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by Charles Carreon
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
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