Expert guidance on implementing quantitative portfolio optimization techniques
In Quantitative Portfolio Optimization: Theory and Practice, renowned financial practitioner Miquel Noguer, alongside physicists Alberto Bueno Guerrero and Julian Antolin Camarena, who possess excellent knowledge in finance, delve into advanced mathematical techniques for portfolio optimization. The book covers a range of topics including mean-variance optimization, the Black-Litterman Model, risk parity and hierarchical risk parity, factor investing, methods based on moments, and robust optimization as well as machine learning and reinforcement technique. These techniques enable readers to develop a systematic, objective, and repeatable approach to investment decision-making, particularly in complex financial markets.
Readers will gain insights into the associated mathematical models, statistical analyses, and computational algorithms for each method, allowing them to put these techniques into practice and identify the best possible mix of assets to maximize returns while minimizing risk. Topics explored in this book include:
Specific drivers of return across asset classes Personal risk tolerance and it#s impact on ideal asses allocation The importance of weekly and monthly variance in the returns of
specific securities
Serving as a blueprint for solving
portfolio optimization problems, Quantitative Portfolio Optimization: Theory and Practice is an essential resource for finance practitioners and individual investors It helps them stay on the cutting edge of modern portfolio theory and achieve the best returns on investments for themselves, their clients, and their organizations.
By:
Miquel Noguer Alonso,
Julian Antolin Camarena,
Alberto Bueno Guerrero
Imprint: John Wiley & Sons Inc
Country of Publication: United States
Dimensions:
Height: 234mm,
Width: 158mm,
Spine: 28mm
Weight: 703g
ISBN: 9781394281312
ISBN 10: 1394281315
Series: Wiley Finance
Pages: 384
Publication Date: 21 February 2025
Audience:
Professional and scholarly
,
Undergraduate
Format: Hardback
Publisher's Status: Active
Contents Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 1 1 Introduction 3 1.1 Evolution of Portfolio Optimization . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 3 1.2 Role of Quantitative Techniques . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 3 1.3 Organization of the Book . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .7 2 History of Portfolio Optimization 9 2.1 Early beginnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Harry Markowitz’s Modern Portfolio Theory (1952) . . . . . . . . . . . . . . 12 2.3 Black-Litterman Model (1990s) . . . . . . . . . . . . . . . ……………………16 2.4 Alternative Methods: Risk Parity, Hierarchical Risk Parity and Machine Learning . . . . . . . . . . . . . . . . . . . … .. . . .. . .. . .. ………. . 21 2.4.1 Risk Parity . . . . . . . . . . . . . . . . . . . . . . . . .. . .. . . .. . . . . . . …...21 2.4.2 Hierarchical Risk Parity . . . . . . . . . . . . . . . . . …………………28 2.4.3 Machine Learning . . . . . . . . . . . . . . . . . . . . . ………………. ...30 2.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . …………………. . . . . . 35 I Foundations of Portfolio Theory 37 3 Modern Portfolio Theory 38 3.1 Efficient Frontier and Capital Market Line . . . . . . . . . . . …………….. 38 3.1.1 Case without riskless asset . . . . . . . . . . . . . . . . . . . .. . . . . . . 39 3.1.2 Case with a riskless asset . . . . . . . . . . . . . . . .. . . . . . . . . . . . 44 3.2 Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 50 3.2.1 Case without riskless asset . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.2 Case with a riskless asset . . . . . . . . . . . . . . . . .. . . . . . . . . . . .54 3.3 Multi-factor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.4 Challenges of Modern Portfolio Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.4.1 Estimation Techniques in Portfolio Allocation . . . . . .. . . . . . .62 3.4.2 Non-Elliptical Distributions and Conditional Value-at- Risk (CVaR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66 3.5 Quantum Annealing in Portfolio Management . . . . . . . . . . . . . . . . . . . . . 68 3.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70 CONTENTS 4 Bayesian Methods in Portfolio Optimization 73 4.1 The Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 The Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80 4.3 The Posterior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.4 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.5 Hierarchical Bayesian Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90 4.6 Bayesian Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.6.1 Gaussian Processes in a Nutshell . . . . . . . . . . . . . . . . . . . . . . . . . .93 4.6.2 Uncertainty Quantification and Bayesian Decision Theory . . . . . 97 4.7 Applications to Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.7.1 GP Regression for Asset Returns . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.7.2 Decision Theory in Portfolio Optimization . . . . . . . . . . . . . . . . . . 100 4.7.3 The Black-Litterman Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103 4.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 II Risk Management 109 5 Risk Models and Measures 110 5.1 Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . 111 5.2 VaR and CVaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 113 5.2.1 VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. . .114 5.2.2 CVaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 116 5.3 Estimation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119 5.3.1 Variance-Covariance Method . . . . . . . . . . . . . . . . . . . . . . . . . .. . .120 5.3.2 Historical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120 5.3.3 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121 5.4 Advanced Risk Measures: Tail Risk and Spectral Measures . . . . . . . . . . . . . .121 5.4.1 Tail Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.4.2 Spectral Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6 Factor Models and Factor Investing 128 6.1 Single and Multi-Factor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.1.1 Statistical Models . