How Trend-Following and Contrarian Rules Shape Chaotic Stock Prices

Summary

This paper analyzes a stock price dynamics model built around moving-average trading rules. It attributes chaotic price behavior to interaction between trend-following and contrarian demand, and describes parameter ranges associated with divergence, chaos, and oscillation. The model is reported to have infinitely many equilibrium points, all unstable. The analysis also derives a Lyapunov exponent in terms of model parameters and investigates how return volatility behaves even when prices are chaotic.

The authors report that volatility quickly approaches a constant, estimate its relationship to model parameters using Monte Carlo simulations, and examine when returns appear independent. A plotted strange attractor and return distribution illustrate complex behavior and fat tails. These are findings about a specified mathematical model; the description does not provide empirical validation on market data or establish that its dynamics explain actual stock returns. The conclusions depend on the model assumptions and parameter choices.

Key ideas

• The model links price chaos to interaction between trend-following and contrarian trading rules.
• Its equilibrium points are described as unstable, with parameter ranges governing divergence, chaos, and oscillation.
• The analysis derives a Lyapunov exponent and relates it to short-term volatility behavior.
• Monte Carlo simulations are used to estimate the model’s converged volatility.
• The described chaotic dynamics are model results, with no empirical validation specified in the text.

Tags

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


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









In Part II of this paper, we concentrate our analysis on the price dynamical model with the moving average rules developed in Part I of this paper. 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 give 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 equilibrium points but all these equilibrium points are unstable. We demonstrate the short-term predictability of the return volatility and derive the detailed formula of the Lyapunov exponent as function 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 show independency and illustrate in details how the independency index changes with the model parameters. Finally, we plot the strange attractor and return distribution of the chaotic price model to illustrate the complex structure and fat-tailed distribution of the returns.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.