On the Price Dynamics of a Two-Dimensional Financial Market Model with Entry Levels

Abstract

This paper aims to extend the model developed by Tramontana et al. By adding trend followers who pay attention to the most recent observed price trend, we formulate a financial market model driven by a new two-dimensional discontinuous piecewise linear (PWL) map with three branches. The dynamic behavior of the mapping system is studied in two cases according to different trend followers’ expectation of the stock price. The existence and stability conditions of periodic attractors and other bounded attractors are derived by using qualitative and quantitative methods, theoretical analysis, and numerical simulation. When trend followers are neutral on the stock market, we present that the basin of locally attracting fixed points can be determined by the preimages of two borderlines. We also prove that one of its surprising features is that model simulations may appear to be chaotic, although only regular dynamics can emerge. When trend followers are bullish or bearish on the stock market, we present the existence conditions of attracting coexistence fixed point, globally and locally attracting fixed point, and periodic and other bounded attractors. The transversal homoclinic theory of flip saddle periodic point is applied to prove the existence of chaotic attractor. We also give the calculation methods of border collision bifurcation (BCB) curves. This paper advances our knowledge of discontinuous PWL systems and reveals the endogenous evolution of bubbles and crashes and excessive volatility in financial markets from a new perspective with new methods.