Price variations in a stock market with many agents

Abstract

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 Levy 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 larger, the market price is generally locked within the price range they define.

On financial markets trading

“…It is shown that such speculative actions make demand and supply balance automatically on average by changing the market price, resembling a kind of self organized criticality. Fig.5 is in total agreement with that: Contrary to [3] we do not need to impose that the number of the shares has to be half of the number of traders in order to get a balance between demand and supply. The three circles in the upper panel indicate the most extreme events taking place in the price evolution: There is a corresponding movement in the book and a peak in the volumes.…”

“…This represents, on our opinion, a realistic upper limit for the market price, since nobody would buy anything if they cannot afford it (maybe the reality is a little bit different; traders stop buying shares when the price is too high not because they do not have money, but because they believe that the price is too far away from a fair 3) Just to give some colors to these numbers, we can imagine the following situation, not too far away from the reality in a big european country. Suppose that 10 million of traders are interested in trading a given new stock and they would like to invest, on average, 3000 EUROs.…”

“…Previous models are all essentially based on the assumption that two different kinds of economic agents are interacting in the market: Some authors [1] call them dealers and savers, others [2] use the names fundamentalists and noise traders (further distinguishing between optimistic and pessimistic), others more [3] say rationals and chartists. If one goes into the details of these models, one sees that fundamentalists follow the premise of the efficient market hypothesis in that they expect the price to follow the fundamental value of the asset.…”

“…It is in fact clear that without the fundamentalists the price would follow just one trend at infinitum and without the noise traders the price would never escape the range of its fundamental value. In [3] this fact becomes clear, since already a 20% of fundamentalists is enough to confine the market price within the range of the rational traders. Just very short deviations occur before common sense was restored.…”

Agent-based simulation of a financial market

“…Others have proposed artificial markets populated with heterogeneous agents characterised by simple trading rules [4][5][6][7][8][9][10][11]. These lend themselves better to analytical modelling while still retaining the ability to capture fundamental features of market behaviour.…”