The Misbehavior of Markets: A Fractal View of Financial Turbulence (英語) ペーパーバック – 2006/3/7
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名著『The Fractal Geometry of Nature』（邦題『フラクタル幾何学』）で自然界について行ったように、マンデルブローは本書でもフラクタル幾何学を用いて、市場の動きを説明する新しくより正確な方法を提示している。IBM株価やドル-ユーロ相場の複雑な変動もいまやシンプルな公式に還元され、従来よりはるかに優れたリスクモデルを導くことができるのだ。マンデルブローはフラクタル・ツールを使って金融市場の真の仕組みを解明するとともに、これまで専門家たちが決して説明しなかった、その気まぐれで危険な（そして不思議に美しい）性質も明らかにする。金融の新しい科学の礎となる貴重な1冊。 --このテキストは、絶版本またはこのタイトルには設定されていない版型に関連付けられています。
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Mandelbrot goes through the models that set up the whole thing: Bachelier, Sharpe, Black-Scholes, and standard portfolio theory. He briefly discusses their power. It's a great, if somewhat sketchy overview of what tools financiers and bankers often use. But in each case, lurking in the background are the assumptions of normality in price movements, and of statistical independence between time periods and between different asset classes.
There is no question that Mandelbrot proves that cotton price fluctuations are badly described by the normal distribution. The quantitative and qualitative information he brings to other asset classes is much less robust. He gives us very good arguments as to why other classes behave as does cotton; but It is hard to say that he brings the same level of quantitative rigor to these. For those of us who want the argument to end with everyone believing the fractal story, it's a bit of a disappointment. What he does do, though, is to describe the Cauchy distribution function which, with some slight generalizations can produce distribution functions that will accurately characterize time series price data whose variation obeys power-laws in the tails of the distribution. The upshot is that anyone with a solid understanding of college level statistics could go on to derive their own Black-Scholes formula.
His publisher appears to have set two rules: 1) no math of any sort in the body of the book, and 2) only simple algebraic equations in the notes. These prohibitions have several consequences. One is that the book is quite readable to anyone, even someone who has not finished eighth grade algebra. A reader can get a vague sense for what Mandelbrot is saying without the math. The flip side is that people who have finished eighth grade algebra may find the arguments hand-wavy when they could be much more solid. Anyone who has a solid background in statistics is likely to be able to fill in the gaps much better, but they will find the arguments fall far short of the kind of proof that one would expect in a 300 page book written by a world-famous mathematician. The people who have studied Black-Scholes, understand its derivation, and use it everyday will likely want a little bit more data and a lot more math before they kill the beast that writes their paychecks. Specifically, they will want a replacement method, which Mandelbrot only hints at.
I found the text here to be a little bit discursive and somewhat repetitive. I often enjoyed his anecdotes, but I did find myself skipping paragraphs, pages, and even chapters. I bought the book knowing that markets have fractal behavior, and hoping to be able to make my own mathematical models based on information in this book. It did allow me to make the intuitive connection between power-law behavior and fractal behavior. And I believe the book has gotten me to the point where I can do all the steps required to price risk and characterize random motions in the prices of assets; although I think a six page monograph that admitted mathematical notation would have been more than sufficient.
My key takeaway from this book is that market participants' tools underassess risk and thus market participants should be wary of becoming model-dependent.
Additionally, supporting research and proofs are in the appendix or on the book's designated website for the more curious readers.
fractals are the by now familiar mathematical objects that display self-similarity when scaled larger or smaller. their progenitors are those weird constructs, such as peano's space-filling curve and the cantor set, that were introduced in the late nineteenth century and subsequently sparked a revolution in logic. all of these animals of pure mathematical fancy were designed to challenge the conventional notions of the time and forced mathematicians to revisit the foundations of their craft. indeed, this line of thought led to the strange notion of non-integer fractional dimensions.
so what does all of this have to do with finance? the dimension of a fractal is given by a power law. a lot of economic and financial data seem to fit power laws as well. fractals are characteristically self-similar. charts of stock prices exhibit self-similarity. yada yada yada and thus, markets are governed by fractals. wait a minute. that's actually not quite logical!
ok, so there are some speculative aspects fueling this enterprise. this is the source of most of the negative criticism mandelbrot receives for this book. in my opinion, laying out some speculative avenues of thought is not a crime. scientists should dare to dream! mandelbrot himself acknowledges that this circle of ideas is merely in its infancy. he hopes others will pursue this path of inquiry and continue his life's work. and just why would anybody pick up that banner? well, because our current understanding of finance is deeply flawed while mandelbrot offers a (very rough) potential alternative.
in the first part of the book, mandelbrot does an outstanding job presenting data contradicting conventional financial theories. the punchline: markets are much riskier than people think. in particular, he attacks the use of the so-called "normal" probability distributions in finance. this foundational attack threatens modern portfolio theory, the capital asset pricing model, the black-scholes formula for pricing options, etc. essentially, all the major developments in finance in the second half of the twentieth century are in jeopardy. some of the creators of these theories have won nobel prizes in economics, so a lot is at stake here. (an understatement!) note that mandelbrot's arguments in part one are valid even if the fractal speculations presented afterward turn out to be unfounded.
