A Generic One-Factor Lévy Model for Pricing Synthetic CDOs

  • Hansjörg Albrecher
  • Sophie A. Ladoucette
  • Wim Schoutens
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


The one-factor Gaussian model is well known not to fit the prices of the different tranches of a collateralized debt obligation (CDO) simultaneously, leading to the implied correlation smile. Recently, other one-factor models based on different distributions have been proposed. Moosbrucker [12] used a one-factor Variance-Gamma (VG) model, Kalemanova et al. [7] and Guégan and Houdain [6] worked with a normal inverse Gaussian (NIG) factor model, and Baxter [3] introduced the Brownian Variance-Gamma (BVG) model. These models bring more flexibility into the dependence structure and allow tail dependence. We unify these approaches, describe a generic one-factor Lévy model, and work out the large homogeneous portfolio (LHP) approximation. Then we discuss several examples and calibrate a battery of models to market data.

Key words

Lévy processes collateralized debt obligation (CDO) credit risk credit default large homogeneous portfolio approximation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    O.E. Barndorff-Nielsen. Normal inverse Gaussian distributions and the modelling of stock returns. Research Report No. 300, Department of Theoretical Statistics, Aarhus University, 1995.Google Scholar
  2. 2.
    O.E. Barndorff-Nielsen. Normal inverse Gaussian distributions and stochastic volatility modelling. Scandinavian Journal of Statistics, 24(1):1–13, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    M. Baxter. Dynamic modelling of single-name credits and CDO tranches. Working Paper - Nomura Fixed Income Quant Group, 2006.Google Scholar
  4. 4.
    J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge University Press, 1996.Google Scholar
  5. 5.
    B. Grigelionis. Processes of Meixner type. Lithuanian Mathematical Journal, 39(1):33–41, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    D. Guégan and J. Houdain. Collateralized debt obligations pricing and factor models: A new methodology using normal inverse Gaussian distributions. Research Report IDHE-MORA No. 07-2005, ENS Cachan, 2005.Google Scholar
  7. 7.
    A. Kalemanova, B. Schmid, and R. Werner. The normal inverse Gaussian distribution for synthetic CDO pricing. Technical Report, 2005.Google Scholar
  8. 8.
    D.B. Madan and F. Milne. Option pricing with VG martingale components. Mathematical Finance, 1(4):39–55, 1991.zbMATHCrossRefGoogle Scholar
  9. 9.
    D.B. Madan and E. Seneta. Chebyshev polynomial approximations and characteristic function estimation. Journal of the Royal Statistical Society, Series B, 49(2):163–169, 1987.MathSciNetGoogle Scholar
  10. 10.
    D.B. Madan and E. Seneta. The Variance-Gamma (V.G.) model for share market returns. Journal of Business, 63(4):511–524, 1990.CrossRefGoogle Scholar
  11. 11.
    D.B. Madan, P.P. Carr, and E.C. Chang. The variance gamma process and option pricing. European Finance Review, 2:79–105, 1998.zbMATHCrossRefGoogle Scholar
  12. 12.
    T. Moosbrucker. Pricing CDOs with correlated variance gamma distributions. Research Report, Department of Banking, University of Cologne, 2006.Google Scholar
  13. 13.
    T. Rydberg. Generalized hyperbolic diffusions with applications towards finance. Research Report No. 342, Department of Theoretical Statistics, Aarhus University, 1996.Google Scholar
  14. 14.
    T. Rydberg. The normal inverse Gaussian Lévy process: Simulations and approximation. Research Report No. 344, Department of Theoretical Statistics, Aarhus University, 1996.Google Scholar
  15. 15.
    T. Rydberg. A note on the existence of unique equivalent martingale measures in a Markovian setting. Finance and Stochastics, 1:251–257, 1997.zbMATHCrossRefGoogle Scholar
  16. 16.
    K. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge University Press, 2000.Google Scholar
  17. 17.
    W. Schoutens. Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics 146. Springer-Verlag, 2000.Google Scholar
  18. 18.
    W. Schoutens. The Meixner process in finance. EURANDOM Report 2001-002, EURANDOM, Eindhoven, 2001.Google Scholar
  19. 19.
    W. Schoutens. Meixner processes: Theory and applications in finance. EURAN-DOM Report 2002-004, EURANDOM, Eindhoven, 2002.Google Scholar
  20. 20.
    W. Schoutens. Lévy Processes in Finance - Pricing Financial Derivatives. John Wiley & Sons, 2003.Google Scholar
  21. 21.
    W. Schoutens and J.L. Teugels. Lévy processes, polynomials and martingales. Communications in Statistics - Stochastic Models, 14(1–2):335–349, 1998.zbMATHMathSciNetGoogle Scholar
  22. 22.
    O. Vasicek. Probability of loss on loan portfolio. Technical Report, KMV Corporation, 1987.Google Scholar

Copyright information

© Birkhäuser Boston 2007

Authors and Affiliations

  • Hansjörg Albrecher
    • 1
    • 2
  • Sophie A. Ladoucette
    • 3
  • Wim Schoutens
    • 3
  1. 1.Austrian Academy of SciencesJohann Radon InstituteAltenbergerstrasse 69Austria
  2. 2.Department of MathematicsGraz University of TechnologySteyrergasse 30Austria
  3. 3.Department of MathematicsKatholieke Universiteit LeuvenBelgium

Personalised recommendations