Incorporating multi-dimensional tail dependencies in the valuation of credit derivatives

Abstract

The need for an accurate representation of tail risk has become increasingly acute in the wake of the credit crisis. We introduce a hyper-cuboid normal mixture copula that permits the representation of complex tail-dependence structures in a multi-dimensional setting. We outline an efficient pattern-recognition calibration methodology that can identify tail dependencies independent of the number of risk factors considered. This model is used to develop a new framework for pricing credit derivative instruments, and we derive semi-analytical and analytical pricing formulae for a first-to-default swap and illustrate with an example valuation. Model assumptions are validated against iTraxx Series 5 equity data over an 8-year period. Identification and representation of tail dependencies is crucial to further the study of contagion dynamics, and our model provides a basis for future research in this area.

Keywords:

• Credit derivatives
• Tail analysis
• Contagion
• Parameter estimation techniques
• Non-Gaussian distributions
• Pattern recognition
• Copulas

Acknowledgements

The authors are deeply grateful for the guidance and support of Malcolm Carmichael, Ellen Stars and Flick Roper. We would also like to thank Dr. Simon Marshall for his assistance in coding the pattern analysis routine and Yimin Liu and Robert Hone for their constructive feedback on earlier drafts.

Notes

†Our assumption on the form of the univariate margins will not impact the marginal form of the copula, although our restriction here will generally affect the results of the optimization.

†Assuming that the higher-dimensional components are themselves normal.

† For an interesting discussion on the topology of mixture distributions with particular reference to modality, see Ray and Lindsay (2005).

‡We first determine the contribution of the kth component to the ith sum ∑ _j=1,i≠j τ _ij : noting that , then , where . We derive ξ noting that the recursive sequence

with A ⁽¹⁾ = [0, 1] yields the matrix ψ = A ⁽ⁿ⁾. Hence the column sum may also be expressed recursively, where

for all i or, equivalently, that . Thus evaluated at i = n the ‘1's counting’ sequence ξ (id: A000120; Sloane 2009) corresponds to the sum of the elements within each column of ψ. Adding to the left-hand side of equation (7) yields the first set of constraints in terms of τ. To obtain the final constraint we trivially note for any i = 1, … , n. Since ψ _R:i W = 1 − p, then if follows from (8) that we require

for an arbitrary i. Let i = n so that the constraint is formulated in terms of all such components that do not have tail elements in the nth dimension. Consequently, we reformulate in terms of all τ _ij , i ≠ j ≠ n, by substituting , which reduces to

†These same relations may also be used to express the set of permissible W with respect to a given τ _T .