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A tractable LIBOR model with default risk

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Abstract

We develop a model for the dynamic evolution of default-free and defaultable interest rates in a LIBOR framework. Utilizing the class of affine processes, this model produces positive LIBOR rates and spreads, while the dynamics are analytically tractable under defaultable forward measures. This leads to explicit formulas for CDS spreads, while semi-analytical formulas are derived for other credit derivatives. Finally, we give an application to counterparty risk.

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References

  1. Brace A., Ga̧tarek D., Musiela M.: The market model of interest rate dynamics. Math. Financ. 7, 127–155 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brigo D., Mercurio F.: Interest Rate Models: Theory and Practice. 2nd edn. Springer, Berlin (2006)

    Google Scholar 

  3. Brémaud P., Yor M.: Changes of filtrations and of probability measures. Z. Wahrsch. Verw. Geb. 45, 269–295 (1978)

    Article  MATH  Google Scholar 

  4. Bielecki T.R., Rutkowski M.: Credit Risk: Modeling, Valuation and Hedging. Springer, Berlin (2002)

    Google Scholar 

  5. Crépey, S., Grbac, Z., Nguyen, H.-N.: A multiple-curve HJM model of interbank risk. Math. Financ. Econ. 6, 155-190 (2012)

    Google Scholar 

  6. Duffie D., Filipović D., Schachermayer W.: Affine processes and applications in finance. Ann. Appl. Probab. 13, 984–1053 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Duffie D., Gârleanu N.: Risk and valuation of collateralized debt obligations. Financ. Anal. J. 57, 41–59 (2001)

    Article  Google Scholar 

  8. Eberlein E., Glau K., Papapantoleon A.: Analysis of Fourier transform valuation formulas and applications. Appl. Math. Financ. 17, 211–240 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Elliott R., Jeanblanc M., Yor M.: On models of default risk. Math. Financ. 10, 179–195 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Eberlein E., Kluge W., Schönbucher Ph. J.: The Lévy LIBOR model with default risk. J. Credit Risk 2, 3–42 (2006)

    Google Scholar 

  11. Filipović D.: A general characterization of one factor affine term structure models. Financ. Stoch. 5, 389–412 (2001)

    Article  MATH  Google Scholar 

  12. Filipović D.: Time-inhomogeneous affine processes. Stoch. Process. Appl. 115, 639–659 (2005)

    Article  MATH  Google Scholar 

  13. Filipović, D., Trolle, A.: The term structure of interbank risk. Preprint, EPFL (2011)

  14. Gatarek D., Bachert P., Maksymiuk R.: The LIBOR Market Model in Practice. Wiley, Chichester (2006)

    Google Scholar 

  15. Grbac, Z.: Credit risk in Lévy LIBOR modeling: rating based approach. PhD thesis, Univ. Freiburg (2010)

  16. Hubalek, F., Kallsen, J.: Variance-optimal hedging and Markowitz-efficient portfolios for multivariate processes with stationary independent increments with and without constraints. Working paper, TU München (2005)

  17. Huge B., Lando D.: Swap pricing with two-sided default risk in a rating-based model. Eur. Financ. Rev. 3, 239–268 (1999)

    Article  MATH  Google Scholar 

  18. Hull J., White A.: The impact of default risk on the prices of options and other derivative securities. J. Banking Financ. 19, 299–322 (1995)

    Article  Google Scholar 

  19. Hurd T.R., Zhou Z.: A Fourier transform method for spread option pricing. SIAM J. Financ. Math. 1, 142–157 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jamshidian F.: LIBOR and swap market models and measures. Financ. Stoch. 1, 293–330 (1997)

    Article  MATH  Google Scholar 

  21. Jeanblanc, M., Le Cam, Y.: Reduced form modelling for credit risk. Working paper (2008)

  22. Jeanblanc, M., Rutkowski, M.: Modelling of default risk: an overview. In: Mathematical Finance: Theory and Practice. Higher Education Press, Beijing (2000)

  23. Johnson H., Stulz R.: The pricing of options with default risk. J. Financ. 42, 267–280 (1987)

    Article  Google Scholar 

  24. Jacod J., Shiryaev A.N.: Limit Theorems for Stochastic Processes. 2nd edn. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  25. Jarrow R.A., Turnbull S.M.: Pricing derivatives on financial securities subject to credit risk. J. Financ. 50, 53–85 (1995)

    Article  Google Scholar 

  26. Kluge, W.: Time-inhomogeneous Lévy processes in interest rate and credit risk models. PhD thesis, Univ. Freiburg (2005)

  27. Kokholm T., Nicolato E.: Sato processes in default modelling. Appl. Math. Financ. 17, 377–397 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Keller-Ressel, M.: Affine processes: Theory and applications to finance. PhD thesis, TU Vienna (2008)

  29. Keller-Ressel, M., Papapantoleon, A., Teichmann, J.: The affine LIBOR models. Math. Finance 2011 (forthcoming)

  30. Lotz C., Schlögl L.: Default risk in a market model. J. Banking Financ. 24, 301–327 (2000)

    Article  Google Scholar 

  31. Papapantoleon A.: Old and new approaches to LIBOR modeling. Stat. Neerlandica 64, 257–275 (2010)

    Article  MathSciNet  Google Scholar 

  32. Papapantoleon, A., Schoenmakers, J., Skovmand, D.: Efficient and accurate log-Lévy approximations to Lévy-driven LIBOR models. J. Comput. Financ. 15(4), 3–44 (2012)

    Google Scholar 

  33. Schönbucher, P.J.: A LIBOR market model with default risk. Working paper, University of Bonn (2000)

  34. Sandmann, K., Sondermann, D., Miltersen, K.R.: Closed form term structure derivatives in a Heath–Jarrow–Morton model with log-normal annually compounded interest rates. In: Proceedings of the Seventh Annual European Futures Research Symposium Bonn, pp. 145–165, (1995). Chicago Board of Trade

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Authors and Affiliations

  1. Départment Mathématiques, Université d’Évry Val d’Essonne, 23 Boulevard de France, 91037, Évry Cedex, France

    Zorana Grbac

  2. Institute of Mathematics, TU Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Antonis Papapantoleon

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  1. Zorana Grbac
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  2. Antonis Papapantoleon
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Correspondence to Antonis Papapantoleon.

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Grbac, Z., Papapantoleon, A. A tractable LIBOR model with default risk. Math Finan Econ 7, 203–227 (2013). https://doi.org/10.1007/s11579-012-0090-5

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  • DOI: https://doi.org/10.1007/s11579-012-0090-5

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