Benoit Mandelbrot Quotes

"In science, all important ideas need names and stories to fix them in the memory. It occurred to me that the market's first wild trait, abrupt change or discontinuity, is prefigured in the Bible tale of Noah. As Genesis relates, in Noah's six hundredth year God ordered the Great Flood to purify a wicked world. Then "were all the fountains of the great deep broken up, and the windows of heaven were opened." Noah survived, of course: He prepared against the coming flood by building a ship strong enough to withstand it. The flood came and went-catastrophic, but transient. Market crashes are like that. The 29.2 percent collapse of October 19, 1987, arrived without warning or convincing reason; and at the time, it seemed like the end of the financial world. Smaller squalls strike more often, with more localized effect. In fact, a hierarchy of turbulence, a pattern that scales up and down with time, governs this bad financial weather. At times, even a great bank or brokerage house can seem like a little boat in a big storm."

A cauliflower shows how an object can be made of many parts, each of which is like a whole, but smaller. Many plants are like that. A cloud is made of billows upon billows upon billows that look like clouds. As you come closer to a cloud you don't get something smooth but irregularities at a smaller scale.

(Since algebra derives from the Arabic jabara = to bind together, fractal and algebra are etymological opposites!)

If you have a hammer, use it everywhere you can, but I do not claim that everything is fractal.

"The market's second wild trait-almost-cycles-is prefigured in the story of Joseph. Pharaoh dreamed that seven fat cattle were feeding in the meadows, when seven lean kine rose out of the Nile and ate them. Likewise, seven scraggly ears of corn consumed seven plump ears. Joseph, a Hebrew slave, called the dreams prophetic: Seven years of famine would follow seven years of prosperity. He advised Pharaoh to stockpile grain for bad times to come. And when all passed as prophesied, "Joseph opened all the storehouses, and sold unto the Egyptians...And all countries came into Egypt to Joseph to buy corn; because that the famine was so sore in all lands." Given the profits he and Pharaoh must have made, one might call Joseph the first international arbitrageur. That pattern, familiar from Hurst's work on the Nile, also appears in markets. A big 3 percent change in IBM's stock one day might precede a 2 percent jump another day, then a 1.5 percent change, then a 3.5 percent move-as if the first big jumps were continuing to echo down the succeeding days' trading. Of course, this is not a regular or predictable pattern. But the appearance of one is strong. Behind it is the influence of long-range dependence in an otherwise random process-or, put another way, a long-term memory through which the past continues to influence the random fluctuations of the present."

Just the opposite appears to happen in the medium term, three to eight years. A stock that was rising over one multi-year stretch has slightly greater odds of falling in the next. A 1988 study by Fama and another economist, Kenneth R. French, documented this. They looked back over the price records of hundreds of stocks and grouped them into portfolios based on their size. They found that about 10 percent of a stock's performance in one eight-year period-that is, there was a small but measurable tendency for a stock doing well in one decade to do poorly in the next. The effect was weaker, but still statistically significant, at shorter time-scales of three to five years. Others have corroborated such findings.

This difficulty — am I a mathematician because my degree says so? Am I an engineer because I'm interested in things? Am I a social scientist because I don't think there's a difference between the turbulence in stock markets in terms of unpredictability? At IBM I wouldn't have to worry about that. The names of departments were totally strange and totally meaningless, so it looked like a promising situation for a short time. As it turned out I was going to spend thirty-five years and twelve days at IBM, almost from the beginning to the day when IBM decided that successful research was no longer going to be carried on in that division.

My ambition was not to create a new field, but I would have welcomed a permanent group of people having interests close to mine and therefore breaking the disastrous tendency towards increasingly well-defined fields. Unfortunately, I failed on this essential point, very badly. Order doesn't come by itself.

Given the profits he and Pharaoh must have made, one might call Joseph the first international arbitrageur.

Most of the world is of great roughness and infinite complexity. However, the infinite sea of complexity includes two islands of simplicity: one of Euclidean simplicity and a second of relative simplicity in which roughness is present but is the same at all scales.

The word fractal, once introduced, had an extraordinary integrating effect upon myself and upon many people around. Initially again it was simply a word to write a book about, but once a word exists one begins to try to define it, even though initially it was simply something very subjective and indicating my field. Now the main property of all fractals, put in very loose terms, is that each part — they're made of parts — each part is like the whole except it is smaller. After having coined this word I sorted my own research over a very long period of time and I realised that I had been doing almost nothing else in my life.

There is a saying that every nice piece of work needs the right person in the right place at the right time. For much of my life, however, there was no place where the things I wanted to investigate were of interest to anyone. So I spent much of my life as an outsider, moving from field to field, and back again, according to circumstances. Now that I near 80, write my memoirs, and look back, I realize with wistful pleasure that on many occasions I was 10, 20, 40, even 50 years "ahead of my time.

If you can distill the essence of GE's stock behavior over the past twenty years, then you can apply it to financial engineering. You can estimate the risk of holding the stock over the next twenty years. You can estimate how many shares of the stock to buy for your portfolio. You can calculate the proper value of options you want to trade on the stock.

Do I claim that everything that is not smooth is fractal? That fractals suffice to solve every problem of science? Not in the least. What I'm asserting very strongly is that, when some real thing is found to be un-smooth, the next mathematical model to try is fractal or multi-fractal. A complicated phenomenon need not be fractal, but finding that a phenomenon is "not even fractal" is bad news, because so far nobody has invested anywhere near my effort in identifying and creating new techniques valid beyond fractals. Since roughness is everywhere, fractals — although they do not apply to everything — are present everywhere. And very often the same techniques apply in areas that, by every other account except geometric structure, are separate.

The Mandelbrot set covers a small space yet carries a large number of different implications. Is it a fitting epitaph? Absolutely.