Mandelbrot and Hudson credit Bachelier with modeling speculative prices probabilistically and relating their evolution to diffusion, while distinguishing that foundational achievement from the later empirical adequacy of his Gaussian model.

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If the market, in effect, does not predict its fluctuations, it does assess them as being more or less likely. And this likelihood can be evaluated mathematically. Opening lines of Théorie de la Spéculation Now, there had been a few earlier mathematical sorties at the market. A French stockbroker, Jules Reynaud, had observed in 1863 that the longer you hold a security, the more you can win or lose on its price variations. He even worked out a formula for it. But, most market analysis looked at stock and bond prices in the conventional way. Something happens and prices react. A story of cause and effect, easy to work out afterwards, difficult to forecast beforehand. But this approach was futile. One can never know everything. Instead, Bachelier tried to estimate the odds that prices will move, then a novel approach. And he did so brilliantly by observing a strange and unexpected analogy between the diffusion of heat, through a substance, and how a bond price wanders up and down.

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Regardless, one cannot help but marvel that the movement of security prices, the motion of molecules, and the diffusion of heat could all be of the same mathematical species. As will be seen, it is one of many such strange liaisons in nature. Bachelier did not stop at theory. He also tested his equations against real prices for options and futures contracts. The theories worked. For instance, he calculated that the buyer of a 45-day option at half a franc has 40% odds of earning a profit. He was uncannily close. Looking back at real trading data, he found 39% of such options had in fact yielded a profit to their buyers. The market, unwittingly, obeys a law which governs it, The law of probability, he concluded. The Efficient Market Alas, Bacalier's economic insights went largely unnoticed for many years. In those days, finance theory was an oxymoron. Finance was a distasteful trade, not a subject fit for academic inquiry. That attitude did not start changing until the crash of 1929.