An Algebraic Framework for the Modeling of Limit Order Books

ArXiv ID: 2406.04969 “View on arXiv”

Authors: Unknown

Abstract

Introducing an algebraic framework for modeling limit order books (LOBs) with tools from physics and stochastic processes, our proposed framework captures the creation and annihilation of orders, order matching, and the time evolution of the LOB state. It also enables compositional settings, accommodating the interaction of heterogeneous traders and different market structures. We employ Dirac notation and generalized generating functions to describe the state space and dynamics of LOBs. The utility of this framework is shown through simulations of simplified market scenarios, illustrating how variations in trader behavior impact key market observables such as spread, return volatility, and liquidity. The algebraic representation allows for exact simulations using the Gillespie algorithm, providing a robust tool for exploring the implications of market design and policy changes on LOB dynamics. Future research can expand this framework to incorporate more complex order types, adaptive event rates, and multi-asset trading environments, offering deeper insights into market microstructure and trader behavior and estimation of key drivers for market microstructure dynamics.

Keywords: Limit Order Book (LOB), Dirac Notation, Gillespie Algorithm, Stochastic Processes, Market Microstructure, Equities (Market Microstructure)

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats
  • Why: The paper introduces a sophisticated algebraic framework using advanced physics and stochastic process concepts like Dirac notation, master equations, and generating functions, indicating high mathematical complexity. However, the empirical evidence is limited to simplified simulations with no real market data, backtests, or implementation details, resulting in low empirical rigor.
  flowchart TD
    A["Research Goal<br>Model Limit Order Books<br>Composably"] --> B["Methodology<br>Algebraic Framework<br>Dirac Notation & Stochastic Processes"]
    B --> C["Input Data<br>Simplified Market Scenarios<br>Heterogeneous Traders"]
    C --> D["Computation<br>Exact Simulation via<br>Gillespie Algorithm"]
    D --> E["Outcomes<br>Impact on Spread, Volatility<br>Liquidity & Market Design"]