Perception of fundamental values and financial market dynamics: mathematical insights from a 2D piecewise linear map. (English) Zbl 1509.37134
Summary: We develop a simple financial market model in which a market maker adjusts the price with respect to orders placed by chartists and fundamentalists. A novel feature of our model is that fundamentalists optimistically (pessimistically) believe in a relatively high (low) fundamental value when the financial market is increasing (decreasing). As it turns out, the dynamics of our model is driven by a two-dimensional discontinuous piecewise linear map for which we provide an in-depth analytical and numerical investigation. Among other things, we obtain in explicit form the boundaries of the periodicity regions associated with attracting cycles with rotation number \(1/n\), \(n\geq 3\). These boundaries correspond to border collision bifurcations of the related cycles. We show that the periodicity regions are organized in a specific period adding structure, and some of the regions may overlap. Several examples of coexisting cycles and their basins of attraction are also presented. Economically, our results offer a new explanation for the boom-bust behavior of actual financial markets.
MSC:
| 37N40 | Dynamical systems in optimization and economics |
| 91G15 | Financial markets |
| 91B55 | Economic dynamics |
| 91B64 | Macroeconomic theory (monetary models, models of taxation) |
Keywords:
discontinuous piecewise linear map; border collision bifurcation; bifurcation structure; financial market model; boom-bust cyclesReferences:
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