ALGORITHMIC AND HIGH-FREQUENCY TRADING The design of trading algorithms requires sophisticated mathematical models, a solid anal- of financial data, and a deep understanding of how markets and exchanges function. In this textbook the authors develop models for algorithmic trading in contexts such as: executing large orders, market making, targeting VWAP and other schedules, trading pairs or collection of assets, and executing in dark pools. These models are grounded on how the exchanges work, whether the algorithm is trading with better informed traders (adverse selection), and the type of information available to market participants at both ultra-high and low frequency. Algorithmic and High-Frequency Trading is the first book that combines sophisticated mathematical modelling, empirical facts and financial economics, taking the reader from basic ideas to the cutting edge of research and practice. If you need to understand how modern electronic markets operate, what information provides a trading edge. and how other market participants may affect the profitability of the algorithms, then this is the book for you. AL VAR o c ARTE A is a Reader in Financial Mathematics at University College London. Before joining UCL he was Associate Professor of Finance at Universidad Carlos III, Madrid-Spain (2009-2012) and from 2002 until 2009 he was a Lecturer (with tenure) in the School of Economics, Mathematics and Statistics at Birkbeck - University of London. He was previously JP Morgan Lecturer in Financial Mathematics at Exeter College, University of Oxford. s EB As TI AN J A IM u NG AL is an Associate Professor and Chair, Graduate Studies in the Department of Statistical Sciences at the University of Toronto where he teaches in the PhD and Masters of Mathematical Finance programs. He consults for major banks and hedge funds focusing on implementing advance derivative valuation engines and algorith­ mic trading strategics. He is also an associate editor for the SIAM Journal on Financial Mathematics, the International Journal of Theoretical and Applied Finance, the journal Risks and the Argo newsletter. Jaimungal is the Vice Chair for the Financial Engineering & Mathematics activity group of SIAM and his research is widely published in academic and practitioner journals. His recent interests include High-Frequency and Algorithmic trading, applied stochastic control, mean-field games, real options, and commodity models and derivative pricing . .r o sf: PEN AL v A is an Associate Professor at the Universidad Carlos Ill in Madrid where he teaches in the PhD and Master in Finance programmes, as well as at the undergraduate level. He is currently working on information models and market microstructure and his research has been published in Econometrica and other top academic journals. ALGORITHMIC AND HIGH-FREQUENCY TRADING ALVARO CARTEA University College London SEBASTIAN JAIMUNGAL University of Toronto JOSE PENALVA Universidad Carlos III de Madrid CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE UNIVERSITY PRESS University Printing Honse, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It forthers the University's mission by disseminating knowledge in the pnrsnit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/978 l l 07091146 © Alvaro Cartea, Sebastian Jaimungal and Jose Penalva 2015 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2015 Printed in the United Kingdom by Bell and Bain Ltd A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Cartea, Alvaro. Algorithmic and high-frequency trading I Alvaro Cartea, Sebastian Jaimungal, Jose Penalva. pages cm Includes bibliographical references and index. ISBN 978-1-107-09114-6 (Hardback: alk. paper) 1. Electronic trading of securities-Mathematical models. 2. Finance-Mathematical models. 