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 130 6.1.2 Macroeconomic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131 6.1.3 Cross Sectional Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133 6.2 Factor Risk and Performance Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.3 Machine Learning in Factor Investing . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 145 6.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7 Market Impact, Transaction Costs and Liquidity 149 7.1 Market Impact Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ….150 7.2 Modeling Transaction Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . …153 7.2.1 Single asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . … 156 CONTENTS 7.2.2 Multiple assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . …..158 7.3 Optimal Trading Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . …...160 7.3.1 Mei, DeMiguel and Nogales (2016) . . . . . . . . . . . . . .. . . . . … .. 161 7.3.2 Skaf and Boyd (2009) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . …..164 7.4 Liquidity Considerations in Portfolio Optimization . . . . . . . . . . . . . . . …...166 7.4.1 MV and Liquidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7.4.2 CAPM and Liquidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 168 7.4.3 APT and Liquidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 170 7.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 172 III Dynamic Models and Control 174 8 Optimal Control 175 8.1 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .175 8.2 Approximate Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 8.3 The Hamilton-Jacobi-Bellman Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 8.4 Sufficiently Smooth Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .179 8.5 Viscosity Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .181 8.6 Applications to Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 8.6.1 Classical Merton Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .185 8.6.2 Multi-Asset Portfolio with Transaction Costs . . . . . . . . . . . . . . . 186 8.6.3 Risk-Sensitive Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . 188 8.6.4 Optimal Portfolio Allocation with Transaction Costs . . . . . . . . . 189 8.6.5 American Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .189 8.6.6 Portfolio Optimization with Constraints . . . . . . . . . . . . . . . . . . . 190 8.6.7 Mean-Variance Portfolio Optimization . . . . . . . . . . . . . . . . . . . .190 8.6.8 Sch¨odinger Control in Wealth Management . . . . . . . . . . . . . . . 191 8.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .193 9 Markov Decision Processes 195 9.1 Fully Observed MDPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9.2 Partially Observed MDPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 199 9.3 Infinite Horizon Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .202 9.4 Finite Horizon Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .206 9.5 The Bellman Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 9.6 Solving the Bellman Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .212 9.7 Examples in Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 9.7.1 An MDP in Multi-Asset Allocation with Transaction Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 9.7.2 A POMDP for Asset Allocation with Regime Switching . . . . . 214 9.7.3 An MDP with Continuous State and Action Spaces for Option Hedging with Stochastic Volatility . . . . . . . . . . . . . . . 215 9.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 CONTENTS 10 Reinforcement Learning 219 10.1 Connections to Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 10.1.1 Policy Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 10.1.2 Value Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 10.1.3 Continuous vs. Discrete Formulations . . . . . . . . . . . . . . . . . . . . .226 10.2 The Environment and The Reward Function . . . . . . . . . . . . . . . . . . . . . . 228 10.2.1 The Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 10.2.2 The Reward Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .232 10.3 Agents Acting in an Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 10.4 State-Action and Value Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .238 10.4.1 Value Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .238 10.4.2 Gradients and Policy Improvement . . . . . . . . . . . . . . . . . . . . .240 10.5 The Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . . . . . . . . . 243 10.6 On-Policy Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 10.7 Off-Policy Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 10.8 Applications to Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 253 10.8.1 Mean-Variance Optimization . . . . . . . . . . . . . . . . . . . . . . . . 253 10.8.2 Reinforcement Learning Comparison with Mean-Variance Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .254 10.8.3 G-Learning and GIRL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 10.8.4 Continuous-time Penalization in Portfolio Optimization . . .259 10.8.5 Reinforcement Learning for Utility Maximization . . . . . . . .260 10.8.6 Continuous-Time Portfolio Optimization with Transaction Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261 10.