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Utility Maximization in a Binomial Model with transaction costs: a Duality Approach Based on the Shadow Price Process

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  • Christian Bayer
  • Bezirgen Veliyev

Abstract

We consider the problem of optimizing the expected logarithmic utility of the value of a portfolio in a binomial model with proportional transaction costs with a long time horizon. By duality methods, we can find expressions for the boundaries of the no-trade-region and the asymptotic optimal growth rate, which can be made explicit for small transaction costs. Here we find that, contrary to the classical results in continuous time, the size of the no-trade-region as well as the asymptotic growth rate depend analytically on the level of transaction costs, implying a linear first order effect of perturbations of (small) transaction costs. We obtain the asymptotic expansion by an almost explicit construction of the shadow price process.

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File URL: http://arxiv.org/pdf/1209.5175
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Paper provided by arXiv.org in its series Papers with number 1209.5175.

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Date of creation: Sep 2012
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Handle: RePEc:arx:papers:1209.5175

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Web page: http://arxiv.org/

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  1. Jaksa Cvitanić & Ioannis Karatzas, 1996. "HEDGING AND PORTFOLIO OPTIMIZATION UNDER TRANSACTION COSTS: A MARTINGALE APPROACH-super-2," Mathematical Finance, Wiley Blackwell, vol. 6(2), pages 133-165.
  2. Jörn Sass, 2005. "Portfolio optimization under transaction costs in the CRR model," Computational Statistics, Springer, vol. 61(2), pages 239-259, 06.
  3. J. Kallsen & J. Muhle-Karbe, 2010. "On using shadow prices in portfolio optimization with transaction costs," Papers 1010.4989, arXiv.org.
  4. Karel Janeček & Steven Shreve, 2004. "Asymptotic analysis for optimal investment and consumption with transaction costs," Finance and Stochastics, Springer, vol. 8(2), pages 181-206, 05.
  5. Magill, Michael J. P. & Constantinides, George M., 1976. "Portfolio selection with transactions costs," Journal of Economic Theory, Elsevier, vol. 13(2), pages 245-263, October.
  6. Yan Dolinsky & Halil Mete Soner, 2011. "Duality and Convergence for Binomial Markets with Friction," Papers 1106.2095, arXiv.org.
  7. Dumas, Bernard & Luciano, Elisa, 1991. " An Exact Solution to a Dynamic Portfolio Choice Problem under Transactions Costs," Journal of Finance, American Finance Association, vol. 46(2), pages 577-95, June.
  8. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-57, August.
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