Benoit Mandelbrot and the wildness of financial markets

[Roger Bagula]

But real-life economic systems, Mandelbrot said, are "dominated by
details"--the extreme cases, and specifically the outer 5 percent, are
just as important as the rest of the data. To prove his point,
Mandelbrot showed a graph of the S&P since 1985, overlaid with the
same data minus the wild swings that constitute the outliers. The two
graphs were completely different, implying that to ignore the extreme
cases is to ignore reality.

_/*"Prices do not have any element of physical inertia,"
*/_
The markets behave more like a crowd of people than a physical system
where there is a contagious element to the behavior.
If someone yells "fire", you can get a stampede._/*
*/_

http://www.sciam.com/blog/60-second-science/post.cfm?id=benoit-mandelbrot-and-the-wildness-2009-03-13

Mar 13, 2009 05:10 PM
<http://www.sciam.com/blog/60-second-science/post.cfm?id=benoit-mandelbrot-and-the-wildness-2009-03-13#comments>
Benoit Mandelbrot and the wildness of financial markets
By John Matson
<http://www.sciam.com/blog/60-second-science/index.cfm?author=1237>
In a lecture at Columbia University this week, famed fractal pioneer
Benoit Mandelbrot once again inveighed against traditional economic
theories, returning at a time of financial malaise to many of the
points he raised in a 1999 /Scientific American/ feature
<http://www.sciam.com/article.cfm?id=multifractals-explain-wall-street>.
(In September 2008, as the U.S. economy began to shake, editor Gary
Stix provided a brief recap of Mandelbrot's article and the ensuing
response from readers in this blog post
<http://www.sciam.com/blog/60-second-science/post.cfm?id=mathematicians-predicted-stock-mark-2008-09-16>.)
Mandelbrot, 84, spoke at the Festival della Matematica, or Mathematics
Festival, an event produced jointly in Rome and New York City by a
consortium of Italian governmental and cultural agencies.
A persistent complaint levied by the Wolf Prize-winning French
mathematician <http://www.wolffund.org.il/full.asp?id=116>: many
economic models ignore dramatic jumps, whether in a commodity's price
or in an index such as the S&P 500, treating them as outliers. But
real-life economic systems, Mandelbrot said, are "dominated by
details"--the extreme cases, and specifically the outer 5 percent, are
just as important as the rest of the data. To prove his point,
Mandelbrot showed a graph of the S&P since 1985, overlaid with the
same data minus the wild swings that constitute the outliers. The two
graphs were completely different, implying that to ignore the extreme
cases is to ignore reality. "I'm extremely visual," Mandelbrot said.
"Often the pictures suggest the deeper truth underlying the formulas."
Mandelbrot also has beef with economists who model prices for shares
or commodities using variations on so-called random walks
<http://www.sciam.com/article.cfm?id=how-randomness-rules-our-world>,
which assume that the price at any given moment depends on what it was
the moment before. But prices, Mandelbrot noted, can be discontinuous,
jumping instantly from one value to another without any graduated
transition--more like a random teleportation. _/*"Prices do not have
any element of physical inertia,"*/_ Mandelbrot said by way of
illustrating the difference between economics and physical science
<http://www.sciam.com/article.cfm?id=the-economist-has-no-clothes>, a
difference that he said is all too often ignored. "A very large part
of economic theory is just physical theory with the words changed," he
said.
Theories grounded in the physical sciences, Mandelbrot said, presume
that the markets harbor elements of randomness, but in a form that he
calls "mild randomness." Mild randomness is embodied by the roulette
wheel at a casino--each spin is random but over time the distribution
of winning numbers averages out. (And, of course, over time the casino
wins out.) He contends that more realistic models of
economics--including, naturally, models based on fractals--are driven
by "wild randomness," wherein things don't average out and individual
freak occurrences matter. This wildness, he said, "imitates real
phenomena in a very strong way."
Mandelbrot took care to note that economics is just one field that he
has investigated in his decades of work on fractals and mathematical
modeling. But it's certainly a timely one: a book on fractal-based
financial risk management that he co-wrote in 2004 ("before things
were bad"), called /The (Mis)Behavior of Markets/
<http://www.amazon.com/Mis-behavior-Markets-Benoit-Mandelbrot/dp/0465043550>,
has recently begun "selling like hotcakes," he said.
For more on the fractal pioneer, see this /Scientific American
/article
<http://www.sciam.com/article.cfm?id=mandelbrot-set-1990-horgan> on
the disputed provenance of the Mandelbrot set from the April 1990
issue--an issue, incidentally, that featured a back-page essay by a
Tennessee senator named Al Gore
<http://www.sciam.com/blog/60-second-science/post.cfm?id=al-gore-nabs-elusive-award-triple-c-2009-02-09>.
/Photo of Mandelbrot at Columbia © John Matson/Scientific American/

