Louis Bachelier was born in Le Havre on 11 March 1870.

He stopped his studies in 1889 on the death of his father and managed the family business. He enrolled in 1992 in the Paris Faculty of sciences and, in 1900, defended his thesis on ‘Theorie de la speculation’.

Bachelier was not aiming at “analysing the causes, which act on the stock prices; this research would be vain and would only lead to errors” (Bachelier 1914 [1993]: 176-7). The evolution of the stock prices cannot be forecast: it is submitted to the law of chance. Bachelier, however, does not discard determinism: the variations of stock prices have causes, but they are unknown and, worse, unknowable. One must thus reason as if chance alone was acting. But while the evolution of prices cannot be determined, it is perhaps possible to find the law of probability, which governs them. “It is precisely because one wants to ignore everything that it is possible to know” (ibid.).

As an illustration of his research, Bachelier examines the 3 per cent rent, which was at that time the object of the greatest number of transactions. As a matured coupon of 0.75 franc was paid monthly, the price of the bond increased every month by 25 cents. It was necessary to eliminate this drift and Bachelier takes into account what he calls the true price, net of the mature coupon. He finds it, moreover, convenient to norm the present market price at zero. When he writes that, at instant t, the price of the bond is x_t this means that x_t is the difference between the true price at t and the actual price of the bond.

Bachelier’s developments rely on three principles: the principles of the mathemati­cal expectation, of indifference and of uniformity. Jules Regnault (1863: 34, original emphasis) in his Calcul des chances, wrote that at the stock exchange:

the mechanism of the game boils down to... two opposite probabilities: the increase or the decrease... At any moment, the probability of the one is not superior to the probability of the other; as we ignore the future result, it is absolutely indifferent to bet for or against,... to buy rather than sell or sell rather than buy.

Bachelier develops this idea. At a given moment, in the market, some speculators expect a rise in price, some others a fall. However, as the equality between supply and demand determines the market price, “it seems that at a given moment, all the speculators must neither believe in a rise nor in a fall since, for any quoted price, there are as many sellers and buyers” (Bachelier 1900: 31-2, original emphasis). The mathematical expectation of any operation is nil.

Regnault (1863: 38) maintained that, at the stock exchange:

the player is always tempted to forecast what should happen on the basis of what happened, so that, after three or four days of falling prices, he tends to believe in a rise the next day, but at some other times he will think that the fall will continue, despite the fact that these various effects are totally independent.

This statement can be seen as the origin of what Bachelier calls the principle of independence: the changes in the market price at a given moment are independent from

the past variations and from the price that is quoted at the given instant. Quotations follow a random walk, a random process without any memory, where the future only depends on the past through the present.

The principle of uniformity states that the instability of the market prices does not vary through time. Bachelier (1912: 286) justifies this by maintaining that the market does not have any reason to believe that the probability of the given varia­tion in prices would be different tomorrow from what it is today.

On these bases, Bachelier determines the law of probability of the price. Let p_{z t}dz be the probability that, at instant t = t₁ + t₂, the price of the bond is included between z and z + dz. We seek the probability that the price z be quoted at epoch t₁ + t₂, the price x having been quoted at epoch t₁. It is equal to the probability p_{x t}dx that in t₁ the price will be x, multiplied by the probability p_z-_xtdz that z will be quoted in t₁ + t₂, knowing that x was quoted in t₁. Hence p_ztdz = pp dxdz. Through integration, the probability that, at instant t, the price will be z is: ²

Bachelier then shows that, if the principle of uniformity is verified and if k is the coef­ficient of instability, a solution is:

At instant t, the bond price follows a standard normal distribution. The probability is a function of time, increases until a certain time and decreases thereafter. In order to study the law of probability followed by the market prices, and as they are the effects of “unanalysable” factors, Bachelier must face the problem of the sum of small independ­ent hazards. He thought that the shocks, which create price fluctuations, are of the same nature and not too scattered, so that their variance is finite. This restrictive hypothesis was the only one possible at that time: the addition of small random magnitudes can generate other laws than the Gaussian distribution, but it is only in the 1930s that Paul Levy and Alexandre Khintchine established this point.

