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Modelling approaches for optimal liquidation under a limit-order book structure

Blair, James

[Thesis]. Manchester, UK: The University of Manchester; 2016.

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Abstract

This thesis introduces a selection of models for optimal execution of financial assets at the tactical level. As opposed to optimal scheduling, which defines a trading schedule for the trader, this thesis investigates how the trader should interact with the order book. If a trader is aggressive he will execute his order using market orders, which will negatively feedback on his execution price through market impact. Alternatively, the models we focus on consider a passive trader who places limit orders into the limit-order book and waits for these orders to be filled by market orders from other traders. We assume these models do not exhibit market impact. However, given we await market orders from other participants to fill our limit orders a new risk is borne: execution risk.We begin with an extension of Guéant et al. (2012b) who through the use of an exponential utility, standard Brownian motion, and an absolute decay parameter were able to cleverly build symmetry into their model which significantly reduced the complexity. Our model consists of geometric Brownian motion (and mean-reverting processes) for the asset price, a proportional control parameter (the additional amount we ask for the asset), and a proportional decay parameter, implying that the symmetry found in Guéant et al. (2012b) no longer exists. This novel combination results in asset-dependent trading strategies, which to our knowledge is a unique concept in this framework of literature. Detailed asymptotic analyses, coupled with advanced numerical techniques (informing the asymptotics) are exploited to extract the relevant dynamics, before looking at further extensions using similar methods.We examine our above mentioned framework, as well as that of Guéant et al. (2012), for a trader who has a basket of correlated assets to liquidate. This leads to a higher-dimensional model which increases the complexity of both numerically solving the problem and asymptotically examining it. The solutions we present are of interest, and comparable with Markowitz portfolio theory. We return to our framework of a single underlying and consider four extensions: a stochastic volatility model which results in an added dimension to the problem, a constrained optimisation problem in which the control has an explicit lower bound, changing the exponential intensity to a power intensity which results in a reformulation as a singular stochastic control problem, and allowing the trader to trade using both market orders and limit orders resulting in a free-boundary problem.We complete the study with an empirical analysis using limit-order book data which contains multiple levels of the book. This involves a novel calibration of the intensity functions which represent the limit-order book, before backtesting and analysing the performance of the strategies.