On the game interpretation of a shadow price process in utility maximization problems under transaction costs

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• Dmitry B. Rokhlin
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Abstract

To any utility maximization problem under transaction costs one can assign a frictionless model with a price process $S^*$, lying in the bid/ask price interval $[\underline S, \bar{S}]$. Such process $S^*$ is called a \emph{shadow price} if it provides the same optimal utility value as in the original model with bid-ask spread. We call $S^*$ a \emph{generalized shadow price} if the above property is true for the \emph{relaxed} utility function in the frictionless model. This relaxation is defined as the lower semicontinuous envelope of the original utility, considered as a function on the set $[\underline S, \bar{S}]$, equipped with some natural weak topology. We prove the existence of a generalized shadow price under rather weak assumptions and mark its relation to a saddle point of the trader/market zero-sum game, determined by the relaxed utility function. The relation of the notion of a shadow price to its generalization is illustrated by several examples. Also, we briefly discuss the interpretation of shadow prices via Lagrange duality.

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File URL: http://arxiv.org/pdf/1112.2406

Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 1112.2406.

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Date of revision: Dec 2011
Handle: RePEc:arx:papers:1112.2406

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1. Koopmans, Tjalling C, 1977. "Concepts of Optimality and Their Uses," American Economic Review, American Economic Association, American Economic Association, vol. 67(3), pages 261-74, June.
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3. Cvitanic, Jaksa & Wang, Hui, 2001. "On optimal terminal wealth under transaction costs," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 223-231, April.
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5. Jaksa Cvitanić & Ioannis Karatzas, 1996. "HEDGING AND PORTFOLIO OPTIMIZATION UNDER TRANSACTION COSTS: A MARTINGALE APPROACH-super-2," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 6(2), pages 133-165.
6. Marco Frittelli & Emanuela Rosazza Gianin, 2011. "On The Penalty Function And On Continuity Properties Of Risk Measures," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., World Scientific Publishing Co. Pte. Ltd., vol. 14(01), pages 163-185.
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Cited by:
1. Christoph Czichowsky & Johannes Muhle-Karbe & Walter Schachermayer, 2012. "Transaction Costs, Shadow Prices, and Duality in Discrete Time," Papers 1205.4643, arXiv.org, revised Jan 2014.

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