Recognizing that robust decision making is vital in risk management, this book provides concepts and algorithms for computing the best decision in view of the worst-case scenario. The main tool used is minimax, which ensures robust policies with guaranteed optimal performance that will improve further if the worst case is not realized. The applications considered are drawn from finance, but the design and algorithms presented are equally applicable to problems of economic policy, engineering design, and other areas of decision making. Critically, worst-case design addresses not only Armageddon-type uncertainty. Indeed, the determination of the worst case becomes nontrivial when faced with numerous--possibly infinite--and reasonably likely rival scenarios. Optimality does not depend on any single scenario but on all the scenarios under consideration. Worst-case optimal decisions provide guaranteed optimal performance for systems operating within the specified scenario range indicating the uncertainty. The noninferiority of minimax solutions--which also offer the possibility of multiple maxima--ensures this optimality. Worst-case design is not intended to necessarily replace expected value optimization when the underlying uncertainty is stochastic. However, wise decision making requires the justification of policies based on expected value optimization in view of the worst-case scenario. Conversely, the cost of the assured performance provided by robust worst-case decision making needs to be evaluated relative to optimal expected values. Written for postgraduate students and researchers engaged in optimization, engineering design, economics, and finance, this book will also be invaluable to practitioners in risk management.

Preface xiiiChapter 1. Introduction to minimax 1
1. Background and Notation 1
1.1. Linear Independence 5
1.2. Tangent Cone, Normal Cone and Epigraph 7
1.3. Subgradiemts and Subdifferentials of Convex Functions 7
2. Continuous Minimax 10
3. Optimality Conditions and Robustness of Minimax 11
3.1. The Haar Condition 13
4. Saddle Points and Saddle Point Conditions 15
References 17
Comments and Notes 18
Chapter 2. A survey of continuous minimax algorithms 23
1. Introduction 23
2. The Algorithm of Chaney 25
3. The Algorithm of Panin 30
4. The Algorithm of Kiwiel 31
References 33
Comments and Notes 34
Chapter 3. Algorithms for computing saddle points 37
1. Computation of Saddle Points 37
1.1. Saddle Point Equilibria 37
1.2. Solution of Systems of Equations 40
2. The Algorithms 42
2.1. A Gradient-based Algorithm for Unconstrained Saddle Points 42
2.2. Quadratic Approximation Algorithm for Constrained Minimax Saddle Points 44
2.3. Interior Point Saddle Point Algorithm for Constrained Problems 45
2.4. Quasi-Newton Algorithm for Nonlinear Systems 49
3. Global Convergence of Newton-type Algorithms 50
4. Achievement of Unit Stepsizes and Superlinear Convergence 54
5. Concluding Remarks 58
References 58
Comments and Notes 59
Chapter 4. A quasi-Newton algorithm for continuous minimax 63
1. Introduction 63
2. Basic Concepts and Definitions 66
3. The quasi-Newton Algorithm 70
4. Basic Convergence Results 76
5. Global Convergence and Local Convergence Rates 81
References 86
Appendix A: Implementation Issues 87
Appendix B: Motivation for the Search Direction 90
Comments and Notes 91
Chapter 5. Numerical experiments with continuous minimax algorithms 93
1. Introduction 93
2. The Algorithms 94
2.1. Kiwiel's Algorithm 94
2.2. Quasi-Newton Methods 95
3. Implementation 96
3.1. Terminology 96
3.2. The Stopping Criterion 97
3.3. Evaluation of the Direction of Descent 97
4. Test Problems 98
5. Summary of the Results 110
5.1. Iterations when is Satisfied 110
5.2. Calculation of Minimum-norm Subgradient 111
5.3. Superlinear Convergence 111
5.4. Termination Criterion and Accuracy of the Solution 112
References 119
Chapter 6. Minimax as a robust strategy for discrete rival scenarios 121
1. Introduction to Rival Models and Forecast Scenarios 121
2. The Discrete Minimax Problem 123
3. The Robust Character of the Discrete Minimax Strategy 125
3.1. Naive Minimax 125
3.2. Robustness of the Minimax Strategy 126
3.3. An Example 128
4. Augmented Lagrangians and Convexification of Discrete Minimax 132
References 137
Chapter 7. Discrete minimax algorithm for nonlinear equality and inequality constrained models 139
1. Introduction 139
2. Basic Concepts 141
3. The Discrete Minimax Algorithm 142
3.1. Inequality Constraints 142
3.2. Quadratic Programming Subproblem 143
3.3. Stepsize Strategy 144
3.4. The Algorithm 145
3.5. Basic Properties 147
4. Convergence of the Algorithm 152
5. Achievement of Unit Stepsizes 156
6. Superlinear Convergence Rates of the Algorithm 162
