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Consistent price systems and face-lifting pricing under transaction costs

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  • Paolo Guasoni
  • Mikl\'os R\'asonyi
  • Walter Schachermayer
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    Abstract

    In markets with transaction costs, consistent price systems play the same role as martingale measures in frictionless markets. We prove that if a continuous price process has conditional full support, then it admits consistent price systems for arbitrarily small transaction costs. This result applies to a large class of Markovian and non-Markovian models, including geometric fractional Brownian motion. Using the constructed price systems, we show, under very general assumptions, the following ``face-lifting'' result: the asymptotic superreplication price of a European contingent claim $g(S_T)$ equals $\hat{g}(S_0)$, where $\hat{g}$ is the concave envelope of $g$ and $S_t$ is the price of the asset at time $t$. This theorem generalizes similar results obtained for diffusion processes to processes with conditional full support.

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    File URL: http://arxiv.org/pdf/0803.4416
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 0803.4416.

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    Date of creation: Mar 2008
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    Publication status: Published in Annals of Applied Probability 2008, Vol. 18, No. 2, 491-520
    Handle: RePEc:arx:papers:0803.4416

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    Web page: http://arxiv.org/

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    1. Andrew W. Lo, 1989. "Long-term Memory in Stock Market Prices," NBER Working Papers 2984, National Bureau of Economic Research, Inc.
    2. HuyËn Pham & Nizar Touzi & Jaksa Cvitanic, 1999. "A closed-form solution to the problem of super-replication under transaction costs," Finance and Stochastics, Springer, Springer, vol. 3(1), pages 35-54.
    3. Kallal, Hedi & Jouini, Elyès, 1995. "Martingales and arbitrage in securities markets with transaction costs," Economics Papers from University Paris Dauphine, Paris Dauphine University 123456789/5630, Paris Dauphine University.
    4. repec:fth:inseep:9513 is not listed on IDEAS
    5. J. Jacod & A.N. Shiryaev, 1998. "Local martingales and the fundamental asset pricing theorems in the discrete-time case," Finance and Stochastics, Springer, Springer, vol. 2(3), pages 259-273.
    6. Christian Bender & Tommi Sottinen & Esko Valkeila, 2008. "Pricing by hedging and no-arbitrage beyond semimartingales," Finance and Stochastics, Springer, Springer, vol. 12(4), pages 441-468, October.
    7. Jouini Elyes & Kallal Hedi, 1995. "Martingales and Arbitrage in Securities Markets with Transaction Costs," Journal of Economic Theory, Elsevier, Elsevier, vol. 66(1), pages 178-197, June.
    8. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 7(1), pages 95-105.
    9. Salopek, D. M., 1998. "Tolerance to arbitrage," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 76(2), pages 217-230, August.
    10. Y.M. Kabanov, 1999. "Hedging and liquidation under transaction costs in currency markets," Finance and Stochastics, Springer, Springer, vol. 3(2), pages 237-248.
    11. Luciano Campi & Walter Schachermayer, 2006. "A super-replication theorem in Kabanov’s model of transaction costs," Finance and Stochastics, Springer, Springer, vol. 10(4), pages 579-596, December.
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