Electronic Journal of Probability

Representation of continuous linear forms on the set of ladlag processes and the hedging of American claims under proportional costs

Jean-Francois Chassagneux and Bruno Bouchard

Full-text: Open access

Abstract

We discuss a d-dimensional version (for làdlàg optional processes) of a duality result by Meyer (1976) between {bounded} càdlàg adapted processes and random measures. We show that it allows to establish, in a very natural way, a dual representation for the set of initial endowments which allow to super-hedge a given American claim in a continuous time model with proportional transaction costs. It generalizes a previous result of Bouchard and Temam (2005) who considered a discrete time setting. It also completes the very recent work of Denis, De Vallière and Kabanov (2008) who studied càdlàg American claims and used a completely different approach.

Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 24, 612-632.

Dates
Accepted: 27 February 2009
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819485

Digital Object Identifier
doi:10.1214/EJP.v14-625

Mathematical Reviews number (MathSciNet)
MR2486816

Zentralblatt MATH identifier
1186.91210

Subjects
Primary: 91B28
Secondary: 60G42: Martingales with discrete parameter

Keywords
Randomized stopping times American options transaction costs

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Chassagneux, Jean-Francois; Bouchard, Bruno. Representation of continuous linear forms on the set of ladlag processes and the hedging of American claims under proportional costs. Electron. J. Probab. 14 (2009), paper no. 24, 612--632. doi:10.1214/EJP.v14-625. https://projecteuclid.org/euclid.ejp/1464819485


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References

  • Bismut J.-M. (1979). Temps d'arrêt optimal, quasi-temps d'arrêt et retournement du temps. Ann. Probab. 7, 933-964.
  • Bouchard B. and H. Pham (2004). Wealth-Path Dependent Utility Maximization in Incomplete Markets. Finance and Stochastics, 8 (4), 579-603.
  • Bouchard B. and E. Temam (2005). On the Hedging of American Options in Discrete Time Markets with Proportional Transaction Costs. Electronic Journal of Probability, 10, 746-760.
  • Bouchard B., N. Touzi and A. Zeghal (2004). Dual Formulation of the Utility Maximization Problem : the case of Nonsmooth Utility. The Annals of Applied Probability, 14 (2), 678-717.
  • Campi L. and W. Schachermayer (2006). A super-replication theorem in Kabanov's model of transaction costs. Finance and Stochastics, 10(4), 579-596.
  • Chalasani P. and S. Jha (2001). Randomized stopping times and American option pricing with transaction costs. Mathematical Finance, 11(1), 33-77.
  • Delbaen F. and W. Shachermayer (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Annalen, 312, 215-250.
  • Dellacherie C. (1972). Capacités et processus stochastiques. Springer-Verlag.
  • Denis E., D. De Vallière and Y. Kabanov (2008). Hedging of american options under transaction costs. preprint.
  • El Karoui N. (1979). Les aspects probabilistes du contrôle stochastique. Ecole d'Eté de Probabilités de Saint Flour IX, Lecture Notes in Mathematics 876, Springer Verlag.
  • El Karoui N. (1982). Une propriété de domination de l'enveloppe de Snell des semimartingales fortes. Sém. prob. Strasbourg, 16, 400-408.
  • Kabanov Y. and G. Last (2002). Hedging under transaction costs in currency markets: a continuous time model. Mathematical Finance, 12, 63-70.
  • Kabanov Y. and C. Stricker (2002). Hedging of contingent claims under transaction costs. Advances in Finance and Stochastics. Eds. K. Sandmann and Ph. Schˆnbucher, Springer, 125-136.
  • Karatzas I. and S. G. Kou (1998). Hedging American contingent claims with constrained portfolios. Finance and Stochastics, 2, 215-258.
  • Karatzas I. and S. E. Shreve (1991). Brownian motion and stochastic calculus. Springer Verlag, Berlin.
  • Karatzas I. et S.E. Shreve (1998), Methods of Mathematical Finance, Springer Verlag.
  • Kindler J. (1983). A simple proof of the Daniell-Stone representation theorem. Amer. Math. Monthly, 90 (3), 396-397.
  • Kramkov D. (1996). Optional decomposition of supermartingales and hedging in incomplete security markets. Probability Theory and Related Fields, 105 (4), 459-479.
  • Kramkov D. and W. Schachermayer (1999). The Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Markets. Annals of Applied Probability, bf 9 3, 904 - 950.
  • Meyer P.A. (1966). Probabilités et potentiel. Hermann, Paris.
  • Meyer P.A. (1976). Un cours sur les intégrales stochastiques. Sém. prob. Strasbourg, 10, 245-400.
  • Rasonyi M. (2003). A remark on the superhedging theorem under transaction costs. Séminaire de Probabilités XXXVII, Lecture Notes in Math., 1832, Springer, Berlin-Heidelberg-New York, 394-398.
  • Schachermayer W. (2004). The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time. Mathematical Finance, 14 (1), 19-48.