* * *

“The sensitivity of speculative prices and the huge volume of securities traded result in an impressive total of gains and losses in each trading day. The changes in wealth represented by these fluctuations have served as a constant lure to men who hope to earn fame and fortune by somehow unraveling the puzzle of price forecasting, who yearn for the discovery of a predictive formula which will unlock those gold-filled vaults. … [U]ntil fairly recently, the study of these prices was the province of the speculator, rather than the academician. … In more recent years, however, economists and statisticians alike have brought their research tools to bear on this subject, not primarily to find an easy road to fortune (though who is to say such a thought did not occur) but to establish the relationship of the securities markets to the ideal constructs of their theories. Primarily because those ideal markets would not offer a road to easy fortune, these academic studies have proven to be more skeptical about the folklore of the market place than those of the professional practitioners. To several of the authors represented in this volume the ‘patterns’ described by some market analysis are mere superstitions. Julian Huxley has argued that mythology, religion, and superstition all flourish when men have to make decisions about matters over which they have no control. Whether or not that is the reason, it is hard to find a practitioner, no matter how sophisticated, who does not believe that by looking at the past history of prices one can learn something about their prospective behavior, while it is almost as difficult to find an academician who believes that such a backward look is of any

__substantial__value.” (pp. 1-2)

“In his paper, Roberts presents briefly and clearly the largely heuristic reasoning that lies behind what has come to be known as the random walk theory of stock prices. The basic proposition depends upon a characteristic of competition in perfect markets: That participants in such a market will eliminate any profits above the bare minimum required to induce them to continue in the market, except for any profits which might accrue to someone who can exercise some degree of market monopoly. There is, for example, no reason why a trader with special information about future events cannot profit from that monopolized knowledge. On the other hand, we should not expect, in such a market, that traders could continue to profit from the use of a formula depending only upon past price data and generally available rules of ‘technical analysis.’ If this is so, all changes in prices should be independent of any past history about a company which is generally available to the trading public. Then, except possibly for a trend which is related to the desired rate of return, future changes in stock prices could just as well be determined by a flip of a coin as by any elaborate analysis of past data.” (p. 2)

“Roberts’ second point, which arises first in the U.S. literature in the writings of H. Working (1934), is the demonstration that a random walk series will, in fact, look very much like an actual stock series…. This seems to be due to a tendency to ascribe to

__sums__of independent random variables, behavior which is typical of the individual random variables themselves.” (p. 3)

“In [Bachelier’s] paper we find the Chapman-Kolmogorov-Smoluchowski equation for continuous stochastic processes, the derivation of the Einstein-Wiener Brownian motion process and the recognition that this process is a solution of the partial differential equation for heat diffusion. The Einstein-Wiener process is the analogue, for continuous time and continuous random variables, of the discrete random walk process. … Most of this theory was later to be developed by the mathematicians who were transforming probability theory into a rigorous discipline, Levy, Kolmogorov, Borel, Khinchine, and Feller. Compared to these standards of rigor, Bachelier’s work was heuristic, and scorn for the heuristics led to an underestimation by contemporaries of the significance of the contributions.” (p. 3)

“[F]or Bachelier the mathematical formulation of the problem was but the first step towards empirically testing the results with the real world. The results are striking, the prices of the options correspond very closely to their calculated expected values, suggesting strongly that the option pricing process is rational and the bond pricing process is an independent increment process. If successive increments were positively correlated, we would expect options to be priced more expensively than Bachelier’s theory implies. If price changes had a tendency to reverse themselves, options should be less expensive. Furthermore, actual option buying proved profitable about as often as the theory predicted.” (pp. 5-6)

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