On the Mechanism of CDOs behind the Current Financial Crisis and Mathematical Modeling with Levy Distributions

H.W. Du, J.L. Wu, W. Yang

.

DOI: 10.4236/iim.2010.22018                

Abstract

This paper aims to reveal the mechanism of Collateralized Debt Obligations (CDOs) and how CDOs extend the current global financial crisis. We first introduce the concept of CDOs and give a brief account of the de-velopment of CDOs. We then explicate the mechanism of CDOs within a concrete example with mortgage deals and we outline the evolution of the current financial crisis. Based on our overview of pricing CDOs in various existing random models, we propose an idea of modeling the random phenomenon with the feature of heavy tail dependence for possible implements towards a new random modeling for CDOs.

Keywords

Collateralized Debt Obligations (CDOs), Cashflow CDO, Synthetic CDO, Mechanism, Financial Crisis, Pricing Models, Levy Stable Distributions

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	J. Hull, “The credit crunch of 2007: What went wrong? Why? What lessons can be learned?” Working Paper, University of Toronto, 2008.
[2]	J. L.Wu and W. Yang, “Pricing CDO tranches in an intensity-based model with the mean-reversion app- roach,” Working Paper, Swansea University, 2009.
[3]	O. Vasicek, “Probability of loss on a loan portfolio,” Working Paper, KMV (Published in Risk, December 2002 with the title Loan Portfolio Value), 1987.
[4]	D. X. Li, “On default correlation: A Copula approach,” Journal of Fixed Income, Vol. 9, No. 4, pp. 43–54, 2000.
[5]	L. Andersen, J. Sidenius, and S. Basu, “All your hedges in one basket,” Risk, pp. 67–72, 2003.
[6]	S. Demarta and A. J. McNeil, “The t copula and related copulas,” International Statistical Review, Vol. 73, No. 1, pp. 111–129, 2005.
[7]	P. Embrechts, F. Lindskog, and A. McNeil, “Modelling dependence with Copulas and applications to risk mana- gement,” In Handbook of Heavy Tailed Distributions in Finance, edited by S. Rachev, Elsevier, 2003.
[8]	R. Frey and A. J. McNeil, “Dependent defaults in models of portfolio credit risk,” Journal of Risk, Vol. 6, No. 1, pp. 59–92, 2003.
[9]	A. Greenberg, R. Mashal, M. Naldi, and L. Schloegl, “Tuning correlation and tail risk to the market prices of liquid tranches,” Lehman Brothers, Quantitative Research Quarterly, 2004.
[10]	R. Mashal and A. Zeevi, “Inferring the dependence stru- cture of financial assets: Empirical evidence and imp- lications,” Working paper, University of Columbia, 2003.
[11]	R. Mashal, M. Naldi, and A. Zeevi, “On the dependence of equity and asset returns,” Risk, Vol. 16, No. 10, pp. 83–87, 2003.
[12]	L. Schloegl and D. O’Kane, “A note on the large homo- geneous portfolio approximation with the student-t Co- pula,” Finance and Stochastics, Vol. 9, No. 4, pp. 577– 584, 2005.
[13]	J. Hull and A. White, “Valuation of a CDO and an n-th to default CDS without Monte Carlo simulation,” Journal of Derivatives, Vol. 12, No. 2, pp. 8–23, 2004.
[14]	A. Cousin and J. P. Laurent, “Comparison results for credit risk portfolios,” Working Paper, ISFA Actuarial School, University of Lyon and BNP-Paribas, 2007.
[15]	P. Sch?nbucher and D. Schubert, “Copula dependent default risk in intensity models,” Working Paper, Bonn University, 2001.
[16]	J. Gregory, and J. P. Laurent, “I will survive,” Risk, Vol. 16, No. 6, pp. 103–107, 2003.
[17]	E. Rogge and P. Sch?nbucher, “Modelling dynamic port- folio credit risk,” Working Paper, Imperial College, 2003.
[18]	D. B. Madan, M. Konikov, and M. Marinescu, “Credit and basket default swaps,” The Journal of Credit Risk, Vol. 2, No. 2, 2006.
[19]	J. P. Laurent and J. Gregory, “Basket default swaps, CDOs and factor copulas,” Journal of Risk, Vol. 7, No. 4, pp. 103–122, 2005.
[20]	A. Friend and E. Rogge, “Correlation at first sight,” Eco- nomic Notes, Vol. 34, No. 2, pp. 155–183, 2005.
[21]	P. Sch?nbucher, “Taken to the limit: Simple and not- so-simple loan loss distributions,” Working Paper, Bonn University, 2002.
[22]	T. Berrada, D. Dupuis, E. Jacquier, N. Papageorgiou, and B. Rémillard, “Credit migration and basket derivatives pricing with copulas,” Journal of Computational Finance, Vol. 10, pp. 43–68, 2006.
[23]	D. Wong, “Copula from the limit of a multivariate binary model,” Working Paper, Bank of America Corporation, 2000.
[24]	Y. Elouerkhaoui,. “Credit risk: Correlation with a diffe- rence,” Working Paper, UBS Warburg, 2003.
[25]	Y. Elouerkhaoui, “Credit derivatives: Basket asympto- tics,” Working Paper, UBS Warburg, 2003.
