By Arnold Kaufmann
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Then Z = F [−1] (U) has the distribution function F . 3. Credit Risk Modeling 27 Deﬁnition 1 (Copula) An n-dimensional copula is a function C : [0, 1]n → [0, 1] that has the following properties: 1. There are random variables U1 , . . , Un with range [0, 1] such that C is their joint distribution function. 2. e. for all i = 1, . . , n, ui ∈ [0, 1] C(1, . . , 1, ui, 1, . . , 1) = ui . A copula function can be used to link arbitrary marginal distribution functions to a multivariate distribution function.
2. e. for all i = 1, . . , n, ui ∈ [0, 1] C(1, . . , 1, ui, 1, . . , 1) = ui . A copula function can be used to link arbitrary marginal distribution functions to a multivariate distribution function. Sklar (1959) focused on the converse. He outlined that any multivariate distribution function can be written in the form of a copula. The marginal distribution functions and the dependence structure can be separated. The dependence structure is completely characterized by the copula function. Theorem (Sklar) Let F denote the joint distribution function of the random variables X1 , .
The increments of N(t) are independent, and 3. for all t < T and n = 0, 1, . . P (N(T ) − N(t) = n) = 1 n! n T λ(s)ds t T exp − λ(s)ds t . 24 Chapter 2. Collateralized Debt Obligations: Structure and Valuation The intensity function is a function of time only. 12): T S(t, T ) = exp − λ(s)ds . t In the inhomogeneous setup we obtain the continuous hazard rate, which now depends on the future point in time: ∂ S(t, T ) = λ(T ) . h(t, T ) = − ∂T S(t, T ) Using the inhomogeneous setup, non-ﬂat term structures of credit spreads can be modeled, because the hazard rate depends on the future point in time T .