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

## Author Info

• 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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 Length: Date of creation: Dec 2011 Date of revision: Dec 2011 Handle: RePEc:arx:papers:1112.2406 Contact details of provider: Web page: http://arxiv.org/

## References

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1. Koopmans, Tjalling C, 1977. "Concepts of Optimality and Their Uses," American Economic Review, American Economic Association, vol. 67(3), pages 261-74, June.
2. Cvitanic, Jaksa & Wang, Hui, 2001. "On optimal terminal wealth under transaction costs," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 223-231, April.
3. Simone Cerreia-Vioglio & Fabio Maccheroni & Massimo Marinacci & Luigi Montrucchio, 2008. "Risk Measures: Rationality and Diversification," Carlo Alberto Notebooks 100, Collegio Carlo Alberto.
4. 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., vol. 14(01), pages 163-185.
5. 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.
6. J. Kallsen & J. Muhle-Karbe, 2010. "On using shadow prices in portfolio optimization with transaction costs," Papers 1010.4989, arXiv.org.
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