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**Algorithms for worst-case design and applications to risk management.**
*(English)*
Zbl 1140.90013

Princeton, NJ: Princeton University Press (ISBN 0-691-09154-4/hbk). xviii, 389 p. (2002).

This book, consisting of 11 chapters, offers concepts and algorithms for finding the best decision having in mind the worst-case scenario. It seeks robust decisions (so important in risk management) by means of minimax solutions which yield a specified guaranteed level of performance in an uncertain world. In some sense, the book extends the notion of optimization that no longer refers to a single scenario (as usually), but to all “rival” scenarios under consideration.

The book starts with underlying notions and results, such as, subgradients, subdifferentials, continuous minimax, robustness of minimax, the Haar condition, and saddle point conditions. Chapter 2 provides a survey of continuous minimax algorithms, including those due to Demyanov, Chaney, Pironneau-Polak, Panin, and Kiwiel. Chapter 3 is devoted to computation of saddle points. Two algorithms for unconstrained problems, and two for constrained are discussed in detail. The chapter ends with results on global convergence of Newton-type algorithms. A novel short proof of \(Q\)-superlinear convergence is also given.

In Chapter 4, a new quasi-Newton algorithm for a continuous minimax problem is derived and discussed carefully, and illustrated with 21 numerical experiments in Chapter 5. This approach extends the steepest descent approach of Panin (1981) and Kiwiel (1987). The descent property of the direction chosen by the algorithm is established, and the local convergence rate is shown to be \(Q\)-superlinear. In Chapter 5, the test problems 1–7, having convex-concave objectives, illustrate the situation when the max-function has a unique maximizer for each \(x\). The test problems 8–16 have convex-convex objectives and refer to the situation when the max-function may have multiple maximizers for each \(x\). The last 5 test problems are more difficult and they basically test the robustness of the algorithms.

Rival models and forecast scenarios are discussed in Chapter 6 alongside with robustness of minimax solutions. Augmented Lagrangians and Finsler’s lemma are employed to obtain the existence of a saddle point. In Chapter 7, the local \(Q\)-superlinear convergence of a sequential quadratic programming algorithm for nonlinear constrained problems is demonstrated. A special attention is given to the situation when both the equality and inequality constraints are linear, which is a dominant case in financial applications, including the classical mean-variance framework studied at the very end of Chapter 7.

In the longest Chapter 8 it is demonstrated how minimax can yield a robust hedging for written European call options. A simulation analysis, with 5 volatility levels and stock time series involving 1250 data for each volatility level, is provided to see when delta hedging outperforms specific strategies called variants. Also, a limited empirical study of 30 options available in the UK is included to illustrate the performance of the minimax hedging applied to real data. In addition, the earlier analysis valid for one-period horizon is next extended to cover the multi-period setting, supplemented by a short simulation study when summarizes the difference in relative performance between various minimax strategies. Finally, an alternative formulation of the minimax strategy is discussed, by making use of asset returns evaluations via Capital Asset Pricing Model, and next illustrated by means of European and American bond options.

Chapter 9 is also devoted to financial applications of minimax approach. This time, in the context of portfolio asset allocation (including a multistage setting), with examples featuring bond portfolios. Several extensions of M-V and benchmark-tracking to a minimax framework are explored. It is shown that mix-forecasting can be approximately addressed within the minimax setting. A number of minimax techniques are studied, all applicable to stock, bonds and currencies. Finally, the following issue is addressed: How far out-of-the money can an option be remaining attractive as an insurance provider for a portfolio, if such use of the option is complemented by an active portfolio management via minimax.

Chapter 10 starts with presentation of immunization models applicable to ALM (asset/liability management) problems. An example of a German institution with a stream of bond liabilities is given to illustrate benefits resulting from the minimax framework. Several measures of A/L risk are examined, and next it is shown how various immunization strategies can be adapted to a minimax framework. Extended stochastic ALM models with multiperiodic objectives and varying performance/benchmarking horizons are presented. They are useful for making a comprehensive evaluation of an ALM strategy, whether it is based on minimax or on standard ALM techniques.

The last Chapter 11 is devoted to robust currency management (CM). It starts (Section 2) with a stretegic CM system which is useful in determining an optimal currency benchmark (OCB) for a pure currency portfolio. Section 3 identifies an OCB that hedges the risk of different currency exposures in the context of general asset portfolios. In Section 4, a generic currency model is discussed which generates signals indicating the tactical currency bet that the currency manager should implement. Section 5 generalizes the framework from Section 4 to discuss the issues of mis-forecasting. The interplay between the strategic currency benchmark and the tactical CM is elaborated. Section 7 focuses on the use of currency options in the management of currency exposures. Also, the use of options as very short-term tools for generating excess returns in currency overlay systems is presented. In the appendix the currency forecasting is addressed.

