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Portfolio Optimization and Model Predictive Control: A Kinetic Approach

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  • Torsten Trimborn
  • Lorenzo Pareschi
  • Martin Frank

Abstract

In this paper, we introduce a large system of interacting financial agents in which each agent is faced with the decision of how to allocate his capital between a risky stock or a risk-less bond. The investment decision of investors, derived through an optimization, drives the stock price. The model has been inspired by the econophysical Levy-Levy-Solomon model (Economics Letters, 45). The goal of this work is to gain insights into the stock price and wealth distribution. We especially want to discover the causes for the appearance of power-laws in financial data. We follow a kinetic approach similar to (D. Maldarella, L. Pareschi, Physica A, 391) and derive the mean field limit of our microscopic agent dynamics. The novelty in our approach is that the financial agents apply model predictive control (MPC) to approximate and solve the optimization of their utility function. Interestingly, the MPC approach gives a mathematical connection between the two opponent economic concepts of modeling financial agents to be rational or boundedly rational. Furthermore, this is to our knowledge the first kinetic portfolio model which considers a wealth and stock price distribution simultaneously. Due to our kinetic approach, we can study the wealth and price distribution on a mesoscopic level. The wealth distribution is characterized by a lognormal law. For the stock price distribution, we can either observe a lognormal behavior in the case of long-term investors or a power-law in the case of high-frequency trader. Furthermore, the stock return data exhibits a fat-tail, which is a well known characteristic of real financial data.

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  • Torsten Trimborn & Lorenzo Pareschi & Martin Frank, 2017. "Portfolio Optimization and Model Predictive Control: A Kinetic Approach," Papers 1711.03291, arXiv.org, revised Feb 2019.
    • Handle: RePEc:arx:papers:1711.03291
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    Cited by:

    1. Martin Frank & Michael Herty & Torsten Trimborn, 2019. "Microscopic Derivation of Mean Field Game Models," Papers 1910.13534, arXiv.org.
    2. Torsten Trimborn & Martin Frank & Stephan Martin, 2017. "Mean Field Limit of a Behavioral Financial Market Model," Papers 1711.02573, arXiv.org.
    3. Maximilian Beikirch & Simon Cramer & Martin Frank & Philipp Otte & Emma Pabich & Torsten Trimborn, 2020. "Robust Mathematical Formulation And Probabilistic Description Of Agent-Based Computational Economic Market Models," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 23(06), pages 1-41, September.
    4. Wang, Lingling & Lai, Shaoyong & Sun, Rongmei, 2022. "Optimal control about multi-agent wealth exchange and decision-making competence," Applied Mathematics and Computation, Elsevier, vol. 417(C).
    5. Maximilian Beikirch & Simon Cramer & Martin Frank & Philipp Otte & Emma Pabich & Torsten Trimborn, 2019. "Robust Mathematical Formulation and Probabilistic Description of Agent-Based Computational Economic Market Models," Papers 1904.04951, arXiv.org, revised Mar 2021.
    6. Trimborn, Torsten & Frank, Martin & Martin, Stephan, 2018. "Mean field limit of a behavioral financial market model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 613-631.

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