Periodic Structure in the Brownian Motion of Stock Prices

Abstract

The internal structure of stock prices is examined by comparison with simple random walks of basic step 1/8, in which the individual price changes ΔP are the step length, and the volume measures the rate at which the steps are taken. It is found that there is definite evidence of periodic in time structure corresponding to intervals of a day, week, quarter, and year, these being simply the cycles of human attention span. The evidence is not in the periodicity of the price sequences P(t), rather in the distribution of the first and second differences of P(t), especially the second moment of ΔP (or variance), and in the rate at which the steps are taken. It is also shown that there is a periodic “space structure” in the price coordinate P, corresponding to the Brownian motion in the presence of equally spaced sites of preferred occupancy and reflection barriers, at the whole numbers. There is also marked evidence of “clustered” activity, the data being analyzed by methods appropriate to cosmic ray bursts, or star counts on astronomical photographs. In general, the picture of price motion as simple random walks is supported qualitatively, quantitatively there are some substantial departures from this simple picture.

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