Mandelbrot & Market Behavior: The Fractals of Life

Professor Benoit Mandelbrot thinks economists and analysts are still underestimating market risk by using the traditional statistical model of normal distribution. A model that presupposes that all information is well-known and that things are very easy to handle,  the famous mathematician explains in an interview with Markit Magazine. His latest work is related to market behavior – an issue held up as one of the most difficult ares in economic theory, recently by the chairman of the US Federal Reserve, Ben Bernanke.

“Most fundamentally, and perhaps most challenging for researchers, the crisis should motivate economists to think further about their modeling of human behavior.”

Ben S. Bernanke




According to Mandelbrot’s theories there are no such thing as economic bubbles – just sudden price movements. “If you call them bubbles you are already judging them; you are saying they are bad, they are something undesirable. They are not undesirable. They are there, and if one does not take account of the possibility of a price going up very suddenly, or going down very suddenly, one takes a risk that is higher than anyone wants,” he says.

“Another issue that clearly needs more attention is the formation and propagation of asset price bubbles,” FED chairman Ban Bernanke said in a speech at Princeton University a few weeks ago.

Adding: “Much of the literature at this point addresses how bubbles persist and expand in circumstances where we would generally think they should not, such as when all agents know of the existence of a bubble or when sophisticated arbitrageurs operate in a market. As it was put by my former colleague, Markus Brunnermeier, a scholar affiliated with the Bendheim center who has done important research on bubbles, “We do not have many convincing models that explain when and why bubbles start.” I would add that we also don’t know very much about how bubbles stop either.”

(Full transcript)

There Are No Bubbles

However, the highly respected professor of mathematics, Benoit Mandelbrot, dismiss the very existence of price bubbles. There are no bubbles – just sudden price movements, he says, and if you’re not taking them into account, you’re making a big mistake.

In an interview with Markit Magazine he explains his recent work of implementing his famous fractal theory (an important element in chaos theory) in understanding market behavior.

“When people don’t want to listen, they don’t listen.”

Here’s the full interview with professor Mandelbrot:

Q: Professor Mandelbrot, your book, “The (Mis)Behavior of Markets” offered a wonderful critique of how traditional finance theory has failed over the years, and more or less predicted back in 2004 that we would face another financial crisis like the one we have just suffered. Do you feel vindicated?

A: Of course, but the delay has been costly. I am a mathematician, and not an economist. I try to go step by step. I don’t want to explain what may not be true, which is very often the case. All of my ideas go back to the 1960s. I developed a description of the financial markets and waited for the world to react. It’s been 40 years. The world has reacted to it in the sense it has bought many copies of my books, but it has ignored the economics chapters. I don’t think I need to say I told you so. It is very ineffective. When people don’t want to listen, they don’t listen.

Q: One of your main themes is that risk is underestimated by many in the financial world. Why do you think they keep missing this essential truth?

A: People think that risk means that if you invest $10, you may get back $11 if you’re lucky, perhaps $10.30, but somewhere close to $10. In fact, if you look at the actual data of trading, not for every price, but for the important prices on the market, large price changes are observed often enough to matter a lot. Such large swings mean the game definitely changes. The financial theory that was developed in 1900 says that big changes do not happen. That theory, which is still taught in business schools, is not correct. It does not describe the behavior of markets.

Q: One of your key insights is that turbulence in financial markets is similar to turbulence in wind tunnels and raging rivers. Obviously they don’t have the same underlying causes but exhibit the same patterns, you believe. What is the implication of this theory for people who work in the financial sector?

A: The implication is to be very careful at taking big risks. But details remain to be worked out. Don’t take as big a risk as possible in the expectation you will be lucky and earn big rewards.

Q: Your book, The Fractal Geometry of Nature, has become a classic of chaos theory. Weather is one area where it comes into play. But the fact is we can’t predict weather accurately yet. All we can do is observe it once it happens. Some specialists say the same thing about financial markets. Do you agree with that?

