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

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  • Kiyoshi Kanazawa
  • Takumi Sueshige
  • Hideki Takayasu
  • Misako Takayasu

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.

Suggested Citation

  • Kiyoshi Kanazawa & Takumi Sueshige & Hideki Takayasu & Misako Takayasu, 2017. "Derivation of the Boltzmann Equation for Financial Brownian Motion: Direct Observation of the Collective Motion of High-Frequency Traders," Papers 1703.06739, arXiv.org, revised Mar 2018.
    • Handle: RePEc:arx:papers:1703.06739
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    Cited by:

    1. Kiyoshi Kanazawa & Hideki Takayasu & Misako Takayasu, 2022. "Exact solution to two-body financial dealer model: revisited from the viewpoint of kinetic theory," Papers 2205.15558, arXiv.org.
    2. Alberto Ciacci & Takumi Sueshige & Hideki Takayasu & Kim Christensen & Misako Takayasu, 2020. "The microscopic relationships between triangular arbitrage and cross-currency correlations in a simple agent based model of foreign exchange markets," PLOS ONE, Public Library of Science, vol. 15(6), pages 1-19, June.
    3. Isao Yagi & Yuji Masuda & Takanobu Mizuta, 2020. "Analysis of the Impact of High-Frequency Trading on Artificial Market Liquidity," Papers 2010.13038, arXiv.org.
    4. Takumi Sueshige & Didier Sornette & Hideki Takayasu & Misako Takayasu, 2019. "Classification of position management strategies at the order-book level and their influences on future market-price formation," PLOS ONE, Public Library of Science, vol. 14(8), pages 1-19, August.
    5. Takumi Sueshige & Kiyoshi Kanazawa & Hideki Takayasu & Misako Takayasu, 2018. "Ecology of trading strategies in a forex market for limit and market orders," PLOS ONE, Public Library of Science, vol. 13(12), pages 1-14, December.
    6. Yuki Sato & Kiyoshi Kanazawa, 2023. "Exact solution to a generalised Lillo-Mike-Farmer model with heterogeneous order-splitting strategies," Papers 2306.13378, arXiv.org, revised Nov 2023.
    7. Boilard, J.-F. & Kanazawa, K. & Takayasu, H. & Takayasu, M., 2018. "Empirical scaling relations of market event rates in foreign currency market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 1152-1161.

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