2025/10/02 by Wakatsuki, Taiki, Kanazawa, Kiyoshi
#Computational Finance (q-fin.CP) #FOS: Economics and business #FOS: Physical sciences #General Finance (q-fin.GN) #Mathematical Finance (q-fin.MF) #Statistical Mechanics (cond-mat.stat-mech) #Trading and Market Microstructure (q-fin.TR)
paper · doi:10.48550/arxiv.2510.01814
The Santa Fe model is an established econophysics model for describing stochastic dynamics of the limit order book from the viewpoint of the zero-intelligence approach. While its foundation was studied by combining a dimensional analysis and a mean-field theory by E. Smith et al. in Quantitative Finance 2003, their arguments are rather heuristic and lack solid mathematical foundation; indeed, their mean-field equations were derived with heuristic arguments and their solutions were not explicitly obtained. In this work, we revisit the mean-field theory of the Santa Fe model from the viewpoint of kinetic theory -- a traditional mathematical program in statistical physics. We study the exact master equation for the Santa Fe model and systematically derive the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchical equation. By applying the mean-field approximation, we derive the mean-field equation for the order-book density profile, parallel to the Boltzmann equation in conventional statistical physics. Furthermore, we obtain explicit and closed expression of the mean-field solutions. Our solutions have several implications: (1)Our scaling formulas are available for both μ→ 0 and μ→ ∞ asymptotics, where μ is the market-order submission intensity. Particularly, the mean-field theory works very well for small μ, while its validity is partially limited for large μ. (2)The ``method of image'' solution, heuristically derived by Bouchaud-Mézard-Potters in Quantitative Finance 2002, is obtained for large μ, serving as a mathematical foundation for their heuristic arguments. (3)Finally, we point out an error in E. Smith et al. 2003 in the scaling law for the diffusion constant due to a misspecification in their dimensional analysis.