7. The Algorithm for Only Linear Constraints 172
References 176
Chapter 8. A continuous minimax strategy for options hedging 179
1. Introduction 179
2. Options and the Hedging Problem 181
3. The Black and Scholes Option Pricing Model and Delta Hedging 183
4. Minimax Hedging Strategy 187
4.1. Minimax Problem Formulation 187
4.2. The Worst-case Scenario 188
4.3. The Hedging Error 189
4.4. The Objective Function 190
4.5. The Minimax Hedging Error 192
4.6. Transaction Costs 193
4.7. The Variants of the Minimax Hedging Strategy 194
4.8. The Minimax Solution 194
5. Simulation 196
5.1. Generation of Simulation Data 196
5.2. Setting Up and Winding Down the Hedge 198
5.3. Summary of Simulation Results 198
6. Illustrative Hedging Problem: A Limited Empirical Study 204
6.1. From Set-up to Wind-down 204
6.2. The Hedging Strategies Applied to 30 Options: Summary of Results 205
7. Multiperiod Minimax Hedging Strategies 207
7.1. Two-period Minimax Strategy 207
7.2. Variable Minimax Strategy 211
8. Simulation Study of the Performance of Different Multiperiod Strategies 213
8.1. The Simulation Structure 213
8.2. Results of the Simulation Study 214
8.3. Rank Ordering 214
9. CAPM-based Minimax Hedging Strategy 215
9.1. The Capital Asset Pricing Model 217
9.2. The CAPM-based Minimax Problem Formulation 218
9.3. The Objective Function 219
9.4. The Worst-case Scenario 221
10. Simulation Study of the Performance of CAPM Minimax 222
10.1 Generation of Simulation Data 222
10.2 Summary of Simulation Results 223
10.3 Rank Ordering 224
11. The Beta of the Hedge Portfolio for CAPM Minimax 226
12. Hedging Bond Options 226
12.1 European Bond Options 226
12.2 American Bond Options 229
13. Concluding Remarks 233
References 235
Appendix A: Weighting Hedge Recommendations, Variant B* 236
Appendix B: Numerical Examples 237
Comments and Notes 244
Chapter 9. Minimax and asset allocation problems 247
1. Introduction 247
2. Models for Asset Allocation Based on Minimax 249
2.1. Model 1: Rival Return Scenarios with Fixed Risk 250
2.2. Model 2: Rival Return with Risk Scenarios 250
2.3. Model 3: Rival Return Scenarios with Independent Rival Risk Scenarios 251
2.4. Model 4: Fixed Return with Rival Benchmark Risk Scenarios 251
2.5. Efficiency 252
3. Minimax Bond Portfolio Selection 252
3.1. The Single Model Problem 253
3.2. Application: Two Asset Allocations Using Different Models 254
3.3. Two-model Problem 256
3.4. Application: Simultaneous Optimization across Two Models 257
3.5. Backtesting the Performance of a Portfolio on the Minimax Frontier 258
4. Dual Benchmarking 261
4.1. Single Benchmark Tracking 261
4.2. Application: Tracking a Global Benchmark against Tracking LIBOR 264
4.3. Dual Benchmark Tracking 266
4.4. Application: Simultaneously Tracking the Global Benchmark and LIBOR 267
4.5. Performance of a Portfolio on the Dual Frontier 269
5. Other Minimax Strategies for Asset Allocation 271
5.1. Threshold Returns and Downside Risk 271
5.2. Further Minimax Index Tracking and Range Forecasts 273
6. Multistage Minimax Portfolio Selection 277
7. Portfolio Management Using Minimax and Options 284
8. Concluding Remarks 288
References 289
Comments and Notes 290
Chapter 10. Asset/liability management under uncertainty 291
1. Introduction 291
2. The Immunization Framework 296
2.1. Interest Rates 296
2.2. The Formulation 296
3. Illustration 300
4. The Asset/Liability (A/L) Risk in Immunization 303
5. The Continuous Minimax Directional Immunization 308
6. Other Immunization Strategies 309
6.1. Univariate Duration Model 309
6.2. Univariate Convexity Model 312
7. The Stochastic ALM Model 1 315
8. The Stochastic ALM Model 2 325
8.1. A Dynamic Multistage Recourse Stochastic ALM Model 325
8.2. The Minimax Formulation of the Stochastic ALM Model 2 330
8.3. A Practical Single-stage Minimax Formulation 333
9. Concluding Remarks 335
References 335
Comments and Notes 337
Chapter 11. Robust currency management 341
1. Introduction 341
2. Strategic Currency Management 1: Pure Currency Portfolios 345
3. Strategic Currency Management 2: Currency Overlay 351
4. A Generic Currency Model for Tactical Management 357
5. The Minimax Framework 359
5.1. Single Currency Framework 359
5.2. Single Currency Framework with Transaction Costs 362
5.3. Multicurrency Framework 363
5.4. Multicurrency Framework with Transaction Costs 365
5.5. Worst-case Scenario 367
5.6. A Momentum-based Minimax Strategy 369
5.7. A Risk-controlled Minimax Strategy 371
6. The Interplay between the Strategic Benchmark and Tactical Management 373
7. Currency Management Using Minimax and Options 374
8. Concluding Remarks 375
References 376
Appendix: Currency Forecasting 376
Comments and Notes 378
Index 381