Some Aspects of the Random walk Model of Stock Market Prices

fits such data very well, where Pt is the price series and .t is a purely random, white noise series such that -t and ,, are uncorrelated for all t * s. This model was originally suggested by Bachelier [1] and re-introduced by Osborne [10]. It is now well founded as indicated bv many of the papers in the book edited by Cootner [2] and by subsequent papers, such as those by Fama [4] and Godfrey, Granger and Morgenstern [6]. The model has been tested for many series, both indices and individual stocks, for many countries and a variety of sampling units. A number of different statistical techniques have been used, and almost without exception the random walk model has been found to be a very adequate model for the series. On some occasions the price series is replaced by the logarithm of the series, particularly when data over a time-span of several years is used. The random walk model is, of course, too simple to have been fully accepted and, in fact, a number of modifications to it have been suggested (see [2]). Thus, for example, Granger and Morgenstern [7] found that the long-term movements in the series cannot be fully explained by such a model. In this paper, two specific aspects of the model will be considered. The first concerns the question of whether or not Zt is a purely random series if individual transactions data is considered. The second concerns the suggestion by Mandelbrot (in [2]), and supported by Fama ;in [2], [5]) that the model is correct but that the variance of Zt is infinite.