3. Speculation-Mathematical models. I. Title. HG4515.95.C387 2015 332.64-dc23 2015018946 ISBN 978-1-107-09114-6 Hardback Additional resources for this publication at www.cambridge.org/9781107091146 Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on snch websites is, or will remain, accurate or appropriate. To my girls, in order of appearance, Victoria, Amaya, Carlota, and Penelope. -A.c. To my parents, Korisha and Paul, and my siblings Shelly, Cristina and especially my brother Curt for his constant injection of excitement and encouragement along the way. -S.J. To Nuria, Daniel, Jose Maria and Adelina. For their patience and encouragement every step of the way, and for never losing faith. -J.P. V Contents Preface How to Read this Book Part I Micmstmcture and Empirical Facts Introduction to Part I l Electronic Markets and the limit Order Book 2 3 1.1 Electronic markets and how they function 1.2 Classifying Market Participants 1.3 Trading in Electronic Markets 1.3.1 Orders and the Exchange 1.3.2 Alternate Exchange Structures 1.3.3 Colocation 1.3.4 Extended Order Types 1.3.5 Exchange Fees 1.4 The Limit Order Book 1.5 Bibliography and Selected Readings A Primer on the Microstrncture of Financial Markets 2.1 Market Making 2.1.l Grossman-Miller Market Making Model 2.1.2 Trading Costs 2.1.3 Measuring Liquidity 2.1.4 Market Making using Limit Orders 2.2 Trading on an Informational Advantage 2.3 Market Making with an Informational Disadvantage 2.3.1 Price Dynamics 2.3.2 Price Sensitive Liquidity Traders 2.4 Bibliography and Selected Readings Empirical and Statistical Evidence: Prices and Returns 3.1 Introduction 3.1.1 The Data 3.1.2 Daily Asset Prices and Returns page xiii XVl 1 3 4 4 6 9 9 10 11 12 13 14 18 19 20 21 24 26 28 30 34 36 37 37 39 39 39 41 v111 Contents 4 3.1.3 Daily Trading Activity 3 .1.4 Daily Price Predictability 3.2 Asset Prices and Returns Intraday 3.3 Interarrival Times 3.4 Latency and Tick Size 3.5 Non-Markovian Nature of Price Changes 3.6 Market Fragmentation 3. 7 Empirics of Pairs Trading 3.8 Bibliography and Selected Readings Empirical and Statistical Evidence: Activity and Market Quality 4.1 Daily Volume and Volatility 4.2 Intraday Activity 4.2.1 Intraday Volume Patterns 4.2.2 Intrasecond Volume Patterns 4.2.3 Price Patterns 4.3 Trading and Market Quality 4.3.l Spreads 4.3.2 Volatility 4.3.3 Market Depth and Trade Size 4.3.4 Price Impact 4.3.5 Walking the LOB and Permanent Price Impact 4.4 Messages and Cancellation Activity 4.5 Hidden Orders 4.6 Bibliography and Selected Readings 42 42 46 48 49 52 54 57 60 61 61 63 65 67 68 69 .71 76 79 81 87 90 95 96 Part 11 Mathematical Tools Introduction to Part II 97 99 5 Stochastic Optimal Control and Stopping 5.1 Introduction 5.2 Examples of Control Problems in Finance 5.2.1 The Merton Problem 5.2.2 The Optimal Liquidation Problem 5.2.3 Optimal Limit Order Placement 100 100 101 101 102 103 5.3 Control for Diffusion Processes 103 5.3.1 The Dynamic Programming Principle 105 5.3.2 Dynamic Programming Equation / Hamilton-Jacobi- Bellman Equation 107 5.3.3 Verification 112 5.4 Control for Counting Processes 113 5.4.1 The Dynamic Programming Principle 114 5.4.2 Dynamic Programming Equation / Hamilton-Jacobi- Bellman Equation 115 5.4.3 Combined Diffusion and Jumps 5.5 Optimal Stopping 5.5.1 The Dynamic Programming Principle 5.5.2 Dynamic Programming Equation 5.6 Combined Stopping and Control 5.7 Bibliography and Selected Readings Contents 1x 120 122 124 124 128 130 Part 111 Algorithmic and High-Frequency Trading Introduction to Part III 131 133 1 9 Optimal Execution with Continuous Trading I 6.1 Introduction 6.2 The Model 6.3 Liquidation without Penalties only Temporary Impact 134 134 135 139 6.4 Optimal Acquisition with Terminal Penalty and Temporary Impact 141 6.5 Liquidation with Permanent Price Impact 144 6.6 Execution with Exponential Utility Maximiser 150 6.7 Non-Linear Temporary Price Impact 152 6.8 Bibliography and Selected Readings 154 6.9 Exercises 155 Optimal Execution with Continuous Trading 11 7.1 Introduction 7.2 Optimal' Acquisition with a Price Limiter 7.3 Incorporating Order Flow 7.3.l Probabilistic Interpretation 7.4 Optimal Liquidation in Lit and Dark Markets 7.4.1 Explicit Solution when Dark Pool Executes in Full 7.5 Bibliography and Selected Readings 7.6 Exercises 158 