Asset Price Dynamics for a Two-Asset Market System

Abstract

We present a mathematical model for a market involving two stocks which are traded within a single homogeneous group of investors who have similar motivations and strategies for trading. It is assumed that the market consists of a fixed amount of cash and stocks (additions in time are not allowed, so the system is closed) and that the trading group is affected by trend and valuation motivations while selling or buying each asset, but follows a strategy in which the buying of an asset depends on the other asset's price while the selling does not. By utilizing these assumptions and basic microeconomics principles, the mathematical model is obtained through a dynamical system approach. We analyze the stability of equilibrium points of the model and determine the conditions on parameters for stability. First, we prove that all equilibria are stable in the absence of a clear emphasis on a trend-based value for each stock. Second, for systems in which the group of traders attaches importance to the valuation of one stock and the trend of the other stock for trading, we establish conditions for stability and show with numerical examples that when instability occurs, it is exhibited by oscillations in the price of both stocks. Moreover, we argue the existence of periodic solutions through a Hopf bifurcation by choosing the momentum coefficient as a bifurcation parameter within this setting. Finally, we give examples and numerical simulations to support and extend the analytical results. One of the key conclusions for economics and finance is the existence of a cyclic behavior in the absence of exogenous factors according to the momentum coefficient. In particular, an equilibrium price that is stable becomes unstable as the trend based trading increases.