Chapman & Hall/CRC FINANCIAL MATHEHATICS SERIES

Transcription

1 Chapman & Hall/CRC FINANCIAL MATHEHATICS SERIES The Financial Mathematics of Market Liquidity From Optimal Execution to Market Making Olivier Gueant röc) CRC Press J Taylor & Francis Croup BocaRaton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business A CHAPMAN & HALL BOOK

2 Preface List of Figures List of Tables xv xxi xxiii I Introduction 1 1 General introduction A brief history of Quantitative Finance From Bachelier to Black, Scholes, and Merton A new paradigm and its consequences The long journey towards mathematicians Quantitative Finance by mathematicians Quantitative Finance today Optimal execution and market making in the extended market microstructure literature The classical literature on market microstructure An extension of the literature on market microstructure Conclusion 13 2 Organization of markets Introduction Stock markets A brief history of stock exchanges From the 19th Century to the 1990s The influence of technology A new competitive landscape: MiFID and Reg NMS Description of the trading environment Introduction Limit order books Dark pools and hidden Orders High-frequency trading 29 ix

3 X 2.3 Bond markets Introduction Bond markets and liquidity Electronification of bond trading Corporate bonds Government bonds Conclusion 35 II Optimal Liquidation 37 3 The Almgren-Chriss framework Introduction A generalized Almgren-Chriss model in continuous time Notations The optimization problem The case of deterministic strategies A unique optimal strategy Characterization of the optimal strategy The case of quadratic execution costs General results Stochastic strategies vs. deterministic strategies Choosing a risk profile The model in discrete time Notations The optimization problem Optimal trading curve Hamiltonian characterization The initial Almgren-Chriss framework Conclusion 63 4 Optimal liquidation with different benchmarks Introduction: the different types of Orders Target Close Orders Target Close orders in the Almgren-Chriss framework Target Close orders as reversed IS orders Concluding remarks on Target Close orders POV orders Presentation of the problem Optimal participation rate A way to estimate risk aversion VWAP orders VWAP orders in the Almgren-Chriss framework... 79

4 xi The model Examples and analysis Other models for VWAP orders Conclusion 89 5 Extensions of the Almgren-Chriss Framework A more complex price dynamics The model Extension of the Hamiltonian system Adding participation constraints The model Towards a new Hamiltonian system What about a minimal participation rate? Portfolio liquidation The model Towards a Hamiltonian system of 2d equations How to hedge the risk of the execution process Conclusion Numerical methods The case of single-stock portfolios A shooting method Examples Final remarks on the Single-asset case The case of multi-asset portfolios Newton's method for smooth Hamiltonian functions Convex duality to the rescue Examples Conclusion Beyond Almgren-Chriss Overview of the literature Models with market orders and the Almgren-Chriss market impact model Models with transient market impact Limit orders and dark pools Optimal execution models in practice The two-layer approach: strategy vs. tactics Child order placement Static models for optimal child order placement 128

5 xii Dynamic models for optimal child order placement Final remarks on child order placement Conclusion 137 Appendix to Chapter 7: Market impact estimation 138 III Liquidity in Pricing Models Block trade pricing Introduction General definition of block trade prices and risk-liquidity premia A first definition A time-independent definition The specific case of single-stock portfolios The value function and its asymptotic behavior Closed-form formula for block trade prices Examples and discussion A straightforward extension A simpler case with POV liquidation Guaranteed VWAP contracts Conclusion Option pricing and hedging with execution costs and market impact Introduction Nonlinearity in option pricing Liquidity sometimes matters The model in continuous time Setup of the model Towards a new nonlinear PDE for pricing The Hamilton-Jacobi-Bellman equation The pricing PDE Comments on the model and the pricing PDE The model in discrete time Setup of the model A new recursive pricing equation Numerical examples A trinomial tree Hedging a call option with physical delivery Conclusion 193

6 xiii 10 Share buy-back Introduction Accelerated Share Repurchase contracts Nature of the problem The model Setup of the model Towards a recursive characterization of the optimal strategy Optimal management of an ASR contract Characterization of the optimal trading strategy and the optimal exercise time Analysis of the optimal behavior Numerical methods and examples A pentanomial-tree approach Numerical examples Conclusion 214 IV Market Making Market making models: from Avellaneda-Stoikov to Gueant- Lehalle, and beyond Introduction The Avellaneda-Stoikov model Framework The Hamilton-Jacobi-Bellman equation and its Solution The Gueant-Lehalle-Fernandez-Tapia formulas Generalization of the Avellaneda-Stoikov model Introduction A general multi-asset market making model Framework Computing the optimal quotes Market making on stock markets Conclusion 241 Mathematical Appendices 243 A Mathematical economics 245 A.l The expected Utility theory 245 A.2 Utility functions and risk aversion 246 A.3 Certainty equivalent and indifference pricing 247

7 xiv B Convex analysis and variational calculus 251 B.l Basic notions of convex analysis 251 B.l.l Definitions and classical properties 251 B.l.2 Subdifferentiability 252 B.l.3 The Legendre-Fenchel transform 253 B.l.4 Generalized convex functions 256 B.2 Calculus of variations 257 B.2.1 Bolza problems in continuous time 257 B.2.2 What about discrete-time problems? 259 Bibliography 265 Index 277