The Misbehavior of Markets: A Fractal View of Financial Turbulence

by Benoît B. Mandelbrot

In an ideal market, security prices fully reflect all relevant information. A financial market is a fair game in which buyer balances seller. Given that, the price at any particular moment must be the “right” one. Buyer and seller may differ in opinion; one may be a bear, and another a bull. But they both agree on the price, or there would be no deal. Multiply this thinking by the millions of daily deals of a bustling market, and you conclude that the general market price must be “right,” as well—that is, that the published price reflects the market’s overall best guess, given the information to hand, of what a stock is likely to profit its owner. And if that is true—and here is the bitterest pill for an investor to swallow—then you cannot beat the market.

Continuity is a common human assumption. If we see a man running at one moment here and a half-hour later there, we assume he has run a line covering all the ground in between. It does not occur to us that he may have stopped to rest and then hitched a ride. The greatest innovation of seventeenth-century mathematics, the calculus, was designed to study continuous change; its cofounder, Gottfried von Leibniz, believed deeply in what he called a “principle of continuity.” Economists often do the same. Continuity is a fundamental assumption of conventional finance. The mathematics of Bachelier, Markowitz, Sharpe, and Black-Scholes all assume continuous change from one price to the next. Without that, their formulae simply do not work.