*Nul n'est prophete en son pays ... .*

The French mathematician, **Louis Bachelier** is now recognised internationally as the father of financial mathematics, but this fame, which he so justly deserved, was a long time coming. The Bachelier Society, named in his honour, is the world-wide financial mathematics society and mathematical finance is now a scientific discipline of its own. The Society held its first World Congress on 2000 in Paris on the hundredth anniversary of Bachelier's celebrated PhD Thesis *Théorie de la Spéculation* [24].

Five years before Einstein's famous 1905 paper [4] on *Brownian Motion*, in which Einstein derived the equation (the partial differential heat/diffusion equation of Fourier) governing Brownian motion and made an estimate for the size of molecules, Bachelier had worked out, for his Thesis, the distribution function for what is now known as the Wiener stochastic process (the stochastic process that underlies Brownian Motion) linking it mathematically with the diffusion equation. The probabilist William Feller had originally called it the Bachelier-Wiener Process. It appears that Einstein in 1905 was ignorant of the work of Bachelier.

Seventy three years before Black and Scholes wrote their famous paper in 1973 [5], Bachelier had derived the price of an option where the share price movement is modelled by a Wiener process and derived the price of what is now called a barrier option (namely the option which depends on whether the share price crosses a barrier). Black and Scholes, following the ideas of Osborne and Samuelson, modelled the share price as a stochastic process known as a Geometric Brownian Motion (with drift).

Louis Bachelier was born in Le Havre in 1870. After education at secondary school in Caen he lost both his parents and had to enter the family business. It was during this period that he seems to have become familiar with the workings of financial markets.

At the age of 22, Bachelier arrived in Paris at the Sorbonne where he followed the lectures of Paul Appell. Joseph Boussinesq and Henri Poincaré (the latter being then aged 38). After some 8 years, in 1900, Bachelier defended his thesis *Théorie de la Spéculation* before these three men, the favourable report being written by no less a figure than Henri Poincaré, one of the most eminent mathematicians in the world at the time.

Quite what his employment was between 1900 and 1914 (when he was drafted into the French Army during the First World War) is not known. It is known, however, that he received occasional scholarships to continue his studies (on the recommendation of Émile Borel (1871-1956)) and he gave lectures as a 'free professor' at the Sorbonne between 1909 and 1914. One of his courses was *Probability calculus with applications to financial operations and analogies with certain questions from physics*. In this course he may have drawn out the similarities between the diffusion of probability (the total probability of one being conserved) and the diffusion equation of Fourier (the total heat-energy being conserved). In 1912 he wrote a book *Calcul des Probabilités* and in 1914 a book *Le Jeu, la Chance et le Hazard*. At the end of the War he obtained an academic position (lecturer) at Besançon then moved to Dijon (1922), then to Rennes (1925).

In 1926 he tried to go back to Dijon by applying for the vacant chair but was turned down on account of a critical report from Paul Lévy (1886-1971), then a professor aged 40 at the École Polytechnique.

Bachelier in his Thesis, in progressing from a 'drunkards' random walk with *n* (discrete) steps in time *t*, each step being of length *d*, to a (continuous) distribution for where the drunkard might be at time *t*, realised that there had to be a relationship between *n* and *d* - *d* equal to (*t*/*n*)^{(1/2)} for the limit process to 'work'.

In a later paper [38] he showed, effectively, that if a random walk on the *y*-axis is represented as a graph in time with the 'drunkard' making *n* steps in time *t*, each step of length *d*, the path was such that the tangent of the path angle {i.e. *d* divided by (*t*/*n*)} became increasingly large {in the ratio (*n*/*t*)^{(1/2)}} as *n* increased. The paths in the time-graph got more and more vertical (up or down) with increasing *n* but the resulting distribution of where the drunkard might be became increasingly regular. Paul Levy thought that Bachelier had made a mistake in his paper by making the tangent of the path (up or down) constant and Bachelier failed to be appointed at Dijon. Bachelier was furious and wrote to Levy, who, apparently, was unrepentant over this calumny.

The algebraic sum of the upwards and downwards steps taken by the drunkard gives the height of the drunkard at time *t* above the origin while the sum of the squares of the steps is equal to *t* and the algebraic and absolute sum of the cubes of the upward and downward steps (and higher powers) become closer and closer to zero. It is these properties of continuity, non-differentiability, infinite 1^{st} order variation, finite 2^{nd} order variation and zero 3^{rd} or higher order variation that gives the drunkard's walk and, in the limit, Brownian Motion some of its unique character and leads to Itô's important Lemma.

It seems extraordinary that Levy was, apparently, unfamiliar with Bachelier's work as Bachelier had by this time (1926) published 3 books and some 13 papers on probability and regarded showing how a continuous distribution could be derived from a discrete distribution as his most important achievement. Levy once told J L Doob that "reading other writers' mathematics gave him physical pain" (see website below) so perhaps it was the case that Levy had never read Bachelier.

Borel, however, must have known Bachelier (he had approved the scholarships to Bachelier). It should be pointed out that Poincaré, who would not have made this mistake over the interpretation of Bachelier's work, had died some 14 years earlier.

It seems that Bachelier, was regarded as being of lesser importance in the eyes of the French mathematical élite (Hadamard, Borel, Lebesgue, Lévy, Baire). His mathematics was not rigorous (it could not be as the mathematical techniques necessary to make it so had not been developed e.g. measure theory and axiomatic probability) although, his results were basically correct.

However, Levy, a few years later, was apparently surprised to find Kolmogorov referring to Bachelier's work. In 1931, Levy wrote a letter of apology to Bachelier and they were reconciled.

Bachelier moved back to Besançon (this time as permanent professor) in 1927 and retired aged 67 in 1937. His last publication was in 1941 and he died in 1946 aged 76.

Bachelier's work is remarkable for herein lie the theory of Brownian Motion (one of the most important mathematical discoveries of the 20^{th} century), the connection between random walks and diffusion, diffusion of probability, curves lacking tangents (non-differentiable functions), the distribution of the Wiener process and of the maximum value attained in a given time by a Wiener process, the reflection principle, the pricing of options including barrier options, the Chapman-Kolmogorov equations in the continuous case,

(namely *f* (*x*_{n}|*x*_{s}) = *f* (*x*_{n}|*x*_{r}) *f* (*x*_{r}|*x*_{s}) *dx*_{r} where *n* > *r* > *s* where *f* are the transition densities of a Markov sequence of random variables) and the seeds of Markov Processes, weak convergence of random variables (i.e. convergence in distribution), martingales and Itô stochastic calculus.

Bachelier's treatment and understanding of the theory of Brownian Motion (originally called Brownian Movement) is more elegant and mathematical than in Einstein's 1905 paper. While Einstein had an unsurpassed 'nose' for physics his nose for mathematics was, by his own admission, not so highly developed.

The work of Bachelier leads on to the work of Wiener (1923), Kolmogorov (1931), Itô(1950), and Black, Scholes and Merton (1973).

Bachelier was ahead of his time and his work was not appreciated in his lifetime. In the light of the enormous importance of international derivative exchanges (where the pricing is determined by financial mathematics) the remarkable pioneering work of Bachelier can now be appreciated in its proper context and Bachelier can now be given his proper place.

**Article by:** David O Forfar Heriot-Watt University.