Prior to Bachelier, many scientists applied mathematics to economic phenomena, but their tools were well known. Cournot is a typical example. Bachelier instead introduces new mathematical tools to deal with questions, which, admittedly, had been broached upon by some predecessors like Regnault, but were left unanswered. “It is above all from the point of view of pure science”, he wrote, “that the theory of speculation was useful... It generated the theory of the radiation of probabilities and the theory of continuous probabilities... If speculation did not exist, one should imagine it in order to better understand the laws of chance” (Bachelier, 1914 [1993]: 177-8). The number of discoveries he made in economics and mathematics is remarkable. He started the analysis of the Brownian motion and of the Markovian processes, introduced the idea of a random walk and of efficient markets. Not surprisingly, such a novel and path­breaking work was not understood for a long time.

Mathematicians were among the first to recognize Bachelier’s contribution to the mathematical theory of the Brownian motion (Kolmogorov 1931: 417). Economists kept silent. The idea was prevailing, in general, that it was impossible to rely on the prob­ability calculus to study moral sciences. “When men are together, they do not decide any more by chance and independently from each other; they interact” (Poincare 1907: 275). It was this interaction that had to be understood.

It is through two apparently independent ways that Bachelier’s work became known to economists. Paul Samuelson (Taqqu 2001: 25) writes that, in the 1950s, Leonard J. Savage drew the attention of his colleagues to the book Le jeu, la chance et le hasard, in which Bachelier intended to spread his ideas among a larger public. Samuelson, while looking for this book in the library of the Massachusetts Institute of Technology (MIT) discovered La theorie de la speculation. He advised one of his students, Richard Kruizenga, to read it.

The idea that the price of financial assets could not be forecast had imposed itself progressively. M.F. Maury Osborne, who wrote in this tradition, showed that the prob­ability distribution of the variations of the logarithms of the prices was precisely that of a particle in a Brownian motion (1959a: 145). When he was writing this paper, Osborne did not know Samuelson or Bachelier, but when he read his predecessors, he acknowl­edged the importance of Theorie de la speculation:

I believe the pioneer work on randomness in economic time series, and yet most modern in viewpoint, is that of Bachelier... [He] proceeds, by quite elegant mathematical methods, directly from the assumption that the expected gain (in francs) at any instant on the Bourse is zero, to a normal distribution of price changes, with dispersion increasing as the square root of the time, in accordance with the Fourier equation of heat diffusion... To him is due credit for major priority on this problem (Osborne 1959b: 808).

As soon as Bachelier’s contribution was recognized, a debate arose and the two hypotheses, on which he relied - the variations of prices are independent random variables, and they follow a normal distribution - were scrutinized. The second was discussed first. Benoit Mandelbrot (1963), while studying the evolution of the price of cotton, showed that it exhibited extreme variations, which could not be explained by a Gaussian distribution: “A great number of small variations of prices ran alongside... huge ones... just as huge legions of poor co-exist with some privileged plutocrats” (Mandelbrot and Hudson 2004 [2005]: 183). To the normal standard distribution, Mandelbrot (1962) and Eugene Fama (1963) proposed to substitute a power law such as the one used by Pareto to express the distribution of income.

Bachelier supposed that the successive variations of prices are independent random variables. Mandelbrot (1963: 418) maintained that this is not so: “large changes tend to be followed by large changes - of either sign - and small changes tend to be followed by small changes”. This scheme, which is neither regular nor predictable, shows the exist­ence of a long-term memory through which the past influences the present. The idea that prices follow a random walk, that the probability distribution of the variations of prices at instant t neither depends on the level of prices at this instant, nor on its past evolution, was thus rejected.

The current mainstream in modern finance is based on the foundations Bachelier laid in 1900. Its critics often rely on Mandelbrot’s propositions. However, while Mandelbrot discards Bachelier’s hypotheses, his admiration for the man is real (Mandelbrot and Hudson 2004 [2005]: 65 ff.).

Alain Beraud

See also:

Financial economics (III); Paul Anthony Samuelson (I).