[Paper Review] A Functional Limit Theorem for Limit Order Books with State Dependent Price Dynamics

Christian Bayer, Ulrich Horst+1 more|arXiv (Cornell University)|May 20, 2014

Economic theories and models12 references3 citations

TL;DR

This paper establishes a functional limit theorem for limit order books with state-dependent price dynamics, showing that as order arrival rates increase and individual order impacts shrink, the joint dynamics of best bid/ask prices and volume densities converge to a coupled system of stochastic differential equations (SDEs) and a stochastic partial differential equation (SPDE). The key contribution is a rigorous scaling limit that captures the feedback of standing volumes on price movements in a continuous-time, infinite-dimensional framework.

ABSTRACT

We consider a stochastic model for the dynamics of the two-sided limit order book (LOB). Our model is flexible enough to allow for a dependence of the price dynamics on volumes. For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges in distribution to a fully coupled SDE-SPDE system when the order arrival rates tend to infinity and the impact of an individual order arrival on the book as well as the tick size tends to zero. The SDE describes the bid/ask price dynamics while the SPDE describes the volume dynamics.

Motivation & Objective

• To develop a probabilistic framework for modeling the joint dynamics of limit order book prices and volumes under state-dependent price impact.
• To establish a functional central limit theorem for the full limit order book, including both price and volume dynamics.
• To extend existing scaling limits by incorporating volume-dependent price dynamics, which are empirically observed but often neglected in prior models.
• To rigorously prove convergence in distribution to a fully coupled SDE-SPDE system under heavy-traffic scaling.
• To provide a mathematically sound foundation for modeling market microstructure with feedback from order book state to price formation.

Proposed method

• Model the limit order book as a continuous-time, infinite-dimensional system with Poisson-distributed order arrivals and cancellations, dependent on the current best bid and ask prices.
• Apply a heavy-traffic scaling where order arrival rates tend to infinity and individual order impacts and tick size tend to zero.
• Use Mitoma’s theorem and tightness criteria in spaces of distributions to establish weak convergence of the scaled processes.
• Characterize the limiting dynamics as a coupled system: SDEs for the best bid and ask prices and an SPDE for the volume density functions.
• Employ time-changed weak convergence and Skorokhod’s lemma to handle convergence of processes with random time changes.
• Leverage functional limit theorem techniques from probability theory to derive the joint convergence of price and volume processes in a distributional sense.

Experimental results

Research questions

• RQ1 How does the joint dynamics of price and volume in a limit order book behave under heavy-traffic scaling when price formation depends on standing order volumes?
• RQ2 Can a functional limit theorem be established for a two-sided limit order book with state-dependent price dynamics, rather than assuming constant or exogenous price processes?
• RQ3 What is the limiting system that describes the convergence of the scaled limit order book process, and how is it characterized mathematically?
• RQ4 Does the feedback from volume density to price dynamics survive in the diffusion limit, and if so, how is it encoded in the limiting system?
• RQ5 Under what conditions does the sequence of scaled limit order book processes converge in distribution to a coupled SDE-SPDE system?

Key findings

• The joint dynamics of best bid and ask prices and volume density functions converge in distribution to a fully coupled system of stochastic differential equations (SDEs) and a stochastic partial differential equation (SPDE).
• The limiting price process is governed by an SDE whose drift and diffusion coefficients depend on the current volume density, capturing state-dependent price impact.
• The volume dynamics are described by an SPDE that evolves in continuous time and is driven by the same underlying Poisson processes as the order flow.
• The convergence is established under a heavy-traffic scaling where individual order impacts and tick size vanish while order arrival rates diverge.
• The limiting system is well-defined in distributional spaces, and tightness is proven using functional analytic tools such as Mitoma’s theorem and C-tightness criteria.
• The result generalizes prior scaling limits by incorporating feedback from the order book state to price dynamics, aligning with empirical observations of volume imbalance effects.