Large variations in stock prices happen with suﬀicient frequency to raise doubts about existing models, which all fail to account for non-Gaussian statistics. We construct simple models of a stock market, and argue that the large variations may be due to a crowd eﬀect, where agents imitate each other’s behavior. The variations over diﬀerent time scales can be related to each other in a systematic way, similar to the Lévy stable distribution proposed by Mandelbrot to describe real market indices. In the simplest, least realistic case, exact results for the statistics of the variations are derived by mapping onto a model of diﬀusing and annihilating particles, which has been solved by quantum ﬁeld theory methods. When the agents imitate each other and respond to recent market volatility, diﬀerent scaling behavior is obtained. In this case the statistics of price variations is consistent with empirical observations. The interplay between “rational” traders whose behavior is derived from fundamental analysis of the stock, including dividends, and “noise traders,” whose behavior is governed solely by studying the market dynamics, is investigated. When the relative number of rational traders is small, “bubbles” often occur, where the market price moves outside the range justiﬁed by fundamental market analysis. When the number of rational traders is large, the market price is generally locked within the price range they deﬁne
Bak, Per; Paczuski, Maya; and Shubik, Martin, "Price Variations in a Stock Market with Many Agents" (1996). Cowles Foundation Discussion Papers. 1380.