Conservation Laws in a Limit Order Book

TL;DR

This paper develops macroscopic models of limit order books to understand how they recover from aggressive trades, revealing a universal t^{1/3} scaling law for order book recovery.

Contribution

It introduces a new class of macroscopic models and uncovers a universal scaling law for order book recovery after liquidity shocks.

Findings

Order book recovery follows a t^{1/3} scaling law.

Models are solved numerically and asymptotically.

Similarity solutions relate to order book formation and recovery.

Abstract

We present a class of macroscopic models of the Limit Order Book to simulate the aggregate behaviour of market makers in response to trading flows. The resulting models are solved numerically and asymptotically, and a class of similarity solutions linked to order book formation and recovery is explored. The main result is that order book recovery from aggressive liquidity taking follows a simple $t^{1/3}$ scaling law.