Summary
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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.
O.E. Barndorff-Nielsen. Normal inverse Gaussian distributions and stochastic volatility modelling. Scandinavian Journal of Statistics, 24(1):1–13, 1997.
M. Baxter. Dynamic modelling of single-name credits and CDO tranches. Working Paper - Nomura Fixed Income Quant Group, 2006.
J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge University Press, 1996.
B. Grigelionis. Processes of Meixner type. Lithuanian Mathematical Journal, 39(1):33–41, 1999.
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.
A. Kalemanova, B. Schmid, and R. Werner. The normal inverse Gaussian distribution for synthetic CDO pricing. Technical Report, 2005.
D.B. Madan and F. Milne. Option pricing with VG martingale components. Mathematical Finance, 1(4):39–55, 1991.
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.
D.B. Madan and E. Seneta. The Variance-Gamma (V.G.) model for share market returns. Journal of Business, 63(4):511–524, 1990.
D.B. Madan, P.P. Carr, and E.C. Chang. The variance gamma process and option pricing. European Finance Review, 2:79–105, 1998.
T. Moosbrucker. Pricing CDOs with correlated variance gamma distributions. Research Report, Department of Banking, University of Cologne, 2006.
T. Rydberg. Generalized hyperbolic diffusions with applications towards finance. Research Report No. 342, Department of Theoretical Statistics, Aarhus University, 1996.
T. Rydberg. The normal inverse Gaussian Lévy process: Simulations and approximation. Research Report No. 344, Department of Theoretical Statistics, Aarhus University, 1996.
T. Rydberg. A note on the existence of unique equivalent martingale measures in a Markovian setting. Finance and Stochastics, 1:251–257, 1997.
K. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge University Press, 2000.
W. Schoutens. Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics 146. Springer-Verlag, 2000.
W. Schoutens. The Meixner process in finance. EURANDOM Report 2001-002, EURANDOM, Eindhoven, 2001.
W. Schoutens. Meixner processes: Theory and applications in finance. EURAN-DOM Report 2002-004, EURANDOM, Eindhoven, 2002.
W. Schoutens. Lévy Processes in Finance - Pricing Financial Derivatives. John Wiley & Sons, 2003.
W. Schoutens and J.L. Teugels. Lévy processes, polynomials and martingales. Communications in Statistics - Stochastic Models, 14(1–2):335–349, 1998.
O. Vasicek. Probability of loss on loan portfolio. Technical Report, KMV Corporation, 1987.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Birkhäuser Boston
About this chapter
Cite this chapter
Albrecher, H., Ladoucette, S.A., Schoutens, W. (2007). A Generic One-Factor Lévy Model for Pricing Synthetic CDOs. In: Fu, M.C., Jarrow, R.A., Yen, JY.J., Elliott, R.J. (eds) Advances in Mathematical Finance. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4545-8_14
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4545-8_14
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4544-1
Online ISBN: 978-0-8176-4545-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)