12:30 am, March 7, 2005
I'm about 80% through The (Mis)Behavior of Markets right now, and have found it extraordinarily interesting, illuminating, and readable.
Prof. Mandelbrot's primary thesis is that current models of financial theory are defective because they are based on a false supposition — that investment outcomes conform to the simple distribution of the bell curve. Now what the hell does that mean? Well, the bell-curve, it turns out, illustrates the distribution of outcomes of all types of events. People's heights, weights, and intelligence are distributed according to a bell curve. For example, if IQs are measured, left to right, high to low, with the numbers of persons having the corresponding IQ graphed vertically, the result is a bell curve. The curve is high in the middle because there are lots of people of average intelligence, and low on the left side, because relatively few people have very low IQs, and low on the corresponding right side, because relatively few people have very high IQs. So that is a bell curve distribution.
Bell curve distributions result from graphing the outcomes of processes like how much money you will make if you bet your friend $1 on every coin flip and flip 100 times. You don't each end up with the same amount you started with. Sometimes you're up, sometimes she is. If you play the game a hundred times, and graph the outcomes, though, you'll get a bell curve distribution. One key factor here is that, regardless of the last flip result, you have an equal likelihood of winning or losing this time. “The price changes, in the language of statistics, form a sequence of independent and identically distributed random variables.”
These types of outcome-sequences are governed by “a random walk” process. To describe a “random walk” process, probability theorists use this metaphor: “Suppose you see a blind drunk staggering across an open field. If you pass by a gain later on, how far will he have gotten? Well, he could go two steps left, three right, four backwards, and so on in an aimless, jagged path. On average — like in the coin toss game — he gets nowhere.” So if you were asked to bet on where the drunk would be in five minutes, you'd probably be most safe saying “right where he is, or within ten feet of there.” And if we asked you, “How much will the dollar be worth tomorrow in euros?” your best answer would be, “Whatever it is worth today.”
Alas, that is not good enough for the investor, who desperately wants a crystal ball in which all tomorrow's prices are revealed. So the quest for financial modeling and projection goes on. The father of the fundamental modeling structure on which most of the world's financial theory is based, according to Mandelbrot, was a Frenchman who really didn't get much encouragement for his work from other number-jugglers in the French academy, which included the formidable Poincare.
In 1900 Louis Bachelier, whom Prof. Mandelbort describes as “a brilliant but undervalued mathematician,” published a thesis on French government bonds that paid interest “in perpetuity,” and derivatives of that debt that had exotic names like “call o' mores” and “contangoes.” Says Mandelbrot, “Bachelier was intimately familiar with the arcana of these markets, and [tried] to develop formula to price these complicated derivatives.” The search lead him to a creative discovery.
"Mandelbrot and Hudson“
Nearly a century before, the great French mathematician Jean Baptiste Joseph Fourier had devised equations to describe the way heat spreads. Bachelier knew the fomulae well from his physics lectures. He adapted them to calculate the probability of bond prices moving up or down, and called the technique ”radiation of probability.“ Strangely, it worked.
So prices move like heat radiates. Hmmmm. That notion is familiar to us now, if we have been reading a little chaos theory. By borrowing a pattern that reflected the dynamics of thermal diffusion, Bachelier threw light on the nature of bond pricing processes. This and related insights have been guiding investment advisors, fund managers, bankers, and traders in making financial decisions. Further evolutions of Bachelier's work have resulted in great refinements of financial models which are, however, often confounded in times of financial turmoil that regularly recur.
This is because, just as Newtonian physics cannot describe quantum phenomena, in the same way, models based on bell-curve outcome distributions cannot be used to describe the full range of financial market movements. Why? Because financial outcomes, in the first place, do not follow a bell-curve of distributions.
By that we mean that there are too many big price moves. If you watch the price changes in various stocks, for example, and you measure each day the percentage of price movement from the previous day, and you graph all those outcomes, you do not get a bell-curve. You get a bell curve with ”fat tails,“ that is — way too many price-changes that are way too large. The financial waters, in other words, are choppier than hell.