[Roger Bagula]

In physics or economics...
http://nobelprize.org/nomination/
Nomination for the Nobel Prizes

Each year the respective Nobel Committees send individual invitations to
thousands of members of academies, university professors, scientists
from numerous countries, previous Nobel Laureates, members of
parliamentary assemblies and others, asking them to submit candidates
for the Nobel Prizes for the coming year. These nominators are chosen in
such a way that as many countries and universities as possible are
represented over time.

[Roger Bagula]

/http://domino.research.ibm.com/comm/pr.nsf/pages/bio.mandelbrot.html
IBM Research
Benoit Mandelbrot

IBM Fellow Emeritus




Few scientists can claim to have started revolutions or generated new
paradigms. IBM Fellow Emeritus Benoit Mandelbrot of the T.J. Watson
Research Center is one of them. With a naturalist's broad view of
science, he has ignored the prevailing boundaries and methods in pursuit
of his vision.

In the process, he has become one of the most versatile mathematicians
in history. More importantly, he has created a new geometry of nature
that is centered in physics but has changed our view of the universe.

Father of fractals
His creation of fractal geometry and the concept that simple rules can
generate infinitely complex structures and behaviors defines a paradigm
rooted in the fact that fractals "are irregular geometric shapes having
identical structure at all scales.'' According to Mandelbrot, their
irregular and complex behavior is echoed from scale to scale.

Mandelbrot's multi-disciplinary explorations began with his doctoral
thesis in 1952, which combined linguistics (a mathematical analysis of
the distribution of words) with the tools of statistical thermodynamics.
In the early 1960s, he moved to study finance, demonstrating that price
fluctuations in markets are not smooth, as economists thought, but are
often choppy, discontinuous and always concentrated in time.

And he showed that wealth acquired on the stock market is typically
acquired on a very small number of favorable periods.

Mathematics of the Nile
At IBM Research, where he joined in 1958, Mandelbrot showed that errors
propagating on telephone lines used to transmit computer information
were not classically random and self-similar over any chosen period of
time. Not only would there always be periods of error-free transmission
and of error-plagued transmission, but it was impossible to find a fine
enough time scale in which that wouldn't be the case.

Mandelbrot had found the same mathematical distribution to hold true in
the field of water resources, and the study of floods and droughts in
the Nile River Basin. He revealed the discontinuous nature of the
universe, and the persistence and the tendency of droughts or floods to
come in clusters.

Dimensions in nature
Mandelbrot's work came to fruition in a seminal 1967 paper in Science
Magazine, titled "How long is the coast of Britain? Statistical
self-similarity and fractional dimensionally?'' In it, Mandelbrot
pointed out that the concept of length was meaningless when trying to
describe something as seemingly concrete as a natural coastline; that
length is dependent on one's choice of measuring stick.

To characterize this ever self-similar and yet infinite complexity,
Mandelbrot introduced into science the concept of fractal dimension; if
a smooth curve had a dimension of one, and a smooth surface a fractal
dimension of two, a coastline, for instance, could be said to have a
fractal dimension somewhere in between. The concept was stunning.

When his 1977 and 1982 books provided an extraordinary list of fractal
phenomena from nature -- mostly from physics, but also from the veins
and arteries of anatomy to hierarchical clustering of stars and galaxies
-- and he communicated them through computer-generated imagery.

As Mandelbrot would later state, he "re-introduced the eye to the study
of mathematics."