[26]	K. Giesecke, “A simple exponential model for dependent defaults,” Journal of Fixed Income, Vol. 13, No. 3, pp. 74–83, 2003.
[27]	L. Andersen and J. Sidenius, “Extensions to the Gaussian copula: Random recovery and random factor loadings,” Journal of Credit Risk, Vol. 1, No. 1, pp. 29–70, 2005.
[28]	L. Schloegl, “Modelling oorrelation skew via mixing Copula and uncertain loss at default,” Presentation at the Credit Workshop, Isaac Newton Institute, 2005.
[29]	J. Hull and A. White, “Valuing credit derivatives using an implied copula approach,” Journal of Derivatives, Vol. 14, No. 2, pp. 8–28, 2006.
[30]	C. Albanese, O. Chen, A. Dalessandro, and A. Vidler, “Dynamic credit correlation modeling,” Working Paper, Imperial College, 2005.
[31]	T. R. Hurd and A. Kuznetsov, “Fast CDO computations in affine Markov chains models,” Working Paper, McMaster University, 2006.
[32]	D. Duffie and N. Garleanu, “Risk and valuation of collateralized debt obligations,” Financial Analysts Jour- nal, Vol. 57, No. 1, pp. 41–59, 2001.
[33]	A. Chapovsky, A. Rennie and P. A. C. Tavares, “Sto- chastic intensity modeling for structured credit exotics,” Working paper, Merrill Lynch International, 2006.
[34]	O. E. Barndorff-Nielsen, “Processes of normal inverse Gaussian type,” Finance and Stochastics, Vol. 2, No. 1, pp. 41–68, 1998.
[35]	E. Eberlein, U. Keller and K. Prause, “New insights into smile, mispricing and value at risk,” Journal of Business, Vol. 71, No. 3, pp. 371–406, 1998.
[36]	A. Kalemanova, B. Schmid and R. Werner, “The Normal Inverse Gaussian distribution for synthetic CDO pricing,” Journal of Derivatives, Vol. 14, No. 3, pp. 80–93, 2007.
[37]	Guégan and Houdain, “Collateralized debt obligations pricing and factor models: A new methodology using Normal Inverse Gaussian distributions,” Note the rech- erche IDHE-MORA, ENS Cachan, No. 007-2005, 2005.
[38]	D. B. Madan and E. Seneta, “The Variance Gamma (V.G.) model for share market returns,” Journal of Business, Vol. 63, No. 4, pp. 511–24, 1990.
[39]	D. B. Madan and F. Milne, “Option pricing with V.G. martingale components,” Mathematical Finance, Vol. 1, No. 4, pp. 39–55, 1991.
[40]	D. B. Madan, P. Carr and E. C. Chang, “The variance gamma process and option pricing,” European Finance Review, Vol. 2, No. 1, pp. 79–105, 1998.
[41]	M. Baxter,. “Lévy process dynamic modeling of single- name credits and CDO tranches,” Working Paper, Nomu- ra International, 2006.
[42]	P. Carr, H. Geman, D. Madan, and M. Yor, “The fine structure of asset returns: An empirical investigation,” Journal of Business, Vol. 75, No. 2, pp. 305–332, 2002.
[43]	H. Albrecher, S. A. Landoucette, and W. Schoutens, “A generic one-factor Lévy model for pricing synthetic CDOs,” In: Advances in Mthematical Fiance, R. J. Elliott et al. (eds.), Birkh?user, Boston, 2007.
[44]	Gupton and Stein, “Losscalc: Moody's model for pre- dicting loss given default(LGD),” Working Paper, Moody’s Investor sevices, 2002.
[45]	G. Graziano and L. C. G. Rogers, “A dynamic approach to the modeling of correlation credit derivatives using markov chains,” Working Paper, University of Cambridge, 2005.
[46]	P. Sch?nbucher, “Portfolio losses and the term structure of loss transition rates: A new methodology for the pri- cing portfolio credit derivatives,” Working Paper, ETH Zurich, 2006.
[47]	M. Walker, “Simultaneous calibration to a range of portfolio credit derivatives with a dynamic discrete-time multi-step loss model,” Working Paper, University of Toronto, 2007.
[48]	F. Longstaff and A. Rajan, “An empirical analysis of the pricing of collateralized debt obligations,” Journal of Finance, Vol. 63, No. 2, pp. 529–563, 2008.
[49]	P. Lévy, “Calcul des probabilités,” Gauther-Villars, 1925.
[50]	P. Lévy, “Théorie de l'addition des variables aléatoires,” Gauther-Villars, 1937.
[51]	G. Samorodnitsky and M. S. Taqqu, “Stable non-Gau- ssian random processes: Stochastic models with infinite variance,” Chapman and Hall/CRC, Boca Raton, 1994.
[52]	M. F. Shlesinger, G. M. Zaslavsky, and U. Frisch (Eds.), “Lévy flights and related topics in physics,” Lecture Notes in Physics, Vol. 450, Springer-Verlag, Berlin, 1995.
[53]	B. Mandelbrot, “The Pareto-Lévy law and the distribution of income,” International Economic Review, Vol. 1, pp. 79–106, 1960.
[54]	V. M. Zolotarev, “One-dimensional stable distributions,” American Mathematical Society, R. I. Providence, 1986.
[55]	A. Mortensen, “Semi-analytical valuation of basket credit derivatives in intensity-based models,” Journal of Deri- vatives, Vol. 13, No. 4, pp. 8–26, 2006.