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Multistage Portfolio Optimization: A Duality Result in Conic Market Models

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  • Robert Bassett
  • Khoa Le

Abstract

We prove a general duality result for multi-stage portfolio optimization problems in markets with proportional transaction costs. The financial market is described by Kabanov's model of foreign exchange markets over a finite probability space and finite-horizon discrete time steps. This framework allows us to compare vector-valued portfolios under a partial ordering, so that our model does not require liquidation into some numeraire at terminal time. We embed the vector-valued portfolio problem into the set-optimization framework, and generate a problem dual to portfolio optimization. Using recent results in the development of set optimization, we then show that a strong duality relationship holds between the problems.

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  • Robert Bassett & Khoa Le, 2016. "Multistage Portfolio Optimization: A Duality Result in Conic Market Models," Papers 1601.00712, arXiv.org, revised Jan 2016.
    • Handle: RePEc:arx:papers:1601.00712
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    References listed on IDEAS

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    1. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    2. Andreas Hamel & Andreas Löhne & Birgit Rudloff, 2014. "Benson type algorithms for linear vector optimization and applications," Journal of Global Optimization, Springer, vol. 59(4), pages 811-836, August.
    3. Walter Schachermayer, 2004. "The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time," Mathematical Finance, Wiley Blackwell, vol. 14(1), pages 19-48, January.
    4. Andreas H. Hamel & Andreas Löhne, 2014. "Lagrange Duality in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 161(2), pages 368-397, May.
    5. Y.M. Kabanov, 1999. "Hedging and liquidation under transaction costs in currency markets," Finance and Stochastics, Springer, vol. 3(2), pages 237-248.
    6. Andreas Löhne & Birgit Rudloff & Firdevs Ulus, 2014. "Primal and dual approximation algorithms for convex vector optimization problems," Journal of Global Optimization, Springer, vol. 60(4), pages 713-736, December.
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