Journal of Applied Probability

Limiting dependence structures for tail events, with applications to credit derivatives

Arthur Charpentier and Alessandro Juri

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Abstract

Dependence structures for bivariate extremal events are analyzed using particular types of copula. Weak convergence results for copulas along the lines of the Pickands-Balkema-de Haan theorem provide limiting dependence structures for bivariate tail events. A characterization of these limiting copulas is also provided by means of invariance properties. The results obtained are applied to the credit risk area, where, for intensity-based default models, stress scenario dependence structures for widely traded products such as credit default swap baskets or first-to-default contract types are proposed.

Article information

Source
J. Appl. Probab. Volume 43, Number 2 (2006), 563-586.

Dates
First available in Project Euclid: 8 July 2006

Permanent link to this document
http://projecteuclid.org/euclid.jap/1152413742

Digital Object Identifier
doi:10.1239/jap/1152413742

Mathematical Reviews number (MathSciNet)
MR2248584

Zentralblatt MATH identifier
1117.62049

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 62H20: Measures of association (correlation, canonical correlation, etc.) 62P05: Applications to actuarial sciences and financial mathematics

Keywords
Copula credit risk dependent defaults dependent risks extreme value theory regular variation tail dependence

Citation

Charpentier, Arthur; Juri, Alessandro. Limiting dependence structures for tail events, with applications to credit derivatives. J. Appl. Probab. 43 (2006), no. 2, 563--586. doi:10.1239/jap/1152413742. http://projecteuclid.org/euclid.jap/1152413742.


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References

  • Aczél, J. (1966). Lectures on Functional Equations and Their Applications. Academic Press, New York.
  • Bäuerle, N. and Müller, A. (1998). Modeling and comparing dependencies in multivariate risk portfolios. ASTIN Bull. 28, 59--76.
  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.
  • Brémaud, P. (1981). Point Processes and Queues. Springer, New York.
  • Clayton, D. G. (1978). A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65, 141--151.
  • Coles, S. G. and Tawn, J. A. (1991). Modelling extreme multivariate events. J. R. Statist. Soc. B 53, 377--392.
  • Coles, S. G. and Tawn, J. A. (1994). Statistical methods for multivariate extremes: an application to structural design. J. R. Statist. Soc. C 43, 1--48 (with discussion).
  • Cook, R. D. and Johnson, M. E. (1981). A family of distributions for modelling non-elliptically symmetric multivariate data. J. R. Statist. Soc. B 43, 210--218.
  • Daul, S., De Giorgi, E., Lindskog, F. and McNeil, A. (2003). Using the grouped $t$-copula. Risk 16, 73--76.
  • Davis, M. and Lo, V. (2001a). Infectious defaults. Quantitative Finance 1, 382--387.
  • Davis, M. and Lo, V. (2001b). Modelling default correlation in bond portfolios. In Mastering Risk. Volume 2 -- Applications, ed. C. Alexander, Financial Times/Prentice Hall, Harlow, pp. 141--151.
  • De Haan, L. and Resnick, S. I. (1977). Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitsth. 40, 317--337.
  • De Haan, L., Omey, E. and Resnick, S. I. (1984). Domains of attraction of regular variation in $\boldsymbol R^d$. J. Multivariate Anal. 14, 17--33.
  • Dhaene, J. and Denuit, M. (1999). The safest dependence structure among risks. Insurance Math. Econom. 25, 11--21.
  • Dhaene, J. and Goovaerts, M. J. (1996). Dependency of risks and stop-loss order. ASTIN Bull. 26, 201--212.
  • Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.
  • Embrechts, P., McNeil, A. J. and Straumann, D. (2002). Correlation and dependence in risk management: properties and pitfalls. In Risk Management: Value at Risk and Beyond, ed. M. A. H. Dempster, Cambridge University Press, pp. 176--223.
  • Fredricks, G. A., Nelsen, R. B. and Rodriguez-Lallena, J. A. (2005). Copulas with fractal supports. Insurance Math. Econom. 37, 42--48.
  • Frees, E. W. and Valdez, E. A. (1998). Understanding relationships using copulas. N. Amer. Actuar. J. 2, 1--25.
  • Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.
  • Juri, A. and Wüthrich, M. V. (2002). Copula convergence theorems for tail events. Insurance Math. Econom. 30, 405--420.
  • Juri, A. and Wüthrich, M. V. (2004). Tail dependence from a distributional point of view. Extremes 6, 213--246.
  • Lando, D. (1998). On Cox processes and credit risky securities. Rev. Derivatives Res. 2, 99--120.
  • Nelsen, R. B. (1999). An Introduction to Copulas. Springer, New York.
  • Resnick, S. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
  • Schönbucher, P. and Schubert, D. (2001). Copula-dependent default risk in intensity models. Preprint. Available at http://www.math.ethz.ch/\tiny$\sim$schonbuc/.
  • Sklar, A. (1959). Fonctions de répartition à $n$ dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229--231.