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

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  • Kabanov, Yuri
  • Lépinette-Denis, Emmanuel

Abstract

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 lying outside the solvency region but ending inside. Such a phenomenon was discovered by M. Rásonyi in the discrete-time framework. In this note, we consider a rather abstract continuous-time setting and prove necessary and sufficient conditions for a property which we call no free lunch of the second 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 (superreplication) 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 Rásonyi. In the last section, we deduce the Rásonyi theorem from our general result by using specific features of discrete-time models.

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Bibliographic Info

Paper provided by Paris Dauphine University in its series Economics Papers from University Paris Dauphine with number 123456789/9714.

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Date of creation: 2012
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Publication status: Published in Finance and Stochastics, 2012, Vol. 16, no. 1. pp. 135-154.Length: 19 pages
Handle: RePEc:dau:papers:123456789/9714

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Related research

Keywords: et-valued processes; Consistent price systems; No free lunch; Arbitrage; Transaction costs;

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References

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  1. Y.M. Kabanov & D.O. Kramkov, 1998. "Asymptotic arbitrage in large financial markets," Finance and Stochastics, Springer, vol. 2(2), pages 143-172.
  2. 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.
  3. 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.
  4. Jouini, Elyes & Napp, Clotilde & Schachermayer, Walter, 2005. "Arbitrage and state price deflators in a general intertemporal framework," Journal of Mathematical Economics, Elsevier, vol. 41(6), pages 722-734, September.
  5. (**), 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.
  6. 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.
  7. 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.
  8. 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.
  9. 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.
  10. 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.
  11. 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.
  12. 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.
  13. Paolo Guasoni & Mikl\'os R\'asonyi & Walter Schachermayer, 2008. "Consistent price systems and face-lifting pricing under transaction costs," Papers 0803.4416, arXiv.org.
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Cited by:
  1. Adrien Nguyen Huu, 2012. "A note on super-hedging for investor-producers," Working Papers hal-00653982, HAL.
  2. Adrien Nguyen Huu, 2011. "A note on super-hedging for investor-producers," Papers 1112.4740, arXiv.org, revised Mar 2012.
  3. Erindi Allaj, 2013. "Implicit transaction costs and the fundamental theorems of asset pricing," Papers 1310.1882, arXiv.org, revised May 2014.
  4. Bruno Bouchard & Emmanuel Lepinette & Erik Taflin, 2013. "Robust no-free lunch with vanishing risk, a continuum of assets and proportional transaction costs," Papers 1302.0361, arXiv.org.
  5. Tomasz R. Bielecki & Igor Cialenco & Rodrigo Rodriguez, 2012. "No-Arbitrage Pricing for Dividend-Paying Securities in Discrete-Time Markets with Transaction Costs," Papers 1205.6254, arXiv.org, revised Jun 2013.

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