The Money Formula. Wilmott Paul

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.