Write down the joint survival function and the

Question

Clayton copula model: Denote by τ₁, . . . , τ_m the default times of m companies. Assume that the τ_i are exponentially distributed with parameters λ_i, i = 1, 2, (P (τ_i ≤ t) = 1 - exp(-λ_it)) and that they follow a copula model with Clayton copula with parameter θ.

a) Write down the joint survival function and the corresponding mixture represen- tation explicitly.

b) Assume that you have a random number generator that is able to generate in- dependent Gamma distributed and uniformly distributed random variables. Write down an algorithm to generate a sample of τ₁, τ_m via the mixture distribution.

c) Assume that λ₁ = ... = λ_m. Write down

lim_θ→∞ P (τ₁ ≤ T, τ_m ≤ T ) and lim_θ→0P(τ₁ ≤ T, τ_m ≤ T ).