6.7 Conclusion
In this chapter we have built a financial market model with heterogeneous adaptively learning agents, fundamentalists and technical traders. The model is an extension of the Brock and Hommes (1998) model in that it extends the set of trading techniques the agents can independently choose from with a realistic moving-average technical trading rule. Moving averages are well known and one of the mostly used technical indicators in financial practice and therefore they deserve to be implemented in heterogeneous agents modeling. Furthermore, the model is derived under the assumption of relative risk aversion, instead of absolute risk aversion as in the Brock and Hommes (1998) case.The model is derived under the assumption of infinitely many agents, who only differ in the forecasting rule they select each period. Under the assumption that each agent has zero market power at each date, that is his individual investment decision will not influence the equilibrium price, it is shown that the fraction of total market wealth invested by all agents according to a certain belief converges in probability to the probability that the belief is chosen by the agents. Under the assumption of zero supply of outside stocks and the use of certain beliefs types it turns out that the price equilibrium formula is exactly the same as in Brock and Hommes (1998), namely that the price is equal to the discounted value of the average expected price and dividends by all agents. Moreover if the moving-average technical trading rule is added to the model, then also risk aversion and expected dispersion of future returns play a role in our model.
In the end, our financial market model is an eight dimensional nonlinear dynamical system. The steady state price is equal to the fundamental value, which is the discounted value of all future dividends. Analytically we derive the eigenvalues of the linearized system and we examine for which parameter values bifurcations occur. It is shown that the system only can exhibit a Hopf bifurcation. We use numerical tools such as delay, phase and bifurcation diagrams, and computation of Lyapunov characteristic exponents to study the local stability around the fundamental steady state. If there is no difference in costs of applying the fundamental or moving-average strategy, then it is found that the intensity of choice parameter, measuring how quickly traders switch beliefs, has no influence on the dynamical behavior. In the presence of costs, if the risk aversion parameter of the fundamental traders is low enough, then these traders always drive prices back to the fundamental steady state for the case the intensity of choice parameter is sufficiently low. For high values of the intensity of choice parameter, even for low risk aversion, quasi periodic price behavior can occur as a consequence of a Hopf bifurcation. If costs of all trader types are set to zero and if more realistic values for the risk aversion parameter are