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Dependence Modeling

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Dependent defaults and losses with factor copula models

Damien Ackerer / Thibault Vatter
Published Online: 2017-12-29 | DOI: https://doi.org/10.1515/demo-2017-0022


We present a class of flexible and tractable static factor models for the term structure of joint default probabilities, the factor copula models. These high-dimensional models remain parsimonious with paircopula constructions, and nest many standard models as special cases. The loss distribution of a portfolio of contingent claims can be exactly and efficiently computed when individual losses are discretely supported on a finite grid. Numerical examples study the key features affecting the loss distribution and multi-name credit derivatives prices. An empirical exercise illustrates the flexibility of our approach by fitting credit index tranche prices.

Keywords: credit portfolio; credit derivatives; discrete Fourier transform; factor copula; random loss; survival models

MSC 2010: 60E05; 60E10; 62H05; 62H20; 65T50; 91G20; 91G40; 91G60


  • [1] Aas, K., C. Czado, A. Frigessi, and H. Bakken (2009). Pair-copula constructions of multiple dependence. Insurance Math. Econom. 44(2), 182-198.Google Scholar

  • [2] Ackerer, D. and D. Filipovic (2016). Linear credit risk models. Swiss Finance Institute Research Paper 16-34. Available at https://ssrn.com/abstract=2782455.Google Scholar

  • [3] Albrecher, H., S. A. Ladoucette, and W. Schoutens (2007). A generic one-factor Lévy model for pricing synthetic CDOs. In Advances in Mathematical Finance, pp. 259-277. Birkhäuser, Boston MA.Google Scholar

  • [4] Altman, E., A. Resti, and A. Sironi (2004). Default recovery rates in credit risk modelling: A review of the literature and empirical evidence. Economic Notes 33(2), 183-208.CrossrefGoogle Scholar

  • [5] Amraoui, S. and S. G. Hitier (2008). Optimal stochastic recovery for base correlation. Available at https://ssrn.com/abstract=2719672.Google Scholar

  • [6] Andersen, L. and J. Sidenius (2004). Extensions to the Gaussian copula: random recovery and random factor loadings. J. Credit Risk 1(1), 29-70.Google Scholar

  • [7] Andersen, L., J. Sidenius, and S. Basu (2003). All your hedges in one basket. RISK 16 (11), 67-72.Google Scholar

  • [8] Bedford, T. and R. M. Cooke (2001). Probability density decomposition for conditionally dependent random variables modeled by vines. Ann. Math. Artif. Intell. 32(1-4), 245-268.Google Scholar

  • [9] Bedford, T. and R. M. Cooke (2002). Vines - a new graphical model for dependent random variables. Ann. Statist. 30(4), 1031-1068.CrossrefGoogle Scholar

  • [10] Bielecki, T. R. and M. Rutkowski (2013). Credit Risk: Modeling, Valuation and Hedging. Springer-Verlag, Berlin.Google Scholar

  • [11] Brigo, D., A. Pallavicini, and R. Torresetti (2007). Calibration of CDO tranches with the dynamical generalized-Poisson loss model. RISK 20(6), 70-75.Google Scholar

  • [12] Brigo, D., A. Pallavicini, and R. Torresetti (2010). Credit Models and the Crisis: A Journey Into CDOs, Copulas, Correlations and Dynamic Models. John Wiley & Sons, Chichester.Google Scholar

  • [13] Burtschell, X., J. Gregory, and J.-P. Laurent (2007). Beyond the Gaussian copula: stochastic and local correlation. J. Credit Risk 3(1), 31-62.Google Scholar

  • [14] Burtschell, X., J. Gregory, and J.-P. Laurent (2009). A comparative analysis of CDO pricing models. J. Derivatives 16(4), 9-37.Google Scholar

  • [15] Carr, P. and D. Madan (1999). Option valuation using the fast Fourier transform. J. Comput. Financ. 2(4), 61-73.Google Scholar

  • [16] Collin-Dufresne, P. (2009). A short introduction to correlation markets. J. Financ. Economet. 7(1), 12-29.Google Scholar

  • [17] Cousin, A. and J.-P. Laurent (2008). An overview of factor models for pricing CDO tranches. In R.Cont (Ed.), Frontiers In Quantitative Finance: Volatility and Credit Risk Modeling, pp. 185-216. John Wiley & Sons, Hoboken.Google Scholar

  • [18] Dufresne, D., J. Garrido, and M. Morales (2009). Fourier inversion formulas in option pricing and insurance. Methodol. Comput. Appl. Probab. 11(3), 359-383.CrossrefGoogle Scholar

  • [19] Embrechts, P., R. Grübel, and S. Pitts (1993). Some applications of the fast Fourier transform algorithm in insurance mathematics. Statist. Neerlandica 47(1), 59-75.Google Scholar

  • [20] Filipovic, D. and M. Larsson (2016). Polynomial di_usions and applications in finance. Finance Stoch. 20(4), 931-972.Google Scholar

  • [21] Filipovic, D., L. Overbeck, and T. Schmidt (2011). Dynamic CDO term structure modeling. Math. Finance 21(1), 53-71.CrossrefGoogle Scholar

  • [22] Fouque, J.-P., R. Sircar, and K. Sølna (2009). Multiname and multiscale default modeling. Multiscale Model. Simul. 7(4), 1956-1978.Google Scholar

  • [23] Giesecke, K. (2008). Portfolio credit risk: Top-down versus bottom-up approaches. In R.Cont (Ed.), Frontiers In Quantitative Finance: Volatility and Credit Risk Modeling, pp. 251-267. John Wiley & Sons, Hoboken.Google Scholar

