Modeling High-Frequency Traders with a Financial Boltzmann Equation

Summary

The paper builds a microscopic account of financial price motion from observations of high-frequency traders in a foreign exchange market. It tracks individual trader trajectories and reports a collective trend-following pattern that conventional order-book models do not capture.

The authors then model trader interactions using ideas from molecular kinetic theory. Applying the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy, they derive Boltzmann-like and Langevin-like descriptions of market dynamics. They report that the model agrees quantitatively with empirical findings at both mesoscopic and market-wide scales. The document gives no details about the data sample, estimation choices, or limits to generalizing beyond the observed market, so those remain open questions.

Key ideas

• Tracking individual HFT trajectories reveals a collective trend-following pattern.
• The model adapts molecular kinetic theory to describe financial market dynamics.
• A hierarchy of equations yields Boltzmann-like and Langevin-like descriptions.
• The authors report quantitative agreement with empirical results at multiple scales.
• The document does not specify the dataset or the model's range of applicability.

Tags

# 1703.06739


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









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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