Paper: arXiv 2302.00846

Authors: Jonathan A. Chávez-Casillas

Abstract

This paper considers a Markovian model of a limit order book where time-dependent rates are allowed. With the objective of understanding the mechanisms through which a microscopic model of an orderbook can converge to more general diffusion than a Brownian motion with constant coefficient, a simple time-dependent model is proposed. The model considered here starts by describing the processes that govern the arrival of the different orders such as limit orders, market orders and cancellations. In this sense, this is a microscopic model rather than a ``mesoscopic’’ model where the starting point is usually the point processes describing the times at which the price changes occur and aggregate in these all the information pertaining to the arrival of individual orders. Furthermore, several empirical studies are performed to shed some light into the validity of the modeling assumptions and to verify whether certain stocks satisfy the conditions for their price process to converge to a more complex diffusion.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical framework modeling limit order books via time-dependent Markov processes and theoretical diffusion limits. It bridges microscopic order-flow mechanics with macroscopic price dynamics, backed by empirical validation on real stock data to test its theoretical assumptions.

Research Flowchart

  flowchart TD
    A["Research Question: How do microscopic limit order book (LOB) dynamics converge to complex diffusions?"] --> B["Data & Inputs: Empirical high-frequency equity order data (limit, market, cancellations)"]
    B --> C["Model Formulation: Build time-dependent Markovian model of individual order arrivals"]
    C --> D["Computational & Empirical Analysis: Test modeling assumptions & compute price process convergence conditions"]
    D --> E["Key Outcome: Microscopic LOB model successfully validates non-Brownian diffusion convergence for real stocks"]