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Algorithms for Worst-Case Design and Applications to Risk Management - Berc Rustem

Algorithms for Worst-Case Design and Applications to Risk Management

Hardcover Published: 15th September 2002
ISBN: 9780691091549
Number Of Pages: 408

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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.

Industry Reviews

"This book will be very helpful to those interested in uncertainty and robust decisions. I recommend it warmly to all practitioners and researchers in economics, environment, engineering design, finance and operations research."--P.M. Pardalos, Journal of Economics "This is minimax made practical, while maintaining theoretical rigor, computational feasibility, and good problem formulation... The book is very comprehensive, and in many places quite detailed. However, it is easy to find a path through the material suited to one's purpose, ranging from a quick overview of this powerful approach to a detailed study of it and the relevant background material. It is an excellent example of how the results of an extensive research program can be translated into a book that is accessible and which is likely to have significant impact in both the optimization and finance communities."--David G. Luenberger, Journal of Economic Dynamics & Control "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."--Zentralblatt MATH

Prefacep. xiii
Introduction to minimaxp. 1
Background and Notationp. 1
Linear Independencep. 5
Tangent Cone, Normal Cone and Epigraphp. 7
Subgradiemts and Subdifferentials of Convex Functionsp. 7
Continuous Minimaxp. 10
Optimality Conditions and Robustness of Minimaxp. 11
The Haar Conditionp. 13
Saddle Points and Saddle Point Conditionsp. 15
Referencesp. 17
Comments and Notesp. 18
A survey of continuous minimax algorithmsp. 23
Introductionp. 23
The Algorithm of Chaneyp. 25
The Algorithm of Paninp. 30
The Algorithm of Kiwielp. 31
Referencesp. 33
Comments and Notesp. 34
Algorithms for computing saddle pointsp. 37
Computation of Saddle Pointsp. 37
Saddle Point Equilibriap. 37
Solution of Systems of Equationsp. 40
The Algorithmsp. 42
A Gradient-based Algorithm for Unconstrained Saddle Pointsp. 42
Quadratic Approximation Algorithm for Constrained Minimax Saddle Pointsp. 44
Interior Point Saddle Point Algorithm for Constrained Problemsp. 45
Quasi-Newton Algorithm for Nonlinear Systemsp. 49
Global Convergence of Newton-type Algorithmsp. 50
Achievement of Unit Stepsizes and Superlinear Convergencep. 54
Concluding Remarksp. 58
Referencesp. 58
Comments and Notesp. 59
A quasi-Newton algorithm for continuous minimaxp. 63
Introductionp. 63
Basic Concepts and Definitionsp. 66
The quasi-Newton Algorithmp. 70
Basic Convergence Resultsp. 76
Global Convergence and Local Convergence Ratesp. 81
Referencesp. 86
Implementation Issuesp. 87
Motivation for the Search Directionp. 90
Comments and Notesp. 91
Numerical experiments with continuous minimax algorithmsp. 93
Introductionp. 93
The Algorithmsp. 94
Kiwiel''s Algorithmp. 94
Quasi-Newton Methodsp. 95
Implementationp. 96
Terminologyp. 96
The Stopping Criterionp. 97
Evaluation of the Direction of Descentp. 97
Test Problemsp. 98
Summary of the Resultsp. 110
Iterations when is Satisfiedp. 110
Calculation of Minimum-norm Subgradientp. 111
Superlinear Convergencep. 111
Termination Criterion and Accuracy of the Solutionp. 112
Referencesp. 119
Minimax as a robust strategy for discrete rival scenariosp. 121
Introduction to Rival Models and Forecast Scenariosp. 121
The Discrete Minimax Problemp. 123
The Robust Character of the Discrete Minimax Strategyp. 125
Naive Minimaxp. 125
Robustness of the Minimax Strategyp. 126
An Examplep. 128
Augmented Lagrangians and Convexification of Discrete Minimaxp. 132
Referencesp. 137
Discrete minimax algorithm for nonlinear equality and inequality constrained modelsp. 139
Introductionp. 139
Basic Conceptsp. 141
The Discrete Minimax Algorithmp. 142
Inequality Constraintsp. 142
Quadratic Programming Subproblemp. 143
Stepsize Strategyp. 144
The Algorithmp. 145
Basic Propertiesp. 147
Convergence of the Algorithmp. 152
Achievement of Unit Stepsizesp. 156
Superlinear Convergence Rates of the Algorithmp. 162
