Limit order books, diffusion approximations and reflected SPDEs: from microscopic to macroscopic models

TL;DR

This paper develops a multi-scale probabilistic framework for limit order books, connecting microscopic, mesoscopic, and macroscopic models, and introduces reflected SPDEs to describe their macroscopic behavior, with applications in market simulation and prediction.

Contribution

It introduces a unified approach linking microscopic to macroscopic limit order book models using diffusion approximations and reflected SPDEs, with calibration and numerical illustration.

Findings

The macroscopic limit is described by reflected SPDEs.

The model can reproduce observed price behaviors.

Calibration procedures are proposed for real data.

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.