| The financial market is modelled as a complex self-organizing system. Three economic agents interact in a simplified economy and seek the maximization of their wealth. Replicator dynamics are used as a myopic behavioral rule to describe how agents learn and benefit from their experiences. Stock price fluctuations result from interactions between economic agents, budget constraints and conservation laws. Time is discrete. Invariant distributions over the state space, that is to say probability measures that remain unchanged by the one-period transition rule, form stochastic equilibria for our composite system. When agents make mistakes, there is a unique stochastic steady state which reflects the average and limit behavior. Convergence of the iterates occurs at a geometric rate in the total variation norm. Interestingly, when the probability of making a mistake tends to zero, the invariant distribution converges weakly to a stochastic equilibrium for the model without mistakes. Most agent-based computational economies heavily rely on simulations. Having adopted a simple representation of financial markets, we have been able to prove the above theoretical results and gain intuition on complexity economics. The impact of simple monetary policies on the limit stock price distribution, such as a decrease of the riskfree rate of interest, has been analyzed. Of interest as well, the limit stock log return distribution presents real-world features (skewed and leptokurtic) that more traditional models usually fail to explain or consider. Our artificial market is incomplete. The bid and ask prices of a vanilla Call option have been computed to illustrate option pricing in our setting. |
