[Paper Review] Limit order books, diffusion approximations and reflected SPDEs: from microscopic to macroscopic models

Ben Hambly, Jasdeep Kalsi+1 more|arXiv (Cornell University)|Aug 21, 2018

Complex Systems and Time Series Analysis27 references5 citations

TL;DR

This paper establishes a rigorous mathematical bridge from microscopic limit order book dynamics to macroscopic stochastic partial differential equations (SPDEs) via diffusion approximations, showing that high-frequency order book behavior converges to reflected SPDEs under scaling limits. The key contribution is a probabilistic macroscopic model that captures stylized market features and enables simulation for execution strategy testing.

ABSTRACT

Motivated by a zero-intelligence approach, the aim of this paper is to connect the microscopic (discrete price and volume), mesoscopic (discrete price and continuous volume) and macroscopic (continuous price and volume) frameworks for the modelling of limit order books, with a view to providing a natural probabilistic description of their behaviour in a high to ultra high-frequency setting. Starting with a microscopic framework, we first examine the limiting behaviour of the order book process when order arrival and cancellation rates are sent to infinity and when volumes are considered to be of infinitesimal size. We then consider the transition between this mesoscopic model and a macroscopic model for the limit order book, obtained by letting the tick size tend to zero. The macroscopic limit can then be described using reflected SPDEs which typically arise in stochastic interface models. We then use financial data to discuss a possible calibration procedure for the model and illustrate numerically how it can reproduce observed behaviour of prices. This could then be used as a market simulator for short-term price prediction or for testing optimal execution strategies.

Motivation & Objective

• To unify microscopic, mesoscopic, and macroscopic models of limit order books under a zero-intelligence framework.
• To derive a continuous-time limit for the order book process as order arrival and cancellation rates increase to infinity.
• To show that the macroscopic limit is described by reflected SPDEs, linking market microstructure to stochastic interface models.
• To provide a calibration procedure using real financial data for practical market simulation.
• To enable short-term price prediction and optimal execution strategy backtesting through a validated model.

Proposed method

• Uses a microscopic model with discrete price levels and infinitesimal order volumes, then scales arrival and cancellation rates to infinity.
• Applies functional central limit theorems to derive a mesoscopic model with continuous volume and discrete price.
• Takes a further limit as tick size tends to zero, leading to a continuous price and volume framework.
• Derives the macroscopic limit as a system of reflected stochastic PDEs driven by space-time white noise.
• Employs Skorokhod representation and weak convergence techniques to prove convergence of rescaled processes.
• Calibrates the model using empirical financial data and validates it numerically against observed price dynamics.

Experimental results

Research questions

• RQ1 How does the limit order book process converge from discrete to continuous price and volume frameworks under high-frequency scaling?
• RQ2 What is the limiting SPDE description of the order book when tick size and order volume scale to zero?
• RQ3 Can the macroscopic SPDE model reproduce observed stylized facts of price formation and order flow?
• RQ4 How can the model be calibrated to real market data for practical use in market simulation?
• RQ5 What is the connection between limit order book dynamics and reflected SPDEs in stochastic interface theory?

Key findings

• The mesoscopic limit of the order book process converges to a system of coupled SDEs under high-frequency scaling of order arrivals and cancellations.
• The macroscopic limit is described by reflected SPDEs, which arise naturally in stochastic interface models and capture the non-negativity and boundary behavior of the order book.
• The convergence of the rescaled microscopic process to the macroscopic SPDE model is established in law in the space of cadlag paths.
• The model can be calibrated to real financial data, allowing numerical reproduction of observed price dynamics and market microstructure features.
• The resulting SPDE model serves as a valid market simulator for short-term price prediction and optimal execution strategy testing.
• The convergence results are proven using Skorokhod representation and continuity arguments on function spaces of censored processes.