Miquel Noguer Alonso, Julian Antolin Camarena, Alberto Bueno Guerrero

Quantitative Portfolio Optimization

Advanced Techniques and Application

• Gebundenes Buch

Produktbeschreibung

PRAISE FOR QUANTITATIVE PORTFOLIO OPTIMIZATIONOPTIMIZATION "This book provides an excellent exposition on portfolio optimization, serving not only as a self-contained guide to this important topic, but also modernizing the field with the latest advances in battle-tested machine learning approaches. The book is well structured and application centric. This is a must read for every quantitative portfolio manager." - Matthew Dixon, FRM, Ph.D., Associate Professor of Applied Math at the Illinois Institute of Technology and an Affiliate Associate Professor of the Stuart School of Business "Quantitative Portfolio Optimization: Advanced Techniques and Applications is an essential guide for anyone seeking to navigate the complex world of modern portfolio management. This book masterfully blends the foundational principles of portfolio theory with cutting-edge advancements in risk management, dynamic models, and control systems. Its integration of machine learning and deep learning offers readers a forward-looking perspective on leveraging AI-driven techniques for optimization. What truly sets this book apart is its comprehensive approach. From theoretical insights to practical backtesting applications, it equips professionals, researchers, and students with the tools to design and refine robust investment strategies. Whether you're delving into the nuances of risk modelling or exploring dynamic portfolio control with the latest AI methodologies, this text is an invaluable resource. This book isn't just about managing portfolios-it's about mastering the art and science behind it. Highly recommended for anyone aiming to achieve excellence in quantitative finance and portfolio optimization." -Daniel Bloch, Director, Quant Finance Limited

---

Hinweis: Dieser Artikel kann nur an eine deutsche Lieferadresse ausgeliefert werden.

Produktdetails

• Verlag: Wiley
• Seitenzahl: 384
• Erscheinungstermin: 29. Januar 2025
• Englisch
• ISBN-13: 9781394281312
• ISBN-10: 1394281315
• Artikelnr.: 70698647

Inhaltsangabe

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