No-arbitrage conditions,scenario trees,and multi-asset financial optimization |
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Authors: | Alois Geyer Michael Hanke Alex Weissensteiner |
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Institution: | 1. WU (Vienna University of Economics and Business) and Vienna Graduate School of Finance, Heiligenstaedter Straße 46-48, 1190 Vienna, Austria;2. University of Innsbruck, Department of Banking and Finance, Universitaetsstr. 15, 6020 Innsbruck, Austria |
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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. |
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Keywords: | Financial optimization Uncertainty modeling Scenario trees Sparse trees |
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