# American Institute of Mathematical Sciences

November  2019, 24(11): 6209-6238. doi: 10.3934/dcdsb.2019136

## Portfolio optimization and model predictive control: A kinetic approach

 1 Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany 2 Department of Mathematics and Computer Science, University of Ferrara, Via Machiavelli 30, I-44121 Ferrara, Italy 3 Karlsruhe Institute of Technology, Steinbuch Center for Computing, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany

Corresponding author: trimborn@igpm.rwth-aachen.de

Received  June 2018 Revised  January 2019 Published  July 2019

In this paper, we introduce a large system of interacting financial agents in which all agents are faced with the decision of how to allocate their 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 [30]. 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 [33] and derive the mean field limit of the 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 opposing 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 the kinetic approach, we can study the wealth and price distribution on a mesoscopic level. The wealth distribution is characterized by a log-normal law. For the stock price distribution, we can either observe a log-normal behavior in the case of long-term investors or a power-law in the case of high-frequency trader. Furthermore, the stock return data exhibit a fat-tail, which is a well known characteristic of real financial data.

Citation: Torsten Trimborn, Lorenzo Pareschi, Martin Frank. Portfolio optimization and model predictive control: A kinetic approach. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6209-6238. doi: 10.3934/dcdsb.2019136
##### References:

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##### References:
Sketch of the modelling process
Example of the value function $U_{\gamma}$ with different reference points
Stock price evolution in the long-term investor case with a constant fundamental price $s^f$ (left figure) and a time varying fundamental price (right figure). In both figures one obtains that the average stock price is above the funcamental value
Quantile-quantile plot of logarithmic stock return distribution (left-hand side) and logarithmic return of fundamental prices (right-hand side). The simulation has been performed in the case of long-term investors and a stochastic fundamental price. The risk tolerance has been set to $\gamma = 0.9$, the scale to $\rho = \frac{5}{8}$ and the random seed is chosen to be $\texttt{rng(767)}$. All further parameters are chosen as reported in section A.4 of the Appendix
Stock price distribution in the long-term investor case. The solid lines are analytical solution, whereas the circles are the numerical result
Distribution of the wealth invested in stocks with a Gaussian fit (solid line). Left figure has a linear scale, whereas the right figure shows the distribution in log-log scale
Distribution of the wealth invested in bonds in the special case $K>0$. The numerical results (circles) are plotted with the corresponding log-normal analytic self-similar solution (solid lines)
Stock price distribution in the high-frequency case (red circles). The fit by the inverse-gamma distribution (solid line) clearly underestimates the tail. This reveals that the full model can create heavier tails than the inverse-gamma distribution
Marginal wealth distributions in the high-frequency investor case. The left hand side illustrates the distribution of investments in stocks and the right-hand side the wealth invested in bonds at $t = 1$
Steady state stock price distribution in the high-frequency investor case (circles) together with the analytically computed steady state of inverse-gamma type (solid line)
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