Second, price changes from one moment to the next are not following a true random walk — they are trending — unpredictably, but they are trending. In other words, the drunk is likely to continue staggering in the same direction for a while, until he starts a new trend.
Because of this basic defect in Bachelier-derived economic analysis, Mandelbrot describes our modern captains of finance as competent sailors, as long as they don't hit a storm. In that case, all bets are off, and the best of them will end up swimming.
So far in the book, Mandelbrot hasn't addressed the issue of how lying and deception affect the financial markets. It would be hard, for example, to properly graph a series of coin toss outcomes if someone kept cheating, or you kept losing count of the number of tosses. In today's financial markets, such problems are common, due to incompetence and fraud. How many ”market corrections" occur when the truth comes out about declining profits, a truth that has been known for many months, concealed with financial manipulations, layoffs, debt-rescheduling, and mumbo jumbo? Although natural processes illuminate the structure of monetary systems, and provide fertile inspiration for mathematical models, human deceptiveness may throw a monkey wrench into the process of understanding this central pillar of human society.
From the UK Telegraph
The geometry of Mandelbrot
(Filed: 17/10/2004)
The founding father of fractal theory warns that another Great Crash is a real threat. Martin Baker meets him
Preposterous though it seems, there is something of the lone gunslinger about Benoit Mandelbrot. He is Sterling Professor of Mathematical Sciences at Yale University. And, yes, he is turning 80 next month. But if ever a man fixed the establishment in the eye and shot from the hip, it is Mandelbrot.
Next week is the 75th anniversary of the Wall Street Crash of 1929 - now consider this extract from the book Mandelbrot is publishing next month: "The financiers and investors of the world are, at the moment, like mariners who heed no weather warnings."
No, no, no. Surely it couldn't happen again? Conventional wisdom tells us that we understand risk so much better now. Those clever hedge fund chappies have got it all sewn up, surely. There has been the occasional disaster, such as the collapse of the huge US hedge fund Long Term Capital Management in the late 1990s. But we all survived. The investment industry has a sophisticated understanding of the riskiness of the market, and by using analytical tools such as chaos theory, the risks are managed and controlled.
Unfortunately not, according to Mandelbrot. And he should know. As the founder of fractal geometry and the discoverer of the Mandelbrot set (pictorially represented as beautiful, complex swirls of coral) Mandelbrot is acknowledged as the father of chaos theory. Here are his views of the current state of play: "A few fund managers have experimented with these concepts [of price dependence, whatever that is, and volatility]. They often call it chaos theory - though strictly speaking that is marketing language riding on the coat-tails of a popular scientific trend. In reality, the mathematics is still young, the research barely begun, and reliable applications still distant."
Mandelbrot has tweaked a few tails in the academic world, too. James Gleik, author of one of the many books on chaos theory, acknowledges Mandelbrot's contribution to the doctrine, while calling him "exasperating and indispensable".
So what is the principal theory of this near-octogenarian, so often described as a maverick? It was clearly a good idea to find out before having lunch with him.
Fractal geometry is a way of describing complex, irregular shapes that repeat themselves in nature. Take a leaf on a fir tree, for example. The leaf itself is a mini-me version of the whole tree. And if you look at the individual bits of the leaf, they look like the leaf that looks like the tree. So you have a complex mathematical formula that describes a pattern that keeps repeating itself. Thus a single formula describes lots and lots of data. This is the kind of thing that makes mathematicians happy.
The practical applications of the theory are that it can be used to model and describe, though not predict, a huge number of complex phenomena such as coastlines, water and air turbulence, galaxy clusters, and the fluctuations of stock markets and commodity prices (there is an hilarious passage in the book describing early research on cotton price fluctuations that reads like a comedy thriller). By describing such phenomena, fractal geometry moves on from Euclidian geometry, which is confined to smooth shapes and planes.