158 159 167 174 175 178 182 182 Optimal Exerntioro with Limit and Market Orders 184 8.1 Introduction 184 8.2 Liquidation with Only Limit Orders 185 8.3 Liquidation with Exponential Utility Maximiser 193 8.4 Liquidation with Limit and Market Orders 196 8.5 Liquidation with Limit and Market Orders Targeting Schedules 206 8.6 Bibliography and Selected Readings 209 8.7 Exercises 209 Targeting Volume 9.1 Introduction 9.2 Targeting Percentage of Market's Speed of Trading 9.2.1 Solving the DPE when Targeting Rate of Trading 212 212 215 216 x Contents 9.2.2 Stochastic Mean-Reverting Trading Rate 9.2.3 Probabilistic Representation 9.2.4 Simulations 9.3 Percentage of Cumulative Volume 9.3.1 Compound Poisson Model of Volume 9.3.2 Stochastic Mean-Reverting Volume Rate 9.3.3 Probabilistic Representation 9.4 Including Impact of Other Traders 9.4.1 Probabilistic Representation 9.4.2 Example: Stochastic Mean-Reverting Volume 9.5 Utility Maximiser 9.5.1 Solving the DPE with Deterministic Volume 9.6 Bibliography and Selected Readings 9.7 Exercises 10 Market Making 10.1 Introduction 10.2 Market Making 10.2.1 Market Making with no Inventory Restrictions 10.2.2 Market Making At-The-Touch 10.2.3 Market Making Optimising Volume 10.3 Utility Maximising Market Maker 10.4 Market Making with Adverse Selection 10.4.1 Impact of Market Orders on Midprice 10.4.2 Short-Term-Alpha and Adverse Selection 10.5 Bibliography and Selected Readings 10.6 Exercises 11 Pairs Trading and Statistical Arbitrage Strategies 11.1 Introduction 11.2 Ad Hoc Bands 11.3 Optimal Band Selection 11.3.1 The Optimal Exit Problem 11.3.2 The Optimal Entry Problem 11.3.3 Double-Sided Optimal Entry-Exit 11.4 Co-integrated Log Prices with Short-Term-Alpha 11.4.1 Model Setup 11.4.2 The Agent's Optimisation Problem 11.4.3 Solving the DPE 11.4.4 Numerical Experiments 11.5 Bibliography and Selected Readings 12 Order Imbalance 12.1 Introduction 220 222 225 227 231 232 233 235 237 238 239 240 243 243 246 246 247 253 254 257 259 261 262 266 271 272 273 273 274 277 278 279 281 283 284 286 288 292 294 295 295 12.2 Intraday Features 12.2.1 A Markov Chain Model 12.2.2 Jointly Modelling Market Orders 12.2.3 Modelling Price Jumps 12.3 Daily Features 12.4 Optimal Liquidation 12.4.1 Optimisation Problem 12.5 Bibliography and Selected Readings 12.6 Exercises Appendix A Stochastic Calculus for Finance A.l Diffusion Processes A.1.1 Brownian Motion A.1.2 Stochastic Integrals A.2 Jump Processes A.3 Doubly Stochastic Poisson Processes A.4 Feynman-Kac and PDEs A.5 Bibliography and Selected Readings Bibliography Glossary Subject index Contents xi 295 297 300 303 305 306 308 313 313 315 315 316 316 319 322 325 326 327 337 342 Preface We have written this book because we feel that existing ones do not provide a sufficiently broad view to address the rich variety of issues that arise when trying to understand and design a successful trading algorithm. This book puts together the diverse perspectives, and backgrounds, of the three authors in a manner that ties together the basic economics, the empirical foundations of high-frequency data, and the mathematical tools and models to create a balanced perspective of algorithmic and high-frequency trading. This book has grown out of the authors' interest in the field of algorith­ mic and high-frequency finance and from graduate courses taught at Univer­ sity College London, University of Toronto, Universidad Carlos III de Madrid, IMPA, and University of Oxford. Readers are expected to have basic knowl­ edge of continuous-time finance, but it assumes that they have no knowledge of stochastic optimal control and stopping. To keep the book self-contained, we include an appendix with the main stochastic calculus tools and results that are needed. The treatment of the material should appeal to a wide audience and it is ideal for a graduate course on Algorithmic Trading at a Master's or PhD level. It is also ideal for those already working in the finance sector who wish to combine their industry knowledge and expertise with robust mathematical