Simplicity to define complexity
According to New Scientist, "Mandelbrot's massive....achievement has
been to convert [a] abstract formalism into a flourishing branch of
applied mathematics.'' Or in the words of the Mathematical Gazette,
"Euclid is replaced as hero by a celestial committee....whose ideas have
condensed into fractals under Mandlebrot's supervision.''

With the introduction of the Mandelbrot set in 1980, he showed that such
complex phenomena could be created and described by simple rules
iterated over and over again, and he set a whole generation of
mathematicians, computer scientists, and even artists to generating and
studying the beautiful images that resulted./

[Roger Bagula]

http://www.sciam.com/article.cfm?id=mandelbrot-set-1990-horgan&print=true

Features - March 13, 2009
Who Discovered the Mandelbrot Set?
Did the father of fractals "discover" his namesake set?

By John Horgan

Editor's note: This article originally appeared in the April 1990 issue
of Scientific American, under the title "Mandelbrot Set-To." We are
posting it now to coincide with our reporting on a talk this week by
Benoit Mandelbrot at Columbia University on fractals and financial
markets. The phrasing of some references to dates has been changed, in
brackets, for clarity.

Who discovered the Mandelbrot set? This is not a trick question--or a
trivial one. The set has been called (in this magazine) "the most
complex object in mathematics." That is debatable, yet it is almost
certainly the most famous such object. The infinitely intricate
computer-generated image of the set serves as an icon for the burgeoning
field of chaos theory and has attracted enormous public attention.

The set is named after Benoit B. Mandelbrot, a mathematician at the IBM
Thomas J. Watson Research Center. He is best known for coining the term
fractal to describe phenomena (such as coastlines, snowflakes, mountains
and trees) whose patterns repeat themselves at smaller and smaller
scales. Mandelbrot claims that he and he alone discovered the Mandelbrot
set--which has fractal properties--about a decade ago. He refers to its
image as his "signature."

Three other mathematicians have challenged his claim. Two maintain that
they independently discovered and described the set at about the same
time as Mandelbrot did. A third asserts that his work on the set not
only predated Mandelbrot's efforts but also helped to guide them. These
assertions have long circulated in the mathematics community but have
only recently surfaced in print.

Mathematicians are not known for priority battles, but Mandelbrot--a
self-described "black sheep "--has often bumped heads with colleagues.
_/*"Were it not for his personality," remarks Robert L. Devaney of
Boston University, who says he admires Mandelbrot's work, "there would
be no controversy."*/_

The scientific stakes are also high. Even those who scorn the set's
popularity acknowledge its mathematical significance. Dennis P. Sullivan
of the City University of New York calls it a "crucible" for testing
ideas about the behavior of dynamical (or nonlinear, or complex, or
chaotic) systems. "It is really quite fundamental," he says.

Part of the charm of the set is that it springs from such a simple
equation: z2 + c. The terms z and c are complex numbers, which consist
of an imaginary number (a multiple of the square root of -1) combined
with a real number. One begins by assigning a fixed value to c, letting
z = 0 and calculating the output. One then repeatedly recalculates, or
iterates, the equation, substituting each new output for z. Some values
of c, when plugged into this iterative function, produce outputs that
swiftly soar toward infinity. Other values of c produce outputs that
eternally skitter about within a certain boundary. This latter group of
c's, or complex numbers, constitutes the Mandelbrot set.

When plotted on a graph consisting of all complex numbers, the members
of the set cluster into a distinctive shape. From afar, it is not much
to look at: it has been likened to a tumor-ridden heart, a beetle, a
badly burned chicken and a warty figure eight on its side.

A closer look reveals that the borders of the set do not form crisp
lines but seem to shimmer like flames. Repeated magnification of the
borders plunges one into a bottomless phantasmagoria of baroque imagery.
Some forms, such as the basic heartlike shape, keep recurring but always
with subtle differences.

Today virtually anyone with a personal computer can "discover" the set.
But [in 1979] computers were much less powerful, and few mathematicians
associated computers with serious mathematics.

Even Mandelbrot has described his first tentative steps toward the set
in 1979 as "mindless fun." He began using a computer to map out Julia
sets, which are generated by plugging complex numbers into iterative
functions. The sets' peculiar properties had been described as early as
1906 by the French mathematician Pierre Fatou. They were named later for
Gaston Julia, who successfully claimed that his work on the sets some
dozen years later had greater significance than Fatou's. Mandelbrot, who
was born [in 1924] in Poland, had read the work of both men and studied
under Julia in the 1940's.

