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How well does any particular average describe normal? How stable, how powerful is an average as an indicator of behavior? When observations wander away from the averages of the past, how likely are they to regress to that average in the future? And if they do regress, do they stop at the average or overshoot it?
It was yet another Frenchman, Louis Bachelier, who was the first to suggest that stock exchange prices might follow a normal distribution. That said, his 1900 doctoral thesis had little impact. It was not until nearly 60 years later that Martin J. Osborne, a physicist working at the Naval Research Laboratory in Washington, put forward the idea of representing the evolution of market prices by means of a distribution of normal probability. He had developed this thesis before becoming acquainted with Bachelier’s work, but once he had done so, he wrote: I believe the pioneer work on randomness in economic time series, and yet most modern in viewpoint, is that of Bachelier also described in less mathematical detail in reference. As reference is rather inaccessible (it is available in the Library of Congress rare book room), it might be well to summarize here. In it Bachelier proceeds, by quite elegant mathematical methods, directly from the assumption that the expected gain (in francs) at any instant on the Bourse is zero, to a normal distribution of price changes, with dispersion increasing as the square root of the time, in accordance with the Fourier equation of heat diffusion. The theory is applied to speculation on rente, an interest-bearing obligation which appeared to be the principal vehicle of speculation at the time, but no attempt was made to analyze the variation of prices into components except for the market discounting of future coupons, or interest payments. The theory was fitted to observations on rente for the years 1894 – 98. There is a considerable quantitative discussion of the expectations from the use of options (puts and calls). He also remarked that the theory was equally applicable to other types of speculation, in stock, commodities, and merchandise. To him is due credit for major priority on this problem.
Bachelier indeed demonstrated that price changes are randomly distributed. That said, if random distribution works reasonably well for the average return on a portfolio, it does not work nearly so well for the return on one asset (or even just one type of asset). If it is true that the price of a share does not follow the law of normal distribution, the average prices represented by a stock market index are, by contrast, described rather adequately by the law of normal distribution.