3-Milestones Page 79 Wednesday, February 4, 2004 12:47 PM
Milestones in Financial Modeling and Investment Management 79elier, “The expectation of the speculator is zero.” The formal concept of a
martingale (i.e., of a process such that its expected value at any moment
coincides with the present value) had not yet been introduced in probabil-
ity theory. In fact, the rigorous notion of conditional probability and fil-
tration (see Chapter 6) were developed only in the 1930s. In formulating
his hypothesis on market behavior, Bachelier relied on intuition.
Bachelier actually went much further. He assumed that stock prices
evolve as a continuous-time Markov process. This was a brilliant intu-
ition: Markov was to start working on these problems only in 1906.
Bachelier established the differential equation for the time evolution of
the probability distribution of prices, noting that this equation was the
same as the heat diffusion equation. Five years later, in 1905, Albert
Einstein used the same diffusion equation for the Brownian motion (i.e.,
the motion of a small particle suspended in a fluid). Bachelier also made
the connection with the continuous limit of random walks, thus antici-
pating the work of the Japanese mathematician Kiyosi Itô at the end of
the 1940s and the Russian mathematician and physicist Ruslan L. Stra-
tonovich on stochastic integrals at the end of the 1950s.
By computing the extremes of Brownian motion, Bachelier computed
the price of several options. He also computed the distributions of a
number of functionals of Brownian motion. These were remarkable
mathematical results in themselves. Formal proof was given only much
later. Even more remarkable, Bachelier established option pricing formu-
las well before the formal notion of absence of arbitrage was formulated.
Though the work of Bachelier was correctly assessed by his advisor
Poincaré, it did not bring him much recognition at the time. Bachelier
succeeded in getting several books on probability theory published, but
his academic career was not very successful. He was offered only minor
positions in provincial towns and suffered a major blow when in 1926,
at the age of 56, he was refused a permanent chair at the University of
Dijon under the pretext (false) that his 1900 thesis contained an error.^5
Bachelier’s work was outside the mainstream of contemporary
mathematics but was too mathematically complex for the economists of
his time. It wasn’t until the formal development of probability theory in
1930s that his ideas became mainstream mathematics and only in the
1960s, with the development of the theory of efficient markets, that his
ideas became part of mainstream finance theory. In an efficient market,
asset prices should, in each instant, reflect all the information available
at the time, and any event that causes prices to move must be unex-(^5) The famous mathematician Paul Levy who, apparently in bona fide, initially en-
dorsed the claim that Bachelier’s thesis contained an error, later wrote a letter of
apology to Bachelier.