Abstract
Recent technological development has enabled researchers to study social phenomena scientifically in detail and financial markets has particularly attracted physicists since the Brownian motion has played the key role as in physics. In our previous report (arXiv:1703.06739; to appear in Phys. Rev. Lett.), we have presented a microscopic model of trend-following high-frequency traders (HFTs) and its theoretical relation to the dynamics of financial Brownian motion, directly supported by a data analysis of tracking trajectories of individual HFTs in a financial market. Here we show the mathematical foundation for the HFT model paralleling to the traditional kinetic theory in statistical physics. We first derive the time-evolution equation for the phase-space distribution for the HFT model exactly, which corresponds to the Liouville equation in conventional analytical mechanics. By a systematic reduction of the Liouville equation for the HFT model, the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchal equations are derived for financial Brownian motion. We then derive the Boltzmann-like and Langevin-like equations for the order-book and the price dynamics by making the assumption of molecular chaos. The qualitative behavior of the model is asymptotically studied by solving the Boltzmann-like and Langevin-like equations for the large number of HFTs, which is numerically validated through the Monte-Carlo simulation. Our kinetic description highlights the parallel mathematical structure between the financial Brownian motion and the physical Brownian motion.
Highlights
The goal of statistical physics is to reveal macroscopic behavior of physical systems from their microscopic setups, and this has been partially achieved in equilibrium and nonequilibrium statistical mechanics [1]
Lett. 120, 138301 (2018)], we presented a microscopic model of high-frequency traders (HFTs) through direct data analyses of individual trajectories of HFTs and revealed its theoretical dynamics by introducing the Boltzmann and Langevin equations for finance
Since our work is based on interdisciplinary techniques in statistical physics and financial market microstructure, we believe that these reviews will help familiarize readers to these fields, the main results in this paper are self-contained
Summary
The goal of statistical physics is to reveal macroscopic behavior of physical systems from their microscopic setups, and this has been partially achieved in equilibrium and nonequilibrium statistical mechanics [1]. The fundamental equations of kinetic theory (i.e., the Boltzmann and Langevin equations) were historically introduced on the basis of phenomenological arguments within the frameworks of nonlinear master equations and stochastic processes [13,14] Their systematic derivations were mathematically developed from analytical mechanics by Bogoliubov-Born-Green-KirkwoodYvon (BBGKY) and van Kampen [14,15,16,17]. On the basis of the “equation of motions” for the HFTs, Boltzmann-like and Langevin-like equations were derived for mesoscopic and macroscopic dynamics, respectively This framework is shown to be consistent with empirical findings, such as HFTs’ trend following, average order-book profile, price movement, and layered order-book structure. The presentation of the main results is written as self-contained, followed by the detailed derivation of the results; readers interested only in the main results may skip their theoretical derivation without referring to the detailed calculation
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