Bifurcation analysis of a single-group asset flow model

Abstract

We study the stability and Hopf bifurcation analysis of an asset pricing model that based on the model introduced by Caginalp and Balenovich in 1999, under the assumption of a xed amount of cash and stock in the system. First, we study stability analysis of equilibrium points. Choosing the momentum coe¢ cient as a bifurcation parameter, we also show that Hopf bifurcation occurs when the bifurcation parameter passes through a critical value. Analytical results are supported by numerical simulations. A key conclusion for economics and nance is the existence of periodic solutions for an interval of the bifurcation parameter, which is the trend-based (or momentum) coe¢ cient. Key words. Asset price dynamics, stability of price dynamics, Hopf bifurcation, price trend, momentum, market dynamics, liquidity, periodic solutions. AMS subject classi cation. : 91B25, 91B26, 91B50, 91G80, 91G99, 34D20, 34C60, 37G15, 37N40 1. Introduction. A central theme in classical nance is that market participants all have access to the same information, and all seek to optimize their returns so that a unique equilibrium price is established (see, for example, [3], [18], [20]). The approach to equilibrium is often assumed to be a process involving some randomness or noise, but otherwise smooth and rapid. Aside from noise, one can expect that prices will not overshoot the equilibrium price since the equation governing the change in price, P , is a rst order in time, i.e., P 0 = F (D=S) where S and D are supply and demand that depend on price but not on the recent price derivative history. As such, there is no mechanism for oscillations or cyclic behavior within this setting. A well known example of cyclic behavior in economics is called the cobweb theorem whereby prices oscillate periodically due to the time lag between supply and demand decisions. Agricultural commodities provide a simple example with a delay between planting and harvesting (see [21] (pages 133-134 gives two agricultural examples: rubber and corn) and [36]). In nancial markets, however, the prevailing theory (at least during latter part of the 20th century), called the e¢ cient market hypothesis (EMH), maintains the existence of in nite arbitrage capital that would quickly exploit any deviations between the trading price and the intrinsic or fundamental value of the asset, which are necessarily unique since the participants have the same information and calculation of future returns. The absence of any delay in information or trading excludes, mathematically, the possibilities of overshooting the equilibrium price or oscillating about it. While policy makers often discuss instabilities in asset prices, classical nance tends to treat these as rare occurrences within a stochastic setting. In particular, much of classical nance is based on the concept that an asset’s price, P (t), is governed by Current address: Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA. Permanent address: Department of Mathematics, TOBB University of Economics and Technology, 06560-Ankara, TURKEY. H. Merdan was supported by TUBITAK (The Scienti c and Technological Research Council of Turkey) yDepartment of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA zDepartment of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA