**5. Concluding remarks**

In a market with fundamentalists and contrarians the fundamental steady state is the unique steady state of the system, which arises for low values of intensity of choice. Memory affects neither the position of this steady state nor its stability. For increasing intensity of choice a primary bifurcation, i.e. a period doubling bifurcation occurs; the steady state becomes unstable and a stable period two-cycle appears. Both the primary bifurcation value and the position of the period two-cycle are independent of the memory. For further increasing intensity of choice a secondary bifurcation, i.e. a supercritical Hopf bifurcation, occurs; the period two-cycle becomes unstable and an attractor appears consisting of two invariant circles around each of the two (unstable) period two-points, one lying above and the other one below the fundamental. For high intensity of choice chaotic asset price dynamics occurs, interspersed with many stable period cycles. Such a bifurcation route to chaos is often called the rational route to randomness.

Memory and Asset Pricing Models with Heterogeneous Beliefs 343

even when there are no costs and memory is infinite, fundamentalist beliefs can not drive out opposite purely biased beliefs, when the intensity of choice to switch strategies is high. Hence, following the argumentation of Brock and Hommes (1998, p. 1260), the market can protect a biased trader from his own folly if he is part of a group of traders whose biases are balanced. In conclusion, both our analytical work and our numerical simulations suggest that biases alone do not trigger chaotic asset price fluctuations. Sensitivity to initial states and irregular switching between different phases seem to be triggered by trend extrapolators; in our case by contrarians. Apparently, some (strong) trend extrapolator beliefs are needed, such as strong trend followers or strong contrarians, in order to trigger chaotic asset price fluctuations. A key feature of our heterogeneous beliefs model is that the irregular fluctuations in asset prices are triggered by a rational choice in prediction strategies, based upon realized profits, *viz*. the observed deviations from the fundamentals are driven by short-run profit seeking. We can also talk about rational animal spirits that, according to Brock and Hommes (1997b), exhibit some qualitative features of asset price fluctuations in the actual financial markets, such as the autocorrelation structure of prices and returns.

*Faculty of Economics, University of Ljubljana, Slovenia & Institute for Economic Research, Ljubljana,* 

I am grateful for very helpful suggestions and comments from Cars H. Hommes, Valentyn Panchenko, Jan Tuinstra and Florian O. O. Wagener from the University of Amsterdam.

Arrowsmith, D. K. and Place, C. M., 1990. *An Introduction to Dynamical Systems*. Cambridge,

Awrejcewicz, J., 1991. *Bifurcation and Chaos in Coupled Oscillators*. Singapore: World Scientific

Awrejcewicz, J. and Lamarque, C.-H., 2003. *Bifurcation and Chaos in Nonsmooth Mechanical* 

Brock, W. A. and Hommes, C. H., 1997a. »A Rational Route to Randomness«. *Econometrica*

Brock, W. A. and Hommes, C. H., 1997b. »Models of Complexity in Economics and Finance« in: C. Heij [et al.]. *System Dynamics in Economic and Financial Models.* New York: John

Brock, W. A. and Hommes, C. H., 1998. »Heterogeneous Beliefs and Routes to Chaos in a Simple Asset Pricing Model«. *Journal of Economic Dynamics and Control,* 22 (8-9), 1235-1274. Brock, W. A., Hommes, C. H. and Wagener, F. O. O., 2005. »Evolutionary Dynamics in Markets with Many Trader Types«. *Journal of Mathematical Economics,* 41, 7-42.

**Author details** 

**Acknowledgement** 

UK: Cambridge University Press.

65 (September), 1059-1095.

*Systems*. Singapore: World Scientific Publishing.

Miroslav Verbič

**6. References** 

Publishing.

Wiley & Sons.

*Slovenia* 

In case of strong contrarians and high intensity of choice it is reasonable to expect that we will obtain a system that is close to having a homoclinic intersection between the stable and unstable manifolds of the fundamental steady state, which indicates the occurrence of chaos. There exists a certain limited interval of memory strength values, for which at a given intensity of choice we are more likely to obtain such a system with more additional memory in the model. A rational choice between fundamentalists' and contrarians' beliefs triggers situations that do not reach fruition due to practical considerations and are thus unattainable, 'castles in the air', as Brock and Hommes (1998, p. 1258) would put it. As a consequence we obtain market instability, characterized by irregular up and down oscillations around the unstable efficient market hypothesis fundamental price. Additional memory lengthens on average the period of this irregular cycle and mainly appears to be stabilizing with regard to asset prices.

In a market with fundamentalists and opposite biases the fundamental steady state is also the unique steady state of the system, arising for low values of intensity of choice. Memory does not affect the position of this steady state, but does affect its stability. For increasing intensity of choice a supercritical Hopf bifurcation occurs; the steady state becomes unstable and an attractor appears. Memory affects the emergence of this bifurcation; the higher the memory strength, the higher the bifurcation value. More memory thus has a stabilizing effect on dynamics. For high intensity of choice the dynamic behaviour is more complex. However, irrespective of the amount of additional memory such a route to complicated dynamics is different from that in the presence of contrarians, for after the bifurcation value only regular (quasi-)periodic fluctuations around the unstable fundamental steady state occur. Consequently, an important finding is that the predominating quasi-periodic dynamics does not seem to evolve to chaotic dynamics.

After the incidence of the bifurcation the higher value of the memory strength parameter causes the dynamics to be less periodic and more quasi-periodic; the dynamics therefore converges on purely quasi-periodic behaviour with increasing memory strength. Opposite biases may cause perpetual oscillations around the fundamental, even without costs for fundamentalists, but can not lead to chaotic movements. Furthermore, in a three-type world, even when there are no costs and memory is infinite, fundamentalist beliefs can not drive out opposite purely biased beliefs, when the intensity of choice to switch strategies is high. Hence, following the argumentation of Brock and Hommes (1998, p. 1260), the market can protect a biased trader from his own folly if he is part of a group of traders whose biases are balanced.

In conclusion, both our analytical work and our numerical simulations suggest that biases alone do not trigger chaotic asset price fluctuations. Sensitivity to initial states and irregular switching between different phases seem to be triggered by trend extrapolators; in our case by contrarians. Apparently, some (strong) trend extrapolator beliefs are needed, such as strong trend followers or strong contrarians, in order to trigger chaotic asset price fluctuations. A key feature of our heterogeneous beliefs model is that the irregular fluctuations in asset prices are triggered by a rational choice in prediction strategies, based upon realized profits, *viz*. the observed deviations from the fundamentals are driven by short-run profit seeking. We can also talk about rational animal spirits that, according to Brock and Hommes (1997b), exhibit some qualitative features of asset price fluctuations in the actual financial markets, such as the autocorrelation structure of prices and returns.
