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
Brace A., Ga̧tarek D., Musiela M.: The market model of interest rate dynamics. Math. Financ. 7, 127–155 (1997)
Brigo D., Mercurio F.: Interest Rate Models: Theory and Practice. 2nd edn. Springer, Berlin (2006)
Brémaud P., Yor M.: Changes of filtrations and of probability measures. Z. Wahrsch. Verw. Geb. 45, 269–295 (1978)
Bielecki T.R., Rutkowski M.: Credit Risk: Modeling, Valuation and Hedging. Springer, Berlin (2002)
Crépey, S., Grbac, Z., Nguyen, H.-N.: A multiple-curve HJM model of interbank risk. Math. Financ. Econ. 6, 155-190 (2012)
Duffie D., Filipović D., Schachermayer W.: Affine processes and applications in finance. Ann. Appl. Probab. 13, 984–1053 (2003)
Duffie D., Gârleanu N.: Risk and valuation of collateralized debt obligations. Financ. Anal. J. 57, 41–59 (2001)
Eberlein E., Glau K., Papapantoleon A.: Analysis of Fourier transform valuation formulas and applications. Appl. Math. Financ. 17, 211–240 (2010)
Elliott R., Jeanblanc M., Yor M.: On models of default risk. Math. Financ. 10, 179–195 (2000)
Eberlein E., Kluge W., Schönbucher Ph. J.: The Lévy LIBOR model with default risk. J. Credit Risk 2, 3–42 (2006)
Filipović D.: A general characterization of one factor affine term structure models. Financ. Stoch. 5, 389–412 (2001)
Filipović D.: Time-inhomogeneous affine processes. Stoch. Process. Appl. 115, 639–659 (2005)
Filipović, D., Trolle, A.: The term structure of interbank risk. Preprint, EPFL (2011)
Gatarek D., Bachert P., Maksymiuk R.: The LIBOR Market Model in Practice. Wiley, Chichester (2006)
Grbac, Z.: Credit risk in Lévy LIBOR modeling: rating based approach. PhD thesis, Univ. Freiburg (2010)
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)
Huge B., Lando D.: Swap pricing with two-sided default risk in a rating-based model. Eur. Financ. Rev. 3, 239–268 (1999)
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)
Hurd T.R., Zhou Z.: A Fourier transform method for spread option pricing. SIAM J. Financ. Math. 1, 142–157 (2010)
Jamshidian F.: LIBOR and swap market models and measures. Financ. Stoch. 1, 293–330 (1997)
Jeanblanc, M., Le Cam, Y.: Reduced form modelling for credit risk. Working paper (2008)
Jeanblanc, M., Rutkowski, M.: Modelling of default risk: an overview. In: Mathematical Finance: Theory and Practice. Higher Education Press, Beijing (2000)
Johnson H., Stulz R.: The pricing of options with default risk. J. Financ. 42, 267–280 (1987)
Jacod J., Shiryaev A.N.: Limit Theorems for Stochastic Processes. 2nd edn. Springer, Berlin (2003)
Jarrow R.A., Turnbull S.M.: Pricing derivatives on financial securities subject to credit risk. J. Financ. 50, 53–85 (1995)
Kluge, W.: Time-inhomogeneous Lévy processes in interest rate and credit risk models. PhD thesis, Univ. Freiburg (2005)
Kokholm T., Nicolato E.: Sato processes in default modelling. Appl. Math. Financ. 17, 377–397 (2010)
Keller-Ressel, M.: Affine processes: Theory and applications to finance. PhD thesis, TU Vienna (2008)
Keller-Ressel, M., Papapantoleon, A., Teichmann, J.: The affine LIBOR models. Math. Finance 2011 (forthcoming)
Lotz C., Schlögl L.: Default risk in a market model. J. Banking Financ. 24, 301–327 (2000)
Papapantoleon A.: Old and new approaches to LIBOR modeling. Stat. Neerlandica 64, 257–275 (2010)
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)
Schönbucher, P.J.: A LIBOR market model with default risk. Working paper, University of Bonn (2000)
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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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
Keywords
- LIBOR rates
- Default risk
- Affine processes
- Affine LIBOR models
- Analytically tractable models
- CDS spread
- Counterparty risk