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Consistent price systems and arbitrage opportunities of the second kind in models with transaction costs

  • Emmanuel Denis

    ()

  • Yuri Kabanov

    ()

In contrast with the classical models of frictionless financial markets, market models with proportional transaction costs, even satisfying usual no-arbitrage properties, may admit arbitrage opportunities of the second kind. This means that there are self-financing portfolios with initial endowments laying outside the solvency region but ending inside. Such a phenomenon was discovered by M. R´asonyi in the discrete-time framework. In this note we consider a rather abstract continuous-time setting and prove necessary and sufficient conditions for the property which we call No Free Lunch of the 2nd Kind, NFL2. We provide a number of equivalent conditions elucidating, in particular, the financial meaning of the property B which appeared as an indispensable “technical” hypothesis in previous papers on hedging (super-replication) of contingent claims under transaction costs. We show that it is equivalent to another condition on the “richness” of the set of consistent price systems, close to the condition PCE introduced by R´asonyi. In the last section we deduce the R´asonyi theorem from our general result using specific features of discrete-time models.

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File URL: http://hdl.handle.net/10.1007/s00780-010-0144-6
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Article provided by Springer in its journal Finance and Stochastics.

Volume (Year): 16 (2012)
Issue (Month): 1 (January)
Pages: 135-154

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Handle: RePEc:spr:finsto:v:16:y:2012:i:1:p:135-154
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  1. Paolo Guasoni & Mikl\'os R\'asonyi & Walter Schachermayer, 2008. "Consistent price systems and face-lifting pricing under transaction costs," Papers 0803.4416, arXiv.org.
  2. 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.
  3. D. Vallière & E. Denis & Y. Kabanov, 2009. "Hedging of American options under transaction costs," Finance and Stochastics, Springer, vol. 13(1), pages 105-119, January.
  4. 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.
  5. Y.M. Kabanov & D.O. Kramkov, 1998. "Asymptotic arbitrage in large financial markets," Finance and Stochastics, Springer, vol. 2(2), pages 143-172.
  6. Clotilde Napp & Elyès Jouini, 2005. "Arbitrage and state price deflators in a general intertemporal framework," Post-Print halshs-00151526, HAL.
  7. Kabanov, Yu. M. & Stricker, Ch., 2001. "The Harrison-Pliska arbitrage pricing theorem under transaction costs," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 185-196, April.
  8. 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.
  9. (**), Christophe Stricker & (*), Miklós Rásonyi & Yuri Kabanov, 2002. "No-arbitrage criteria for financial markets with efficient friction," Finance and Stochastics, Springer, vol. 6(3), pages 371-382.
  10. Grigoriev Pavel G., 2005. "On low dimensional case in the fundamental asset pricing theorem with transaction costs," Statistics & Risk Modeling, De Gruyter, vol. 23(1/2005), pages 33-48, January.
  11. Napp, Clotilde & Jouini, Elyès, 2005. "Arbitrage and state price deflators in a general intertemporal framework," Economics Papers from University Paris Dauphine 123456789/345, Paris Dauphine University.
  12. Yuri M. Kabanov & Günter Last, 2002. "Hedging under Transaction Costs in Currency Markets: a Continuous-Time Model," Mathematical Finance, Wiley Blackwell, vol. 12(1), pages 63-70.
  13. Dimitri De Vallière & Yuri Kabanov & Christophe Stricker, 2007. "No-arbitrage criteria for financial markets with transaction costs and incomplete information," Finance and Stochastics, Springer, vol. 11(2), pages 237-251, April.
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