Chaos, border collisions and stylized empirical facts in an asset pricing model with heterogeneous agents


An asset pricing model with chartists, fundamentalists and trend followers is considered. A market maker adjusts the asset price in the direction of the excess demand at the end of each trading session. An exogenously given fundamental price discriminates between a bull market and a bear market. The buying and selling orders of traders change moving from a bull market to a bear market. Their asymmetric propensity to trade leads to a discontinuity in the model, with its deterministic skeleton given by a two-dimensional piecewise linear dynamical system in discrete time. Multiple attractors, such as a stable fixed point and one or more attracting cycles or cycles and chaotic attractors, appear through border collision bifurcations. The multi-stability regions are underlined by means of two-dimensional bifurcation diagrams, where the border collision bifurcation curves are detected in analytic form at least for basic cycles with symbolic sequences \({\hbox {LR}}^{n}\) and \({\hbox {RL}}^{n}\). A statistical analysis of the simulated time series of the asset returns, generated by perturbing the deterministic dynamics with a random walk process, indicates that this is one of the simplest asset pricing models which are able to replicate stylized empirical facts, such as excess volatility, fat tails and volatility clustering.

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    An alternative modeling choice is represented by assuming that this type of trader adopts with probability one-half the trading strategy employed in the bull market and with probability one-half the trading strategy employed in the bear market every time the market price coincides with the fundamental price, i.e., he adopts the trend following strategy using an extrapolation rate which is the average of the extrapolation rates used in the bear and the bull markets. This alternative modeling choice would impact the dynamics of the deterministic skeleton of the model only in a zero-measure subset of the state space as discussed in the next section.


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Work developed in the framework of the research project on Models of behavioral economics for sustainable development of the Department of Economics, Society, Politics (DESP), University of Urbino. Davide Radi acknowledges the support of the Czech Science Foundation (GACR) under project [20-16701S] and VSB-TU Ostrava under the SGS project SP2020/11.

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Anufriev, M., Gardini, L. & Radi, D. Chaos, border collisions and stylized empirical facts in an asset pricing model with heterogeneous agents. Nonlinear Dyn 102, 993–1017 (2020).

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  • Piecewise linear asset pricing model
  • Heterogeneous traders
  • Chaos and border collision bifurcations
  • Stylized empirical facts of asset returns