PRICE VARIATIONS IN A STOCK-MARKET WITH MANY AGENTS

Large variations in stock prices happen with sufficient frequency to r aise doubts about existing models, which all fail to account for non-G aussian statistics. We construct simple models of a stock market, and argue that the large variations may be due to a crowd effect, where ag ents imitate each other's behavior. The variations over different time scales can be related to each other in a systematic way, similar to t he Levy stable distribution proposed by Mandelbrot to describe real ma rket 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 quan tum field theory methods. When the agents imitate each other and respo nd 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 who se behavior is derived from fundamental analysis of the stock, includi ng dividends, and ''noise traders'', whose behavior is governed solely by studying the market dynamics and the behavior of other traders, is investigated. When the relative number of rational traders is small, ''bubbles'' often occur, where the market price moves outside the rang e justified by fundamental market analysis. When the number of rationa l traders is larger, the market price is generally locked within the p rice range they define.