If you are going to use probability to model a financial market, then you had better use the right kind of probability. Real markets are wild. Their … - Benoit Mandelbrot

"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."

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.

"There is a rich vein of jokes about economists and their assumptions. Take the old one about the engineer, the physicist, and the economist. They find themselves shipwrecked on a desert island with nothing to eat but a sealed can of beans. How to get at them? The engineer proposes breaking the can open with a rock. The physicist suggests heating the can in the sun, until it bursts. The economist's approach: "First, assume we have a can opener....