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 262 IV Machine Learning and Deep Learning 265 11 Deep Learning in Portfolio Management 266 11.1 Neurons and Activation Functions . . . . . . . . . . . . . . . .. . . . . . . . . . . . 266 11.2 Neural Networks and Function Approximation . . . . . . . . . . . . . . . . . . 269 11.3 Review of Some Important Architectures . . . . . . . . . . . . . . .. . . . . . . . 273 11.4 Physics-Informed Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 11.5 Applications to Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . .292 11.5.1 Dynamic Asset Allocation Using the Heston Model . . . . . . 292 11.5.2 Option-Based Portfolio Insurance Using the Bates Model . .293 11.5.3 Factor Learning Approach to Generative Modeling of Equities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 11.6 The Case for and Against Deep Learning . . . . . . . . . . . . . . . . . . . . . . 296 11.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 298 12 Graph Based Portfolios 300 12.1 Graph Theory Based Portfolios . . . . . . . . . . . . . . . . . 300 12.1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .300 12.1.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 CONTENTS 12.2 Equations and Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 12.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .302 12.3 Hierarchical Risk Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 12.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .309 13 Sensitivity-based Portfolios 310 13.1 Modelling Portfolios Dynamics with PDEs . . . . . . . . . . . . . . . . . . . . . . 312 13.2 Optimal Drivers Selection: Causality and Persistence . . . . . . . . . . . . . . 313 13.3 AAD Sensitivities Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .319 13.3.1 Optimal Network Selection . . . . . . . . . . . . . . . . . . . . . . . 319 13.3.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .320 13.3.3 Sensitivity Distance Matrix . . . . . . . . . . . . . . . . . . . . . . . .320 13.4 Hierarchical Sensitivity Parity . . . . . . . . . . . . . . . . . . . . . . .322 13.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 13.5.1 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 13.5.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 13.5.3 Short-to-medium investments . . . . . . . . . . . . . . . . . . . . . 324 13.5.4 Long-term investments . . . . . . . . . . . . . . . . . . . . . . . . . . 328 13.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 V Backtesting 333 14 Backtesting in Portfolio Management 334 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . …………….. .. . . . . ..334 14.2 Data Preparation and Handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 14.3 Implementation of Trading Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 335 14.4 Types of Backtests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 14.4.1 Walk-Forward Backtest . . . . . . . . . . . . . . . . . . . . . . . . 336 14.4.2 Resampling Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 14.4.3 Monte Carlo Simulations and Generative Models . . . . 337 14.5 Performance Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .337 14.6 Avoiding Common Pitfalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 14.7 Advanced Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 14.8 Case Study: Applying Backtesting to a Real-World Strategy . . . . . . . 339 14.9 Impact of Market Conditions on Backtest Results . . . . . . . . . . . . . . . .340 14.10 Integration with Portfolio Management . . . . . . . . . . . . . . . . . . . . . .. . 340 14.11 Tools and Software for Backtesting . . . . . . . . . . . . . . . . . . . . . . .. . . 341 14.12 Regulatory Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 14.13Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 15 Scenario Generation 344 15.1 Historical Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 15.2 Bootstrapping Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 15.3 Copula Based Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 CONTENTS 15.4 Risk Factor Model Based Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . .345 15.5 Time Series Model Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .346 15.6 Variational Autoencoders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 15.7 Generative Adversarial Networks (GANs) . . . . . . . . . . . . . . . . . .. . . .347 Appendices 348 15.8 Software and Tools for Portfolio Optimization . . . . . . . . . . . . . . . . . 348
MIQUEL NOGUER ALONSO is a financial markets practitioner with 25+ years of experience in asset management. He is the Founder of the Artificial Intelligence Finance Institute and serves as Head of Development at Global AI. He is also the co-editor of the Journal of Machine Learning in Finance. JULIÁN ANTOLÍN CAMARENA holds a Bachelor’s, Master’s and a PhD in physics. For his Master’s he worked on the foundations of quantum mechanics examining alternative quantization schemes and their application to exotic atoms to discover new physics. His PhD dissertation work was on computational and theoretical optics, electromagnetic scattering from random surfaces, and nonlinear optimization. He then went on to a postdoctoral stint with the U.S. Army Research Laboratory working on inverse reinforcement learning for human-autonomy teaming. ALBERTO BUENO GUERRERO has two Bachelor’s degrees in physics and economics, and a PhD in banking and finance. Since he got his doctorate, he has dedicated himself to research in mathematical finance. His work has been presented at various international conferences and published in journals such as Quantitative Finance, Journal of Derivatives, Journal of Mathematics, and Chaos, Solitons and Fractals. His article “Bond Market Completeness Under Stochastic Strings with Distribution-Valued Strategies” has been considered a feature article in Quantitative Finance.
