One-dimensional maps with two discontinuity points and three linear branches: mathematical lessons for understanding the dynamics of financial markets

We develop a simple financial market model with heterogeneous interacting speculators. The dynamics of our model is driven by a one-dimensional discontinuous piecewise linear map, having two discontinuity points and three linear branches. On the one hand, we study this map analytically and numerically to advance our knowledge about such dynamical systems. In particular, not much is known about discontinuous maps involving three branches. On the other hand, we seek to improve our understanding of the functioning of financial markets. We find, for instance, that such maps can generate complex bull and bear market dynamics.

[1]  L. Gardini,et al.  The bull and bear market model of Huang and Day : Some extensions and new results , 2013 .

[2]  James A. Yorke,et al.  BORDER-COLLISION BIFURCATIONS FOR PIECEWISE SMOOTH ONE-DIMENSIONAL MAPS , 1995 .

[3]  Soumitro Banerjee,et al.  Robust Chaos , 1998, chao-dyn/9803001.

[4]  F. Tramontana,et al.  One-Dimensional Discontinuous Piecewise-Linear Maps and the Dynamics of Financial Markets , 2013 .

[5]  Tönu Puu,et al.  Business Cycle Dynamics : Models and Tools , 2006 .

[6]  Joanne Locke,et al.  The emergence of , 2016 .

[7]  Laura Gardini,et al.  Degenerate bifurcations and Border Collisions in Piecewise Smooth 1D and 2D Maps , 2010, Int. J. Bifurc. Chaos.

[8]  Laura Gardini,et al.  Global Bifurcations in a Three-Dimensional Financial Model of Bull and Bear Interactions , 2010 .

[9]  J. Rosser.  Handbook of Research on Complexity , 2011 .

[10]  James A. Yorke,et al.  Border-collision bifurcations including “period two to period three” for piecewise smooth systems , 1992 .

[11]  Laura Gardini,et al.  On the complicated price dynamics of a simple one-dimensional discontinuous financial market model with heterogeneous interacting traders , 2010 .

[12]  Carl Chiarella,et al.  Heterogeneity, Market Mechanisms, and Asset Price Dynamics , 2008 .

[13]  Klaus Reiner Schenk-Hoppé,et al.  Handbook of Financial Markets: Dynamics and Evolution , 2009 .

[14]  Frank Westerhoff,et al.  Exchange Rate Dynamics: A Nonlinear Survey , 2009 .

[15]  Michael Schanz,et al.  On multi-parametric bifurcations in a scalar piecewise-linear map , 2006 .

[16]  Richard H. Day,et al.  Bulls, bears and market sheep , 1990 .

[17]  Michael Schanz,et al.  Period Adding in Piecewise Linear Maps with Two Discontinuities , 2012, Int. J. Bifurc. Chaos.

[18]  Financial Crisis and Interacting Heterogeneous Agents , 2009 .

[19]  C. Hommes,et al.  CHAPTER 4 – Complex Evolutionary Systems in Behavioral Finance , 2009 .

[20]  Ping Chen,et al.  Nonlinear dynamics and evolutionary economics , 1993 .

[21]  Laura Gardini,et al.  The emergence of bull and bear dynamics in a nonlinear model of interacting markets. , 2009 .

[22]  Michael Schanz,et al.  Border-Collision bifurcations in 1D Piecewise-Linear Maps and Leonov's Approach , 2010, Int. J. Bifurc. Chaos.

[23]  Laura Gardini,et al.  Border Collision bifurcations in 1D PWL Map with One Discontinuity and Negative Jump: Use of the First Return Map , 2010, Int. J. Bifurc. Chaos.

[24]  Laura Gardini,et al.  Center Bifurcation for a Two-Dimensional Piecewise Linear Map , 2006 .

[25]  F. Westerhoff,et al.  The Use of Agent-Based Financial Market Models to Test the Effectiveness of Regulatory Policies , 2008 .

[26]  Laura Gardini,et al.  Nonlinear Dynamics in Economics, Finance and Social Sciences , 2010 .

[27]  Michael Schanz,et al.  Multi-parametric bifurcations in a piecewise–linear discontinuous map , 2006 .

[28]  Laura Gardini,et al.  A simple financial market model with chartists and fundamentalists: Market entry levels and discontinuities , 2011, Math. Comput. Simul..

[29]  J. Keener.  Chaotic behavior in piecewise continuous difference equations , 1980 .

[30]  Thomas Lux,et al.  Stochastic behavioral asset pricing models and the stylized facts , 2008 .