Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T15:09:23.780Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Applications Multivariate analysis Distribution theory

Published online by Cambridge University Press:  14 July 2016

Arthur Charpentier  and
Alessandro Juri
Arthur Charpentier*
Affiliation:
ENSAE/CREST
Alessandro Juri*
Affiliation:
UBS AG, Zürich
*
∗Postal address: Laboratoire de Finance et Assurance, ENSAE/CREST, Timbre J120, 3 avenue Pierre Larousse, FR-92245 Malakoff Cedex, France. Email address: [email protected]
∗∗Postal address: Credit Risk Control, UBS AG, PO Box, CH-8098 Zürich, Switzerland. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Copulacredit riskdependent defaultsdependent risksextreme value theoryregular variationtail dependence

MSC classification

Primary: 62E20: Asymptotic distribution theory
Secondary: 62H20: Measures of association (correlation, canonical correlation, etc.) 62P05: Applications to actuarial sciences and financial mathematics
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 2 , June 2006 , pp. 563 - 586
Copyright
© Applied Probability Trust 2006 

References

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