Finance and Stochastics

, Volume 17, Issue 2, pp 325–354 | Cite as

The dual optimizer for the growth-optimal portfolio under transaction costs

  • S. Gerhold
  • J. Muhle-KarbeEmail author
  • W. Schachermayer


We consider the maximization of the long-term growth rate in the Black–Scholes model under proportional transaction costs as in Taksar et al. (Math. Oper. Res. 13:277–294, 1988). Similarly as in Kallsen and Muhle-Karbe (Ann. Appl. Probab. 20:1341–1358, 2010) for optimal consumption over an infinite horizon, we tackle this problem by determining a shadow price, which is the solution of the dual problem. It can be calculated explicitly up to determining the root of a deterministic function. This in turn allows one to explicitly compute fractional Taylor expansions, both for the no-trade region of the optimal strategy and for the optimal growth rate.


Transaction costs Growth-optimal portfolio Shadow price 

Mathematics Subject Classification (2010)

91B28 91B16 60H10 

JEL Classification



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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Institute for Mathematical Methods in EconomicsVienna University of TechnologyWienAustria
  2. 2.Departement MathematikETH ZürichZürichSwitzerland
  3. 3.Fakultät für MathematikUniversität WienWienAustria

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