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Admissible Trading Strategies under Transaction Costs

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  • Walter Schachermayer
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    Abstract

    A well known result in stochastic analysis reads as follows: for an $\mathbb{R}$-valued super-martingale $X = (X_t)_{0\leq t \leq T}$ such that the terminal value $X_T$ is non-negative, we have that the entire process $X$ is non-negative. An analogous result holds true in the no arbitrage theory of mathematical finance: under the assumption of no arbitrage, a portfolio process $x+(H\cdot S)$ verifying $x+(H\cdot S)_T\geq 0$ also satisfies $x+(H\cdot S)_t\geq 0,$ for all $0 \leq t \leq T$. In the present paper we derive an analogous result in the presence of transaction costs. A counter-example reveals that the consideration of transaction costs makes things more delicate than in the frictionless setting.

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    File URL: http://arxiv.org/pdf/1308.1492
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    Paper provided by arXiv.org in its series Papers with number 1308.1492.

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    Date of creation: Aug 2013
    Date of revision: May 2014
    Handle: RePEc:arx:papers:1308.1492

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    References

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    1. W. Schachermayer, 1994. "Martingale Measures For Discrete-Time Processes With Infinite Horizon," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 4(1), pages 25-55.
    2. repec:fth:inseep:9513 is not listed on IDEAS
    3. Luciano Campi & Walter Schachermayer, 2006. "A super-replication theorem in Kabanov’s model of transaction costs," Finance and Stochastics, Springer, vol. 10(4), pages 579-596, December.
    4. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
    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. Kallal, Hedi & Jouini, Elyès, 1995. "Martingales and arbitrage in securities markets with transaction costs," Economics Papers from University Paris Dauphine 123456789/5630, Paris Dauphine University.
    7. Campi, Luciano & Schachermayer, Walter, 2006. "A super-replication theorem in Kabanov’s model of transaction costs," Economics Papers from University Paris Dauphine 123456789/5455, Paris Dauphine University.
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    Cited by:
    1. Christoph Czichowsky & Walter Schachermayer, 2014. "Duality Theory for Portfolio Optimisation under Transaction Costs," Papers 1408.5989, arXiv.org.
    2. Christoph Czichowsky & Walter Schachermayer & Junjian Yang, 2014. "Shadow prices for continuous processes," Papers 1408.6065, arXiv.org.

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