Bulut H, Merdan H, Swigon D
Department of Mathematics, TOBB University of Economics and Technology, 06560 Ankara, Turkey.
Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA.
Chaos. 2019 Feb;29(2):023114. doi: 10.1063/1.5046925.
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.
我们提出了一个针对包含两只股票的市场的数学模型，这两只股票在具有相似交易动机和策略的单一同质投资者群体中进行交易。假设市场由固定数量的现金和股票组成（不允许随时间增加，因此系统是封闭的），并且交易群体在买卖每种资产时受到趋势和估值动机的影响，但遵循一种策略，即一种资产的买入取决于另一种资产的价格，而卖出则不然。通过利用这些假设和基本微观经济学原理，通过动态系统方法获得了数学模型。我们分析了模型平衡点的稳定性，并确定了稳定性的参数条件。首先，我们证明在没有明确强调每只股票基于趋势的价值的情况下，所有平衡点都是稳定的。其次，对于交易群体在交易中重视一只股票的估值和另一只股票的趋势的系统，我们建立了稳定性条件，并通过数值例子表明，当出现不稳定性时，两只股票的价格都会出现振荡。此外，我们通过在这种情况下选择动量系数作为分岔参数，通过霍普夫分岔论证了周期解的存在。最后，我们给出例子和数值模拟来支持和扩展分析结果。经济学和金融学的一个关键结论是，根据动量系数，在没有外部因素的情况下存在循环行为。特别是，随着基于趋势的交易增加，原本稳定的均衡价格会变得不稳定。
