Preface xiii
Acknowledgements xv
About the Authors xvii

CHAPTER 1 Introduction 1
1.1 Evolution of Portfolio Optimization 1
1.2 Role of Quantitative Techniques 1
1.3 Organization of the Book 4

CHAPTER 2 History of Portfolio Optimization 7
2.1 Early beginnings 7
2.2 Harry Markowitz's Modern Portfolio Theory (1952) 9
2.3 Black-Litterman Model (1990s) 13
2.4 Alternative Methods: Risk Parity, Hierarchical Risk Parity and Machine Learning 19
2.5 Notes 31

PART ONE Foundations of Portfolio Theory

CHAPTER 3 Modern Portfolio Theory 35
3.1 Efficient Frontier and Capital Market Line 35
3.2 Capital Asset Pricing Model 48
3.3 Multifactor Models 54
3.4 Challenges of Modern Portfolio Theory 59
3.5 Quantum Annealing in Portfolio Management 65
3.6 Mean-Variance Optimization with CVaR Constraint 67
3.7 Notes 70

CHAPTER 4 Bayesian Methods in Portfolio Optimization 73
4.1 The Prior 75
4.2 The Likelihood 79
4.3 The Posterior 80
4.4 Filtering 83
4.5 Hierarchical Bayesian Models 87
4.6 Bayesian Optimization 89
4.7 Applications to Portfolio Optimization 96
4.8 Notes 103

PART TWO Risk Management

CHAPTER 5 Risk Models and Measures 107
5.1 Risk Measures 107
5.2 VaR and CVaR 109
5.3 Estimation Methods 116
5.4 Advanced Risk Measures: Tail Risk and Spectral Measures 118
5.5 Notes 123

CHAPTER 6 Factor Models and Factor Investing 125
6.1 Single and Multifactor Models 126
6.2 Factor Risk and Performance Attribution 135
6.3 Machine Learning in Factor Investing 141
6.4 Notes 144

CHAPTER 7 Market Impact, Transaction Costs, and Liquidity 145
7.1 Market Impact Models 145
7.2 Modeling Transaction Costs 148
7.3 Optimal Trading Strategies 155
7.4 Liquidity Considerations in Portfolio Optimization 161
7.5 Notes 167

PART THREE Dynamic Models and Control

CHAPTER 8 Optimal Control 171
8.1 Dynamic Programming 171
8.2 Approximate Dynamic Programming 171
8.3 The Hamilton-Jacobi-Bellman Equation 172
8.4 Sufficiently Smooth Problems 174
8.5 Viscosity Solutions 176
8.6 Applications to Portfolio Optimization 180
8.7 Notes 187

CHAPTER 9 Markov Decision Processes 189
9.1 Fully Observed MDPs 191
9.2 Partially Observed MDPs 192
9.3 Infinite Horizon Problems 194
9.4 Finite Horizon Problems 198
9.5 The Bellman Equation 200
9.6 Solving the Bellman Equation 203
9.7 Examples in Portfolio Optimization 205
9.8 Notes 207

CHAPTER 10 Reinforcement Learning 209
10.1 Connections to Optimal Control 211
10.2 The Environment and The Reward Function 217
10.3 Agents Acting in an Environment 223
10.4 State-Action and Value Functions 225
10.5 The Policy 230
10.6 On-Policy Methods 233
10.7 Off-Policy Methods 235
10.8 Applications to Portfolio Optimization 238
10.9 Notes 247

PART FOUR Machine Learning and Deep Learning

CHAPTER 11 Deep Learning in Portfolio Management 253
11.1 Neurons and Activation Functions 253
11.2 Neural Networks and Function Approximation 256
11.3 Review of Some Important Architectures 259
11.4 Physics-Informed Neural Networks 269
11.5 Applications to Portfolio Optimization 276
11.6 The Case for and Against Deep Learning 280
11.7 Notes 282

CHAPTER 12 Graph-based Portfolios 285
12.1 Graph Theory-Based Portfolios 285
12.2 Graph Theory Portfolios: MST and TMFG 285
12.3 Hierarchical Risk Parity 289
12.4 Notes 294

CHAPTER 13 Sensitivity-based Portfolios 295
13.1 Modeling Portfolios Dynamics with PDEs 296
13.2 Optimal Drivers Selection: Causality and Persistence 297
13.3 AAD Sensitivities Approximation 303
13.4 Hierarchical Sensitivity Parity 307
13.5 Implementation 307
13.6 Conclusion 315

PART FIVE Backtesting

CHAPTER 14 Backtesting in Portfolio Management 319
14.1 Introduction 319
14.2 Data Preparation and Handling 319
14.3 Implementation of Trading Strategies 320
14.4 Types of Backtests 321
14.5 Performance Metrics 322
14.6 Avoiding Common Pitfalls 323
14.7 Advanced Techniques 323
14.8 Case Study: Applying Backtesting to a Real-World Strategy 324
14.9 Impact of Market Conditions on Backtest Results 324
14.10 Integration with Portfolio Management 325
14.11 Tools and Software for Backtesting 325
14.12 Regulatory Considerations 326
14.13 Conclusion 326

CHAPTER 15 Scenario Generation 329
15.1 Historical Scenarios 329
15.2 Bootstrapping Scenarios 330
15.3 Copula-Based Scenarios 330
15.4 Risk Factor Model-Based Scenarios 330
15.5 Time Series Model Scenarios 331
15.6 Variational Autoencoders 331
15.7 Generative Adversarial Networks (GANs) 332

Appendix 333
A.1 Software and Tools for Portfolio Optimization 333

Bibliography 335
Index 357