The Financial Mathematics of Market Liquidity: From Optimal Execution to Market Making

Olivier Gueant



This book is among the first to present the mathematical models most commonly used to solve optimal execution problems and market making problems in finance. The Financial Mathematics of Market Liquidity: From Optimal Execution to Market Making presents a general modeling framework for optimal execution problems–inspired from the Almgren-Chriss approach–and then demonstrates the use of that framework across a wide range of areas.

The book introduces the classical tools of optimal execution and market making, along with their practical use. It also demonstrates how the tools used in the optimal execution literature can be used to solve classical and new issues where accounting for liquidity is important. In particular, it presents cutting-edge research on the pricing of block trades, the pricing and hedging of options when liquidity matters, and the management of complex share buy-back contracts.

What sets this book apart from others is that it focuses on specific topics that are rarely, or only briefly, tackled in books dealing with market microstructure. It goes far beyond existing books in terms of mathematical modeling–bridging the gap between optimal execution and other fields of Quantitative Finance.

The book includes two appendices dedicated to the mathematical notions used throughout the book. Appendix A recalls classical concepts of mathematical economics. Appendix B recalls classical tools of convex analysis and optimization, along with central ideas and results of the calculus of variations.

This self-contained book is accessible to anyone with a minimal background in mathematical analysis, dynamic optimization, and stochastic calculus. Covering post-electronification financial markets and liquidity issues for pricing, this book is an ideal resource to help investment banks and asset managers optimize trading strategies and improve overall risk management.

Table of Contents

I: INTRODUCTION

General Introduction
A Brief History of Quantitative Finance
Optimal Execution and Market Making in the Extended Market Microstructure Literature
Conclusion

Organization of Markets
Introduction
Stock Markets
Bond Markets
Conclusion

II: OPTIMAL LIQUIDATION

The Almgren-Chriss Framework
Introduction
A Generalized Almgren-Chriss Model in Continuous Time
The Model in Discrete Time
Conclusion

Optimal Liquidation with Different Benchmarks
Introduction: the Different Types of Orders
Target Close Orders
POV Orders
VWAP Orders
Conclusion

Extensions of the Almgren-Chriss Framework
A More Complex Price Dynamics
Adding Participation Constraints
Portfolio Liquidation
Conclusion

Numerical Methods
The Case of Single-Stock Portfolios
The Case of Multi-Asset Portfolios
Conclusion

Beyond Almgren-Chriss
Overview of the Literature
Optimal Execution Models in Practice
Conclusion

III: LIQUIDITY IN PRICING MODELS

Block Trade Pricing
Introduction
General Definition of Block Trade Prices and Risk-Liquidity Premium
The Specific Case of Single-Stock Portfolios
A Simpler Case with POV Liquidation
Guaranteed VWAP Contracts
Conclusion

Option Pricing and Hedging with Execution Costs and Market Impact
Introduction
The Model in Continuous Time
The Model in Discrete Time
Numerical Examples
Conclusion

Share Buy-Back
Introduction
The Model
Optimal Management of an ASR Contract
Numerical Methods and Examples
Conclusion

IV: MARKET MAKING

Market Making: From Avellaneda-Stoikov to Guéant-Lehalle, and Beyond
Introduction
The Avellaneda-Stoikov Model
Generalization of the Avellaneda-Stoikov Model
Market Making on Stock Markets
Conclusion

Appendices:

Mathematical Economics
The Expected Utility Theory
Utility Functions and Risk Aversion
Certainty Equivalent and Indifference Pricing

Convex Analysis and Variational Calculus
Basic Notions of Convex Analysis
Calculus of Variation