Limit order book models and optimal trading strategies vorgelegt von Marcel H¨oschler Master of Science aus Frechen Von der Fakult¨atII { Mathematik und Naturwissenschaften der Technischen Universit¨atBerlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften { Dr. rer. nat. { genehmigte Dissertation Vorsitzender: Prof. Dr. Harry Yserentant Berichter: Prof. Dr. Peter Bank Prof. Dr. Bruno Bouchard Tag der wissenschaftlichen Aussprache: 7.7.2011 Berlin 2011 D 83 Zusammenfassung In den letzten Jahren haben fast alle großen B¨orsen elektronische Orderb¨ucher eingef¨uhrt.Diese sammeln eingehende Limitorder und f¨uhren Marktorder automatisch zum bestm¨oglichen Preis aus. Durch die Einf¨uhrungvon Orderb¨uchern haben sich Handelsstrategien stark ver¨andert;dies liegt zum einen an der sehr viel h¨oherenHandelsgeschwindigkeit sowie an den verschiedenen Ordertypen, aus denen die H¨andlernun ausw¨ahlenk¨onnen. Es stellt sich daher die Frage, welcher Ordertyp unter welchen Umst¨andenverwendet werden sollte, oder - allgemeiner - ob und wie optimale Handelsstrategien gefunden werden k¨onnen.W¨ahrendeinige dieser Fragen in der Wirtschafts- und ¨okonometrischen Literatur betrachtet wurden, fehlt oft noch eine strenge mathematische Behandlung. In dieser Arbeit entwickeln wir geeignete mathematische Modelle und finden die angemessenen mathematischen Werkzeuge, um diese Fragen zu beantworten. Im ersten Teil entwickeln wir ein mathematisches Modell f¨urein dynamisches, zeitstetiges Orderbuch. Innerhalb dieses Modells untersuchen wir, wie der aktuelle Zustand des Orderbuchs seine kurzfristige zuk¨unftigeEntwicklung bestimmt. Insbesondere analysieren wir die Verteilung des Ausf¨uhrungszeitpunktseiner Limitorder. Da automatisierte Micro-Trader die Order in- nerhalb von Millisekunden platzieren m¨ussen,bestimmen wir eine N¨aherungsformelf¨urdie Laplace-Transformation und die Momente des Ausf¨uhrungszeitpunkts,die sehr effizient berech- net werden kann. Anschließend testen wir das Modell mit realen Hochfrequenz-Orderbuchdaten und zeigen, dass wichtige Eigenschaften sehr gut durch das Modell wiedergegeben werden. Im zweiten Teil dieser Arbeit analysieren wir optimale Handelsstrategien in Orderb¨uchern. Zun¨achst bleiben im Rahmen des Modells aus Teil I und berechnen die optimale Handelsstrate- gie, wenn der H¨andlernur Marktorder benutzt. Danach berechnen wir optimale Handelsstrate- gien mit sowohl Markt- als auch Limitordern in einem vereinfachten Orderbuchmodell. Schließlich betrachten wir das Problem des Kaufs einer einzelnen Aktie. Der H¨andler platziert eine Lim- itorder zu Beginn der Handelsperiode. Nun gilt es, den optimalen Zeitpunkt zu finden, wenn die Limitorder in eine Marktorder umgewandelt werden sollte, falls sie noch nicht ausgef¨uhrt wurde. Wir zeigen, wie dieser Zeitpunkt vom spread abh¨angigist, d.h. dem zus¨atzlichen Preis, den man bei der Umwandlung der Limit- zur Marktorder zahlen muss. 1 Summary In the last few years, almost all major stock exchanges have introduced electronic limit order books, which collect incoming limit orders and automatically match market orders against the best available limit order. The introduction of limit order books has significantly changed trading strategies as the speed of trading increased dramatically and traders have the choice between different order types. This automatically raises the question which order type should be used under which circumstances, and more generally, if and how optimal trading strategies can be found. While some of these questions have been considered in the economic and econometric literature, a rigorous mathematical treatment of is often still lacking. In this thesis we develop suitable mathematical frameworks and find appropriate mathematical tools to address these questions. In the first part, we propose a mathematical model for a dynamic, continuous time limit order book. Within this model, we study how the current state of the order book determines its short-time evolution. In particular, we analyse the distribution of the time-to-fill of a limit order. Since automated microtraders have to place orders within milliseconds, we also propose approximate formulae for the Laplace transform and the moments of the time-to-fill that can be computed very efficiently. Finally, we test the model with real-world high-frequency order book data and show that important properties are well reproduced by the model. In the second part of this thesis, we analyse optimal trading strategies in limit order books. We first remain in the setting of the model of part I, and compute optimal liquidation strategies when the trader is restricted to use only market orders. Next, we compute optimal liquidation strategies with both market and limit orders in a simplified order book model. Finally, we turn to the problem of buying a single share. The trader places a limit order at the beginning of the trading period. The question is to find the optimal time when the limit order should be converted to a market order if it has not been filled yet. We show how this time depends on the spread, i.e. the additional price that is charged when converting the limit to a market order. 