Kinetic modeling of financial market models

• Kinetische Modellierung von Finanzmarktmodellen

Trimborn, Torsten; Frank, Martin (Thesis advisor); Herty, Michael Matthias (Thesis advisor); Pareschi, Lorenzo (Thesis advisor)

Aachen (2017, 2018)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2017

Abstract

The modeling of financial markets has a long tradition in economics and has developed into a significant and highly accepted field of research. Especially the financial crises of the last decade, have demonstrated the incapability of many standard financial market models to account for financial crashes or merely reproduce them. Stylized facts, observations in statistical data, are considered to play a prominent role in the emergence of financial crashes. More recent financial market models such as agent-based models can reproduce these stylized facts. They share many similarities with models from particle physics and substantially differ from classical financial market models. Agent-based models indicate that behavioral aspects in investors' investment decisions account for the existence of stylized facts. The disadvantage of these models, however, is the necessity to verify any results of the microscopic model by means of Monte Carlo simulations, which have the drawback of a slow convergence rate. Furthermore, it has been shown that many stylized facts in agent-based models are mere numerical artifacts. These disadvantages can be overcome by time continuous kinetic financial market models which can be studied analytically. Thus, steady state distributions can be analyzed and the origins of stylized facts can be proven.The centerpiece of this work is concerned with the derivation and the analysis of two kinetic financial market models. Econophysical agent-based models will be our starting point as they can replicate stylized facts. Our analysis will provide new insights on the origin of stylized facts as well as helping us to characterize the distribution of wealth and stock prices. We will derive a portfolio model of interacting financial agents. The agents face an optimization problem on the microscopic level. Model predictive control will be applied to simplify and solve the utility maximization before we derive the mean field limit. We will study the wealth and stock price distributions in detail and obtain analytically steady state solutions. The wealth distribution is always characterized by a lognormal distribution function. For the stock price distribution, we can either observe lognormal behavior in the case of long-term investors or a power-law in the case of high-frequency traders. The second model is a behavioral asset pricing model depicting agents's investment decisions either solely based on a trading strategy or additionally influenced by the behavioral herding pressure. We will show that quantitatively, the kinetic limit is an appropriate limit of the original econophysical model. Moreover, we can quantify that the emergence of stylized facts can be attributed to herding pressure and analyze the deterministic version of our kinetic system. A further aspect this work is concerned with is the theory of mean field limits. In the case of deterministic ODEs, we rigorously derive the mean field equations for a simple financial market model. Additionally, we study the case of a microscopic differential game, a new field of research called mean field games. We derive the mean field game limit system for a large class of ODE models. In fact, we obtain a genuinely new class of models which has never been discussed in literature before. The last aspect we will present in this work is the SABCEMM software, a simulation tool for agent-based models and the first simulator specialized for econophysical agent-based computational financial market models. This tool is designed to enable an unbiased comparison between different models and to minimize the amount of coding needed in order to create new models. This code is written so efficiently that it enables a standard laptop to compute simulations with up to several millions of agents. We can conclude that kinetic theory is appropriate to approximate agent-based financial market models and that it allows for the discovery of new results in comparison to microscopic agent-based market models.