We present a macroscopic portfolio model which considers the time evolution of the stock price and the investments in bonds and stocks . The asset allocation between bonds and stocks is determined by an combination of a fundamentalist or chartist strategy. The stock price is determined by the inflow or outflow of stock investments. The 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. We want to point out that we derive the model from microscopic agent dynamics. On the microscopic level each financial agent is faced with an optimization problem, where each agent seeks to find a Nash equilibrium solution. We use model predictive control to approximate the control problem. This allows us to give a precise mathematical definition of rational and bounded rational financial agents. Moreover the approximation scheme of the microscopic optimal control problem gives a natural connections between rational and bounded rational agents. In fact the model can be regarded as  the result of the simplest approximation of the optimal control problem and thus considers financial agents of maximum boundedness. Mathematically, the model can be regarded as the moment model of the recently introduced mesoscopic kinetic portfolio model(Trimborn, Pareschi, Frank: Portfolio Optimization and Model Predictive Control: A Kinetic Approach, arXiv:1711.03291) .
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 . The stock price evolution is driven by the aggregated investment decision of all agents. In fact, we are faced with a differential game since all agents aim to invest optimal. Mathematically such a problem is ill posed 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 analytically. We select the running cost to be the expected missed revenue of an agent and we assume quadratic transaction costs. More precisely the expected revenues are determined by a combination of a fundamentalist or 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. Simulations reveal, that our model is able to replicate the most prominent features of financial markets, namely booms and crashes .
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present a macroscopic portfolio model which considers the time evolution of the stock price and the investments
introduce a microscopic model of interacting financial agents, where each agent is characterized by two portfolios; money invested
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stocks
money invested in stocks. Furthermore, each agent is faced with an optimization problem in order to determine the optimal asset allocation
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The asset allocation between bonds and stocks is determined by an combination of a fundamentalist or chartist strategy.
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is determined by the inflow or outflow of stock investments. The 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. We want to point out that we derive the model from microscopic agent dynamics. On the microscopic level each financial agent is faced with an optimization problem, where each agent seeks to find a Nash equilibrium solution. We
evolution is driven by the aggregated investment decision of all agents. In fact, we are faced with a differential game since all agents aim to invest optimal. Mathematically such a problem is ill posed 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
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allows us to give
enables us to derive
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definition of rational and bounded rational financial agents. Moreover the approximation scheme of the microscopic optimal control problem gives a natural connections between rational and bounded rational agents. In fact the model
characterization of the degree of rationality of a financial agent. This is a novel concept in portfolio optimization and
A
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 analytically. We select the running cost to be
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result of the simplest approximation of the optimal control problem and thus considers financial agents of maximum boundedness. Mathematically, the model can be regarded as the moment model of the recently introduced mesoscopic kinetic portfolio model(Trimborn, Pareschi, Frank: Portfolio Optimization and Model Predictive Control: A Kinetic Approach, arXiv:1711.03291)
expected missed revenue of an agent and we assume quadratic transaction costs. More precisely the expected revenues are determined by a combination of a fundamentalist or 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. Simulations reveal, that our model is able to replicate the most prominent features of financial markets, namely booms and crashes