3 4 Contents 1 Introduction 9 1.1 The limit order book . .9 1.2 Economic background . 11 1.2.1 Empirical studies on limit order book . 11 1.2.2 Dynamic limit order book models . 12 1.2.3 Optimal trading in limit order books . 12 1.3 Mathematical and economic results . 13 1.3.1 Part I: Limit order book models . 13 1.3.2 Part II: Optimal trading strategies in limit order books . 14 1.4 Acknowledgements . 16 I Limit order book models 19 2 A general framework for limit orderbook models 21 2.1 Modelling framework . 21 2.1.1 Representation of limit order books . 22 2.1.2 Dynamics of limit order book models . 25 2.1.3 Proportional placing/cancelling of limit orders . 29 2.2 Dimension reduction . 30 2.3 Linear orderbook models and fundamental examples . 32 2.4 Proofs . 34 3 Order flow in limit order books 39 3.1 State-dependent behaviour of the order flow . 39 3.2 Choice of order flow processes . 42 4 Analysis of order book 45 4.1 Basic order book properties . 45 4.2 Time-to-fill . 48 4.2.1 Problem definition . 48 4.2.2 Formulation as Dirichlet problem and viscosity solutions . 49 4.3 Asymptotic analysis of time-to-fill . 50 4.3.1 Motivation . 50 4.3.2 Modified Dirichlet problem . 51 4.3.3 Asymptotic approximation . 52 4.3.4 Error estimation . 53 4.3.5 Discussion of asymptotic approximation . 55 4.4 Proofs . 61 5 6 Contents 5 Model calibration and test 67 5.1 Method overview . 67 5.2 The data . 68 5.3 Calibration . 69 5.3.1 Estimation of volatility parameters . 69 5.3.2 Estimation of CIR process . 71 5.3.3 Estimation of OU process . 72 5.3.4 Estimation of midquote price . 73 5.3.5 Calibration algorithm . 73 5.3.6 Calibration results . 74 5.4 Computation of time-to-fill/first passage time . 75 5.5 Comparison of time-to-fill/first passage times . 77 II Optimal trading strategies in limit order books 83 6 Optimal trading strategies with market orders 85 6.1 Model assumptions and problem formulation . 85 6.2 Derivation of candidate strategy using Euler-Lagrange method . 88 6.2.1 Buying region . 90 6.2.2 Waiting region . 91 6.2.3 Characterization of optimal strategy . 92 6.3 Derivation and verification of value function . 94 6.4 Proofs . 96 7 Optimal trading using market and limit orders with partial filling 99 7.1 Model assumptions and trading costs . 99 7.2 Problem formulation . 101 7.3 Optimal strategies in infinite-time horizon . 103 7.3.1 Trading in a pure market order market . 106 7.3.2 Trading in a pure limit order market . 107 7.3.3 Trading in a market without limit order impact . 110 7.3.4 Trading in a market with limit order impact . 112 7.4 Optimal pure buy strategies in infinite-time horizon . 117 7.5 Optimal strategies with finite horizon . 118 7.5.1 Explicit solution in markets without limit order impact . 118 7.5.2 The general case . 119 7.6 Proofs . 122 8 Optimal 'Peg-Cross' strategies 133 8.1 Model assumptions and problem formulation . 133 8.2 Optimal strategies without market impact . 136 8.2.1 Geometric Brownian motion . 138 8.2.2 CIR process . 140 8.2.3 Calibration and test . 143 8.3 Optimal strategies with market impact . 146 8.4 Proofs . 151 Contents 7 A Matlab code 161 A.1 Calibration of 3-dimensional model . 161 A.2 Computation of first passage time . 164 8 Contents Chapter 1 Introduction 1.1 The limit order book Stock market trading has changed dramatically in the last two decades. Next to globalization and deregulation of the financial market, technological innovation has been one of the main drivers of these changes: While traditionally a market maker collected buy and sell orders and provided liquidity by setting bid and ask quotes, nowadays most exchanges work with order- driven systems. These fully automated electronic trading platforms collect and match orders. Electronic Communication Networks (ECN) aggregate incoming limit orders at each price level. They constitute the overall liquidity and are made available to all market participants in the limit order book (LOB) by financial market data providers, see for example figure 1.1. Market orders are automatically executed against the best available limit orders. The automated electronic matching of orders significantly increases the speed of trading. It now often only takes a few milli-seconds from sending an order to its execution. Figure 1.1: Bloomberg screenshot of Vodafone limit order book, with sizes of limit orders stored on different price ticks. The bid side (red) displays buy offers, and the ask side (blue) displays sell offers. Most major exchanges such as Deutsche B¨orse,London Stock Exchange, Nasdaq, NYSE, Paris 9 10 Introduction Bourse and Tokyo Stock Exchange rely on either pure electronic limit order trading, or on hybrid trading systems combining order- and quote-driven markets.