Skip to main content
SpringerLink
Log in

Cross hedging with stochastic correlation

  • Published:
Finance and Stochastics Aims and scope Submit manuscript

Abstract

This paper is concerned with the study of quadratic hedging of contingent claims with basis risk. We extend existing results by allowing the correlation between the hedging instrument and the underlying of the contingent claim to be random itself. We assume that the correlation process ρ evolves according to a stochastic differential equation with values between the boundaries −1 and 1. We keep the correlation dynamics general and derive an integrability condition on the correlation process that allows to describe and compute the quadratic hedge by means of a simple hedging formula that can be directly implemented. Furthermore, we show that the conditions on ρ are fulfilled by a large class of dynamics. The theory is exemplified by various explicitly given correlation dynamics.

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. Ankirchner, S., Imkeller, P., Reis, G.: Pricing and hedging of derivatives based on non-tradeable underlyings. Math. Finance 20, 289–312 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. Boortz, C.: Modelling correlation risk. Master thesis. Technische Universität Berlin (2008). http://www.qpl.db.com/en/binaer_view.asp?binaernr=83

  3. Davis, M.H.A.: Optimal hedging with basis risk. In: Kabanov, R.L.Y., Stoyanov, J. (eds.) From Stochastic Calculus to Mathematical Finance, pp. 169–187. Springer, Berlin (2006)

    Chapter  Google Scholar 

  4. Durrett, R.: Probability: Theory and Examples, 2nd edn. Duxbury Press, Belmont (1996)

    Google Scholar 

  5. El Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Finance 7, 1–71 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. Frei, C., Schweizer, M.: Exponential utility indifference valuation in two Brownian settings with stochastic correlation. Adv. Appl. Probab. 40, 401–423 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Frei, C., Schweizer, M.: Exponential utility indifference valuation in a general semimartingale model. In: Delbaen, F., Rásonyi, M., Stricker, C. (eds.) Optimality and Risk—Modern Trends in Mathematical Finance. The Kabanov Festschrift, pp. 49–86. Springer, Berlin (2009)

    Chapter  Google Scholar 

  8. Henderson, V.: Valuation of claims on nontraded assets using utility maximization. Math. Finance 12, 351–373 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hulley, H., McWalter, T.A.: Quadratic hedging of basis risk. Research Paper Series 225, Quantitative Finance Research Centre, University of Technology, Sydney, June 2008. http://www.business.uts.edu.au/qfrc/research/research_papers/rp225.pdf

  10. Imkeller, P., Réveillac, A., Richter, A.: Differentiability of quadratic BSDEs generated by continuous martingales and hedging in incomplete markets. arXiv:0907.0941

  11. Karlin, S., Taylor, H.M.: A Second Course in Stochastic Processes. Academic Press, New York (1981)

    MATH  Google Scholar 

  12. Monoyios, M.: Performance of utility-based strategies for hedging basis risk. Quant. Finance 4, 245–255 (2004)

    Article  MathSciNet  Google Scholar 

  13. Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Probability and its Applications. Springer, Berlin (2006)

    MATH  Google Scholar 

  14. Protter, P.E.: Stochastic Integration and Differential Equations, 2nd edn. Applications of Mathematics, vol. 21. Springer, Berlin (2004). Stochastic Modelling and Applied Probability

    MATH  Google Scholar 

  15. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Grundlehren der Mathematischen Wissenschaften, vol. 293. Springer, Berlin (1999)

    MATH  Google Scholar 

  16. Schweizer, M.: A guided tour through quadratic hedging approaches. In: Jouini, E., Cvitanić, J., Musiela, M. (eds.) Option Pricing, Interest Rates and Risk Management, pp. 538–574. Cambridge Univ. Press, Cambridge (2001)

    Google Scholar 

  17. Schweizer, M.: Local risk-minimization for multidimensional assets and payment streams. Banach Cent. Publ. 83, 213–229 (2008)

    Article  MathSciNet  Google Scholar 

  18. van Emmerich, C.: Modelling correlation as a stochastic process. Technical report, Bergische Universität Wuppertal, March 2006. http://www-num.math.uni-wuppertal.de//Files/amna_06_03.pdf

Download references

Author information

Authors and Affiliations

  1. Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Stefan Ankirchner

  2. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Gregor Heyne

Authors
  1. Stefan Ankirchner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gregor Heyne
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gregor Heyne.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ankirchner, S., Heyne, G. Cross hedging with stochastic correlation. Finance Stoch 16, 17–43 (2012). https://doi.org/10.1007/s00780-010-0148-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00780-010-0148-2

Keywords

Mathematics Subject Classification (2000)

JEL Classification

Search

Navigation