The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 723 Wednesday, February 4, 2004 1:12 PM

Credit Risk Modeling and Credit Default Swaps 723

where

pA( C∩B)= pB()– pA( ∩B)

pA∩B

C

( )= pA()– pA( ∩B)

pA( C∩BC)= 1 – pB()– pA( ∩BC)

The default correlation is

cov 1( A, (^1) B) pB( A )pA()– pA()pB()
--------------------------------------------- = ------------------------------------------------------------------------------------
var 1( A)var 1( B) pA()( 1 – pA()pB())( 1 – pB())
For example, suppose that A is a large automobile manufacturer
and B is a small auto part supplier. Assume their joint default distribu-
tion is given as follows:
Firm A
0 1

Firm B 0 80% 0% 80%
1 10% 10% 20%
90% 10% 100%
In this example where A defaults should bankrupt B but not vice
versa, B contains A and
pA( ∩B)= pA()
The dependency of the part supplier on the auto manufacturer is
pA( ∩B) pA()
pB( A)= ------------------------ = ------------ = 100%
pA() pA()
and the dependency of the auto manufacturer on the part supplier is
pA( ∩B) pA()
pA( B) = ------------------------ = ------------ = 50%
pB() pB()