The discrete choice model determines on the basis of the fitnesses of the beliefs with which probabilities the moving average and fundamental beliefs are chosen by the agents. The memory parameter, η, we choose to be equal to 0.25. We choose the intensity of choice parameter, β, to be equal to 250. The minimum probabilities with which the fundamental belief, mfund, and the moving average belief, mma are chosen, we set equal to 0.01.
6.6.3 Model simulations
Bifurcations
We have seen in equation (6.45) that in the case of risk neutrality of the fundamental traders, i.e. a=0, there is locally always convergence to the fundamental steady state. If a=4, then for β=0 the dynamical system exhibits quasi periodic behavior and no change in the dynamics occurs by increasing β. Only for a<0.456 changes in the dynamical behavior can be observed by varying β. Therefore we set the risk aversion parameter a initially low (0.42), so that the local dynamics around the steady state is dependent on the intensity of choice parameter β.For a=0.42 figure 6.3a shows the bifurcation diagram with respect to β. A Hopf bifurcation occurs at βH=635. Figure 6.3b shows the corresponding largest LCE plot. Before the Hopf bifurcation occurs the largest LCE is clearly smaller than zero, indicating convergence to the steady state. After the Hopf bifurcation occurred, the largest LCE is close to zero, indicating quasi periodic dynamical behavior. Thus for costs and low risk aversion for the fundamental traders and low intensity of choice for all traders, the price locally converges to the fundamental value. However for high intensity of choice, traders quickly change to the most profitable strategy and the moving-average trading strategy can survive in the market even for low risk aversion of the fundamental traders. Price fluctuations are then driven by the evolutionary dynamics between the two different beliefs.
If the costs for the fundamental traders decrease to zero, then locally when varying β there is always convergence to the fundamental steady state, for low risk aversion. Fundamental expectations then dominate the moving-average strategy. Hence, costs can cause the fundamental steady state to become unstable, even if the risk aversion of fundamental traders is low. In the case of no costs and β=250, figure 6.4a shows the bifurcation diagram with respect to the parameter a, when a is varied between 0.1 and 5. Figure 6.4b shows the corresponding largest LCE plot. At aH=0.456 a Hopf bifurcation occurs and the dynamics shows quasi periodic behavior after the Hopf bifurcation. Hence, if funda-