Risk objectives in two-stage stochastic programming models

Dupačová, Jitka

Kybernetika, Tome 44 (2008) no. 2, pp. 227-242

Abstract (VO)

In applications of stochastic programming, optimization of the expected outcome need not be an acceptable goal. This has been the reason for recent proposals aiming at construction and optimization of more complicated nonlinear risk objectives. We will survey various approaches to risk quantification and optimization mainly in the framework of static and two-stage stochastic programs and comment on their properties. It turns out that polyhedral risk functionals introduced in Eichorn and Römisch [Eich-Ro] have many convenient features. We shall complement the existing results by an application of contamination technique to stress testing or robustness analysis of stochastic programs with polyhedral risk objectives with respect to the underlying probability distribution. The ideas will be illuminated by numerical results for a bond portfolio management problem.

Détail

Classification : 90C15, 90C39, 91B28, 91B30, 93E20
Keywords: two-stage stochastic programs; polyhedral risk objectives; robustness; contamination; bond portfolio management problem

Bibliographie

[1] Acerbi C.: Spectral measures of risk: A coherent representation of subjective risk aversion. J. Bank. Finance 26 (2002), 1505–1518

[2] Ahmed S.: Convexity and decomposition of mean-risk stochastic programs. Math. Programming A 106 (2006), 447–452 | MR | Zbl

[3] Artzner P., Delbaen F., Eber, J., Heath D.: Coherent measures of risk. Math. Finance 9 (1999), 203–228. See also , pp. 145–175 | MR | Zbl

[4] Bertocchi M., Dupačová, J., Moriggia V.: Sensitivity analysis of a bond portfolio model for the Italian market. Control Cybernet. 29 (2000), 595–615 | Zbl

[5] Bertocchi M., Moriggia, V., Dupačová J.: Horizon and stages in applications of stochastic programming in finance. Ann. Oper. Res. 142 (2006), 63–78 | MR | Zbl

[6] Dempster M. A. H., ed.: Risk Management: Value at Risk and Beyond. Cambridge Univ. Press, Cambridge 2002 | MR | Zbl

[7] Dupačová J.: Stability in stochastic programming with recourse – contaminated distributions. Math. Programing Stud. 27 (1986), 133–144 | MR | Zbl

[8] Dupačová J.: Stability and sensitivity analysis in stochastic programming. Ann. Oper. Res. 27 (1990), 115–142 | MR

[9] Dupačová J.: Postoptimality for multistage stochastic linear programs. Ann. Oper. Res. 56 (1995), 65–78 | MR | Zbl

[10] Dupačová J.: Scenario based stochastic programs: Resistance with respect to sample. Ann. Oper. Res. 64 (1996), 21–38 | MR | Zbl

[11] Dupačová J.: Reflections on robust optimization. In: Stochastic Programming Methods and Technical Applications (K. Marti and P. Kall, eds.), LNEMS 437, Springer, Berlin 1998, pp. 111–127 | MR | Zbl

[12] Dupačová J.: Stress testing via contamination. In: Coping with Uncertainty. Modeling and Policy Issues (K. Marti et al., eds.), LNEMS 581, Springer, Berlin 2006, pp. 29–46 | MR | Zbl

[13] Dupačová J.: Contamination for multistage stochastic programs. In: Prague Stochastics 2006 (M. Hušková and M. Janžura, eds.), Matfyzpress, Praha 2006, pp. 91–101. See also SPEPS 2006-06

[14] Dupačová J., Bertocchi, M., Moriggia V.: Testing the structure of multistage stochastic programs. Submitted to Optimization | Zbl

[15] Dupačová J., Hurt, J., Štěpán J.: Stochastic Modeling in Economics and Finance, Part II. Kluwer Academic Publishers, Dordrecht 2002 | MR

[16] Dupačová J., Polívka J.: Stress testing for VaR and CVaR. Quantitative Finance 7 (2007), 411–421 | MR | Zbl

[17] Eichhorn A., Römisch W.: Polyhedral risk measures in stochastic programming. SIAM J. Optim. 16 (2005), 69–95 | MR | Zbl

[18] Eichhorn A., Römisch W.: Mean-risk optimization models for electricity portfolio management. In: Proc. 9th International Conference on Probabilistic Methods Applied to Power Systems (PMAPS 2006), Stockholm 2006

[19] Eichhorn A., Römisch W.: Stability of multistage stochastic programs incorporating polyhedral risk measures. To appear in Optimization 2008 | MR | Zbl

[20] Föllmer H., Schied A.: Stochastic Finance. An Introduction in Discrete Time. (De Gruyter Studies in Mathematics 27). Walter de Gruyter, Berlin 2002 | MR | Zbl

[21] Kall P., Mayer J.: Stochastic Linear Programming. Models, Theory and Computation. Springer-Verlag, Berlin 2005 | MR | Zbl

[22] Mulvey J. M., Vanderbei R. J., Zenios S. A.: Robust optimization of large scale systems. Oper. Res. 43 (1995), 264–281 | MR | Zbl

[23] Ogryczak W., Ruszczyński A.: Dual stochastic dominance and related mean-risk models. SIAM J. Optim. 13 (2002), 60–78 | MR | Zbl

[24] Pflug G. Ch.: Some remarks on the Value-at-Risk and the Conditional Value-at-Risk. In: Probabilistic Constrained Optimization, Methodology and Applications (S. Uryasev, ed.), Kluwer Academic Publishers, Dordrecht 2001, pp. 272–281 | MR | Zbl

[25] Pflug G. Ch., Römisch W.: Modeling, Measuring and Managing Risk. World Scientific, Singapur 2007 | MR | Zbl

[26] Rockafellar R. T., Uryasev S. : Conditional value-at-risk for general loss distributions. J. Bank. Finance 26 (2001), 1443–1471

[27] Rockafellar R. T., Uryasev, S., Zabarankin M.: Generalized deviations in risk analysis. Finance Stochast. 10 (2006), 51–74 | MR | Zbl

[28] Römisch W.: Stability of stochastic programming problems. Chapter 8 in , pp. 483–554 | MR

[29] Römisch W., Wets R. J-B.: Stability of $\varepsilon $-approximate solutions to convex stochastic programs. SIAM J. Optim. 18 (2007), 961–979 | MR | Zbl

[30] Ruszczyński A., Shapiro A., eds.: Handbook on Stochastic Programming. Handbooks in Operations Research & Management Science 10, Elsevier, Amsterdam 2002

[31] Ruszczyński A., Shapiro A.: Optimization of risk measures. Chapter 4 in: Probabilistic and Randomized Methods for Design under Uncertainty (G. Calafiore and F. Dabbene, eds.), Springer, London 2006, pp. 121–157 | Zbl