Quantitative Portfolio Optimization

Contents   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 Contents

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      2.4.1 Risk Parity  19      2.4.2 Hierarchical Risk Parity  26      2.4.3 Machine Learning  27 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       3.1.1 Case Without Riskless Asset  35       3.1.2 Case With a Riskless Asset  41 3.2 Capital Asset Pricing Model  48       3.2.1 Case Without Riskless Asset  48       3.2.2 Case With a Riskless Asset  52 3.3 Multifactor Models  54 3.4 Challenges of Modern Portfolio Theory  59       3.4.1 Estimation Techniques in Portfolio Allocation  60       3.4.2 Non-Elliptical Distributions and Conditional Value-at-Risk (CVaR)  63 3.5 Quantum Annealing in Portfolio Management  65 3.6 Mean-Variance Optimization with CVaR Constraint  67       3.6.1 Problem Formulation  67       3.6.2 Optimization Problem  68       3.6.3 Clarification of Optimization Classes  68       3.6.4 Numerical Example  69 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       4.6.1 Gaussian Processes in a Nutshell  90      4.6.2 Uncertainty Quantification and Bayesian Decision Theory  94 4.7 Applications to Portfolio Optimization  96      4.7.1 GP Regression for Asset Returns  96      4.7.2 Decision Theory in Portfolio Optimization  96      4.7.3 The Black-Litterman Model  99 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       5.2.1 VaR  110      5.2.2 CVaR  112 5.3 Estimation Methods  116      5.3.1 Variance-Covariance Method  116      5.3.2 Historical Simulation  116      5.3.3 Monte Carlo Simulation  117 5.4 Advanced Risk Measures: Tail Risk and Spectral Measures  118      5.4.1 Tail Risk Measures  118      5.4.2 Spectral Measures  120 5.5 Notes 123
CHAPTER 6   Factor Models and Factor Investing  125   6.1 Single and Multifactor Models  126       6.1.1 Statistical Models  127       6.1.2 Macroeconomic Models  128       6.1.3 Cross-sectional Models  130 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       7.2.1 Single Asset  151       7.2.2 Multiple Assets  154 7.3 Optimal Trading Strategies  155       7.3.1 Mei, DeMiguel, and Nogales (2016)  156       7.3.2 Skaf and Boyd (2009)  159 7.4 Liquidity Considerations in Portfolio Optimization  161       7.4.1 MV and Liquidity  162       7.4.2 CAPM and Liquidity  163       7.4.3 APT and Liquidity  165 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       8.6.1 Classical Merton Problem  180       8.6.2 Multi-asset Portfolio with Transaction Costs  181       8.6.3 Risk-sensitive Portfolio Optimization  183       8.6.4 Optimal Portfolio Allocation with Transaction Costs  184       8.6.5 American Option Pricing  184       8.6.6 Portfolio Optimization with Constraints  184       8.6.7 Mean-variance Portfolio Optimization  185       8.6.8 Schodinger Control in Wealth Management  185 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       9.7.1 An MDP in Multi-asset Allocation with Transaction Costs  205       9.7.2 A POMDP for Asset Allocation with Regime Switching  205       9.7.3 An MDP with Continuous State and Action Spaces for Option Hedging with Stochastic Volatility  206 9.8 Notes  207   CHAPTER  10   Reinforcement Learning  209   10.1 Connections to Optimal Control  211        10.1.1 Policy Iteration  212        10.1.2 Value Iteration  214        10.1.3 Continuous vs. Discrete Formulations  215 10.2 The Environment and The Reward Function  217          10.2.1 The Environment  217          10.2.2 The Reward Function  220 10.3 Agents Acting in an Environment  223 10.4 State-Action and Value Functions  225         10.4.1 Value Functions  226         10.4.2 Gradients and Policy Improvement  227 10.5 The Policy  230 10.6 On-Policy Methods  233 10.7 Off-Policy Methods  235 10.8 Applications to Portfolio Optimization  238        10.8.1 Mean-variance Optimization  238        10.8.2 Reinforcement Learning Comparison with Mean-variance Optimization  239        10.8.3 G-Learning and GIRL  241        10.8.4 Continuous-time Penalization in Portfolio Optimization  244        10.8.5 Reinforcement Learning for Utility Maximization  246        10.8.6 Continuous-time Portfolio Optimization with Transaction Costs  246 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         11.5.1 Dynamic Asset Allocation Using the Heston Model  276         11.5.2 Option-Based Portfolio Insurance Using the Bates Model  277         11.5.3 Factor Learning Approach to Generative Modeling of Equities  278 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        12.1.1 Literature Review  285 12.2 Graph Theory Portfolios: MST and TMFG  285        12.2.1 Equations and Formulas  286        12.2.2 Results  287 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         13.3.1 Optimal Network Selection  304         13.3.2 Sensitivity Analysis  304         13.3.3 Sensitivity Distance Matrix  304 13.4 Hierarchical Sensitivity Parity  307 13.5 Implementation  307         13.5.1 Datasets  307         13.5.2 Experimental Setup  308         13.5.3 Short-to-medium Investments  309         13.5.4 Long-term Investments  312 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         14.4.1 Walk-forward Backtest  321         14.4.2 Resampling Method  321         14.4.3 Monte Carlo Simulations and Generative Models  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