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No-arbitrage conditions, scenario trees, and multi-asset financial optimization. (English) Zbl 1188.91242

Summary: Many numerical optimization methods use scenario trees as a discrete approximation for the true (multi-dimensional) probability distributions of the problem’s random variables. Realistic specifications in financial optimization models can lead to tree sizes that quickly become computationally intractable. In this paper we focus on the two main approaches proposed in the literature to deal with this problem: scenario reduction and state aggregation. We first state necessary conditions for the node structure of a tree to rule out arbitrage. However, currently available scenario reduction algorithms do not take these conditions explicitly into account. State aggregation excludes arbitrage opportunities by relying on the risk-neutral measure. This is, however, only appropriate for pricing purposes but not for optimization. Both limitations are illustrated by numerical examples. We conclude that neither of these methods is suitable to solve financial optimization models in asset-liability or portfolio management.

MSC:

91G80 Financial applications of other theories
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[1] Birge, J. R.; Louveaux, F., Introduction to Stochastic Programming (1997), Springer · Zbl 0892.90142
[2] Bradley, S. P.; Crane, D. B., A dynamic model for bond portfolio management, Management Science, 19, 2, 139-151 (1972)
[3] Cariño, D. R.; Ziemba, W. T., The Russell-Yasuda-Kasai model: An asset/liability model for a Japanese insurance company using multistage stochastic programming, Interfaces, 24, 1, 29-49 (1994)
[4] Cox, J. C.; Ross, S. A.; Rubinstein, M., Option pricing: a simplified approach, Journal of Financial Economics, 7, 229-263 (1979) · Zbl 1131.91333
[5] Cvitanic, J.; Zapatero, F., Introduction to the Economics and Mathematics of Financial Markets (2004), MIT Press · Zbl 1103.91018
[6] Dempster, M. A.H.; Germano, M.; Medova, E. A.; Rietbergen, M. I.; Sandrini, F.; Scrowston, M., Designing minimum guaranteed return funds, Quantitative Finance, 7, 2, 245-256 (2007) · Zbl 1278.91133
[7] Duffie, D., Dynamic Asset Pricing Theory (2001), Princeton University Press · Zbl 1140.91041
[8] Dupačová, J.; Consigli, G.; Wallace, S. W., Scenarios for multistage stochastic programs, Annals of Operations Research, 100, 25-53 (2000) · Zbl 1017.90068
[9] Dupačová, J.; Gröwe-Kuska, N.; Römisch, W., Scenario reduction in stochastic programming: An approach using probability metrics, Mathematical Programming, Series A, 95, 493-511 (2003) · Zbl 1023.90043
[10] Ferstl, R.; Weissensteiner, A., Cash management using multi-stage stochastic programming, Quantitative Finance, 10, 2, 209-219 (2010) · Zbl 1198.91224
[11] Geyer, A.; Ziemba, W. T., The innovest Austrian pension fund financial planning model InnoALM, Operations Research, 56, 4, 797-810 (2008) · Zbl 1167.90704
[12] Geyer, A.; Hanke, M.; Weissensteiner, A., Life-cycle asset allocation and optimal consumption using stochastic linear programming, Journal of Computational Finance, 12, 4, 29-50 (2009) · Zbl 1184.91207
[13] Gondzio, J.; Kouwenberg, R., High performance computing for asset-liability management, Operations Research, 49, 879-891 (2001) · Zbl 1163.90548
[14] Harrison, M. J.; Kreps, D. M., Martingale and arbitrage in multiperiod securities markets, Journal of Economic Theory, 20, 381-408 (1979) · Zbl 0431.90019
[15] Harrison, J. M.; Pliska, S. R., Martingales and stochastic integrals in the theory of continuous trading, Stochastic Processes and their Applications, 11, 215-260 (1981) · Zbl 0482.60097
[16] Heitsch, H.; Römisch, W., Scenario reduction algorithms in stochastic programming, Computational Optimization and Applications, 24, 187-206 (2003) · Zbl 1094.90024
[17] Høyland, K.; Wallace, S. W., Generating scenario trees for multistage decision problems, Management Science, 47, 2, 295-307 (2001) · Zbl 1232.91132
[18] Høyland, K.; Kaut, M.; Wallace, S. W., A heuristic for moment-matching scenario generation, Computational Optimization and Applications, 24, 169-185 (2003) · Zbl 1094.90579
[19] Jobst, N. J.; Zenios, S. A., On the simulation of portfolios of interest rate and credit risk sensitive securities, European Journal of Operational Research, 161, 2, 298-324 (2005) · Zbl 1066.91058
[20] Kaut, M., 2003. Updates to the published version of ‘A Heuristic for Moment-matching Scenario Generation by K. Høyland, M. Kaut, and S.W. Wallace’, working paper.; Kaut, M., 2003. Updates to the published version of ‘A Heuristic for Moment-matching Scenario Generation by K. Høyland, M. Kaut, and S.W. Wallace’, working paper.
[21] Klaassen, P., Comment on “Generating Scenario Trees for Multistage Decision Problems”, Management Science, 48, 1512-1516 (2002) · Zbl 1232.91135
[22] Klaassen, P., Discretized reality and spurious profits in stochastic programming models for asset/liability management, European Journal of Operations Research, 101, 374-392 (1997) · Zbl 0929.91029
[23] Klaassen, P., Financial asset-pricing theory and stochastic programming models for asset/liability management: A synthesis, Management Science, 44, 1, 31-48 (1998) · Zbl 1008.91503
[24] Kusy, M. I.; Ziemba, W. T., A bank asset and liability management model, Operations Research, 34, 3, 356-376 (1986)
[25] Pflug, G. C., Optimal scenario tree generation for multiperiod financial planning, Mathematical Programming, 89, 2, 251-271 (2001) · Zbl 0987.91034
[26] Ruszczinsky, A.; Shapiro, A., Stochastic Programming (2003), Elsevier
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.