# Formation Years

Bachelier defended his thesis "Theory of Speculation" in 1900 in Paris. Let us say a few words about this extraordinary thesis.

The problem investigated by Bachelier is described in less than a page.

The stock market is subject to innumerable random influences, and so it is unreasonable to expect a mathematically precise forecast of stock prices. However, we can try to establish the law of the changes in stock prices over a fixed period of time. The determination of this law was the subject of Bachelier's thesis. The thesis was not particularly original. Since the early nineteenth century, people had applied probability theory to study exchange rates. In his thesis, Bachelier intended to revisit this issue from several viewpoints  taken from physics and probability theory.

The first method he used is the method adopted by Einstein, 5 years later, to determine the law of BM in a physical context. It consists of studying the integral equation that governs the probability that the change in price is y at time t, under 2 natural assumptions:

1. the change in price during 2 separate time intervals is independent.

2. the expectation of the change in price is zero.

The resulting equation is a homogeneous version of the diffusion equation, now known as the Kolmogorov equation, in which Bachelier asserts that the appropriate solution is given by a centered Gaussian law with variance proportional to time t. He proved a statement already proposed by Regnault in 1860 that the expectation of the absolute change in price after time t is proportional to the square root of t.

But this first method did not seem to satisfy Bachelier, since he proposed a second method, which was further developed in the 1930s by the Moscow School: the approximation of the law of BM by an infinite sequence of coin flips, properly normalized. Since the change in price over a given period of time is the result of a very large number of independent random variables, it is not surprising that this change in price is Gaussian. But the extension to this approximation to a continuous-time version is not straightforward.

Bachelier states and prepares the way to the first known version of a theorem:

Let {X1, X2, ..., Xn, ...} be a sequence of independent random variables taking values 1 or -1 with probability 1/2.

If we let Sn = X1+ X+ ... + Xand let [x] denote the integer part of a real number x, then: in law as , where is a strandard BM.

This second equation naturally leads to the previous solution. But it is still not sufficient.

Bachelier proposed a third method, the "radiation of probability". Bachelier was aware of the "method of Laplace", which gives the fundamental solution of the heat equation, a solution that has exactly the form given by the first and second methods used by Bachelier.

Bachelier explains the curious intersection between the theory of heat and the prices of annuities on the Paris stock exchange. This is his third method:

Consider the fame of flipping a fair coin an infinite number of times and set f(n, x) = P(Sn = x).

It has been known since ages that: Substracting f(n, x) from both sides, we obtain: It suffices to take the unit 1 to be infinitely small to obtain heat equation: whose solution is the law of centered Gaussian RV with variance n.