Mandelbrot's early computer images served to confirm his suspicion that
Julia sets have fractal properties. He says he began producing
recognizable pictures of the Mandelbrot set--which in a sense is a
generalized version of all Julia sets--in late 1979. Mandelbrot
subsequently displayed images of the set and elaborated on its
significance in speeches, papers and books. This discovery and his other
work in fractals were also celebrated in the media, in numerous books
(notably the best-seller Chaos, by former New York Times reporter James
Gleick) and in IBM advertisements.

No one denies that Mandelbrot's pictures and descriptions spurred other
mathematicians to study the set. Two prominent examples are John H.
Hubbard of Cornell University and Adrien Douady of the University of
Paris. In the early 1980's--in the course of proving that tiny "islands"
surrounding the main body of the set are linked to it by infinitesimal
filaments--they named the set after Mandelbrot. "Mandelbrot was the
first one to produce pictures of it, using a computer, and to start
giving a description of it," Douady wrote in 1986.

Douady now says, however, that he and other mathematicians began to
think that Mandelbrot took too much credit for work done by others on
the set and in related areas of chaos. "He loves to quote himself,"
Douady says, "and he is very reluctant to quote others who aren't dead."

[In the fall of 1989] Steven G. Krantz of Washington University aired
some of these grievances in the Mathematical Intelligencer, a quarterly
journal. The main point of his article was that fractals,
computer-generated graphics and other "popular" mathematical phenomena
associated with Mandelbrot have contributed little of substance to
mathematics, especially in comparison to the publicity they have garnered.

This view--and its opposite, which holds that Mandelbrot's "popular"
work has been a stimulating force in mathematics--had been voiced
before. Krantz introduced a new element into the debate, however, by
stating that the Mandelbrot set "was not invented by Mandelbrot but
occurs explicitly in the literature a couple of years before the term
'Mandelbrot set' was coined." He cited a paper by Robert Brooks and J.
Peter Matelski published in the proceedings of a 1978 conference at
Stony Brook, N.Y.

Sure enough, the paper contains the famous z2 + c formula and a crude
but unmistakable computer printout of the set's basic image. Brooks and
Matelski say they did not actually present the paper at the 1978
conference, but they did circulate it as a preprint in early 1979.
Brooks, who is now at the University of California at Los Angeles, also
presented the paper at Harvard University in the spring of that year.
(Mandelbrot, who held an appointment at Harvard then, says he did not
hear Brooks speak and only saw the paper years later.) The paper was not
published, however, until early 1981.

In a rebuttal to Krantz's article, called "Some 'Facts' that Evaporate
upon Examination," Mandelbrot noted that he "fully published" on the
Mandelbrot set before Brooks and Matelski did. (Mandelbrot's paper,
published in the December 26, 1980, Annals of the New York Academy of
Sciences, features a function and image that are variants of those now
associated with the Mandelbrot set, which Mandelbrot did not publish
until 1982.)

Mandelbrot also suggested that even if Brooks and Matelski's publication
had preceded his, they still could not be considered discoverers of the
set, because they did not appreciate its significance. "[They were]
close to something that was to prove special, but they gave no thought
to the picture," he wrote.

Brooks retorted in the following issue of the Intelligencer: "I don't
know how he can be so sure of what we gave thought to and what we
didn't." Brooks says he respects Mandelbrot's achievements as a
popularizer and does not object to the set's being named after him. "It
makes more sense than 'the thing with the big cardioid,'" he says,
recalling how he and Matelski had referred to the set. "I just wish
Mandelbrot were a little more gentlemanly."

Matelski, who works at the Hartford Graduate Center in Connecticut,
notes that neither he nor Brooks asked Krantz to credit them with having
discovered the Mandelbrot set. (Krantz confirms that another
mathematician drew his attention to their paper.) But now that the issue
has become public, Matelski says he and Brooks should be acknowledged as
co-discoverers with Mandelbrot.

