The Misbehavior of Markets: A Fractal View of Risk, Ruin and Reward


Benoit B. Mandelbrot, one of the century's most influential mathematicians, is world-famous for making mathematical sense of a fact everybody knows but that geometers from Euclid on down had never assimilated: Clouds are not round, mountains are not cones, coastlines are not smooth. To these classic lines we can now add another example: Markets are not the safe bet your broker may claim. In his first book for a general audience, Mandelbrot, with co-author Richard L. Hudson, shows how the dominant way of thinking ...

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Benoit B. Mandelbrot, one of the century's most influential mathematicians, is world-famous for making mathematical sense of a fact everybody knows but that geometers from Euclid on down had never assimilated: Clouds are not round, mountains are not cones, coastlines are not smooth. To these classic lines we can now add another example: Markets are not the safe bet your broker may claim. In his first book for a general audience, Mandelbrot, with co-author Richard L. Hudson, shows how the dominant way of thinking about the behavior of markets-a set of mathematical assumptions a century old and still learned by every MBA and financier in the world-simply does not work. As he did for the physical world in his classic The Fractal Geometry of Nature, Mandelbrot here uses fractal geometry to propose a new, more accurate way of describing market behavior. The complex gyrations of IBM's stock price and the dollar-euro exchange rate can now be reduced to straightforward formulae that yield a far better model of how risky they are. With his fractal tools, Mandelbrot has gotten to the bottom of how financial markets really work, and in doing so, he describes the volatile, dangerous (and strangely beautiful) properties that financial experts have never before accounted for. The result is no less than the foundation for a new science of finance.

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Editorial Reviews

Soundview Executive Book Summaries
A Fractal View of Risk, Ruin and Reward
The work of mathematical genius Benoit Mandelbrot has been turning heads since the early 1960s. As an award-winning scientist, he has always broken from the pack of others to originate groundbreaking ideas that simplify complex problems and forces, such as the forces that create complexity in financial analysis. One of these ideas is fractal geometry (a scientific discovery which makes computer animation possible).

A fractal is a geometric shape that can be broken into smaller parts, each a small-scale echo of the whole, such as the branches of a tree or the bifurcations of a river. Fractal geometry goes beyond the smooth lines and planes of circles and squares and applies wherever roughness is present. Studying roughness, Mandelbrot found fractal order where others had seen only disorder, and changed our view of nature as well as the irregular charts of a stock index or exchange rate. In The (Mis)Behavior of Markets, Mandelbrot and Wall Street Journal editor Richard L. Hudson explore marketplace risk and how Mandelbrot's fractal theories can be used to predict marketplace outcomes and make the market more secure.

According to the authors, fractal finance is a tool that can be used to reduce the way a specific price varies to a small number of mathematical theories and models. With these powerful tools, an analyst can model how an asset behaves, process various possible scenarios with computer models, and evaluate the risk of an investment with more accuracy. Prices of trade goods and exchange rates, for example, might not be able to be predicted with complete accuracy, but the authors write that they can be measured and characterized, and their volatility can be forecast.

Shaky Business
Throughout The (Mis)Behavior of Markets, the authors explain that much of "what passes for orthodoxy in economics and finance proves, on closer examination, to be shaky business." As someone who has always disrespected "received wisdom," Mandelbrot writes that his ideas about economics come from observation, and not abstract theory. As a practical and objective observer of the patterns of financial markets, he has collected 10 "obvious" facts. He calls these his "Ten Heresies of Finance." They include:

  1. Markets Are Turbulent. After studying the patterns found in wind tunnels and ocean currents, Mandelbrot applied his "multifractal" math to the analysis of financial markets. He writes, "The tell-tale traces of turbulence are plainly there, in the price charts. It has the turbulent parts that scale up to echo the whole." The normal expectations of the bell curve do not capture its changes, and only the metaphor of turbulence can be used to describe it.
  2. Markets Are Very, Very Risky - More Risky Than the Standard Theories Imagine. Turbulence is dangerous because it can swing wildly and suddenly. Not only is it hard to predict, but it is harder to protect against, and hardest of all to engineer and profit from.
  3. Market "Timing" Matters Greatly. Big Gains and Losses Concentrate Into Small Packages of Time. Since news events like corporate earnings releases and inflation reports help drive prices, big news can cause big market action, and that action concentrates in small slices of time. Particulars matter more than averages, the authors point out, so getting the timing right can produce giant windfalls, such as the two weeks in 1992 when George Soros made $2 billion by betting against the British pound.
  4. Prices Often Leap, Not Glide. That Adds to the Risk. The conceptual difference between economics and classic physics is the capacity for jumps, or discontinuity. In financial markets, news can compel many investors to act all at once, now and instantaneously.