A: The book wasn’t speaking about wheather or anything specific. It merely tells us the world is very complicated. You know, this science developed in the simplest parts of the world and was widely accepted. But now key people have become very arrogant in thinking that they know how markets behave and they actually do not know. There are many statements about finance by well-known scholars that I don’t think could be justified on the basis of the evidence.

“ It shows that large price movements happen only rarely, but often enough to matter a great deal.”

Q: You came up with your theory of how market prices work by studying the cotton market over a hundred years. How did you happen to choose this area for study, which yielded such an important conclusion that prices in financial markets aren’t really random even though they seem to be?

A: It was just a lucky accident. When I started my work on cotton, I wouldn’t say it was sexy, that’s not the right term, but people don’t expect a mathematician to write about cotton. Cotton has been a commodity for a very long time and data about cotton sales have been gathered in Cairo over this period. There was an old Englishman in Cairo who was very brilliant, and he had very little to do so he studied science as an avocation. So there is a great deal of data about cotton and I was lucky enough to hear about it. Cotton is among the most variable in price of all the commodities. Wheat’s price is less variable than cotton’s is, but cotton is easier to study. So there was all this data in one place, it was not expensive for me to access.

Q: Yes, and in the process you found out something extremely interesting about financial markets. That cotton prices were not distributed over a bell curve, as modern financial theory believes prices for various assets are distributed. You found the distribution was not normal but was better described by the theories of French mathematician Paul Pierre Lévy. What was the significance of that?

A: It shows that large price movements happen only rarely, but often enough to matter a great deal. In my book, I am careful to show examples of what data that are ruled by a bell curve really look like. They never go up enormously, they never go down enormously. The bell curve became very well known around 1800 thanks to (German scientist Carl Friedrich) Gauss, a very great man, who was studying simple phenomena in astronomy. When he got a very large variation in his data, he thought it was an error of measurement. Things didn’t go up and down very much. It was a mistake. But in phenomena like prices, that’s not so. Every so often, not so rarely, prices change dramatically, and today prices move much more quickly and these changes are much more important. But it has always been like that. There are stories in the Merchant of Venice by Shakespeare, and even much older books than that, which talked about the existence of a category of people, bankers, who knew very well from experience that ships sometimes went safely on a long trip and sometimes didn’t. And when they didn’t return, it was a big loss to their business. A single loss could very well sink a big company. That was well-known hundreds of years ago and it wasn’t a matter of a mathematical dispute.

Q: This leads to the problem you describe of what happens when there is not a bell curve distribution of prices. You get so-called fat tails, which are extreme variations from normal distributions. The Black-Scholes model of options pricing, on the other hand, is based on a normal distribution using a bell curve. What does your theory tell you about large swings in prices?

A: You buy something for a certain price and find that a minute later it’s worth one half of what you paid for it. No one expected things to change that much. Given what the market knows about a particular commodity, it doesn’t expect that so great a change in price could happen.

“There are stories in the Merchant of Venice by Shakespeare, and even much older books than that, which talked about the existence of a category of people, bankers, who knew very well from experience that ships sometimes went safely on a long trip and sometimes didn’t. And when they didn’t return, it was a big loss to their business. A single loss could very well sink a big company. That was well-known hundreds of years ago and it wasn’t a matter of a mathematical dispute.”

Q: Is that what leads to financial market bubbles?

A: It’s a little subtler than that. That is really the first step in my theory of bubbles. But there are sudden changes of prices. If you call them bubbles you are already judging them; you are saying they are bad, they are something undesirable. They are not undesirable. They are there, and if one does not take account of the possibility of a price going up very suddenly, or going down very suddenly, one takes a risk that is higher than anyone wants.

Q: This leads into your criticism of the famous Efficient Markets Theory, which is still popular in the financial markets and used to structure many types of investments. It maintains that there aren’t any bubbles. You say this theory died under the weight of real data. Why doesn’t this theory work – don’t you believe that the market knows all the information that it is possible to know?

A: No, all the information about a market is never known. I haven’t worked in the securities business but in the world of commodities. Even still, I know that in the realm of securities, there are many companies that begin with a large degree of optimism, but they then hit a sudden downdraft that they didn’t allow for and they vanish. It happens all the time.

