with Anon84, except to extend the analogy further: no mathematics can explain the markets in a predictive way such that people don't loose money. If that were the case, everyone would make money and then you'd have, oh, wait, a bubble. Here's a direct analogy:

Black-Sholes trading : Russian Financal Crisis of 1998

Rocket on infinite trajectory : Giant black obelisk appearing in the flight path right in front of the rocket

There are things that can be done in markets that are not possible in the space-time continuum.

Buyer be warry.

--------------

Quant, btw, is a pun. It's quantitative economics, sure, but it's also "Quantum", because it has been, since I was an undergrad in the '90s, the easiest place for a physics grad to get a job. To see how tight the relationship is, Google for quant and search the first article you find for "Black-Scholes". The two are inextricable.

===========================================

Second: portfolios target 95%

and neglect the 5% that makes us

human

===========================================

You see, the markets aren't the space-time continuum. They are full of nasty discontinuities, legal, psychological, and otherwise. Thus hedge funds are always looking for 'smart people motivated by curiosity', which translates roughly to "We find these damn discontinuities in our trades, and we need someone who can just focus on the math and not worry about how much the fund is leveraged and whether or not our seed money came from Russian mafiosos and how much return we promised to the sovereign wealth fund of Ethiopia (if you'd rather keep race out of the argument, please replace Ethiopia with Iceland)"

And that over-the-top quote points out the issue: the mathematical descriptive theories make assumptions, but the real world contains the entire, real space and the real consequences, and real people and the real environment will bear those consequences. If you wipe out Ethiopia's national wealth, then they have no money until they dig up some more gold. Little babies die, fathers abandon their families looking for work, homes are lost. But, hey, at least the price of gas went down. Good thing America has so well insulated itself from the consequences of its actions.

The problem of "fat tails" has been known in finance for a long time, and there are plenty of models in which the distribution of price increments has a higher kurtosis than a Gaussian distribution. Interestingly enough, fractal models are not commonly used at all. Stochastic vol models are a lot more widespread. I'm a trader, not a quant, so I can't comment on why fractals were never in favor -- anyone else know?

"Editor's Note: This story was originally published in the February 1999 edition of Scientific American. We are posting it in light of recent news involving Lehman Brothers and Merrill Lynch."

And Mandlebrot's original research was done in the 1960's before Black, Scholes and Merton were getting prizes and destroying hedge funds in the 1990s. Moreover, Mandlebrot was certainly renowned by the 1970's. You would think people would pay attention to his ideas.

Just goes to show the appeal of research which tells people what they want to hear..

Interesting article. If this was originally written in 1999 then someone must have tried to create a fractal generator on historical market data for the Dow or Nasdaq. I wonder if it helped anyone prepare for the current stock market troubles.

Hilarity: when I loaded it, on the page was one of Chrysler's "thanks for your investment" ads.

Lost me when he confused the normal distribution with the plain English meaning of normal.

From http://www.andrews.edu/~calkins/math/webtexts/stat06.htm:

  Normal in statistics generally refers to the gaussian distribution or the "normal" way we would expect errors to be distributed.

When Mandelbrot says: "Granted, the bell curve is often described as normal—or, more precisely, as the normal distribution. But should financial markets then be described as abnormal?", he has reason. But I don't think he even meant it as a very serious remark.

Yes, I do believe he was making a pun.

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