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Kinetic Theory Derives Brownian Models of HFT and Price Dynamics

Article arXiv papers · Author: Kiyoshi Kanazawa et al.
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Summary

This work develops a kinetic-theory foundation for a microscopic model of trend-following high-frequency traders. Starting from the model’s exact phase-space evolution equation, analogous to the Liouville equation in mechanics, it reduces the description through a hierarchy of equations for financial Brownian motion.

Assuming molecular chaos, the authors derive Boltzmann-like dynamics for the order book and Langevin-like dynamics for prices. They study the large-trader-population behavior analytically and report numerical validation with Monte Carlo simulation. The account frames financial and physical Brownian motion as having parallel mathematical structures. Its results rely on the model and the molecular-chaos assumption; the excerpt does not provide details of the underlying market data analysis or demonstrate that the equations describe markets beyond this setting.

Key ideas

  • The paper models trend-following HFTs using a microscopic framework.
  • An exact phase-space evolution equation is the starting point for the kinetic reduction.
  • A hierarchy of equations is derived for financial Brownian motion.
  • The molecular-chaos assumption leads to Boltzmann-like order-book and Langevin-like price equations.
  • Large-population behavior is studied and checked with Monte Carlo simulation.

Tags

High-frequency tradingTrend followingMarket microstructureStatistics
Full text
# Kinetic Theory for Finance Brownian Motion from Microscopic Dynamics


# Kinetic Theory for Finance Brownian Motion from Microscopic Dynamics









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.

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