Topics in Statistical Simulation pp 333-341 | Cite as
Simulating from the Copula that Generates the Maximal Probability for a Joint Default Under Given (Inhomogeneous) Marginals
Abstract
Starting from two default times with given univariate distribution functions, the copula which maximizes the probability of a joint default can be computed in closed form. This result can be retrieved from Markov-chain theory, where it is known under the terminology “maximal coupling”, but typically formulated without copulas. For inhomogeneous marginals the solution is not represented by the comonotonicity copula, opposed to a common modeling (mal-)practice in the financial industry. Moreover, a stochastic model that respects the marginal laws and attains the upper-bound copula for joint defaults can be inferred from the maximal-coupling construction. We formulate and illustrate this result in the context of copula theory and motivate its importance for portfolio-credit risk modeling. Moreover, we present a sampling strategy for the “maximal-coupling copula”.
Notes
Acknowledgements
The authors would like to thank Alfred Müller and an anonymous referee for valuable remarks.
References
- 1.den Hollander, W.T.F.: Probability Theory: Coupling. Lecture Notes Leiden University. Available at http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf (2010)
- 2.Embrechts, P.: Copulas: a personal view. J. Risk Insur. 76, 639–650 (2009)CrossRefGoogle Scholar
- 3.Embrechts, P., Puccetti, G.: Bounds for functions of dependent risks. Financ. Stoch. 10(3), 341–352 (2006)CrossRefMATHMathSciNetGoogle Scholar
- 4.Embrechts, P., Puccetti, G.: Bounds for functions of multivariate risks. J. Multivar. Anal. 97(2), 526–547 (2006)CrossRefMATHMathSciNetGoogle Scholar
- 5.Embrechts, P., Höing, P.A., Puccetti, G.: Worst VaR scenarios. Insur. Math. Econ. 37(1), 115–134 (2005)CrossRefMATHGoogle Scholar
- 6.Genest, C., Nešlehová, J.: A primer on copulas for count data. Astin Bull. 37, 475–515 (2007)CrossRefMATHMathSciNetGoogle Scholar
- 7.Marshall, A.W.: Copulas, marginals and joint distributions. In: Rüschendorf, L., Schweizer, B., Taylor, M.D. (eds.) Distributions with Fixed Marginals and Related Topics, pp. 213–222. Institute of Mathematical Statistics, Hayward (1996)CrossRefGoogle Scholar
- 8.Mikosch, T.: Copulas: tales and facts. Extremes 9, 55–62 (2006)CrossRefMathSciNetGoogle Scholar
- 9.Morini, M.: Mistakes in the market approach to correlation: a lesson for future stress-testing. In: Wehn, C.S., et al. (eds.) Rethinking Valuation and Pricing Models, pp. 331–359. Academic, Oxford (2012)Google Scholar
- 10.Puccetti, G., Rüschendorf, L.: Bounds for joint portfolios of dependent risks. Stat. Risk Model. 29(2), 107–132 (2012)CrossRefMathSciNetGoogle Scholar
- 11.Puccetti, G., Rüschendorf, L.: Computation of sharp bounds on the distribution of a function of dependent risks. J. Comput. Appl. Math. 236(7), 1833–1840 (2012)CrossRefMATHMathSciNetGoogle Scholar
- 12.Puccetti, G., Rüschendorf, L.: Sharp bounds for sums of dependent risks. J. Appl. Probab. 50(1), 42–53 (2013)CrossRefMATHMathSciNetGoogle Scholar
- 13.Puccetti, G., Rüschendorf, L.: Computation of sharp bounds on the expected value of a supermodular function of risks with given marginals. Working Paper (2013)Google Scholar
- 14.Schönbucher, P.J., Schubert, D.: Copula-dependent default risk in intensity models. Working Paper. Available at http://www.defaultrisk.com/pp_corr_22.htm (2001)
- 15.Sklar, M.: Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut de Statistique de L’Université de Paris 8, 229–231 (1959)MathSciNetGoogle Scholar
- 16.Thorisson, H.: Coupling, Stationarity, and Regeneration. Springer, New York (2000)CrossRefMATHGoogle Scholar