Kinetic theory for financial Brownian motion from microscopic dynamics

Author

Kanazawa, Kiyoshi and Sueshige, Takumi and Takayasu, Hideki and Takayasu, Misako

Abstract

Recent technological development has enabled researchers to study social phenomena in detail, and financial markets have attracted the attention of physicists particularly since key concepts in Brownian motion are applicable to the description of financial systems. In our previous Letter [Kanazawa et al., Phys. Rev. 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. However, the formulation therein was rather heuristic and a more mathematically exact derivation is necessary from the microscopic dynamics of the HFT model. We hereby establish the mathematical foundation of kinetic theory for financial Brownian motion in a manner parallel to traditional statistical physics. We first derive the exact time-evolution equation for the phase-space distribution for the HFT model, corresponding to the Liouville equation in analytical mechanics. By a systematic reduction of the Liouville equation for finance, the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchal equations are generalized for financial Brownian motion. We then derive the Boltzmann and Langevin equations for the order book and the price dynamics by assuming molecular chaos. The asymptotic solutions to these equations are presented for a large number of HFTs, which qualitatively reveal how the strategies of traders at the microscopic level impact the macroscopic dynamics of market price. Our theoretical prediction was numerically examined via Monte Carlo simulations. Our kinetic description highlights the parallel mathematical structure between the financial and physical Brownian motions by a straightforward extension of statistical mechanics.