Jump diffusion approximation for the price dynamics of a fully state dependent limit order book model

TL;DR

This paper derives a macroscopic jump diffusion model for a limit order book that accounts for large, unforeseen price movements by including non-scaled price changes, extending previous models.

Contribution

It introduces a new limit order book model that incorporates large price jumps not scaled with tick size, and proves convergence to a jump diffusion and fluid process system.

Findings

Model captures large, unforeseen price movements.

Convergence to jump diffusion and fluid processes established.

Extends existing models by including non-scaled price changes.

Abstract

We study a microscopic limit order book model, in which the order dynamics depend on the current best bid and ask price and the current volume density functions, simultaneously, and derive its macroscopic high-frequency dynamics. As opposed to the existing literature on scaling limits for limit order book models, we include price changes which do not scale with the tick size in our model to account for large price movement, being for example triggered by highly unforeseen events. Our main result states that, when the size of an individual limit order and the tick size tend to zero while the order arrival rate tends to infinity, the microscopic limit order book model dynamics converge to two one-dimensional jump diffusion processes describing the prices coupled with two infinite dimensional fluid processes describing the standing volumes at the buy and sell side.