Managing Synthetic CDO Tranches using Base Correlations

Gaussian Copula Model (con’t)
Standard Model Assumptions: One-factor Gaussian copula model
F
i
( t)
= Pr[ τ
i
< t]
⇔ Φ(
Zi
) = F(
ti
)
Two name example: condition on the market factor
Z
i
= ρ Z + 1− ρε
i
X
1
⎧1
= ⎨
⎩0
prob p ( z)
1
prob (1 − p ( z))
1
X
2
⎧1
= ⎨
⎩0
prob p
2
( z)
prob (1 − p
2
( z))
Loss Distribution
⎧0
⎪
l1
L = ⎨
⎪l2
⎪
⎩l1
+ l
2
(1 − p ( z))(1
− p
p ( z)(1
− p
1
2
p ( z)
p
1
1
2
2
1
( z)
2
( z))
p ( z)(1
− p ( z))
( z))
Tranche Pricing
Derived from portfolio loss distribution
⎧0
L < a
⎪
L T
= ⎨L
− a a < L < b
⎪
⎩b
− a b < L
Many names …
Use FFT (fast Fourier transforms) or recursive method (Andersen, Sidenius, Basu (2003)) to
compute probabilities
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