Stochastic models for the Limit Order Book

Abstract

The aim of this thesis is to outline a new approach to the Limit Order Book's (LOB) modelling. This is accomplished starting from the existing literature and then proposing several models with different features and complexity levels, the aim being to compute some quantities of interest. Chapter 1 is an introduction to the LOB. It explains what this object is, why we want to model it and what has been done so far in literature. We also outline the general ideas behind this work. Chapter 2 is focused on the avalanche approach: orders in the LOB accumulate on some levels and get executed when the price process crosses such values. This idea, as it will be explained, belongs to my supervisor Dr Rheinlander. This model uses the theory related to the local time of a Brownian motion. Chapter 3 defines a model introducing a Poisson process for incoming orders and cancellations. The framework outlined in this way is used to calculate quantities of interest: a simulation technique is described and a refinement of the model is also presented, in order to be consistent with empirical behaviours observed in the markets. Finally, in Chapter 4, the local time approach is once again taken into considerations and two models are proposed using downcrossings and excursions. It is also shown how to link the two frameworks