Kinetic modeling of financial markets

Originally developed for measuring the heterogeneity of wealth measures, inequality indices are quantitative scores that take values in the unit interval, with the zero score characterizing perfect equality. In this paper, we draw attention to a new inequality index, based on the Fourier transform, which exhibits a number of interesting properties that make it very promising in applications. As a by-product, it is shown that other inequality measures, including Gini and Pietra indices can be fruitfully expressed in terms of the Fourier transform, which allows to enlighten properties in a new and simple way.

Economic modelling and financial markets.- Agent-based models of economic interactions.- On kinetic asset exchange models and beyond: microeconomic formulation,trade network, and all that.- Microscopic and kinetic models in financial markets.- A mathematical theory for wealth distribution.- Tolstoy's dream and the quest for statistical equilibrium in economics and the social sciences.- Social modelling and opinion formation.- New perspectives in the equilibrium statistical mechanics approach to social and economic sciences.- Kinetic modelling of complex socio-economic systems.- Mathematics and physics applications in sociodynamics simulation: the case of opinion formation and diffusion.- Global dynamics in adaptive models of collective choice with social influence.- Modelling opinion formation by means of kinetic equations.- Human behavior and swarming.- On the modelling of vehicular traffic and crowds by kinetic theory of active particles.- Particle, kinetic, and hydrodynamic models of swarming.- Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints.- Statistical physics and modern human warfare.- Diffusive and nondiffusive population models.

We derive a mesoscopic description of the behavior of a simple financial market where the agents can create their own portfolio between two investment alternatives: a stock and a bond. The model is derived starting from the Levy-Levy-Solomon microscopic model (Levy et al. in Econ. Lett. 45:103–111, 1994; Levy et al. in Microscopic Simulation of Financial Markets: From Investor Behavior to Market Phenomena, Academic Press, San Diego, 2000) using the methods of kinetic theory and consists of a linear Boltzmann equation for the wealth distribution of the agents coupled with an equation for the price of the stock. From this model, under a suitable scaling, we derive a Fokker-Planck equation and show that the equation admits a self-similar lognormal behavior. Several numerical examples are also reported to validate our analysis.

In this paper, we discuss the large--time behavior of solution of a simple kinetic model of Boltzmann--Maxwell type, such that the temperature is time decreasing and/or time increasing. We show that, under the combined effects of the nonlinearity and of the time--monotonicity of the temperature, the kinetic model has non trivial quasi-stationary states with power law tails. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution. The same idea is applied to investigate the large-time behavior of an elementary kinetic model of economy involving both exchanges between agents and increasing and/or decreasing of the mean wealth. In this last case, the large-time behavior of the solution shows a Pareto power law tail. Numerical results confirm the previous analysis.

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. We derive a moment model which is able to replicate the most prominent features of the financial markets: oscillatory price behavior, booms and crashes. 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.

In the past decade there has been a growing interest in agent-based econophysical financial market models. The goal of these models is to gain further insights into stylized facts of financial data. We derive the mean field limit of the econophysical model by Cross, Grinfeld, Lamba and Seaman (Physica A, 354) and show that the kinetic limit is a good approximation of the original model. Our kinetic model is able to replicate some of the most prominent stylized facts, namely fat-tails of asset returns, uncorrelated stock price returns and volatility clustering. Interestingly, psychological misperceptions of investors can be accounted to be the origin of the appearance of stylized facts. The mesoscopic model allows us to study the model analytically. We derive steady state solutions and entropy bounds of the deterministic skeleton. These first analytical results already guide us to explanations for the complex dynamics of the model.

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. Thus, we consider a differential game since all agents aim to invest optimal 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 explicitly. We select the running cost to be the expected missed revenue of an agent which are determined by a combination of a fundamentalist and 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. The conducted simulations reveal that the model shares many dynamical properties with existing models in literature. Thus, our 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. Mathematically, the model can be regarded as the moment model of the recently introduced mesoscopic kinetic portfolio model (Trimborn et al. in Portfolio optimization and model predictive con trol: a kinetic approach, arXiv:1711.03291, 2017).