  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.
