be measured (the mean and the standard deviation), and it can be applied to a wide range of phenomena. The “Law of Unreason,” as its Victorian popularizer Francis Galton called it, would find perhaps its greatest application in mastering, or appearing to master, the chaos of the markets.29
The desire to bring order out of chaos, and to see the hidden pattern in the noise, is basic to human nature. In mathematics, even chaos theory is not so much about chaos as about showing that what appears to be wild and unruly behavior can actually be explained by a simple equation. As the field's founder, French mathematician Henri Poincaré, told one of his PhD students: “what is chance for the ignorant is not chance for the scientists. Chance is only the measure of our ignorance.”30
The student who earned this rebuke was called Louis Bachelier. His mistake, perhaps, was choosing a thesis subject that was a little too chaotic – the buying and selling of securities that took place within the mock-Greek temple building of the Paris Exchange, or Bourse. He was awarded a good but undistinguished grade on his 1900 dissertation, entitled Théorie de la Spéculation, and his work failed to unite the academic community in a frenzy of excitement (it took him 27 years to find a permanent job).31
Bachelier began his thesis as follows (imaginary editorial remarks in italics):
The influences which determine the movements of the Stock Exchange are innumerable. Events past, present or even anticipated, often showing no apparent connection with its fluctuations, yet have repercussions on its course. (I had a dog like that once)
Beside fluctuations from, as it were, natural causes, artificial causes are also involved. The Stock Exchange acts upon itself and its current movement is a function not only of earlier fluctuations, but also of the present market position. (Chased its own tail)
The determination of these fluctuations is subject to an infinite number of factors: it is therefore impossible to expect a mathematically exact forecast. Contradictory opinions in regard to these fluctuations are so divided that at the same instant buyers believe the market is rising and sellers that it is falling. (Sounds like we're wasting our time, people)
Undoubtedly, the Theory of Probability will never be applicable to the movements of quoted prices and the dynamics of the Stock Exchange will never be an exact science. (Thought this was a science exam?)
However, it is possible to study mathematically the static state of the market at a given instant, that is to say, to establish the probability law for the price fluctuations that the market admits at this instant. Indeed, while the market does not foresee fluctuations, it considers which of them are more or less probable, and this probability can be evaluated mathematically. (Too much on finance! – this was a real comment on Bachelier's thesis by France's leading probability theorist, Paul Lévy)
Bachelier's starting assumption, which he called his “Principle of Mathematical Expectation,” was that the mathematical expectation of a speculator is zero. As in the random walks of Figure 2.1, some bets will win, and others will lose, but these cancel out in the long run. Note that we are referring here to the mathematical chances of success – a speculator's psychological expectations may be very different. He then assumed that prices move in a random walk, with price changes following a normal distribution, and referred to what he called the “Law of Radiation (or Diffusion) of Probability,” which described how the future price became more uncertain as you went further into the future. The results are very similar to Figure 2.3 (the displacements at each iteration were there set to plus or minus a fixed amount, in this case 1, rather than being normally distributed, but the effects are almost identical over large enough times). From this, he derived a method for pricing options, which grant the purchaser the right to buy or sell an asset at a fixed price at some time in the future. As discussed further later, the technique he developed is essentially a special case of the ones commonly used today.
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