Contents -- Preface -- Chapter 1. Introduction to minimax -- 1 Background and Notation -- 1.1 Linear Independence -- 1.2 Tangent Cone, Normal Cone and Epigraph -- 1.3 Subgradiemts and Subdifferentials of Convex Functions -- 2 Continuous Minimax -- 3 Optimality Conditions and Robustness of Minimax -- 3.1 The Haar Condition -- 4 Saddle Points and Saddle Point Conditions -- References -- Comments and Notes -- Chapter 2. A survey of continuous minimax algorithms -- 1 Introduction -- 2 The Algorithm of Chaney -- 3 The Algorithm of Panin -- 4 The Algorithm of Kiwiel -- References -- Comments and Notes -- Chapter 3. Algorithms for computing saddle points -- 1 Computation of Saddle Points -- 1.1 Saddle Point Equilibria -- 1.2 Solution of Systems of Equations -- 2 The Algorithms -- 2.1 Gradient-based Algorithm for Unconstrained Saddle Points -- 2.2 Quadratic Approximation Algorithm for Constrained Minimax Saddle Points -- 2.3 Interior Point Saddle Point Algorithm for Constrained Problems -- 2.4 Quasi-Newton Algorithm for Nonlinear Systems -- 3 Global Convergence of Newton-type Algorithms -- 4 Achievement of Unit Stepsizes and Superlinear Convergence -- 5 Concluding Remarks -- References -- Comments and Notes -- Chapter 4. A quasi-Newton algorithm for continuous minimax -- 1 Introduction -- 2 Basic Concepts and Definitions -- 3 The quasi-Newton Algorithm -- 4 Basic Convergence Results -- 5 Global Convergence and Local Convergence Rates -- References -- Appendix A: Implementation Issues -- Appendix B: Motivation for the Search Direction d -- Comments and Notes -- Chapter 5. Numerical experiments with continuous minimax algorithms -- 1 Introduction -- 2 The Algorithms -- 2.1 Kiwiel's Algorithm -- 2.2 Quasi-Newton Methods -- 3 Implementation -- 3.1 Terminology -- 3.2 The Stopping Criterion -- 3.3 Evaluation of the Direction of Descent -- 4 Test Problems.

5 Summary of the Results -- 5.1 Iterations when is Satisfied -- 5.2 Calculation of Minimum-norm Subgradient -- 5.3 Superlinear Convergence -- 5.4 Termination Criterion and Accuracy of the Solution -- References -- Chapter 6. Minimax as a robust strategy for discrete rival scenarios -- 1 Introduction to Rival Models and Forecast Scenarios -- 2 The Discrete Minimax Problem -- 3 The Robust Character of the Discrete Minimax Strategy -- 3.1 Naive Minimax -- 3.2 Robustness of the Minimax Strategy -- 3.3 An Example -- 4 Augmented Lagrangians and Convexification of Discrete Minimax -- References -- Chapter 7. Discrete minimax algorithm for nonlinear equality and inequality constrained models -- 1 Introduction -- 2 Basic Concepts -- 3 The Discrete Minimax Algorithm -- 3.1 Inequality Constraints -- 3.2 Quadratic Programming Subproblem -- 3.3 Stepsize Strategy -- 3.4 The Algorithm -- 3.5 Basic Properties -- 4 Convergence of the Algorithm -- 5 Achievement of Unit Stepsizes -- 6 Superlinear Convergence Rates of the Algorithm -- 7 The Algorithm for Only Linear Constraints -- References -- Chapter 8. A continuous minimax strategy for options hedging -- 1 Introduction -- 2 Options and the Hedging Problem -- 3 The Black and Scholes Option Pricing Model and Delta Hedging -- 4 Minimax Hedging Strategy -- 4.1 Minimax Problem Formulation -- 4.2 The Worst-case Scenario -- 4.3 The Hedging Error -- 4.4 The Objective Function -- 4.5 The Minimax Hedging Error -- 4.6 Transaction Costs -- 4.7 The Variants of the Minimax Hedging Strategy -- 4.8 The Minimax Solution -- 5 Simulation -- 5.1 Generation of Simulation Data -- 5.2 Setting Up and Winding Down the Hedge -- 5.3 Summary of Simulation Results -- 6 Illustrative Hedging Problem: A Limited Empirical Study -- 6.1 From Set-up to Wind-down -- 6.2 The Hedging Strategies Applied to 30 Options: Summary of Results.