Why We Like This Book
The (Mis)Behavior of Markets re-evaluates the standard tools of financial theory and injects them with the insights of a man who uses simple explanations to dissolve the false assumptions that have caused many investors to underestimate the risks involved in the market. By pointing out the flaws in previous thinking, and using scientific research and ideas to ground his theories, Mandelbrot once again changes the way we think. Copyright © 2004 Soundview Executive Book Summaries

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Product Details

  • ISBN-13: 9780465043552
  • Publisher: Basic Books
  • Publication date: 8/2/2004
  • Pages: 328
  • Product dimensions: 6.40 (w) x 9.54 (h) x 1.22 (d)

Meet the Author

Benoit B. Mandelbrot is Sterling Professor of Mathematical Sciences at Yale University and a Fellow Emeritus at IBM's Thomas J. Watson Laboratory. He is the inventor of fractal geometry, whose most famous example, the Mandelbrot Set, has been replicated on millions of posters, T-shirts, and record albums. He was a leading figure in James Gleick's Chaos and has received the Wolf Prize in Physics, the Japan Prize in science and technology, and awards from the U.S. National Academy of Sciences, the IEEE, and numerous universities in the U.S. and abroad. His books include Fractals: Form, Chance and Dimension, which was later expanded into the classic The Fractal Geometry of Nature, which has sold more than 200,000 copies. This is his first book for lay readers on finance, a subject he has studied since the 1960s. He lives in Scarsdale, New York. Richard L. Hudson was the managing editor of the Wall Street Journal's European edition for six years, and a Journal reporter and editor for twenty-five years. He is a 1978 graduate of Harvard University and a 1991 Knight Fellow of MIT. He lives in Brussels, Belgium.

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Table of Contents

Prelude : introducing a maverick in science : Benoit Mandelbrot, the "father" of fractals, has made a career of going against the prevailing fashions in science
Ch. I Risk, ruin, and reward 3
Ch. II By the toss of a coin or the flight of an arrow? 25
Ch. III Bachelier and his legacy 43
Ch. IV The house of modern finance 59
Ch. V The case against the modern theory of finance 79
Pictorial essay : images of the abnormal 88
Ch. VI Turbulent markets : a preview 111
Ch. VII Studies in roughness : a fractal primer 123
Pictorial essay : a fractal gallery 132
Ch. VIII The mystery of cotton 147
Ch. IX Long memory, from the Nile to the marketplace 173
Ch. X Noah, Joseph, and market bubbles 197
Ch. XI The multifractal nature of trading time 207
Ch. XII Ten heresies of finance 225
Ch. XIII In the lab 253
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Interviews & Essays

An Interview with Richard L. Hudson

Q. What is this book saying?

A. People can lose money unnecessarily in financial markets. One reason is that the standard financial theories, as taught in business schools around the world, are fundamentally wrong. They underestimate the real risk of stock, bond, commodity and other asset markets. By contrast, a fractal financial theory -- based on Mandelbrot's iconoclastic mathematical and economic research over the past 40 years -- promises a new kind of finance: safer and sounder. It can also help us understand better how markets really work; how we can better evaluate risk; and, at the most fundamental level, how it is that we gain and lose fortunes with such frightening rapidity.

Q. What's new about this book?

A. It is the first full exposition of fractal finance by the field's founder. It is an event in scientific publishing - much like Mandelbrot's famous 1982 manifesto on fractals generally, The Fractal Geometry of Nature. It is accessible to the general reader and will spark a new round of research and analysis in the fundamentals of finance. This book is also timely. We live in an age of risk: Of asset bubbles, currency crises, global boom and bust, war and terrorism. This book - much as Burton G. Malkiel's 1973 A Random Walk Down Wall Street spoke for an earlier age - coincides with the new spirit of risk assessment and risk management that dominates finance and business today. And it is of a rare breed of book in which a scientist uses deep thought but plain English to offer a new and absorbing way for us all to look at the world. It will change how many think about risk, ruin and reward. Q. Who is Mandelbrot? A. Benoit B. Mandelbrot is one of the leading mathematicians and scientists of our age -- best-known as the founder of a branch of mathematics called fractal geometry. He first came to popular attention as a protagonist in New York Times journalist James Gleick's 1987 bestseller, Chaos: Making a New Science. His impact on the scientific world is uniquely broad. His published theories have transformed mathematics itself, fluid dynamics, Internet file compression, geology, cosmology, and economics. Mandelbrot came to Princeton as the last post-doc of famed Hungarian mathematician John von Neumann. After 35 years at IBM, he retired as IBM Fellow Emeritus and is now Sterling Professor of Mathematical Sciences at Yale University. His work has earned numerous academy fellowships, honorary doctorates, and awards, including Japan's top scientific prize in 2003. On receiving the prestigious Wolf Prize for Physics in 1993, he was cited for "having changed our view of nature."

