Abstract
We present a simple one-dimensional discontinuous piecewise-linear agent-based financial market model in which prices evolve with respect to the trading activity of heterogeneous speculators. In line with empirical evidence, speculators rely on technical or fundamental trading rules to determine their orders. The general setup that comes out of our assumptions can be subdivided into various sub-models. We survey some analytical results obtained for these sub-models and illustrate how their deterministic skeletons are able to produce some important stylized facts of financial markets, including bubbles, crashes and excess volatility. We also develop and calibrate a stochastic version of the model that matches the dynamics of actual financial markets quite well. In fact, simulated returns are virtually unpredictable and display features like volatility clustering and long memory effects.
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Notes
- 1.
Due to (14), our stochastic model is closely related to the deterministic model of Tramontana et al. (2011a). They exclude random influences and assume that the slopes in the bull and bear market regions are identical. As we will see, analytical results about such deterministic models help explain how our stochastic model works.
- 2.
Our model is related to that of Westerhoff and Franke (2012). Technically, their approach consists of only our inner regime. On the one hand, this makes their model simpler than ours. On the other hand, they have to fine-tune their parameters very carefully. To generate bubble dynamics, but to prevent complete price explosions at the same time, they have to set their average slope parameter to slightly below one. Our model is more robust in the sense that it works for a somewhat larger parameter space. For instance, we can allow for a more unstable inner regime when the outer regimes guarantee an eventual end of bubbles.
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Acknowledgements
This work is dedicated to Laura Gardini whom FT has known since 2004 and whom FW met for the first time in 2005. Since then, Laura has never ceased to amaze us with regard to her ideas about nonlinear dynamical systems. We hope to have the pleasure of collaborating with her for many years to come.
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Tramontana, F., Westerhoff, F. (2013). One-Dimensional Discontinuous Piecewise-Linear Maps and the Dynamics of Financial Markets. In: Bischi, G., Chiarella, C., Sushko, I. (eds) Global Analysis of Dynamic Models in Economics and Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29503-4_9
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