Dynamical Models of Stock Prices Based on Technical Trading Rules---Part II: Analysis of the Model

In Part II of this study, we concentrate our analysis on the price dynamical model with the moving average rules developed in Part I. By decomposing the excessive demand function, we reveal that it is the interplay between trend-following and contrarian actions that generates the price chaos and gives parameter ranges for the price series to change from divergence to chaos and to oscillation. We prove that the price dynamical model has an infinite number of equilibriums, but all these equilibriums are unstable. We demonstrate the short-term predictability of the price volatility and derive the detailed formulas of the Lyapunov exponent as functions of the model parameters. We show that although the price is chaotic, the volatility converges to some constant very quickly at the rate of the Lyapunov exponent. We extract the formula relating the converged volatility to the model parameters based on Monte Carlo simulations. We explore the circumstances under which the returns are uncorrelated and illustrate in detail as to how the correlation index changes with the model parameters. Finally, we plot the strange attractor and the return distribution of the chaotic price series to illustrate the complex structure and the fat-tailed distribution of the returns.

I. Introduction

The analysis of the price dynamical models developed in Part I of this paper should aim at not only the properties of the models but the meanings of these properties in terms of financial economics as well. Specifically, we will show how the price dynamics of the models contribute to our understanding of four fundamental issues in financial economics: equilibrium, volatility, return predictability, and return independence.