Algorithms for Worst Case Design and Applications to Risk Management

by Rustem, Berc

• ISBN13: 9780691091549
• ISBN10: 0691091544
• Format: Hardcover
• Copyright: 2002-08-26
• Publisher: Princeton Univ Pr

Summary

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.

Table of Contents

Preface	xiii
Introduction to minimax	1	(22)
Background and Notation	1	(9)
Linear Independence	5	(2)
Tangent Cone, Normal Cone and Epigraph	7	(1)
Subgradiemts and Subdifferentials of Convex Functions	7	(3)
Continuous Minimax	10	(1)
Optimality Conditions and Robustness of Minimax	11	(4)
The Haar Condition	13	(2)
Saddle Points and Saddle Point Conditions	15	(8)
References	17	(1)
Comments and Notes	18	(5)
A survey of continuous minimax algorithms	23	(14)
Introduction	23	(2)
The Algorithm of Chaney	25	(5)
The Algorithm of Panin	30	(1)
The Algorithm of Kiwiel	31	(6)
References	33	(1)
Comments and Notes	34	(3)
Algorithms for computing saddle points	37	(26)
Computation of Saddle Points	37	(5)
Saddle Point Equilibria	37	(3)
Solution of Systems of Equations	40	(2)
The Algorithms	42	(8)
A Gradient-based Algorithm for Unconstrained Saddle Points	42	(2)
Quadratic Approximation Algorithm for Constrained Minimax Saddle Points	44	(1)
Interior Point Saddle Point Algorithm for Constrained Problems	45	(4)
Quasi-Newton Algorithm for Nonlinear Systems	49	(1)
Global Convergence of Newton-type Algorithms	50	(4)
Achievement of Unit Stepsizes and Superlinear Convergence	54	(4)
Concluding Remarks	58	(5)
References	58	(1)
Comments and Notes	59	(4)
A quasi-Newton algorithm for continuous minimax	63	(30)
Introduction	63	(3)
Basic Concepts and Definitions	66	(4)
The quasi-Newton Algorithm	70	(6)
Basic Convergence Results	76	(5)
Global Convergence and Local Convergence Rates	81	(12)
References	86	(1)
Appendix A: Implementation Issues	87	(3)
Appendix B: Motivation for the Search Direction d	90	(1)
Comments and Notes	91	(2)
Numerical experiments with continuous minimax algorithms	93	(28)
Introduction	93	(1)
The Algorithms	94	(2)
Kiwiel's Algorithm	94	(1)
Quasi-Newton Methods	95	(1)
Implementation	96	(2)
Terminology	96	(1)
The Stopping Criterion	97	(1)
Evaluation of the Direction of Descent	97	(1)
Test Problems	98	(12)
Summary of the Results	110	(11)
Iterations when < xf(xk, y), d> ≥ - ζ is Satisfied	110	(1)
Calculation of Minimum-norm Subgradient	111	(1)
Superlinear Convergence	111	(1)
Termination Criterion and Accuracy of the Solution	112	(7)
References	119	(2)
Minimax as a robust strategy for discrete rival scenarios	121	(18)
Introduction to Rival Models and Forecast Scenarios	121	(2)
The Discrete Minimax Problem	123	(2)
The Robust Character of the Discrete Minimax Strategy	125	(7)
Naive Minimax	125	(1)
Robustness of the Minimax Strategy	126	(2)
An Example	128	(4)
Augmented Lagrangians and Convexification of Discrete Minimax	132	(7)
References	137	(2)
Discrete minimax algorithm for nonlinear equality and inequality constrained models	139	(40)
Introduction	139	(2)
Basic Concepts	141	(1)
The Discrete Minimax Algorithm	142	(10)
Inequality Constraints	142	(1)
Quadratic Programming Subproblem	143	(1)
Stepsize Strategy	144	(1)
The Algorithm	145	(2)
Basic Properties	147	(5)
Convergence of the Algorithm	152	(4)
Achievement of Unit Stepsizes	156	(6)
Superlinear Convergence Rates of the Algorithm	162	(10)
The Algorithm for Only Linear Constraints	172	(7)
References	176	(3)
A continuous minimax strategy for options hedging	179	(68)
Introduction	179	(2)
Options and the Hedging Problem	181	(2)
The Black and Scholes Option Pricing Model and Delta Hedging	183	(4)
Minimax Hedging Strategy	187	(9)
Minimax Problem Formulation	187	(1)
The Worst-case Scenario	188	(1)
The Hedging Error	189	(1)
The Objective Function	190	(2)
The Minimax Hedging Error	192	(1)
Transaction Costs	193	(1)
The Variants of the Minimax Hedging Strategy	194	(1)
The Minimax Solution	194	(2)
Simulation	196	(8)
Generation of Simulation Data	196	(2)
Setting Up and Winding Down the Hedge	198	(1)
Summary of Simulation Results	198	(6)
