Theory of Speculation

At the stock market, probability radiates like heat. This "demonstrates" the role of Gaussian laws in problems related to the stock market.

This point would be developed by Kolmogorov in a famous article in 1931. In fact, the first and third methods used by Bachelier are intrinsically linked: the Kolmogorov equation for any regular Markov process is equivalent to a partial differential equation of parabolic type.

In all regular Markovian schemes that are continuous, probability radiates like heat from a fire fanned by the thousand winds of chance. And further work, exploiting this real analogy, would transform not only the theory of Markov processes but also the theory of Fourier equations and parabolic equations.

Now, having determined the law of price changes, all calculations of financial products involving time follow easily. But Bachelier did not stop there. He proposed a general theory of speculation integrating all stock market products that could be proposed to clients, whose expected value at maturity (therefore whose price) can be calculated using general formulas resulting from theory.

The most remarkable product that Bachelier priced was based on the maximum value of a stock during the period between its purchase and a maturity date; In this case, one must determine the law of the maximum of a stock price over some interval of time. It involves knowing a priori the law of the price over an infinite time interval, but it was not known at that time how to easily calculate the integrals of functions of an infinite number of variables.

Let us explain the reasing used by Bachelier as an example of his methods of analysis.

Bachelier proceeds in 2 different ways.

1. The first way was based on the second method developed in his thesis.

It consists in discretizing time in steps of ∆t, and introducing a change in price at each step of +- ∆x.

Bachelier wanted to calculate the probability that before time t = n∆t, the game (price) exceeds a given value c = m∆x. Let n = m + 2p.

Bachelier proposed to first calculate the probability that the price c is reached for the first time at exactly time t. To this end, he uses the gambler's ruin argument: the probability is equal to (m/n) C_n^p2^-n, which Bachelier obtained from the ballot formula of Bertrand.

It suffices to then pass properly to the limit so that . One then obtains the probability that the price exceeds c before t. Bachelier then noted that this probability is equal to twice the probability that the price exceeds c at time t.

The result is Bachelier's formula for the law of the maximum M_t of the price B_t over the interval [0,t]:

Bachelier had to justify this formula in a simple way to understand why it holds. He used the argument that "the price cannot pass the threshold c over a time interval t of length t without having done so previously" and hence that:

where alpha is the probability that the price c, having been attained before time t, is greater than c at time t. The latter probability is obviously 1/2, due to the symmetry of the sample paths that go above and that remain below c by time t. It was the first example of the use of the reflection principle in probability theory.

It seems that in 1900, Bachelier saw very clearly how to model the continuous movement of stock prices and he established new computational techniques, derived notably from the classical techniques involving infinite sequences of fair coins flips. He provided an intermediate mathematical argument to explain a new class of functions that reflected the vagaries of the market.

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