Price Dynamics for Fundamentalists – Risky Asset – Chartists via Modeling

Substituting (1), (2) and (3) to (4) from here, the dynamical system can be obtained as

pt+1 − pt = θN[(1 − κ)(exp(α(p − pt)) − 1) + κ(exp(β(1 − µ)(pet − pt)) − 1)] pet+1 − pet = µ(pt − pet ) —– (5)

In the following discussion we highlight the impact of increases in the total number of traders n on the price fluctuation that is defined as the price increment,

rt = pt+1 − pt

We first restrict ourselves to investigating the following set of parameters:

α = 3, β = 1, µ = 0.5, κ = 0.5, θ = 0.001 —– (6)

It is clear that the two-dimensional map (5) has a unique equilibrium with pet = pt = p, namely (p¯e , p¯) = (p, p), given the above conditions. Elementary computations show that for our map (5) the sufficient condition for the local stability of the fixed point p is given as

N < (2(2 − µ))/(θ[α(2 − µ)(1 − κ) + 2β(1 − µ)κ]) —– (7)

From (7) it follows that, starting from a small number of traders N inside the stability region, when the number of traders N increases, a loss of stability may occur via a flip bifurcation. We shall now look more globally into the effect of increases in the number of traders on the price dynamics. Figure 1 shows a bifurcation diagram of the price increments rt with N as the bifurcation parameter under the set of parameters (6). For the convenience of illustration, Figure 1 is drawn using θN as the bifurcation parameter. This figure suggests the following bifurcation scenario.


The price increments rt converge to 0 when the number of traders N is small. In other words, the price converges to the fundamental price p when the active traders are few. However the price dynamics become unstable when the number of traders N exceeds about 1000, and chaotic behavior of the price increments occurs after infinitely many period-doubling bifurcations. If N is further increased, then the price increments rt become more regular again after infinitely many period-halving bifurcations. A stable 2 orbit occurs for an interval of N-values. However as N is further increased, the behavior of the price increment rt becomes once again chaotic, and the prices diverge. Let us investigate closely the characteristics of chaos that are observed in the parameter interval (2000 < N < 4000). Figure 2 shows a series of price increments rt with N = 4000 and the set of parameters (6). The figure shows apparently the characteristic of intermittent chaos, that is, a long laminar phase, where the price fluctuations behave regularly, is interrupted from time to time by chaotic bursts. 



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