Armed with this very basic understanding of his work, I meet the man himself in a fantastically noisy French restaurant. Mandelbrot is tall, fairly robust, and has the thick-lensed spectacles of academic cliche. He has a Clouseau-esque accent, despite having worked at IBM in the US since 1958. But there is little of Peter Sellers about this man. His mind is like a steel trap.
It turns out the publication of The (Mis)Behaviour of Markets - A Fractal View of Risk, Ruin and Reward is not entirely designed to upset the financial community. Mandelbrot, after all, is a teacher with a didactic urge: "Part of my business may be to return mathematics and geometry to its role as an instrument to organise and understand the patterns of nature."
At this point readers may be crying out that markets are supposed to be rational phenomena, not natural ones. According to the efficient market hypothesis (a phrase coined by one of Mandelbrot's students, apparently) only rationally relevant information is priced into an asset or market. So what use is a formula that describes the natural world?
Mandelbrot's point is that, whatever the causal factors that go into price movements, markets and prices behave as if they are natural phenomena. He says: "My purpose is always to observe the symptoms and have a model of what is being seen. In the case of markets, it is frightening because there are so many people of great brilliance and extraordinary greed who work there. They don't understand the market, but they understand the numbers."
It is easy to see why Mandelbrot has a reputation for arrogance. He is, simply, very clever indeed, and is impatient with those who aren't. His fractal theory identifies three states of randomness - mild, slow and wild - and he believes that this model describes market behaviour far better than any other theories of randomness.
If he is hard on the financial community, it is because he believes investment managers and advisers are failing investors: "A stockbroker wrote me a very plaintive letter asking why I was giving stockbrokers such a hard time. His argument was that what he did was right 98 percent of the time. Why bother about the events that occur in the rest of the time? The answer is that those events are the ones that really count."
It is said that no one hurries like an old man, and Mandelbrot knows that at 79, time is precious. This only exacerbates his impatience with the financial community: "It is quite clear that some portfolios that were declared to be free of risk turned out not to be. They are very good for 90 percent or more of the time, but at the critical moment, they fail. They are just dreadful. Given the inter-connectedness of things, they may lead to very, very embarrassing complications for the whole world."
His best attempt to save the world, or at least make society aware of its incomprehension of the riskiness of the markets it depends on so much, is probably contained in a book that is surprisingly entertaining (much credit must go to Richard Hudson, the former European bureau chief of The Wall Street Journal, and co-author).
Now Mandelbrot is writing his memoirs - "purely what I remember. I won't research and check dates unless I absolutely have to". They should make compelling reading. Born in Warsaw in 1924, his family moved to Paris in 1936. As a Jew he was lucky to survive the Nazis, and had to move constantly. One possible benefit was the lack of a conventional education (he was tutored by an artist uncle who was also a professor of mathematics).
Mandelbrot says he inherited his independent tendencies from a father who saved his own life by refusing to stay on the road with fellow Jews who had been liberated from an internment camp during the war. Mandelbrot senior set out on his own and took refuge in a wood; those who stayed on the road were strafed by Stuka fighter planes. Following his father's untimely death, the youthful Mandelbrot also learned about the market, selling the crude clothing his father had made to eke out a living at distressed seller prices. "It put food in our mouths," he says.
Mandelbrot was a brilliant student and held a variety of academic positions, before resigning a post in France in 1958 in order to work in IBM's famous ideas factory (a group of oddball intellectuals paid to come up with great innovations). He was already seen as a cross-disciplinarian and a maverick and had created for himself "a very hostile intellectual environment. France does not like people not to belong".
For the last five years he has been at Yale, feted for the long-delayed publication of his paper on fractal geometry. And now, in his memoirs, he's turning those fearsome analytical powers on himself: "There are many things I begin to understand better now."
• The (Mis)behaviour of Markets: A Fractal View of Risk, Ruin and Reward by Benoit B Mandelbrot and Richard L Hudson is published by Profile Books, £18.99