7 Multiperiod Minimax Hedging Strategies -- 7.1 Two-period Minimax Strategy -- 7.2 Variable Minimax Strategy -- 8 Simulation Study of the Performance of Different Multiperiod Strategies -- 8.1 The Simulation Structure -- 8.2 Results of the Simulation Study -- 8.3 Rank Ordering -- 9 CAPM-based Minimax Hedging Strategy -- 9.1 The Capital Asset Pricing Model -- 9.2 The CAPM-based Minimax Problem Formulation -- 9.3 The Objective Function -- 9.4 The Worst-case Scenario -- 10 Simulation Study of the Performance of CAPM Minimax -- 10.1 Generation of Simulation Data -- 10.2 Summary of Simulation Results -- 10.3 Rank Ordering -- 11 The Beta of the Hedge Portfolio for CAPM Minimax -- 12 Hedging Bond Options -- 12.1 European Bond Options -- 12.2 American Bond Options -- 13 Concluding Remarks -- References -- Appendix A: Weighting Hedge Recommendations, Variant B* -- Appendix B: Numerical Examples -- Comments and Notes -- Chapter 9. Minimax and asset allocation problems -- 1 Introduction -- 2 Models for Asset Allocation Based on Minimax -- 2.1 Model 1: Rival Return Scenarios with Fixed Risk -- 2.2 Model 2: Rival Return with Risk Scenarios -- 2.3 Model 3: Rival Return Scenarios with Independent Rival Risk Scenarios -- 2.4 Model 4: Fixed Return with Rival Benchmark Risk Scenarios -- 2.5 Efficiency -- 3 Minimax Bond Portfolio Selection -- 3.1 The Single Model Problem -- 3.2 Application: Two Asset Allocations Using Different Models -- 3.3 Two-model Problem -- 3.4 Application: Simultaneous Optimization across Two Models -- 3.5 Backtesting the Performance of a Portfolio on the Minimax Frontier -- 4 Dual Benchmarking -- 4.1 Single Benchmark Tracking -- 4.2 Application: Tracking a Global Benchmark against Tracking LIBOR -- 4.3 Dual Benchmark Tracking -- 4.4 Application: Simultaneously Tracking the Global Benchmark and LIBOR.

4.5 Performance of a Portfolio on the Dual Frontier -- 5 Other Minimax Strategies for Asset Allocation -- 5.1 Threshold Returns and Downside Risk -- 5.2 Further Minimax Index Tracking and Range Forecasts -- 6 Multistage Minimax Portfolio Selection -- 7 Portfolio Management Using Minimax and Options -- 8 Concluding Remarks -- References -- Comments and Notes -- Chapter 10. Asset/liability management under uncertainty -- 1 Introduction -- 2 The Immunization Framework -- 2.1 Interest Rates -- 2.2 The Formulation -- 3 Illustration -- 4 The Asset/Liability (A/L) Risk in Immunization -- 5 The Continuous Minimax Directional Immunization -- 6 Other Immunization Strategies -- 6.1 Univariate Duration Model -- 6.2 Univariate Convexity Model -- 7 The Stochastic ALM Model 1 -- 8 The Stochastic ALM Model 2 -- 8.1 A Dynamic Multistage Recourse Stochastic ALM Model -- 8.2 The Minimax Formulation of the Stochastic ALM Model 2 -- 8.3 A Practical Single-stage Minimax Formulation -- 9 Concluding Remarks -- References -- Comments and Notes -- Chapter 11. Robust currency management -- 1 Introduction -- 2 Strategic Currency Management 1: Pure Currency Portfolios -- 3 Strategic Currency Management 2: Currency Overlay -- 4 A Generic Currency Model for Tactical Management -- 5 The Minimax Framework -- 5.1 Single Currency Framework -- 5.2 Single Currency Framework with Transaction Costs -- 5.3 Multicurrency Framework -- 5.4 Multicurrency Framework with Transaction Costs -- 5.5 Worst-case Scenario -- 5.6 A Momentum-based Minimax Strategy -- 5.7 A Risk-controlled Minimax Strategy -- 6 The Interplay between the Strategic Benchmark and Tactical Management -- 7 Currency Management Using Minimax and Options -- 8 Concluding Remarks -- References -- Appendix: Currency Forecasting -- Comments and Notes -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O.

P -- Q -- R -- S -- T -- U -- V -- W -- Y.

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.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2018. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.