models for algorithmic trading. We welcome comments! Please send them to algo.trading.book©gmail.com. Brief guide to the contents This book is organised into three parts that take the reader from the work­ ings of electronic exchanges to the economics behind them, then to the relevant mathematics, and finally to models and problems of algorithmic trading. Part I starts with a description of the basic elements of electronic markets and the main ways in which people participate in the market: as active traders exploiting an informational advantage to profit from possibly fleeting profit op­ portunities, or as market makers, simultaneously offering to buy and sell at advantageous prices. A textbook on algorithmic trading would be incomplete if the development of strategies was not motivated by the information that market participants see in electronic markets. Thus it is necessary to devote space to a discussion of xiv Preface data and empirical implications. The data allow us to present the context which determines the ultimate fate of an algorithm. By looking at prices, volumes, and the details of the limit order book, the reader will get a basic overview of some of the key issues that any algorithm needs to account for, such as the information in trades, properties of price movements, regularities in the intraday dynamics of volume, volatility, spreads, etc. Part II develops the mathematical tools for the analysis of trading algorithms. The chapter on stochastic optimal control and stopping provides a pragmatic approach to material which is less standard in financial mathematics textbooks. It is also written so that readers without previous exposure to these techniques equip themselves with the necessary tools to understand the mathematical mod­ els behind some algorithmic trading strategies. Part III of the book delves into the modelling of algorithmic trading strategies. The first two chapters are concerned with optimal execution strategies where the agent must liquidate or acquire a large position over a pre-specified window and trades continuously using only market orders. Chapter 6 covers the classical execution problem when the investor's trades impact the price of the asset and also adjusts the level of urgency with which she desires to execute the programme. In Chapter 7 we develop three execution models where the investor: i) carries out the execution programme as long as the price of the asset does not breach a critical boundary, ii) incorporates order flow in her strategy to take advantage of trends in the midprice which are caused by one-sided pressure in the buy or sell side of the market, and iii) trades in both a lit venue and a dark pool. In Chapter 8 we assume that the investor's objective is to execute a large position over a trading window, but employs only limit orders, or uses both limit and market orders. Moreover, we show execution strategies where the investor also tracks a particular schedule as part of the liquidation programme. Chapter 9 is concerned with execution algorithms that target volume-based schedules. We develop strategies for investors who wish to track the overall vol­ ume traded in the market by targeting: Percentage of Volume, Percentage of Cumulative Volume, and Volume Weighted Average Price, also known as VWAP. The final three chapters cover various topics in algorithmic trading. Chapter 10 shows how market makers choose where to post limit orders in the book. The models that are developed look at how the strategies depend on different factors including the market maker's aversion to inventory risk, adverse selection, and short-term lived trends in the dynamics of the midprice. Finally, Chapter 11 is devoted to statistical arbitrage and pairs trading, and Chapter 12 shows how information on the volume supplied in the limit order book is employed to improve execution algorithms. Style of the book In choosing the content and presentation of the book we have tried to provide a rigorous yet accessible overview of the main foundational issues in market