Among scientists, Mandelbrot is also famous as a maverick -- a sometimes prickly advocate of new ideas and change in the substance and process of science. Now a robust 79 years old (he turns 80 on November 20, 2004), he brushes aside criticism: "I have been a lone rider so often and for so long, that I'm not even bothered by it anymore." Or as a mathematically minded friend put it: He moves orthogonally -- at right angles -- to every fashion.

Q. What are fractals?

A. A fractal, a term Mandelbrot coined from the Latin for "broken," is a geometric shape that can indeed be broken into smaller parts, each a small-scale echo of the whole. Think of a cauliflower: Each floret can be broken off and is, itself, a cauliflower in miniature. Fractal geometry is about spotting such repeating patterns, analyzing them, quantifying them, and manipulating them. The pattern can take many forms. It can be a concrete shape as with the cauliflower, or an abstract statistical pattern as with the pattern of variation in a stock-price chart. Either way, the fractal math is a powerful technique to find and handle the basic patterns hidden in the complexity of the real world. It is a way to simplify the complex and to devise simple rules that generate stunning "forgeries" of nature. Its real-world applications include compressing computer files for transmission over the Internet, fitting micro-antennae efficiently onto tiny radio chips, tracking the electronic "flicker" noise behind many computer errors, and analyzing the turbulence inside a Boeing or Airbus wind tunnel. It has also been adopted by Hollywood to draw artificial landscapes, as in Star Trek II: The Wrath of Khan.

Q. So what's wrong with the standard theories of finance?

A. Over the 20th century, economists have erected a mighty edifice of financial theory. From it, we have Modern Portfolio Theory, by which stock portfolios can be built; the Capital Asset Pricing Model, to price securities and value new factories or acquisitions; the Black-Scholes formula to price stock and other options contracts; and the Efficient Market Hypothesis -- the economic theory behind the growth of stock-index funds and other forms of "passive" investment strategies. They all trace their origins back to a maverick French mathematician, Louis Bachelier, whose Paris doctoral thesis in 1900 pioneered the use of probability theory to model the way markets work. The problem is that all these modern models adopt Bachelier's original assumption that the familiar "bell curve" can describe how much prices vary up or down not in very few big movements, but lots of little movements constrained within a fairly narrow range. In fact, as Mandelbrot was first to demonstrate in 1962 and other economists have since confirmed, prices vary much more wildly than the bell curve. Instead, they follow power-law distributions and change discontinuously. Sharp price swings -- crashes and booms -- are far more common than the standard models assume, and so large that their effects overwhelm those of all the small changes. Price changes in the past affect markets today; they are not "independent" from one another, as the standard models assume. In short, markets are riskier than generally assumed.

So what? It means that the math behind a lot of the most common financial calculations -- how to balance a portfolio of stocks and bonds, how to decide whether an acquisition is priced right, how to hedge a dollar/sterling currency exposure -- can be dangerously off-base. It can, for instance, underestimate the risk of going broke by several orders of magnitude. Some of these problems are well known to the financial pros; an extensive subindustry has evolved to "fix" the flaws (for instance, you can buy off-the-shelf software to try to correct the "smile" in an options-price curve.) But many of these fixes are themselves wrong. And the basic theories continue to be standard teaching at most business schools around the world. What's needed is a new start, a new foundation for finance.

Q. So what is fractal finance?

A. In a nutshell: Fractal finance is a powerful tool for reducing the way a particular price varies -- a cotton price, a stock chart or a dollar-Euro exchange rate -- to a few simple mathematical ideas. With them, an analyst can model how an asset behaves, play "what-if" scenarios on a computer, and more accurately evaluate the risk of an investment. It cannot predict prices: That may be impossible in most circumstances. But it can measure, characterize, and forecast the volatility of an asset. Two key mathematical parameters describe how an asset price behaves. One is its Alpha, or A -- a measure of how wildly the prices vary (the "fat tails" of the distribution curve describing the price changes). The other is its H, or Hurst, exponent -- a measure of how much the prices tend to trend (their "infinite memory" or long-term dependence). For instance, Mandelbrot and some of his Yale students in 1997 applied this math to modeling the dollar-Deutschemark exchange rate successfully.

