A mathematical framework for modelling order book dynamics

Paper: arXiv 2302.01169

Authors: Rama Cont, Pierre Degond, Lifan Xuan

Abstract

We present a general framework for modelling the dynamics of limit order books, built on the combination of two modelling ingredients: the order flow, modelled as a general spatial point process, and market clearing, modelled via a deterministic mass transport operator acting on distributions of buy and sell orders. At the mathematical level, this corresponds to a natural decomposition of the infinitesimal generator describing the evolution of the limit order book into two operators: the generator of the order flow and the clearing operator. Our model provides a flexible framework for modelling and simulating order book dynamics and studying various scaling limits of discrete order book models. We show that our framework includes previous models as special cases and yields insights into the interplay between order flow and price dynamics.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces an advanced, rigorous mathematical framework using spatial point processes and mass transport operators to model order book dynamics. While highly theoretical and novel in its continuous operator decomposition, it lacks empirical validation, real-world data backtesting, and open-source implementation code.

Research Flowchart

  flowchart TD
    A[Goal: General Framework for Limit Order Book Dynamics] --> B[Data Inputs: Buy/Sell Orders & Spatial Point Process Flow]
    B --> C[Method: Decompose Infinitesimal Generator]
    C --> D[Operator 1: Order Flow Dynamics]
    C --> E[Operator 2: Market Clearing via Mass Transport]
    D & E --> F[Simulate Dynamics & Scaling Limits]
    F --> G[Outcome: Unifies Prior Models & Uncovers Price Dynamics]