Brownian trading excursions and avalanches

TL;DR

This paper models the limit order book using a stochastic heat equation framework, classifies trade types, and analyzes order avalanches, providing insights into the timing and distribution of trading events.

Contribution

It introduces a novel Poisson point process for trading excursions and derives Laplace transforms for event timings, advancing understanding of order avalanche distributions.

Findings

Order avalanches follow a specific distribution derived from the model.

The trading excursion process is characterized as a Poisson point process.

Results generalize previous studies on Parisian options.

Abstract

We study a parsimonious but non-trivial model of the latent limit order book where orders get placed with a fixed displacement from a center price process, i.e.\ some process in-between best bid and best ask, and get executed whenever this center price reaches their level. This mechanism corresponds to the fundamental solution of the stochastic heat equation with multiplicative noise for the relative order volume distribution. We classify various types of trades, and introduce the trading excursion process which is a Poisson point process. This allows to derive the Laplace transforms of the times to various trading events under the corresponding intensity measure. As a main application, we study the distribution of order avalanches, i.e.\ a series of order executions not interrupted by more than an $\varepsilon$-time interval, which moreover generalizes recent results about Parisian options.