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No-arbitrage conditions, scenario trees, and multi-asset financial optimization

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  • Geyer, Alois
  • Hanke, Michael
  • Weissensteiner, Alex

Abstract

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.

Suggested Citation

  • Geyer, Alois & Hanke, Michael & Weissensteiner, Alex, 2010. "No-arbitrage conditions, scenario trees, and multi-asset financial optimization," European Journal of Operational Research, Elsevier, vol. 206(3), pages 609-613, November.
    • Handle: RePEc:eee:ejores:v:206:y:2010:i:3:p:609-613
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    5. Staino, Alessandro & Russo, Emilio, 2015. "A moment-matching method to generate arbitrage-free scenarios," European Journal of Operational Research, Elsevier, vol. 246(2), pages 619-630.
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    8. Patrizia Beraldi & Maria Bruni, 2014. "A clustering approach for scenario tree reduction: an application to a stochastic programming portfolio optimization problem," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 934-949, October.
    9. Isha Chopra & Dharmaraja Selvamuthu, 2020. "Scenario generation in stochastic programming using principal component analysis based on moment-matching approach," OPSEARCH, Springer;Operational Research Society of India, vol. 57(1), pages 190-201, March.
    10. Consiglio, Andrea & Tumminello, Michele & Zenios, Stavros A., 2015. "Designing and pricing guarantee options in defined contribution pension plans," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 267-279.
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    12. Consiglio, Andrea & Carollo, Angelo & Zenios, Stavros A., 2014. "Generating Multi-factor Arbitrage-Free Scenario Trees with Global Optimization," Working Papers 13-35, University of Pennsylvania, Wharton School, Weiss Center.
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    14. Owadally, Iqbal & Jang, Chul & Clare, Andrew, 2021. "Optimal investment for a retirement plan with deferred annuities allowing for inflation and labour income risk," European Journal of Operational Research, Elsevier, vol. 295(3), pages 1132-1146.
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