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Mean-field theory of the Santa Fe model revisited: a systematic derivation from an exact BBGKY hierarchy for the zero-intelligence limit-order book model

Taiki Wakatsuki, Kiyoshi Kanazawa

Abstract

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 $μ\to 0$ and $μ\to \infty$ 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.

Mean-field theory of the Santa Fe model revisited: a systematic derivation from an exact BBGKY hierarchy for the zero-intelligence limit-order book model

Abstract

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 and asymptotics, where is the market-order submission intensity. Particularly, the mean-field theory works very well for small , while its validity is partially limited for large . (2)The ``method of image'' solution, heuristically derived by Bouchaud-Mézard-Potters in Quantitative Finance 2002, is obtained for large , 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.

Paper Structure

This paper contains 74 sections, 201 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setup description of the Santa Fe model and its reveiw
  3. Mathematical notation
  4. Setup description of the Santa Fe model
  5. Limit-order submission
  6. Cancellation of an existing limit order
  7. Market-order submission
  8. Focus of this work: the continuous asymptotic limit of the Santa Fe model
  9. Review of the original mean-field theory in E. Smith et al. 2003
  10. Review of the original dimensional analysis
  11. Review of the original mean-field theory
  12. The goal of this paper
  13. Formulation: state space and path-level stochastic dynamics
  14. State space: fixed coordinate system and relative coordinate system
  15. Fixed price coordinate system
  16. ...and 59 more sections

Figures (10)

  • Figure 1: Limit-order submission dynamics in the Santa Fe model. A new limit order is uniformly submitted from the opposite best price with the intensity $\lambda \Delta$. Note that we introduce a cut-off length $L<\infty$, beyond which new limit-order submission does not occur, and take the $L\to \infty$ limit later.
  • Figure 2: Order cancellation dynamics in the Santa Fe model. All the orders have the same cancellation intensity $v$. Again, we introduce a cut-off length $L<\infty$, beyond which no cancellation occurs, and take the $L\to \infty$ limit later.
  • Figure 3: Market-order dynamics in the Santa Fe model. A best-price order is executed by a market order and is removed.
  • Figure 4: Schematic of the fixed price coordinate with the tick size $\Delta=10$.
  • Figure 5: Schematic of the relative ask price coordinate from the best bid price.
  • ...and 5 more figures

Theorems & Definitions (9)

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