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“Most of the work that follows the Moore paper stresses, in one way or another, the deviation of stock prices from the Einstein-Wiener process. The Alexander-Larson-Cootner-Steiger papers all question the independence hypotheses. The Fama-Mandelbrot material questions the assumption of Gaussian increments. The Osborne paper examines the stationarity of the process. In raising these questions it was necessary to invent subtle new tests, or to apply more esoteric probability distribution theory.” (p. 189)

“The central novel idea [in Larson’s essay] was the use of a test based on the range of a random walk to test in a more sensitive and powerful manner the resemblance of stock price series to a Wiener-Einstein type of Brownian motion model. If changes in stock prices tended to be succeeded by changes in the same direction, the range of segments of stock price series would tend to be greater, on the average, than segments drawn from a random walk. On the other hand, if changes tended to reverse themselves, the range would be less than that of random walk segments. The power of the range test lies in its sensitivity to nonlinear dependence.” (pp. 190-191)

“Cootner’s work (1962) divides into two parts. One part, like Alexander’s papers, is devoted to the testing of a decision rule, in this case one which is commonly referred to by stock market professionals [buy if current price is above the 200-day moving average, sell short if it is less than the moving average; reverse to close]. This model, unlike Alexander’s, is applied to individual common stocks, but like Alexander’s, it does substantially better than random buying of stocks on a gross basis—before deducting commission costs. This alone is evidence of either dependence or nonstationarity, although it is not evidence of market imperfections large enough to permit profitable mechanical trading rules.” (p. 192)

“The second part of Cootner’s paper is an attempt to characterize the observed deviations from a Gaussian diffusion process. The hypothesis suggested is that stock prices behave like a restricted random walk, a hypothesis which would explain (1) certain aspects of behavior of the kurtosis, (2) part of the fluctuating behavior of the serial correlations, and (3) the success of the Alexander-Cootner decision rules.” (p. 192)

“Osborne’s second paper (1963) … deals with differences in the behavior of low and high price stocks, the distribution of trading volume, ‘seasonality’ of trading patterns and prices, and the tendency of buyers and sellers to ‘prefer’ certain prices to others. All of these deviations from a random walk have interesting implications for the theory of stock price formation.

“Osborne finds, for example, that the occurrence of a transaction in (say) a given stock is not independent of past history of trading. That is, trading tends to come in ‘bursts:’ If recent trading has been heavy, it is likely to continue large. This observation can help to explain a phenomenon noted by Alexander (1964) and possibly one by Mandelbrot (1963). Alexander has found that the price changes in different subsets of the 1928 to 1962 period are significantly different from one another. This suggests that stock prices are a nonstationary process, which would be hard to analyze. Now Osborne’s results indicate that the observed pattern of behavior could result from a stationary distribution of price changes per transaction and a temporally dependent pattern of volume changes, a process which would be more amenable to study. A second possibility arises from some earlier work by Berger and Mandelbrot (1963) which suggests that stochastic processes that appear to have ‘bursts’ of occurrences may be represented by a model of independent increments which are distributed according to what Mandelbrot calls a Pareto-Levy distribution. Briefly, Berger and Mandelbrot argue that the individual ‘trades’ (transmission errors in their formulation) are in fact independent of each other, but the appearance of clustering is given by a larger probability of long intervals between trades than would be suggested by a Poisson model.” (pp. 193-94)

“Alexander argues, with considerable merit, that there are two separate issues involved in the study of stock prices. The one he chooses to study is whether, in point of fact, stock prices are a random walk, and his conclusion is that they certainly are not. A quite different question is whether or not stock prices are a sufficiently close approximation of a random walk, where the standard of ‘closeness’ is an economic one—that market participants cannot improve their performance by acting on the regularities.” (p. 194)

“The most striking material that Alexander brings to bear on the random walk question is … his investigation into the sources of the dependence of stock price changes. Such dependence does not appear to stem from the distribution of daily price changes themselves, since price series constructed by randomly rearranging the order of actual changes do not show the same nonrandom behavior.

“The ‘profits’ shown by the filter rule seem to derive from a tendency for ‘swings’—intervals between turning points defined by the filters—to persist slightly longer than would be expected in a random walk, for most ranges of durations of ‘swings.’ Since there is no similar tendency for

__daily__price changes to come in runs, the Alexander data suggest that the nonrandomness of price changes is a complex nonlinear process.” (pp. 194-95)

“Mandelbrot’s main empirical argument is of two parts. The Pareto-Levy distribution would be expected to have more extreme values than would be expected from a Gaussian distribution. This is undoubtedly true from all empirical evidence, and to this degree the Pareto-Levy distributions clearly describe a phenomenon which is not explicable in a Gaussian random walk. The second underpinning of Mandelbrot’s empirical argument is the instability of the sequential second moment of changes in speculative prices. If we designate a series of squared deviations of successive price changes from their mean, the means of the successive (or partial) cumulative sums of those squared deviations are what we call the sequential second moment.” (p. 197)

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