"You don't have to fully exploit the mineral resources of a continent to
discover it," Matelski was quoted as saying in the Hartford Courant, a
newspaper that reported on the dispute in December [1989]. "All you have
to do is kneel down and kiss the beach."

A subtly different claim of precedence has been made by Hubbard, who is
now considered one of the world's experts on the Mandelbrot set. In
1976, he explains, he began using a computer to map out sets of complex
numbers generated by an iterative process known as Newton's method.
Hubbard says he did not realize it then, but he had found a different
way of generating the Mandelbrot set.

In late 1978 one of Hubbard's graduate students, Frederick Kochman,
approached Mandelbrot at a conference and showed him Hubbard's pictures.
Mandelbrot "didn't seem very interested," Kochman recalls. Yet shortly
thereafter Mandelbrot wrote a letter to Hubbard inviting him to IBM to
discuss his work. In the letter, which Hubbard kept, Mandelbrot wrote:
"When sampling the works of Fatou and Julia, I had thought of doing
these things myself, but had not mustered the courage. Nevertheless I
can claim that I was awaiting your pictures for a long long time."

Hubbard says he went to IBM early in 1979 and, while there, told
Mandelbrot how to program a computer to plot the output of iterative
functions. Hubbard concedes that he did not appreciate the full
significance of his own images then and that they showed only pieces of
the Mandelbrot set. He also admits that Mandelbrot developed a superior
method for generating images. Nevertheless, Hubbard says he was and
continues to be "outraged" that Mandelbrot did not give him credit in
the 1980 paper and later writings. "It was a breach of mathematical
ethics," he asserts.

Mandelbrot recalls seeing "one impressively early drawing of a Julia
set" by Hubbard but denies that it contributed to his own discovery. In
response to Hubbard and Douady's charge that he is stingy in granting
credit, Mandelbrot says he has also been accused of overcitation. He
adds that his failure to cite the early finding of Brooks and Matelski
might have spared them "derision" for "their failure to do anything with
it."

What about the suggestion of Hubbard, Matelski and Brooks that the true
discoverer of the Mandelbrot set is Fatou, who was the first to define
the set and speculate about its properties? Brooks even proposes that
"if Fatou had had access to modern computing facilities, he could have
and would have drawn pretty much the same pictures that Matelski,
Mandelbrot and I did." Mandelbrot calls such speculation pointless and
insists that Fatou's definition of the Mandelbrot set does not
constitute discovery. "Definition counts for nothing," he says. "You
have to say why something is important."

Other mathematicians familiar with the case are somewhat bemused. "It
seems strange to me that there should be such a fuss," remarks John
Milnor of Princeton University. He maintains that neither Brooks and
Matelski nor Mandelbrot did anything mathematically important. "Hubbard
and Douady are the first ones to really obtain some sharp results," he
says, "and let us know something about the set."

The dispute over precedence, Milnor suggests, may spring from a clash of
different mathematical cultures. "In pure mathematics," he explains,
"there is a tradition of letting others praise your work." Mandelbrot,
he notes, is in applied mathematics.

"Mathematical developments don't take place single-handedly," William P.
Thurston of Princeton points out, "and it's pretty common that things
are not named after the first person to develop them. The Mandelbrot set
follows that pattern." He suggests, however, that no one would begrudge
recognizing Mandelbrot's achievements if he would reciprocate more
himself. "He could be a bit more magnanimous," Thurston says.

Sullivan, who has also been acclaimed for his studies of the Mandelbrot
set, calls himself "sort of a defender of Mandelbrot." Mandelbrot
deserves to have the set named after him, Sullivan says, because his
efforts brought the set to the attention of both the public and of the
pure-mathematics community.

The fact that it was only "by coincidence" that the set proved later to
be mathematically significant, Sullivan says, in no way diminishes
Mandelbrot's achievement. "That's the wonderful thing about
mathematics," he adds. "Even amateurs can make important contributions."

So who did discover the Mandelbrot set? Sullivan calls the question
meaningless. Perhaps. Sheldon Axler, editor of the Intelligencer, plans
to publish a letter pointing out that the Hungarian mathematician F.
Riesz reported on work related to the set in 1952 .

The final answer, if pursued, seems likely to recede in a blur of ever
finer detail.