J. Appl. Prob. 43, 563–586 (2006) Printed in Israel  Applied Probability Trust 2006 LIMITING DEPENDENCE STRUCTURES FOR TAIL EVENTS, WITH APPLICATIONS TO CREDIT DERIVATIVES ARTHUR CHARPENTIER, ∗ ENSAE/CREST ALESSANDRO JURI, ∗∗ UBS AG, Zürich 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. Keywords: Copula; credit risk; dependent defaults; dependent risks; extreme value theory; regular variation; tail dependence 2000 Mathematics Subject Classification: Primary 62E20 Secondary 62H20; 62P05 1. Introduction The reasons for studying and modeling dependencies in finance and insurance are of dif- ferent types. One motivation is that independence assumptions, which are typical of many stochastic models, are often due more to convenience than to the nature of the problem at hand. Furthermore, there are situations where the neglect of dependence effects may incur a (dramatic) risk underestimation (see, e.g. Bäuerle and Müller (1998) and Daul et al. (2003)). Besides this, widely used scalar dependence or risk measures such as linear correlation, tail- dependence coefficients, and value at risk generally do not provide a satisfactory description of the underlying dependence structure and have severe limitations when used for measuring (portfolio) risk outside the Gaussian world (see, e.g. Embrechts et al. (2002) and Juri and Wüthrich (2004) for counterexamples). Taking care of dependencies therefore becomes important in extending standard models to provide more efficient risk management. However, relaxing the independence assumption yields much less tractable models. It is therefore not surprising that only recently, i.e. within the last ten years, has the mathematical literature on the risk management of dependent risks undergone significant development. The main message of much of this research is the following (see, e.g. Dhaene and Goovaerts (1996), Dhaene and Denuit (1999), Frees and Valdez (1998), Joe (1997), Schönbucher and Schubert (2001), and Juri and Wüthrich (2002), (2004), among others). It is (intuitively) clear that the probabilistic mechanism governing the interactions Received 3 November 2004; revision received 30 January 2006. ∗ Postal address: Laboratoire de Finance et Assurance, ENSAE/CREST, Timbre J120, 3 avenue Pierre Larousse, FR-92245 Malakoff Cedex, France. Email address: arthur.charpentier@ensae.fr ∗∗ Postal address: Credit Risk Control, UBS AG, PO Box, CH-8098 Zürich, Switzerland. Email address: alessandro.juri@ubs.com 563