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of a market fluctuation”15– that is, a move upward or downward in stock prices. Recognizing that fluctuations over time are virtually impossible to interpret, he begins by concentrating on the market at a given instant, promising to establish “the law of probability of price changes consistent with the market at that instant.”16 This approach leads him into more profound investigations: the theory of probability and the analysis of particles in space subject to random shocks.

      In view of the originality and brilliance of Bachelier’s analysis of financial markets, we might expect him to have been a man of stature in his own time. It is easy to picture him as an inspiring professor at the Sorbonne, or perhaps lured from France to Harvard or Oxford. We can note his large following of students, who, having gleaned so much wisdom, will go on to make their own mark in the study of finance, uncertainty, and random behavior. Or perhaps we can visualize him as a fabulously successful investor, a forerunner of Keynes, combining financial acumen with theoretical innovation.

      The truth is far different. Bachelier was a frustrated unknown in his own time. When he presented his dissertation to the committee at the Sorbonne, they awarded it merely “mention honorable,” a notch below the “mention très honorable” that was essential for anyone hoping to find a job in the academic world. It was long time before Bachelier finally won an appointment, and even then it was only at the provincial university at Besancon. Besancon is about as provincial as provincial France can get.

      Some of the difficulty seems to have stemmed from a mathematical error in a paper he published in 1913 – a slip that haunted him for many years. As late as 1929, when he applied for a position at the University at Geneva, a Professor Gevrey was still scandalized by the error, and, after consulting Paul Levy, another expert in the field, Gevry had Bachelier blackballed from the University. Levy later recognized the value of Bachelier’s pioneering work, and the two became reconciled.

      Bachelier’s real problem, however, was that he had chosen an odd topic for his dissertation. He was convinced that the financial markets were a rich source of data for mathematicians and students of probability. Twenty years after writing his dissertation, he remarked that his analysis had embodied “images taken from natural phenomena.. a strange and unexpected linkage and a starting point for great progress.” His superiors did not agree. Although Poincarè, his teacher, wrote that “M. Bachelier has evidenced an original and precise mind,” he also observed that “The topic is somewhat remote from those our candidates are in the habit of treating.”17

      Benoit Mandelbrot, the pioneer of fractal geometry and one of Bachelier’s great admirers, recently suggested that no one knew where to pigeonhole Bachelier’s findings. There was no ready means to retrieve them, assuming that someone wanted to. Sixty years were to pass before anyone took the slightest notice of his work.

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      The key to Bachelier’s insight is his observation, expressed in a notably modern manner, that “contradictory opinions concerning [market] changes diverge so much that at the same instant buyers believe in a price increase and sellers believe in a price decrease.”18 Convinced that there is no basis for believing that – on the average – either sellers or buyers consistently know any more about the future than the other, he arrived at an astonishing conjecture: “It seems that the market, the aggregate of speculators, at a given instant can believe in neither a market rise nor a market fall, since, for each quoted price, there are as many buyers as sellers.”19 (emphasis added)

      The fond hopes of home buyers in California during the 1980s provide a vivid example of Bachelier’s perception. Those buyers were willing to pay higher and higher prices for houses “because values could only go up.”20 This myopic view implied that the people who were selling the houses were systematically ignorant or foolish. Clearly they were not.

      Prices in markets that deal in bets on the future are, at any given instant, as likely to rise as they are to fall – as California real-estate prices have demonstrated. That means that a speculator has an equal chance of winning or losing at each moment in time. Now comes the real punch, in Bachelier’s words and with his own emphasis: “The mathematical expectation of the speculator is zero.”21 He describes this condition as a “fair game.”

      Here Bachelier is not just playing a logical trick by setting unrealistic assumptions so tightly that no other result is possible. He knows too much about the marketplace to resort to something that deceitful. In a disarmingly simple but perceptive statement about the nature of security markets, he sums up his case: The probability of a rise in price at any moment is the same as the probability of a fall in price, because “Clearly the price considered most likely by the market is the true current price: if the market judged otherwise, it would quote not this price, but another price higher or lower.”22

      Under these conditions, prices will move, in either direction, only when the market has reason to change its mind about what the “price considered most likely”23 is going to be. But no one knows which way the market will jump when it changes its mind; hence the probabilities are 50 percent for a rise and 50 percent for a fall.

      This conclusion led Bachelier to another important insight. The size of a market fluctuation tends to grow larger as the time horizon stretches out. In the course of a minute, fluctuations will be small – less than a point in most instances. During a full day’s trading, moves of a full point are not unusual. As the time horizon moves from a day to a week to a month to a year and then to a series of years, the range within which prices swing back and forth will grow ever wider.

      But how rapidly will the range expand? Bachelier answered that question with a set of mathematical equations demonstrating that “this interval [will be] proportional to the square root of time.”24 This prediction has held up with stunning precision.

      Stock prices in the United States over the past sixty-odd years have behaved almost exactly as Bachelier said they would. Two-thirds of the time, they have moved within a range of 59 percent on either side of their average level in the course of a month. But the range in the course of a year has not been 72 percent, or twelve times as much; rather, it has averaged around 20 percent, or about three and a half times the monthly range. The square root of 12 is 3.46!

      If stock prices vary according to the square root of time, they bear a remarkable resemblance to molecules randomly colliding with one another as they move in space. An English physicist named Robert Brown discovered this phenomenon early in the nineteenth century, and it is generally known as Brownian motion. Brownian motion was a critical ingredient of Einstein’s theory of the atom. The mathematical formula that describes this phenomenon was one of Bachelier’s crowning achievements.

      Over time, in the literature on finance, Brownian motion came to be called the random walk, which someone once described as the path a drunk might follow at night in the light of a lamppost. No one knows who first used this expression, but it became increasingly familiar among academics during the 1960s, much to the annoyance of financial practitioners. Eugene Fama of the University of Chicago, one of the first and most enthusiastic proponents of the concept, tells me that random walk “is an ancient statistical term; nobody alive can claim it.”25 In later years, the primary focus of research on capital markets was on determining whether or not the random walk is a valid description of security price movements.

      Bachelier himself, hardly a modest man, ended his dissertation with this flat statement: “It is evident that the present theory resolves the majority of problems in the study of speculation by the calculus of probability.”26

      Despite its importance, Bachelier’s thesis was lost until it was rediscovered quite by accident in the 1950s by Jimmie Savage, a mathematical statistician

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