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Derivation of the Boltzmann Equation for Financial Brownian Motion: Direct Observation of the Collective Motion of High-Frequency Traders

Kiyoshi Kanazawa1,2,*, Takumi Sueshige2, Hideki Takayasu1,3, and Misako Takayasu1,2

  • 1Institute of Innovative Research, Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku, Yokohama 226-8502, Japan
  • 2Department of Mathematical and Computing Science, School of Computing, Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku, Yokohama 226-8502, Japan
  • 3Sony Computer Science Laboratories, 3-14-13 Higashi-Gotanda, Shinagawa-ku, Tokyo 141-0022, Japan

  • *Corresponding author. kanazawa.k.ae@m.titech.ac.jp
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Phys. Rev. Lett. 120, 138301 – Published 27 March, 2018

DOI: https://doi.org/10.1103/PhysRevLett.120.138301

Abstract

A microscopic model is established for financial Brownian motion from the direct observation of the dynamics of high-frequency traders (HFTs) in a foreign exchange market. Furthermore, a theoretical framework parallel to molecular kinetic theory is developed for the systematic description of the financial market from microscopic dynamics of HFTs. We report first on a microscopic empirical law of traders’ trend-following behavior by tracking the trajectories of all individuals, which quantifies the collective motion of HFTs but has not been captured in conventional order-book models. We next introduce the corresponding microscopic model of HFTs and present its theoretical solution paralleling molecular kinetic theory: Boltzmann-like and Langevin-like equations are derived from the microscopic dynamics via the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy. Our model is the first microscopic model that has been directly validated through data analysis of the microscopic dynamics, exhibiting quantitative agreements with mesoscopic and macroscopic empirical results.

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Synopsis

Financial Brownian Motion

Published 27 March, 2018

Using data on the activity of individual financial traders, researchers have devised a microscopic financial model that can explain macroscopic market trends.

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Supplemental Material

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Phys. Rev. Lett. 120, 138301– Published 27 March, 2018
  • Received 10 May 2017
  • Revised 24 January 2018
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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