Skip to main content
SpringerLink
Log in

Dual representation of superhedging costs in illiquid markets

  • Published:
Mathematics and Financial Economics Aims and scope Submit manuscript

Abstract

This paper studies superhedging of contingent claims in illiquid markets where trading costs may depend nonlinearly on the traded amounts and portfolios may be subject to constraints. We give dual expressions for superhedging costs of financial contracts where claims and premiums are paid possibly at multiple points in time. Besides classical pricing problems, this setup covers various swap and insurance contracts where premiums are paid in sequences. Validity of the dual expressions is proved under new relaxed conditions related to the classical no-arbitrage condition. A new version of the fundamental theorem of asset pricing is given for unconstrained models with nonlinear trading costs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aubin J.-P.: Mathematical Methods of Game and Economic Theory, Volume 7 of Studies in Mathematics and its Applications. North-Holland Publishing Co, Amsterdam (1979)

    Google Scholar 

  2. Çetin U., Jarrow R.A., Protter P.: Liquidity risk and arbitrage pricing theory. Financ. Stoch. 8(3), 311–341 (2004)

    Article  MATH  Google Scholar 

  3. Cvitanić J., Karatzas I.: Convex duality in constrained portfolio optimization. Ann. Appl. Probab. 2(4), 767–818 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dalang R.C., Morton A., Willinger W.: Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stoch. Stoch. Rep. 29(2), 185–201 (1990)

    MathSciNet  MATH  Google Scholar 

  5. Delbaen F., Schachermayer W.: The Mathematics of Arbitrage. Springer Finance. Springer, Berlin (2006)

    Google Scholar 

  6. Dempster M.A.H., Evstigneev I.V., Taksar M.I.: Asset pricing and hedging in financial markets with transaction costs: an approach based on the Von Neumann–Gale model. Ann. Financ. 2(4), 327–355 (2006)

    Article  MATH  Google Scholar 

  7. Dermody J.C., Rockafellar R.T.: Cash stream valuation in the face of transaction costs and taxes. Math. Financ. 1(1), 31–54 (1991)

    Article  MATH  Google Scholar 

  8. Dermody J.C., Rockafellar R.T.: Tax basis and nonlinearity in cash stream valuation. Math. Financ. 5(2), 97–119 (1995)

    Article  MATH  Google Scholar 

  9. Ekeland I., Temam R.: Convex analysis and variational problems (Translated from the French, Studies in Mathematics and its Applications, Vol. 1). North-Holland Publishing Co., Amsterdam (1976)

    Google Scholar 

  10. Evstigneev I.V., Schürger K., Taksar M.I.: On the fundamental theorem of asset pricing: random constraints and bang-bang no-arbitrage criteria. Math. Financ. 14(2), 201–221 (2004)

    Article  MATH  Google Scholar 

  11. Föllmer H.: On the representation of semimartingales. Ann. Probab. 1, 580–589 (1973)

    Article  MATH  Google Scholar 

  12. Föllmer H., Schied A.: Convex measures of risk and trading constraints. Financ. Stoch. 6(4), 429–447 (2002)

    Article  MATH  Google Scholar 

  13. Föllmer, H., Schied, A.: Stochastic Finance, volume 27 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, extended edition. An introduction in discrete time (2004)

  14. Haugh M.B., Kogan L.: Pricing American options: a duality approach. Oper. Res. 52(2), 258–270 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ingersoll, J.E.: Theory of Financial Decision Making. Studies in Financial Economics. Rowman & Littlefield Publishers, Inc., Lanham, MD (1987)

  16. Jacod J., Shiryaev A.N.: Local martingales and the fundamental asset pricing theorems in the discrete-time case. Financ. Stoch. 2(3), 259–273 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jaschke S., Küchler U.: Coherent risk measures and good-deal bounds. Financ. Stoch. 5(2), 181–200 (2001)