“Pareto observed that if you looked at the distribution of income in the kingdoms of Germany where he was getting his data, a small group of the wealthiest, just a few per cent, had most of the total income.”

Q: In place of the old theories about financial markets and price swings, you have come up with your own parameters. Tell us how those work.

A: The first of two parameters is inspired by the work of (Italian economist Vilfredo) Pareto, who studied the distribution of incomes, not of price changes. Pareto observed that if you looked at the distribution of income in the kingdoms of Germany where he was getting his data, a small group of the wealthiest, just a few per cent, had most of the total income. If you took a large number of people there would be an increasing chance of finding some rich people. So the question is: how many rich people would you find? For that there was a mathematical distribution theory that was the favored idea all the way back to the 1800’s. By the time the 1900’s came around, it was very well known, but it was completely wrong. In fact, the reality was that the richest people were much richer than they would be if income was ruled by a normal distribution. But if you use the distribution of incomes that was discovered by Pareto just before 1900, you find that it fits very well. It provided an uncanny approximation of the order of magnitude of the richest person in a country or a profession. The other key parameter governs the distribution in time. If you speak of individual incomes, like Pareto did, you find time is not involved. But we want to follow the variation of a price in time and often it is very strong. Pareto’s work had led to the idea that prices followed what was called Brownian motion, an idea which was proposed in 1900 by (French mathematician Louis Jean-Baptiste) Bachelier. Until I did my work, people had assumed prices followed this Brownian motion, meaning they were very simple and this is the source of the notion that prices moved up and down only a little and were not very risky.

“I had students who at first followed me, even when I found it hard to follow myself, but then they preferred to go back to the teachings of Bachelier in 1900, based on the normal distribution of prices.”

Q: Do you think the financial markets still underestimate risk?

A: I think so. When I came up with my work in the 1960’s, no one expected the real world to be so complicated. When I came up with work in the 1960’s, no one expected the real world to be so complicated. I had students who at first followed me, even when I found it hard to follow myself, but then they preferred to go back to the teachings of Bachelier in 1900, based on the normal distribution of prices. (He is referring mainly to Eugene Fama, one of Mandelbrot’s students, who as a finance professor at the University of Chicago Booth School of Business became the author of the modern Efficient Markets Theory that Mandelbrot now views as invalid). They believed that all information is well-known, that things are very easy to handle, and, of course, that was a mistake. At that time, I tried to repeat my arguments and told them this is going to lead to trouble very soon. Of course I was laughed at, but I did not just go sit in a corner unhappy. I did work on things other than prices. I felt that the evidence was so clear that prices would come back to the forefront whether I spent my life working on this topic or not. In fact, I spent a very small part of my life working on prices, but the world hasn’t changed.

“Everything the current theory of financial markets uses is essentially Bachelier’s idea on the normal distribution of prices. Or Bachelier with some little modifications. Then there is my alternative theory, which is that prices are not normally distributed.”

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Q: One of the crowning achievements of your career was to coin the now famous term fractals and make them immensely popular with your computer-generated depictions known as the Mandelbrot Set. You applied fractals to such things as the shoreline of Britain, but say they also apply to financial markets. Do you still feel fractals have a useful role in looking at the financial world?

A: I think that has been the thing to look at for the last 50 years. There are not many other possibilities. Everything the current theory of financial markets uses is essentially Bachelier’s idea on the normal distribution of prices. Or Bachelier with some little modifications. Then there is my alternative theory, which is that prices are not normally distributed.

Q: But you make clear that you can’t use fractals to predict prices, that you can’t make a million dollars in the stock market using the Mandelbrot theory. So practically speaking, what’s the advantage of them?

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A: The advantage perhaps is to reduce your probability of being ruined financially because you have misjudged risk. For example, you can use fractals to compare two portfolios with different goals and by using fractals you can choose a portfolio for a different level of risk. My theory is less defective than popular market theory and may help the world avoid or mitigate the agonies that we are now experiencing.

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Download a copy of the original interview here.

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