Market clearing by maximum entropy in agent models of stock markets

Abstract

We replace the conventional market clearing process by maximizing the information entropy. At fixed return agents optimize their demands using an utility with a statistical tolerance against deviations from the deterministic maximum of the utility which leads to information entropies for the agents. Interactions are described by coupling the agents to a large system, called 'market'. The main problem in economic markets is the absence of the analogue of the first law of thermodynamics (energy conservation in physics). We solve this problem by restricting the utilities to be at most quadratic in the demand (ideal gas approximation). Maximizing the sum of agent and market entropy serves to eliminate the unknown properties of the latter. Assuming a stochastic volatility decomposition for the return we derive an expression for the pdf of the return which is in excellent agreement with the daily DAX data. The pdf exhibits a power law behaviour with an integer tail index equal to the number of agent groups. With the assumption of an Ornstein Uhlenbeck model for the risk aversity parameters of the agents the autocorrelation for the absolute return is well described up to time lags of 2.5 years.