mandelbrot uses plain language and analogies in his exposition throughout the book. he purposefully avoided equations, but he partially makes up for it through the use of pictures. mandelbrot was a very visual thinker and it shows in this book. for example, on p.179 mandelbrot offers a diagram of what "removing the trend" means in hurst's research. stare at the picture for a little while and the meaning should become clear to anyone with an interest in math and science. similarly, mandelbrot doesn't really explain how multifractal time works since the given father-mother-child analogy is fuzzy at best. however, the "fractal market cube" diagram on p.214 explains the concept of multifractal time in one picture. anyone familiar with projections should be able to understand this diagram without any problems. this compromise approach of offering analogies for a general audience while providing supplementary mathematical content in the pictures is suitable for an introductory book aimed at a wide audience, in my opinion.
the best feature of this book for me was the autobiographical chronicling of a sharp mathematical mind at work. mandelbrot was able to see patterns and connections between seemingly unrelated fields and then he pursued these links relentlessly over decades of time. his individuality and perseverance allowed him to carry on even when the rest of the establishment were pursuing contrary ideas. mandelbrot also doesn't hide the moments when he was in the dark or when he saw connections that turned out to be trickier than his first instinct suggested. after all, this train of thought spanned a lifetime. and amazingly, some of his greatest insights came from pure serendipity. mandelbrot received a major breakthrough from reading a paper that was pulled out of a garbage can!
in the interest of fairness, there are some relatively minor oversights in this book. this was the only real negative i could think of and it's easily forgivable. for example, mandelbrot incorrectly states that peter lynch's stellar performance as manager of fidelity's magellan fund was most significant when the fund was small. it's actually the opposite: market impact costs become a burden when a mutual fund grows too large, making it much easier to outperform the market when a fund's assets are small, especially with lynch's trading style. in spite of this minor criticism, i found this book to be a page turner written by an obviously extraordinary thinker.
it's always a good idea to read the masters. if you want to understand the spirit of passive investing, read jack bogle. if you want to partake in value investing, read ben graham. and if you want to know why the house of modern finance might stand on shaky foundations, read mandelbrot. read, think, then judge for yourself. lastly, if you were hoping to make a fortune from fractals, read the following quote from p.6 of the book:
"i see a pattern in these price movements -- not a pattern, to be sure, that will make anybody rich; i agree with the orthodox economists that stock prices are probably not predictable in any useful sense of the term."
This book had so much information, that I can only write about the things that left me the deepest impressions.
1. Could this book have been rewritten as a discussion of the limitations of modeling? There are echoes of books that I have read before (notably, Paul Ormerod's "Butterfly Economics") about how the assumptions that go into econometric modeling are not true. The authors seems to go back and forth between showing the limits of academic Economics (one thing) and Finance (a very different thing). I wonder if the book could have been better put together in the aforementioned way and made to flow a bit more smoothly.
2. This guy has a *humongous* ego a la Nassim Taleb. I noticed this right away between pages 5-7 when I went back and counted the word "I" 22 times over those 3 pages.
3. The author did a yeoman's work measuring price change data from 1916-2003. That alone earned my respect and made this book worth reading. 87 years. 365 days per year. That works out to 31,755 data points to deal with.
4. I see where Taleb got his ideas of "Mediocristan" and "Extremistan" from. It's just that he babbled on about those concepts for much longer than Mandelbrot (who did it in something like 3 pages). (I find the earliest date of publication of "The Black Swan" to be something like 2007-- three years before this book came out.) Maybe Mandelbrot was also not the first one to notice that the Gaussian Distribution is not appropriate for many/ most issues of finance. But just the same, this foreshadows Taleb's work, and even makes it look a bit derivative.
5. I had heretofore never thought about what fractals represented graphically. This book started with the basics of that idea, but I might have liked more discussion thereon.
6. The book was very easy and light to read for about the first 3/4. In the last 1/4, the pages just dragged on by. There are also some problems with the organization. He put "In the Lab" and "Ten Heresies of Finance" at the back of the book. But since those are the problem to be answered, why didn't they go in the front of the book? Even after ALL THAT discussion, he comes up with (p. 255): (a) he doesn't know and; (b) that the problem needs further research? Huh?
7. There were some interesting lines of reasoning opened up by this book (that fractals can be used to describe market behavior), and the case might have even been more solid than what it seems. But, then again, there may have been nonsense uttered with seeming profundity (economists are very good at this). After it was all said and done, this was just not settled for me. It may be that Economics is neither Gaussian nor fractal but simply chaotic.
8. The book does not do much talking about prediction. And that is important. For example, on p. 219, he says that "prices scale as the model predicts," but now what? Yes, you can take the known data and go back and make a model for them, but what about using the model to go forward to predict prices? And better yet, why not work through a few different data sets and demonstrate how your model predicts where the other ones failed (and *why* they failed) in prediction? A question on prediction that arose during the book: A physicist might believe that he can predict the future given knowledge of the position, velocity, and direction of each particle as of a given time. But that information is not forthcoming. Is Benoit saying that he could predict prices in that same way with fractals? And that it's just a matter of limited information?
9. If there is a correct calculation to be made, and IF someone finds the correct way to make it, then won't other people find the same solution? And then won't the price on the stock fall back to reality? And then won't we be right back where we started?
This is worth the secondhand purchase price.
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