The book starts with underlying notions and results, such as, subgradients, subdifferentials, continuous minimax, robustness of minimax, the Haar condition, and saddle point conditions. Chapter 2 provides a survey of continuous minimax algorithms, including those due to Demyanov, Chaney, Pironneau-Polak, Panin, and Kiwiel. Chapter 3 is devoted to computation of saddle points. Two algorithms for unconstrained problems, and two for constrained are discussed in detail. The chapter ends with results on global convergence of Newton-type algorithms. A novel short proof of \(Q\)-superlinear convergence is also given.

In Chapter 4, a new quasi-Newton algorithm for a continuous minimax problem is derived and discussed carefully, and illustrated with 21 numerical experiments in Chapter 5. This approach extends the steepest descent approach of Panin (1981) and Kiwiel (1987). The descent property of the direction chosen by the algorithm is established, and the local convergence rate is shown to be \(Q\)-superlinear. In Chapter 5, the test problems 1–7, having convex-concave objectives, illustrate the situation when the max-function has a unique maximizer for each \(x\). The test problems 8–16 have convex-convex objectives and refer to the situation when the max-function may have multiple maximizers for each \(x\). The last 5 test problems are more difficult and they basically test the robustness of the algorithms.

Rival models and forecast scenarios are discussed in Chapter 6 alongside with robustness of minimax solutions. Augmented Lagrangians and Finsler’s lemma are employed to obtain the existence of a saddle point. In Chapter 7, the local \(Q\)-superlinear convergence of a sequential quadratic programming algorithm for nonlinear constrained problems is demonstrated. A special attention is given to the situation when both the equality and inequality constraints are linear, which is a dominant case in financial applications, including the classical mean-variance framework studied at the very end of Chapter 7.

In the longest Chapter 8 it is demonstrated how minimax can yield a robust hedging for written European call options. A simulation analysis, with 5 volatility levels and stock time series involving 1250 data for each volatility level, is provided to see when delta hedging outperforms specific strategies called variants. Also, a limited empirical study of 30 options available in the UK is included to illustrate the performance of the minimax hedging applied to real data. In addition, the earlier analysis valid for one-period horizon is next extended to cover the multi-period setting, supplemented by a short simulation study when summarizes the difference in relative performance between various minimax strategies. Finally, an alternative formulation of the minimax strategy is discussed, by making use of asset returns evaluations via Capital Asset Pricing Model, and next illustrated by means of European and American bond options.

Chapter 9 is also devoted to financial applications of minimax approach. This time, in the context of portfolio asset allocation (including a multistage setting), with examples featuring bond portfolios. Several extensions of M-V and benchmark-tracking to a minimax framework are explored. It is shown that mix-forecasting can be approximately addressed within the minimax setting. A number of minimax techniques are studied, all applicable to stock, bonds and currencies. Finally, the following issue is addressed: How far out-of-the money can an option be remaining attractive as an insurance provider for a portfolio, if such use of the option is complemented by an active portfolio management via minimax.

Chapter 10 starts with presentation of immunization models applicable to ALM (asset/liability management) problems. An example of a German institution with a stream of bond liabilities is given to illustrate benefits resulting from the minimax framework. Several measures of A/L risk are examined, and next it is shown how various immunization strategies can be adapted to a minimax framework. Extended stochastic ALM models with multiperiodic objectives and varying performance/benchmarking horizons are presented. They are useful for making a comprehensive evaluation of an ALM strategy, whether it is based on minimax or on standard ALM techniques.

The last Chapter 11 is devoted to robust currency management (CM). It starts (Section 2) with a stretegic CM system which is useful in determining an optimal currency benchmark (OCB) for a pure currency portfolio. Section 3 identifies an OCB that hedges the risk of different currency exposures in the context of general asset portfolios. In Section 4, a generic currency model is discussed which generates signals indicating the tactical currency bet that the currency manager should implement. Section 5 generalizes the framework from Section 4 to discuss the issues of mis-forecasting. The interplay between the strategic currency benchmark and the tactical CM is elaborated. Section 7 focuses on the use of currency options in the management of currency exposures. Also, the use of options as very short-term tools for generating excess returns in currency overlay systems is presented. In the appendix the currency forecasting is addressed.

Reviewer: Leszek S. Zaremba (Warszawa)