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A Random Walk with Louis Bachelier

In response to:

He Conceived the Mathematics of Roughness from the May 23, 2013 issue

To the Editors:

In my recent review of Benoit Mandelbrot’s posthumous memoir The Fractalist [NYR, May 23], I spent a few paragraphs discussing the subversive work that Mandelbrot did in economics, particularly in the theory of financial markets. I noted that the orthodox model of financial markets—the one taught in business schools today, the one sharply criticized by Mandelbrot during his lifetime—was originally proposed in 1900 by a Frenchman named Louis Bachelier, who, I parenthetically remarked, “copied it from the physics of a gas in equilibrium.” Several readers have written to the Review maintaining—correctly, I think—that I failed to do justice to Bachelier’s originality.

While he was a doctoral student in mathematics at the Sorbonne, Bachelier derived his financial model by considering the movement of bond prices on the Paris Bourse. His striking conclusion was that such prices followed what came to be known as a “random walk”: buffeted by unpredictable bits of breaking news, they meandered in a way that was impossible to foretell. This is the nub of what is today called the Efficient Market Hypothesis. Owing to the frenzied activity of competing traders, this hypothesis holds, share prices on Wall Street and other financial markets instantly adjust to all available information; since any information that might be used to predict tomorrow’s stock price is already reflected in today’s price, it is futile to attempt to “beat the market.”

In creating his model, Bachelier managed, perhaps unwittingly, to shed light on a very different phenomenon: “Brownian motion,” the unpredictably jiggly behavior of particles suspended in a liquid or gas (named after Robert Brown, an English botanist who in 1827 observed how pollen particles in water seemed to dart around at random, as if they were alive). It is Albert Einstein who is usually credited with initiating the theory of Brownian motion. In 1905, the same miraculous year he produced his special theory of relativity, Einstein also published a paper showing how Brownian motion could be explained by the shoving actions of individual water molecules. This marked an important moment in the history of science: Einstein had produced decisive evidence that matter was discrete—made of molecules and atoms—not continuous. But Bachelier, studying a social phenomenon (the Paris Bourse) rather than a physical one, had hit upon the mathematical essence of Brownian motion five years before Einstein. Stock prices shifted about under the influence of random shocks much the way physical particles did.

So it was unfair of me to say that Bachelier “copied” his financial model from physics. (I first wrote about Bachelier’s precedence over Einstein in a May 1995 Lingua Franca column titled “Motion Sickness: A Random Walk from Paris to Wall Street.”) But one critical assumption made by Bachelier—and still central to financial orthodoxy—can indeed be traced to physics: namely, that stock and bond price fluctuations follow the well-behaved randomness embodied by the classic bell curve. Although the price movement of a given stock on a given day is virtually random and hence impossible to predict, such movements over time fall into a stable frequency pattern, the way repeated tosses of a coin fall into a 50-50 pattern of heads and tails.

In the case of stocks and bonds, the well-behaved bell curve pattern would apply if the great majority of price fluctuations were modest (corresponding to the fat central part of the bell), and extreme swings (corresponding to the bell’s low tails on the left and right) were rare. The bell curve distribution—also called “normal” or “Gaussian”—characterizes the range of velocities of the molecules in a gas at equilibrium; that was established in 1860 by the great physicist James Clerk Maxwell.

Does it also characterize price fluctuations in financial markets, as Bachelier’s model supposed? Mandelbrot said no: he insisted that financial markets displayed not the “mild” randomness of the bell curve, but the “wild” randomness associated with his fractals; that is, extreme events were more common than orthodox models assumed. Even Bachelier (as Robert W. Dimand observed in his letter) came to doubt, as early as 1914, his own assumption that price changes in financial markets are normally distributed. Yet (as Ross Levin observed in his letter) that was the very assumption behind the blow-up of Long-Term Capital Management in 1998 and the credit default option debacle of 2007–2009. On the evidence, markets are less well behaved than molecules.

Finally, Jeremy Bernstein—who has written admirably on these matters (see, for example, his article “Bachelier” in the American Journal of Physics, May 2005)—points out that it was misleading of me to have written that Long-Term Capital Management was “founded by two economists who had won Nobel Prizes.” The prime mover behind this hedge fund was John Meriwether, formerly an arbitrageur at Salomon Brothers. The Nobel laureates in question—Robert C. Merton and Myron Scholes—were recruited by Meriwether when he started LTCM in 1994, and are thus more accurately referred to as founding partners.

Jim Holt
New York City

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