• limit order book

The Limit Order Book is a digital queuing system in which buy and sell orders are stored, with the aim to facilitate trades between market participants. It is an example of a dynamical complex system and it has been a subject of many studies by economists, mathematicians, and physicists for the past twenty years. The aim of this thesis is to create a realistic Limit Order Book simulator, able to reproduce its empirical properties.
In the first part of this work I explain what is a Limit Order Book and how it works, then I present a brief overview of the literature focused on its mathematical modeling and numerical simulations. After that I present an original work, in which I analyze and compare the empirical properties of the Limit Order Book of Tesla stocks traded in the NASDAQ exchange and electrical power futures from the European energy market, focusing on highlighting the differences between the two assets.
Following that I present one of the first statistical LOB models called the Zero Intelligence model, which assumes no rationality behind traders actions, being a purely mechanistic model in which the arrival of orders is described by independent Poisson processes. Through numerical simulations of the model, I show that some properties of empirical Limit Order Books, namely the spread (i.e. the difference between the buy order with the highest price and the sell order with the lowest price) and the volatility can be reproduced quite well by the model, meaning that that they can be traced to the stochastic nature of the Limit Order Book. The model though is based on many simplifying assumptions, such as the complete lack of correlation of the order-flow, i.e. the arrival of different type of orders, and the independence of the probability of placing and order at a certain price level from its distance from the best prices. Both of those assumptions are not verified in empirical data, therefore the ZI model is only a first order approximation to build a Limit Order Book simulator.
For this reason, I develop an original LOB model called the Ratio model that tries to overcome the above limitations of the ZI model, using some regularities of the order flow that have been observed in the literature. In particular I use the fact that orders are placed according to a bi-modal distribution centered around the best prices, that orders closer to the best prices are more likely to be canceled, that the spread greatly affects what type of order is placed and that orders have long range correlations in their sign (+1 for buy orders and -1 for sell orders ). Through numerical simulations I show that the Ratio model is able to reproduce some properties of empirical Limit Order Books with high fidelity and that it outperforms the ZI model in almost every aspect. On the other hand since the model uses solely the precedent state of the book to simulate the arrival of a new order, it is not able to faithfully reproduce temporal dependencies of empirical Limit Order Books.
To create a Limit Order Book simulator that can replicate the correlations in the order-flow, I used Recurring Neural Networks to build a predictor that guesses the next order type given the recent past history of the book. Adding this predictor to the Ratio model I create an original Limit Order Book simulator, called the Deep Ratio model, that retains all the properties of the previous model and it is able to reproduce temporal dependencies in the book. Comparing the Deep Ratio model to the ZI and Ratio models, I show that the former has better performances and that, apart from the spread distribution, it is able to replicate many properties of empirical Limit Order Books.