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

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.

arXiv comment5 pages, 5 figures + Appendices

References in corpus (12)

• Traffic and Related Self-Driven Many-Particle Systems
• Scaling of the distribution of price fluctuations of individual companies
• Price Variations in a Stock Market With Many Agents
• Boltzmann and hydrodynamic description for self-propelled particles
• Statistical theory of the continuous double auction
• Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis
• Statistical properties of stock order books: empirical results and models
• Simple model of a limit order-driven market
• A quantitative model of trading and price formation in financial markets
• Exact microscopic analysis of a thermal Brownian motor
• Brownian refrigerator
• Solvable Stochastic Dealer Models for Financial Markets

Cited by in corpus (17)

• Social physics
• Emergence of molecular friction in liquids: bridging between the atomistic and hydrodynamic pictures
• Crossover from linear to square-Root market impact
• Non-Gaussian noise without memory in active matter
• Kinetic Theory for Finance Brownian Motion from Microscopic Dynamics
• Can we infer microscopic financial information from the long memory in market-order flow?: a quantitative test of the Lillo-Mike-Farmer model
• Order book model with herd behavior exhibiting long-range memory
• How does latent liquidity get revealed in the limit order book?
• Analysis of the Impact of High-Frequency Trading on Artificial Market Liquidity
• Adjoint DSMC for Nonlinear Spatially-Homogeneous Boltzmann Equation With a General Collision Model
• The microscopic relationships between triangular arbitrage and cross-currency correlations in a simple agent based model of foreign exchange markets
• Quantitative statistical analysis of order-splitting behaviour of individual trading accounts in the Japanese stock market over nine years
• Stochastic Simulation of Emission Spectra and Classical Photon Statistics of Quantum Dot Superluminescent Diodes
• Anomalous diffusion and price impact in the fluid-limit of an order book
• Exact solution to two-body financial dealer model: revisited from the viewpoint of kinetic theory
• Exact solution to a generalised Lillo-Mike-Farmer model with heterogeneous order-splitting strategies
• The Brownian Motion in an Ideal Quantum Qas