The Algorithm for Only Linear Constraintsp. 172
Referencesp. 176
A continuous minimax strategy for options hedgingp. 179
Introductionp. 179
Options and the Hedging Problemp. 181
The Black and Scholes Option Pricing Model and Delta Hedgingp. 183
Minimax Hedging Strategyp. 187
Minimax Problem Formulationp. 187
The Worst-case Scenariop. 188
The Hedging Errorp. 189
The Objective Functionp. 190
The Minimax Hedging Errorp. 192
Transaction Costsp. 193
The Variants of the Minimax Hedging Strategyp. 194
The Minimax Solutionp. 194
Simulationp. 196
Generation of Simulation Datap. 196
Setting Up and Winding Down the Hedgep. 198
Summary of Simulation Resultsp. 198
Illustrative Hedging Problem: A Limited Empirical Studyp. 204
From Set-up to Wind-downp. 204
The Hedging Strategies Applied to 30 Options: Summary of Resultsp. 205
Multiperiod Minimax Hedging Strategiesp. 207
Two-period Minimax Strategyp. 207
Variable Minimax Strategyp. 211
Simulation Study of the Performance of Different Multiperiod Strategiesp. 213
The Simulation Structurep. 213
Results of the Simulation Studyp. 214
Rank Orderingp. 214
CAPM-based Minimax Hedging Strategyp. 215
The Capital Asset Pricing Modelp. 217
The CAPM-based Minimax Problem Formulationp. 218
The Objective Functionp. 219
The Worst-case Scenariop. 221
Simulation Study of the Performance of CAPM Minimaxp. 222
Generation of Simulation Datap. 222
Summary of Simulation Resultsp. 223
Rank Orderingp. 224
The Beta of the Hedge Portfolio for CAPM Minimaxp. 226
Hedging Bond Optionsp. 226
European Bond Optionsp. 226
American Bond Optionsp. 229
Concluding Remarksp. 233
Referencesp. 235
Weighting Hedge Recommendations, Variant Bp. 236
Numerical Examplesp. 237
Comments and Notesp. 244
Minimax and asset allocation problemsp. 247
Introductionp. 247
Models for Asset Allocation Based on Minimaxp. 249
Model 1: Rival Return Scenarios with Fixed Riskp. 250
Model 2: Rival Return with Risk Scenariosp. 250
Model 3: Rival Return Scenarios with Independent Rival Risk Scenariosp. 251
Model 4: Fixed Return with Rival Benchmark Risk Scenariosp. 251
Efficiencyp. 252
Minimax Bond Portfolio Selectionp. 252
The Single Model Problemp. 253
Application: Two Asset Allocations Using Different Modelsp. 254
Two-model Problemp. 256
Application: Simultaneous Optimization across Two Modelsp. 257
Backtesting the Performance of a Portfolio on the Minimax Frontierp. 258
Dual Benchmarkingp. 261
Single Benchmark Trackingp. 261
Application: Tracking a Global Benchmark against Tracking LIBORp. 264
Dual Benchmark Trackingp. 266
Application: Simultaneously Tracking the Global Benchmark and LIBORp. 267
Performance of a Portfolio on the Dual Frontierp. 269
Other Minimax Strategies for Asset Allocationp. 271
Threshold Returns and Downside Riskp. 271
Further Minimax Index Tracking and Range Forecastsp. 273
Multistage Minimax Portfolio Selectionp. 277
Portfolio Management Using Minimax and Optionsp. 284
Concluding Remarksp. 288
Referencesp. 289
Comments and Notesp. 290
Asset/liability management under uncertaintyp. 291
Introductionp. 291
The Immunization Frameworkp. 296
Interest Ratesp. 296
The Formulationp. 296
Illustrationp. 300
The Asset/Liability (A/L) Risk in Immunizationp. 303
The Continuous Minimax Directional Immunizationp. 308
Other Immunization Strategiesp. 309
Univariate Duration Modelp. 309
Univariate Convexity Modelp. 312
The Stochastic ALM Model 1p. 315
The Stochastic ALM Model 2p. 325
A Dynamic Multistage Recourse Stochastic ALM Modelp. 325
The Minimax Formulation of the Stochastic ALM Model 2p. 330
A Practical Single-stage Minimax Formulationp. 333
Concluding Remarksp. 335
Referencesp. 335
Comments and Notesp. 337
Robust currency managementp. 341
Introductionp. 341
Strategic Currency Management 1: Pure Currency Portfoliosp. 345
Strategic Currency Management 2: Currency Overlayp. 351
A Generic Currency Model for Tactical Managementp. 357
The Minimax Frameworkp. 359
Single Currency Frameworkp. 359
Single Currency Framework with Transaction Costsp. 362
Multicurrency Frameworkp. 363
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691091549
ISBN-10: 0691091544
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 408
Published: 15th September 2002
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.2  x 3.18
Weight (kg): 0.48