Application of Stochastic Control in Optimal Execution Algorithms by Luis Eduardo Pav´onTinoco (CID: 01393260) Department of Mathematics Imperial College London London SW7 2AZ United Kingdom Thesis submitted as part of the requirements for the award of the MSc in Mathematics and Finance, Imperial College London, 2017-2018 Declaration The work contained in this thesis is my own work unless otherwise stated. Signature and date: 2 Acknowledgements I would like to express my sincere gratitude to Dr. Mikko Pakkanen, my research supervisor, for his multiple advice and guidance during this project. Besides my supervisor, I would like to thank PhD candidate Leandro S´anchez-Betancourt for his insightful comments and discussions. This work would not have been possible without their support. The completion of my master's could not have been achieved without the support of my friends, Angel,´ Andr´es,Dafne, Fernanda, Fernando, Ra´ul,Ricardo, and Ruben. You have always been a major source of support when things get a bit discouraging. I would like to show my gratitude to \Banco de Mexico" (Banxico) for its financial support during my master's degree. Last but not the least, I would like to thank my family: My mom for being the light of my life, my brothers who are my main source of inspiration, and Eduardo, Juan and Reyes for their unconditional support. 3 To my mother, Leonor. 4 Abstract In this thesis I study optimal execution models under different scenarios, using Stochastic Control as the main mathematical tool. Many features of electronic markets are also discussed, such as volume, volatility, and liquidity. Regarding the execution problems, I first solve a model with terminal and running penalty, I then move to a more realistic setup, with stochastic volatility and liquidity, which is solved and calibrated with real data, I also study a double liquidation problem, this involves modelling a joint liquidation in the equity market and in the foreign exchange market. Finally, I solve the double liquidation problem assuming ambiguity aversion in the mid-price drift. Keywords: High Frequency Trading, Round-Trip Cost, Stochastic Control, Optimal Execution Models, Double Liquidation Problem, Ambiguity Aversion. 5 Contents 1 Introduction 8 2 Electronic Markets 9 2.1 Assets traded in electronic markets . .9 2.2 Market participants in an electronic market . .9 2.3 Trading in electronic markets . 10 2.4 Intraday market patterns . 13 2.4.1 Trading volume patterns . 14 2.4.2 Intraday volatility pattern . 15 2.4.3 Liquidity: bid-ask spread and round trip cost . 16 2.4.4 Market impact pattern . 19 2.5 Relationship between volatility and round trip cost. 20 3 Stochastic Control 22 3.1 Portfolio optimisation problem . 22 3.2 Dynamic programming principle . 23 3.3 Hamilton-Jacobi-Bellman equation . 25 3.4 Verification theorem . 26 3.5 Portfolio optimisation solution . 28 4 Optimal Executions in the Basic Model 29 4.1 Basic model . 29 4.1.1 Liquidation problem assuming only temporary impact . 29 4.1.2 Optimal acquisition with terminal penalty and only temporary impact . 31 4.1.3 Optimal liquidation with permanent impact . 32 5 Optimal Execution Strategy with Stochastic Volatility and Liquidity 35 5.1 Model . 35 5.2 Analysis of Partial Differential Equation . 37 5.3 Extended model . 38 5.4 Calibration of Partial Differential Equation using Starbucks Stock . 39 6 Double Liquidation Problem 41 6.1 Problem formulation . 41 7 Robust Double Liquidation Problem 44 7.1 General model . 44 6 7.2 Double liquidation problem with ambiguity aversion. 46 Conclusion 48 A Numerical Solution of Partial Differential Equation 49 A.1 Finite difference method . 49 B Code 51 B.1 Code for section 2 . 51 B.2 Code for section 5 . 56 B.3 Code for section 6 . 57 B.4 Code for section 7 . 58 7 8 1 Introduction The world of finance and how to trade a security have changed dramatically in the last ten years, new developments such as machine learning and on-line platforms have revolutionised the way in which banks and brokerage houses execute orders. A leading example is algorithmic trading, this new trading technique allows machines to trade any financial security establishing pre-defined rules, optimising profits, minimising price impact, or reading an alpha-signal. Almgren and Chriss [2] introduced the design of optimal execution problems, in recent years this approach has been developed in a more general setup. In this thesis I study various models for the optimal execution problem, using stochastic control as the primary mathematical tool. In chapter 2, I discuss how the electronic market works, market participants and some financial variables such as volume, volatility, and liquidity. In chapter 3 and 4, I develop the theory behind of stochastic control using as motivation the optimisation portfolio problem introduced by Merton (1971) in his work [20]. I also analyse the execution models in the basic setup, with a particular emphasis in inventory penalties that can be considered in the optimisation model. All models presented previously assume that volatility and liquidity are fixed during the execu- tion process. In chapter 5, I discuss the liquidation problem with stochastic volatility and liquidity, this problem was introduced by Almgren in his work [1]. I also calibrate this model using data from Starbucks stock. In chapter 6, I explain and solve double liquidation problem which involves an execution process in the equity market and simultaneously another execution process in the foreign exchange market. My contribution for this model is to introduce a closed-form solution; As far as I know, this problem had not been solved in closed-form before. Finally, in chapter 7 I extend the double liquidation problem, making it robust to misspecifi- cation. This technique is the so called ambiguity aversion, and for this specific model, I am able to find a closed-form solution. 9 2 Electronic Markets Electronic markets are defined by the U.S Security and Exchange Commission (SEC) as \profes- sional traders acting in a proprietary capacity that generate a large number of trades on a daily basis"; these markets have some characteristics such as: 1. Use programs to execute orders in high speed. 2. Submission of numerous orders which can be cancelled after submission. 3. The market participants seek to close their positions at the end of the day. These markets have grown dramatically in the last ten years1, due to this fact the market micro-structure has also changed, for instance, it can be observed a higher trading volume, the bid-ask spread for large-cap stocks have been tightened, and an increment in the large-cap stocks' volatility at the end of the trading day. 2.1 Assets traded in electronic markets According to Cartea et al. [10], shares are the most common asset traded in electronic markets, also known as equity, shares are issued by companies to raise money through an Initial Public Offer (IPO) and are listed and traded in an exchange (for example, the New York Stock Exchange (NYSE), the Nasdaq, the London Stock Exchange (LSE) and the Tokyo Stock Exchange (TSE)). The investor receives one proportion of the corporation's profits as dividend and has the right to intervene in the corporate decisions, if and only if these shares are ordinary, which are the most common in the market2. In electronic markets also are traded financial contracts such as commodities, currencies, real state contracts, and derivatives. Usually, these assets are found in the form of mutual funds or exchange-traded funds (ETFs). A mutual fund is an investment vehicle that tracks an index, and collects money from different investors. On the other hand, when an investor buys an ETF, she delegates also her money to a portfolio manager. However, the main differences between a mutual fund and an ETF are that an exchange-traded fund generates the same return as a specific index (e.g., S&P500) and if the investor wants to close her participation in the fund, the issuer could give to the investor a basket of securities which has had the same performance as the ETF. 2.2 Market participants in an electronic market The understanding of an electronic market is based on analysing its participants. Every market participant should has as a central purpose to generate profits, but the way that they produce 1It is estimated that electronic markets exceeded 50% of total volume in U.S equity. 2There exists another kind of shares called preferred stock. In this case, the holder cannot be part of company's decisions and receives a pre-arrange income, but is considered as equity from the legal point of view. 2.3 Trading in electronic markets 10 them is different. The main market participants are the following: • Corporate issuer: As I mention previously, they are corporations which raise money via IPO. The reason for this transaction depends entirely on the origin or necessities of the corporation. Another feature of this market participants is that they can increase or reduce the supply of their shares using a secondary share offering (SSO), shares buybacks or converted bonds. • Financial management companies: They are responsible for creating funds such as mutual funds or ETFs. These market participants can be divided into: { long-term investors: based on \the fundamental value". { short-term investors: the leading example is an ETF which seeks to replicate an index. • Fundamental traders: These kinds of traders work using sources of information to make a market decision. That is to say, if news are released then they try to understand the implications in the stock's dynamics due to these news. They use sources of information such as economic reports, political factors, and rumours, among others.