  • [24] Gregory, J. and J.-P. Laurent (2003). I will survive. Risk 16(6), 103-107. Google Scholar

  • [25] Guillaume, F., P. Jacobs, and W. Schoutens (2009). Pricing and hedging of CDO-squared tranches by using a one factor Lévy model. Int. J. Theor. Appl. Finance 12(5), 663-685.CrossrefGoogle Scholar

  • [26] Heiss, F. and V.Winschel (2008). Likelihood approximation by numerical integration on sparse grids. J. Econometrics 144(1), 62-80.Google Scholar

  • [27] Herbertsson, A. (2008). Pricing synthetic CDO tranches in a model with default contagion using the matrix-analytic approach. J. Credit Risk 4(4), 3-35.Google Scholar

  • [28] Hofert, M. and M. Scherer (2011). CDO pricing with nested Archimedean copulas. Quant. Finance 11(5), 775-787.Google Scholar

  • [29] Hull, J. and A. White (2004). Valuation of a CDOand an n-th to default CDSwithout Monte Carlo simulation. J. Derivatives 12(2), 8-23.Google Scholar

  • [30] Hull, J. and A. White (2010). The risk of tranches created from mortgages. Financ. Analysts J. 66(5), 54-67.Google Scholar

  • [31] Hull, J. C. and A. D. White (2006). Valuing credit derivatives using an implied copula approach. J. Derivatives 14(2), 8-28.Google Scholar

  • [32] Jakob, K. and M. Fischer (2014). Quantifying the impact of different copulas in a generalized CreditRisk+ framework. Depend. Model. 2(1), 1-21.Google Scholar

  • [33] Kalemanova, A., B. Schmid, and R. Werner (2007). The normal inverse Gaussian distribution for synthetic CDO pricing. J. Derivatives 14(3), 80-94.Google Scholar

  • [34] Krekel, M. (2008). Pricing distressed CDOs with base correlation and stochastic recovery. Rev. Derivatives Res. 13(3), 219-244.Google Scholar

  • [35] Krupskii, P. and H. Joe (2013). Factor copula models for multivariate data. J. Multivariate Anal. 120, 85-101.CrossrefGoogle Scholar

  • [36] Krupskii, P. and H. Joe (2015). Structured factor copula models: theory, inference and computation. J.Multivariate Anal. 138, 53-73.Google Scholar

  • [37] Laurent, J.-P. and J. Gregory (2005). Basket default swaps, CDOs and factor copulas. J. Risk 7(4), 1-20.Google Scholar

  • [38] Li, D. X. (2000). On default correlation: A copula function approach. J. Fixed Income 9(4), 43-54.Google Scholar

  • [39] Li, D. X. and M. H. Liang (2005). CDO squared pricing using Gaussian mixture model with transformation of loss distribution. Available at https://ssrn.com/abstract=890766.Google Scholar

  • [40] Mai, J.-F., P. Olivares, S. Schenk, and M. Scherer (2014). A multivariate default model with spread and event risk. Appl.Math. Finance 21(1), 51-83.CrossrefGoogle Scholar

  • [41] Mai, J.-F., M. Scherer, and R. Zagst (2012). CIID frailty models and implied copulas. In P. Jaworski, F. Durante, andW. K. Härdle (Eds.), Copulae in Mathematical and Quantitative Finance, pp. 201-230. Springer, Heidelberg.Google Scholar

  • [42] McNeil, A. J., R. Frey, and P. Embrechts (2015). Quantitative Risk Management. Concepts, Techniques and Tools. Revised edition. Princeton University Press.Google Scholar

  • [43] Mortensen, A. (2006). Semi-analytical valuation of basket credit derivatives in intensity-based models. J. Derivatives 13(4), 8-26.Google Scholar

  • [44] Nelsen, R.B. (2006). An Introduction to Copulas. Second edition. Springer, New York.Google Scholar

  • [45] Oh, D. H. and A. J. Patton (2013). Simulated method of moments estimation for copula-based multivariate models. J. Amer. Statist. Assoc. 108(502), 689-700.Google Scholar

  • [46] Oh, D. H. and A. J. Patton (2017). Modeling dependence in high dimensionswith factor copulas. J. Bus. Econom. Statist. 35(1), 139-154.CrossrefGoogle Scholar

  • [47] Sancetta, A. and S. Satchell (2004). The Bernstein copula and its applications to modeling and approximations of multivariate distributions. Econometric Theory 20(3), 535-562.CrossrefGoogle Scholar

  • [48] Schepsmeier, U. and J. Stöber (2014). Derivatives and Fisher information of bivariate copulas. Statist. Papers 55(2), 525-542.CrossrefGoogle Scholar

  • [49] Schloegl, L. and D. O’Kane (2005). A note on the large homogeneous portfolio approximation with the Student-t copula. Finance Stoch. 9(4), 577-584.Google Scholar

  • [50] Schönbucher, P. J. and D. Schubert (2001). Copula-dependent default risk in intensity models. Available at https://ssrn.com/abstract=301968.Google Scholar

  • [51] Vasicek, O. (2002). The distribution of loan portfolio value. RISK 15(12), 160-162.Google Scholar

  • [52] Zhu, S. H. and M. Pykhtin (2007). A guide to modeling counterparty credit risk. GARP Risk Review July-August. Available at https://ssrn.com/abstract=1032522.Google Scholar

About the article

Received: 2017-09-13

Accepted: 2017-12-10

Published Online: 2017-12-29

Published in Print: 2017-12-20

Citation Information: Dependence Modeling, Volume 5, Issue 1, Pages 375–399, ISSN (Online) 2300-2298, DOI: https://doi.org/10.1515/demo-2017-0022.

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© 2018. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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