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A macroscopic portfolio model: from rational agents to bounded rationality

机译：宏观投资组合模型：从理性主体到有限理性

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摘要

We introduce a microscopic model of interacting financial agents, where each agent is characterized by two portfolios; money invested in bonds and money invested in stocks. Furthermore, each agent is faced with an optimization problem in order to determine the optimal asset allocation. Thus, we consider a differential game since all agents aim to invest optimal and we introduce the concept of Nash equilibrium solutions to ensure the existence of a solution. Especially, we denote an agent who solves this Nash equilibrium exactly a rational agent. As next step we use model predictive control to approximate the control problem. This enables us to derive a precise mathematical characterization of the degree of rationality of a financial agent. This is a novel concept in portfolio optimization and can be regarded as a general approach. In a second step we consider the case of a fully myopic agent, where we can solve the optimal investment decision of investors explicitly. We select the running cost to be the expected missed revenue of an agent which are determined by a combination of a fundamentalist and chartist strategy. Then we derive the mean field limit of the microscopic model in order to obtain a macroscopic portfolio model. The novelty in comparison to existent macroeconomic models in literature is that our model is derived from microeconomic dynamics. The resulting portfolio model is a three dimensional ODE system which enables us to derive analytical results. The conducted simulations reveal that the model shares many dynamical properties with existing models in literature. Thus, our model is able to replicate the most prominent features of financial markets, namely booms and crashes. In the case of random fundamental prices the model is even able to reproduce fat tails in logarithmic stock price return data. Mathematically, the model can be regarded as the moment model of the recently introduced mesoscopic kinetic portfolio model (Trimborn et al. in Portfolio optimization and model predictive con trol: a kinetic approach, arXiv:1711.03291, 2017).
机译：我们介绍了相互作用的金融代理商的微观模型，其中每个代理商都有两个投资组合。债券投资和股票投资。此外，每个代理商都面临优化问题，以确定最佳资产分配。因此，由于所有代理商都旨在投资最优，因此我们考虑了一种差分博弈，并且我们引入了纳什均衡解的概念以确保解的存在。特别地，我们表示一个能完全解决此纳什均衡问题的智能体。下一步，我们使用模型预测控制来近似控制问题。这使我们能够得出金融代理人合理程度的精确数学特征。这是投资组合优化中的一个新颖概念，可以视为一种通用方法。在第二步中，我们考虑完全近视代理的情况，在这里我们可以明确解决投资者的最佳投资决策。我们选择运行成本作为代理商的预期错失收入，这是由原教旨主义和宪章主义策略共同决定的。然后，我们得出微观模型的平均场极限，以获得宏观投资组合模型。与文献中现有的宏观经济模型相比，新颖之处在于我们的模型源自微观经济动力学。最终的投资组合模型是一个三维ODE系统，使我们能够得出分析结果。进行的仿真表明，该模型与文献中的现有模型具有许多动力学特性。因此，我们的模型能够复制金融市场最突出的特征，即繁荣和崩溃。在基本价格随机的情况下，该模型甚至能够在对数股票价格收益数据中重现胖尾巴。从数学上讲，该模型可以视为最近引入的介观动力学投资组合模型的矩模型（Trimborn等人在投资组合优化和模型预测控制：动力学方法中，arXiv：1711.03291，2017）。

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