The Variation of Certain Speculative Prices

The classic model of the temporal variation of speculative prices (Bachelier 1900) assumes that successive changes of a price Z(t) are independent Gaussian random variables. But, even if Z(t) is replaced by log Z(t),this model is contradicted by facts in four ways, at least: (1) Large price changes are much more frequent than predicted by the Gaussian; this reflects the “excessively peaked” (“leptokurtic”) character of price relatives, which has been well-established since at least 1915. (2) Large practically instantaneous price changes occur often, contrary to prediction, and it seems that they must be explained by causal rather than stochastic models. (3) Successive price changes do not “look” independent, but rather exhibit a large number of recognizable patterns, which are, of course, the basis of the technical analysis of stocks. (4) Price records do not look stationary, and statistical expressions such as the sample variance take very different values at different times; this nonstationarity seems to put a precise statistical model of price change out of the question.