Second order approximations for limit order books

TL;DR

This paper develops a second order approximation for infinite dimensional limit order book models, capturing price and volume fluctuations under different scaling regimes, with applications to optimal portfolio liquidation.

Contribution

It introduces a second order approximation framework for limit order books that accounts for price-volume dynamics and their fluctuations under various scaling regimes.

Findings

Second order approximation captures price and volume fluctuations.

Different scaling regimes lead to different stochastic models.

Results can be used for confidence intervals in portfolio liquidation.

Abstract

In this paper we derive a second order approximation for an infinite dimensional limit order book model, in which the dynamics of the incoming order flow is allowed to depend on the current market price as well as on a volume indicator (e.g.~the volume standing at the top of the book). We study the fluctuations of the price and volume process relative to their first order approximation given in ODE-PDE form under two different scaling regimes. In the first case we suppose that price changes are really rare, yielding a constant first order approximation for the price. This leads to a measure-valued SDE driven by an infinite dimensional Brownian motion in the second order approximation of the volume process. In the second case we use a slower rescaling rate, which leads to a non-degenerate first order approximation and gives a PDE with random coefficients in the second order approximation for the volume process. Our results can be used to derive confidence intervals for models of optimal portfolio liquidation under market impact.