But there's much more to fractal finance than a few new tools. With his analysis, Mandelbrot has shown that:

  1. Bubbles, such as the Internet boom, are an inevitable part of markets. They will always happen, and will probably always make fools of us, as a natural part of the way prices vary.
  2. Markets are turbulent. That isn't a colorful metaphor: It's mathematical fact. Mandelbrot first developed "multi-fractal" math to analyze a wind tunnel or an ocean current and then applied it to financial markets.
  3. Markets are inherently deceptive. Even the most substantial-looking patterns can, in fact, be the product of mere fractal chance. That explains why most people miss market trends or imagine them where none exist -- to their own financial grief.
  4. Time in markets is relative. The best way to model prices is to put into the math what traders know instinctively: "Trading time" speeds up and slows down as the action mounts or subsides. And the risks of trading look relatively similar whether you are a short-fuse Internet trader or a patient pension-fund investor.
  5. Contrary to economic dogma, markets in all times and places work much the same way. Prices jump or fall abruptly, rather than glide up and down continuously. Price changes tend to cluster together - big changes beget more big changes. And throughout, the classical economic doctrine of that there is some "inherent value" -- while perhaps true in some cases -- is a singularly useless concept for investment decision. What matters more is arbitrage: Taking advantage of perhaps-fleeting price differences.
Q. What does the economics establishment think about this?

A. Though Mandelbrot taught economics for a year at Harvard, he has long been a controversial figure in the field. He first came to prominence in 1962 when he published research showing that cotton prices, at least, did not fit the standard financial models. Neither did other commodities, various securities, and interest rates. One economist-critic of the time groused that "Mandelbrot, like Prime Minister Churchill before him, promises us not utopia but blood, sweat, toil and tears." But since then, many of Mandelbrot's ideas have passed into the cannon of economics: His finding that prices vary wildly, that price movements are dependent from one period to another, that you can model options or other derivatives using so-called "time deformation" processes, and many others. Mandelbrot is often credited with pioneering the application of physics techniques to economics -- what is today called "econophysics." The main thrust of his theories, his multifractal analysis of markets, remains highly controversial. But his influence in economics is nevertheless profound. As MIT economist and Nobelist Paul A. Samuelson put it: "On the scroll of great non-economists who have advanced economics by quantum leaps, next to John von Neumann we read the name of Benoit Mandelbrot."

Q. So will this book make anybody rich?

A. No. This is not an investment how-to book. To be sure: A handful of fund managers around the world are now playing with some of his ideas in live markets. One, in Paris, manages $742 million and achieved a return in 2002 of 29 percent and in 2003 of 8 percent. But fractal finance is simply not yet "cooked" enough to be a practical investing tool. Indeed, this book is in the genre of popular science. It analyzes financial markets as if they were a "black box" to be studied. It is a cry for more research -- and for fresh thinking about the way we win and lose money.

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  • Anonymous

    Posted November 6, 2004

    Essential simplification of the understanding of markets

    The fundamental fractal concept applied to the behavior of markets offers most promising insights. This book takes you on the right track because as you may see from Savov¿s theory of interaction the texture of reality is made of multiscale fractal like 3D-spiral swirls of basic matter that contract and expand, i.e. show ups and downs like everything around us, including prices. Therefore Mandelbrot¿s fractal eye on the dynamics of the markets has its much deeper justification in theory of interaction terms.

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  • Anonymous

    Posted October 13, 2004

    A Must-Read!

    Finance is a difficult and recondite subject, perhaps second only to mathematics in its inability to inspire excitement in most readers. Yet Benoit Mandelbrot and Richard L. Hudson, co-authors of this book, manage to turn financial math into a great yarn, full of interesting characters and dramatic events. Some of what the book actually says will be old news to market professionals, but it says it quite interestingly. Mandelbrot did some of his most important financial work in the 1960s, but his ideas about leptokurtosis (which deals with the shape of probability functions), fractals (which deal with repetitive patterns) and such have received quite a bit of subsequent attention in trading rooms and in the finance departments of major universities. So, perhaps, it is merely a dramatic device that this book presents Mandelbrot as a solitary, clear-thinking prophet struggling against a blind and hostile economic orthodoxy. That presentation certainly succeeds as drama ¿ the story races along and the reader keeps rooting harder and harder for Mandelbrot to win. The co-authors have spun an excellent saga that says important things in a new way. We think every investor, every business journalist and every financial professional ought to read this book.

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