Skip to main content
SpringerLink
Log in

Local Risk-Minimization for Defaultable Claims with Recovery Process

  • Published:
Applied Mathematics & Optimization Aims and scope Submit manuscript

Abstract

We study the local risk-minimization approach for defaultable claims with random recovery at default time, seen as payment streams on the random interval 〚0,τ∧T〛, where T denotes the fixed time-horizon. We find the pseudo-locally risk-minimizing strategy in the case when the agent information takes into account the possibility of a default event (local risk-minimization with G-strategies) and we provide an application in the case of a corporate bond. We also discuss the problem of finding a pseudo-locally risk-minimizing strategy if we suppose the agent obtains her information only by observing the non-defaultable assets.

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. Biagini, F., Cretarola, A.: Quadratic hedging methods for defaultable claims. Appl. Math. Optim. 56(3), 425–443 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Biagini, F., Cretarola, A.: Local risk-minimization for defaultable markets. Math. Finance 19(4), 669–689 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bielecki, T.R., Jeanblanc, M.: Indifference pricing of defaultable claims. In: Carmona, R. (ed.) Indifference Pricing, Theory and Applications, Chap. 6, pp. 211–240. Princeton University Press, Princeton (2009)

    Google Scholar 

  4. Bielecki, T.R., Rutkowski, M.: Credit Risk: Modelling, Valuation and Hedging, 2nd edn. Springer-Finance. Springer, Berlin (2004)

    Google Scholar 

  5. Bielecki, T.R., Jeanblanc, M., Rutkowski, M.: Pricing and hedging of credit risk: Replication and mean-variance approaches. II. In: Yin, G., Zhang, Q. (eds.) Mathematics of Finance. Contemporary Mathematics, vol. 351, pp. 55–64. AMS, Providence (2004)

    Chapter  Google Scholar 

  6. Bielecki, T.R., Jeanblanc, M., Rutkowski, M.: Hedging of defaultable claims. In: Carmona, R., Cinlar, E., Ekeland, I., Jouini, E., Touzi, N. (eds.) Paris-Princeton Lectures on Mathematical Finance 2003. Lecture Notes in Mathematics, vol. 1847, pp. 1–132. Springer, Berlin (2004)

    Chapter  Google Scholar 

  7. Bielecki, T.R., Jeanblanc, M., Rutkowski, M.: Pricing and hedging of credit risk: Replication and mean-variance approaches. I. In: Yin, G., Zhang, Q. (eds.) Mathematics of Finance. Contemporary Mathematics, vol. 351, pp. 37–53. AMS, Providence (2004)

    Chapter  Google Scholar 

  8. Bielecki, T.R., Jeanblanc, M., Rutkowski, M.: Hedging of credit derivatives in models with totally unexpected default. In: Akahori, J., Ogawa, S., Watanabe, S. (eds.) Stochastic Processes and Applications to Mathematical Finance. Proceedings of the 5th Ritsumeikan International Symposium, Ritsumeikan University, Japan, 3–6 March 2005, pp. 35–100. World Scientific, Singapore (2006)

    Chapter  Google Scholar 

  9. Bielecki, T.R., Jeanblanc, M., Rutkowski, M.: Pricing and trading credit default swaps in a hazard process model. Ann. Appl. Probab. 18(6), 2495–2529 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bielecki, T.R., Jeanblanc, M., Rutkowski, M.: Credit Risk Modeling. Osaka University CSFI Lecture Notes Series, vol. 2. Osaka University Press, Osaka (2009)

    Google Scholar 

  11. Choulli, T., Stricker, C.: Deux applications de la décomposition de Galtchouk-Kunita-Watanabe. In: Séminaire de Probabilités XXX. Lecture Notes in Mathematics, vol. 1626, pp. 12–23. Springer, Berlin (1996)

    Chapter  Google Scholar 

  12. Delbaen, F., Schachermayer, W.: The existence of absolutely continuous local martingale measures. Ann. Appl. Probab. 5(4), 926–945 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dellacherie, C., Meyer, P.A.: Probabilities and Potential. Mathematics Studies, vol. 29. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  14. Dellacherie, C., Meyer, P.A.: Probabilities and Potential B: Theory of Martingales. Mathematics Studies, vol. 72. North-Holland, Amsterdam (1982)

    MATH  Google Scholar 

  15. Dellacherie, C., Maisonneuve, B., Meyer, P.A.: Probabilités et potentiel. Chapitres XVII–XXIV. Hermann, Paris (1992)

    Google Scholar 

  16. Duffie, D.: Credit risk modeling with affine processes. J. Bank. Finance 29, 2751–2802 (2005). Cattedra Galileana Lectures, Scuola Normale, Pisa, April 2002

    Article  Google Scholar 

  17. Filipović, D.: Interest rate models. Lecture notes, LMU University of München (2006)

  18. Föllmer, H., Schweizer, M.: Hedging of contingent claims under incomplete information. In: Davis, M.H.A., Elliot, R.J. (eds.) Applied Stochastic Analysis, pp. 389–414. Gordon and Breach, London (1991)

    Google Scholar 

  19. Jacod, J.: Grossissement initial, hypothèse \((\mathcal{H}')\) et théorème de Girsanov. In: Séminaire de Calcul Stochastique 1982–83. Lecture Notes in Mathematics, vol. 1118. Springer, Berlin (1987)

    Google Scholar 

  20. Jeanblanc, M., Le Cam, Y.: Immersion property and credit risk modelling. In: Optimality and Risk—Modern Trends in Mathematical Finance, pp. 99–132. Springer, Berlin (2009)

    Chapter  Google Scholar 

  21. Jeanblanc, M., Le Cam, Y.: Progressive enlargement of filtration with initial times. Stoch. Process. Appl. 119(8), 2523–2543 (2009)

    Article  MATH  Google Scholar 

  22. Jeanblanc, M., Rutkowski, M.: Default risk and hazard process. In: Mathematical Finance—Bachelier Congress, 2000, pp. 281–312. Springer, Berlin (2002). MR1960569

    Google Scholar 

  23. Jeulin, T.: Semi-martingales et grossissements d’une filtration. Lecture Notes in Mathematics, vol. 833. Springer, Berlin (1980)

    Google Scholar 

  24. Kusuoka, S.: A remark on default risk models. Adv. Math. Econ. 1, 69–82 (1999)

    Article  MathSciNet  Google Scholar 

  25. Schweizer, M.: On the minimal martingale measure and the Föllmer-Schweizer decomposition. Stoch. Anal. Appl. 13, 573–599 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, LMU, Theresienstraße 39, 80333, Munich, Germany

    Francesca Biagini

  2. Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, via Vanvitelli 1, 06123, Perugia, Italy

    Alessandra Cretarola

Authors
  1. Francesca Biagini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alessandra Cretarola
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francesca Biagini.

Additional information

Communicating Editor: Bernt Øksendal.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Biagini, F., Cretarola, A. Local Risk-Minimization for Defaultable Claims with Recovery Process. Appl Math Optim 65, 293–314 (2012). https://doi.org/10.1007/s00245-011-9155-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-011-9155-8

Keywords

Search

Navigation