Demonstrating Chaos on Financial Markets through a Discrete Logistic Price Dynamics

The paper highlights the role that speculation plays in making stock price fluctuation chaotic. The positive feedback produce by speculative behavior determines the general dynamics of stock prices. The price dynamics is described by a logistic equation. This logistic equation is also known as Verhulst equation. This equation was originally developed to describe the dynamic behavior of population of an organism. A discrete form of the Verhulst equation called as Ricker model is done to simulate the price dynamics. The simulation of the iterative process in the Ricker model demonstrates that speculation can produce chaos. By varying the value of the parameter describing speculation, the price dynamics becomes chaotic for sufficiently high degree of speculation. The extreme sensitivity to initial condition of a chaotic system produced the so-called butterfly effect. A simulation of the butterfly effect is done using two exactly identical discrete logistic equations. The equations differed only in their initial values by a very minute amount. It shows how two exactly identical dynamical systems quickly behave very differently even if the difference in their initial conditions is so infinitesimally small. The implication of the butterfly effect in doing experiments in the physical world is analyzed. The presence of butterfly effect in a chaotic system raises the issue of measurement errors in the conduct of physical experiments. No matter how accurate the scientific device used in the experiment, it is still subject to measurement errors. Butterfly effect tremendously magnifies the measurement errors over a short span of time. This implies that long-term prediction in a chaotic system is impossible