Illustrative Hedging Problem: A Limited Empirical Study	204	(3)
From Set-up to Wind-down	204	(1)
The Hedging Strategies Applied to 30 Options: Summary of Results	205	(2)
Multiperiod Minimax Hedging Strategies	207	(6)
Two-period Minimax Strategy	207	(4)
Variable Minimax Strategy	211	(2)
Simulation Study of the Performance of Different Multiperiod Strategies	213	(2)
The Simulation Structure	213	(1)
Results of the Simulation Study	214	(1)
Rank Ordering	214	(1)
CAPM-based Minimax Hedging Strategy	215	(7)
The Capital Asset Pricing Model	217	(1)
The CAPM-based Minimax Problem Formulation	218	(1)
The Objective Function	219	(2)
The Worst-case Scenario	221	(1)
Simulation Study of the Performance of CAPM Minimax	222	(4)
Generation of Simulation Data	222	(1)
Summary of Simulation Results	223	(1)
Rank Ordering	224	(2)
The Beta of the Hedge Portfolio for CAPM Minimax	226	(1)
Hedging Bond Options	226	(7)
European Bond Options	226	(3)
American Bond Options	229	(4)
Concluding Remarks	233	(14)
References	235	(1)
Appendix A: Weighting Hedge Recommendations, Variant B*	236	(1)
Appendix B: Numerical Examples	237	(7)
Comments and Notes	244	(3)
Minimax and asset allocation problems	247	(44)
Introduction	247	(2)
Models for Asset Allocation Based on Minimax	249	(3)
Model 1: Rival Return Scenarios with Fixed Risk	250	(1)
Model 2: Rival Return with Risk Scenarios	250	(1)
Model 3: Rival Return Scenarios with Independent Rival Risk Scenarios	251	(1)
Model 4: Fixed Return with Rival Benchmark Risk Scenarios	251	(1)
Efficiency	252	(1)
Minimax Bond Portfolio Selection	252	(9)
The Single Model Problem	253	(1)
Application: Two Asset Allocations Using Different Models	254	(2)
Two-model Problem	256	(1)
Application: Simultaneous Optimization across Two Models	257	(1)
Backtesting the Performance of a Portfolio on the Minimax Frontier	258	(3)
Dual Benchmarking	261	(10)
Single Benchmark Tracking	261	(3)
Application: Tracking a Global Benchmark against Tracking LIBOR	264	(2)
Dual Benchmark Tracking	266	(1)
Application: Simultaneously Tracking the Global Benchmark and LIBOR	267	(2)
Performance of a Portfolio on the Dual Frontier	269	(2)
Other Minimax Strategies for Asset Allocation	271	(6)
Threshold Returns and Downside Risk	271	(2)
Further Minimax Index Tracking and Range Forecasts	273	(4)
Multistage Minimax Portfolio Selection	277	(7)
Portfolio Management Using Minimax and Options	284	(4)
Concluding Remarks	288	(3)
References	289	(1)
Comments and Notes	290	(1)
Asset/liability management under uncertainty	291	(50)
Introduction	291	(5)
The Immunization Framework	296	(4)
Interest Rates	296	(1)
The Formulation	296	(4)
Illustration	300	(3)
The Asset/Liability (A/L) Risk in Immunization	303	(5)
The Continuous Minimax Directional Immunization	308	(1)
Other Immunization Strategies	309	(6)
Univariate Duration Model	309	(3)
Univariate Convexity Model	312	(3)
The Stochastic ALM Model 1	315	(10)
The Stochastic ALM Model 2	325	(10)
A Dynamic Multistage Recourse Stochastic ALM Model	325	(5)
The Minimax Formulation of the Stochastic ALM Model 2	330	(3)
A Practical Single-stage Minimax Formulation	333	(2)
Concluding Remarks	335	(6)
References	335	(2)
Comments and Notes	337	(4)
Robust Currency management	341	(40)
Introduction	341	(4)
Strategic Currency Management 1: Pure Currency Portfolios	345	(6)
Strategic Currency Management 2: Currency Overlay	351	(6)
A Generic Currency Model for Tactical Management	357	(2)
The Minimax Framework	359	(14)
Single Currency Framework	359	(3)
Single Currency Framework with Transaction Costs	362	(1)
Multicurrency Framework	363	(2)
Multicurrency Framework with Transaction Costs	365	(2)
Worst-case Scenario	367	(2)
A Momentum-based Minimax Strategy	369	(2)
A Risk-controlled Minimax Strategy	371	(2)
The Interplay between the Strategic Benchmark and Tactical Management	373	(1)
Currency Management Using Minimax and Options	374	(1)
Concluding Remarks	375	(6)
References	376	(1)
Appendix: Currency Forecasting	376	(2)
Comments and Notes	378	(3)
Index	381