Doubly Stochastic CDO Term Structures
This paper provides a general framework for doubly stochastic term structure models for portfolio of credits, such as collateralized debt obligations (CDOs). We introduce the defaultable (T, x)-bonds, which pay one if the aggregated loss process in the underlying pool of the CDO has not exceededx at maturityT, and zero else. Necessary and sufficient conditions on the stochastic term structure movements for the absence of arbitrage are given. Moreover, we show that any exogenous specification of the forward rates and spreads volatility curve actually yields a consistent loss process and thus an arbitrage-free family of (T, x)-bond prices. For the sake of analytical and computational efficiency we then develop a tractable class of affine term structure models.
KeywordsAffine term structure collateralized debt obligations loss process single tranche CDO term structure of forward spreads
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- 1.N. Bennani, The forward loss model: A dynamic term structure approach for the pricing of portfolio credit derivatives, Working Paper, 2005.Google Scholar
- 6.R. Cont and A. Minca, Recovering portfolio default intensities implied by CDO quotes, Financial Engineering Report No. 2008-01, Columbia University Center for Financial Engineering, 2008.Google Scholar
- 7.R. Cont and I. Savescu, Forward equations for portfolio credit derivatives, In:R.Cont, editor, Frontiers in Quantitative Finance: Volatility and Credit Risk Modeling, Wiley Finance Series, chapter 11, pages 269–293. John Wiley& Sons, Inc., Hoboken, New Jersey, 2009.Google Scholar
- 9.P. Ehlers and P. Schönbucher, Pricing interest rate-sensitive credit portfolio derivatives, Working Paper, ETH Zurich, 2006.Google Scholar
- 11.S.N. Ethier and T.G. Kurtz, Markov Processes. Characterization and Convergence, John Wiley & Sons, 1986.Google Scholar
- 13.J. Jacod and P. Protter, Quelques remarques sur un nouveau type d’équations différentielles stochastiques, Seminar on Probability, XVI, Lecture Notes in Math., 920 (1982), 447–458, Springer, Berlin.Google Scholar
- 14.J. Jacod and A.N. Shiryaev, Limit Theorems for Stochastic Processes, Springer, 1987.Google Scholar
- 16.J.P. Laurent and J. Gregory, Basket default swaps, CDOs and factor copulas, Journal of Risk, 7 (2005), 103–122.Google Scholar
- 17.A. McNeil, R. Frey, and P. Embrechts, Quantitative Risk Management: Concepts, Techniques and Tools, Princeton University Press, 2005.Google Scholar
- 18.P. Schönbucher, Portfolio losses and the term structure of loss transition rates: A new methodology for the pricing of portfolio credit derivatives, Working Paper, ETHGoogle Scholar
- 19.Zürich, 2005.Google Scholar