    Article  MATH  Google Scholar 

  18. Jouini E., Kallal H.: Arbitrage in securities markets with short-sales constraints. Math. Financ. 5(3), 197–232 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jouini E., Kallal H.: Martingales and arbitrage in securities markets with transaction costs. J. Econ. Theory 66(1), 178–197 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jouini E., Napp C.: Arbitrage and investment opportunities. Financ. Stoch. 5(3), 305–325 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jouini E., Napp C., Schachermayer W.: Arbitrage and state price deflators in a general intertemporal framework. J. Math. Econ. 41(6), 722–734 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kabanov Y.M.: Hedging and liquidation under transaction costs in currency markets. Financ. Stoch. 3(2), 237–248 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kabanov Y.M., Safarian M.: Markets with Transaction Costs. Mathematical Theory. Springer Finance. Springer, Berlin (2009)

    Google Scholar 

  24. Kaval K., Molchanov I.: Link-save trading. J. Math. Econ. 42, 710–728 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kreher D.: Hedging of portfolio-valued claims under convex transaction costs and portfolio constraints. Master’s thesis, Humboldt-Universität zu Berlin (2009)

    Google Scholar 

  26. Malo, P., Pennanen, T.: Reduced form modeling of limit order markets. Quant. Financ. (to appear)

  27. Mas-Collel A., Whinston M.D., Green J.R.: Microeconomic Theory. Oxford University Press, New York (1995)

    Google Scholar 

  28. Napp C.: The Dalang–Morton–Willinger theorem under cone constraints. Special issue on equilibrium with asymmetric information. J. Math. Econ. 39(1–2), 111–126 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pennanen T.: Arbitrage and deflators in illiquid markets. Financ. Stoch. 15(1), 57–83 (2011)

    Article  MathSciNet  Google Scholar 

  30. Pennanen T.: Convex duality in stochastic optimization and mathematical finance. Math. Oper. Res. 36(2), 340–362 (2011)

    Article  MathSciNet  Google Scholar 

  31. Pennanen T.: Superhedging in illiquid markets. Math. Financ. 21(3), 519–540 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Pennanen T., Penner I.: Hedging of claims with physical delivery under convex transaction costs. SIAM J. Financ. Math. 1, 158–178 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Rockafellar R.T.: Integrals which are convex functionals. Pac. J. Math. 24, 525–539 (1968)

    MathSciNet  MATH  Google Scholar 

  34. Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, NJ (1970)

  35. Rockafellar R.T.: Conjugate Duality and Optimization. Society for Industrial and Applied Mathematics, Philadelphia, Pa (1974)

    Book  MATH  Google Scholar 

  36. Rockafellar R.T., Wets R.J.-B.: Variational Analysis, volume 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1998)

    Google Scholar 

  37. Rogers L.C.G.: Monte Carlo valuation of American options. Math. Financ. 12(3), 271–286 (2002)

    Article  MATH  Google Scholar 

  38. Rokhlin D.B.: An extended version of the Dalang–Morton–Willinger theorem under convex portfolio constraints. Theory Probab. Appl. 49(3), 429–443 (2005)

    Article  MathSciNet  Google Scholar 

  39. Rokhlin D.B.: The Kreps-Yan theorem for L ∞. Int. J. Math. Math. Sci. 17, 2749–2756 (2005)

    Article  MathSciNet  Google Scholar 

  40. Rokhlin D.B.: Martingale selection problem and asset pricing in finite discrete time. Electron. Commun. Probab. 12, 1–8 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. Schachermayer W.: A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time. Insurance Math. Econ. 11(4), 249–257 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  42. Schachermayer W.: The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Financ. 14(1), 19–48 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, King’s College London, London, UK

    Teemu Pennanen

Authors
  1. Teemu Pennanen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Teemu Pennanen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pennanen, T. Dual representation of superhedging costs in illiquid markets. Math Finan Econ 5, 233–248 (2011). https://doi.org/10.1007/s11579-012-0061-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11579-012-0061-x

Keywords

Search

Navigation