Brownian Motion in the Stock Market

TLDR

The steady‑state distribution of log price ratios follows a Brownian‑motion–like Gaussian, derived under trading conditions and the Weber‑Fechner law. The study shows that stock prices and money values behave like particles in statistical equilibrium, with a Brownian‑motion distribution that predicts a 3–5 % annual rise in expected price and reciprocal price, independent of inflation.

Abstract

It is shown that common-stock prices, and the value of money can be regarded as an ensemble of decisions in statistical equilibrium, with properties quite analogous to an ensemble of particles in statistical mechanics. If Y = log e [P(t + r)/P 0 (t)], where P(t + r) and P 0 (t) are the price of the same random choice stock at random times t + r and t, then the steady state distribution function of Y is [Formula: see text], which is precisely the probability distribution for a particle in Browman motion, if σ is the dispersion developed at the end of unit time. A similar distribution holds for the value of money, measured approximately by stock-market indices. Sufficient, but not necessary conditions to derive this distribution quantitatively are given by the conditions of trading, and the Weber-Fechner law. A consequence of the distribution function is that the expectation values for price itself ℰ(P) = ∫ 0 ∞ Pφ(Y)(dY/dP) dP increases, with increasing time interval τ, at a rate of 3 to 5 per cent per year, with increasing fluctuation, or dispersion, of P. This secular increase has nothing to do with long-term inflation, or the growth of assets in a capitalistic economy, since the expected reciprocal of price, or number of shares purchasable in the future, per dollar, increases with τ in an identical fashion.