The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 724 Wednesday, February 4, 2004 1:12 PM

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724 The Mathematics of Financial Modeling and Investment Management

The default correlation is

pB( A)pA()– pA()pB()

pA()( 1 – pA()pB())( 1 – pB())

10% – 10% ×20%

= ---------------------------------------------------------------------------

10% ×90% ×20% ×80%

0.08 2

= ----------------------= ---

0.0144^3

This examples demonstrates that perfect dependency does not imply

perfect correlation. To reach perfect correlation, p(A) = p(B). Similarly,

perfectly negative dependency does not necessarily mean perfect nega-

tive correlation. To see that, consider the following example:

Firm A

0 1

Firm B 0 70% 10% 80%

1 20% 0% 20%

90% 10% 100%

It is clear that given A defaults, B definitely survives: pB

C

( A) = 1 ,

and pB( A) = 0. But the default correlation is only –0.25. To reach

perfect negative correlation of –100%, p(A) + p(B) = 1.

The reason that perfect dependency does not result in perfect corre-

lation is because correlation alone is not enough to identify a unique

joint distribution. Only a normal distribution family can have a uniquely

identified joint distribution when a correlation matrix is identified. This

is not true for other distribution families.^44

Having now defined default correlation, one can begin to show how

it relates to the pricing of credit default baskets.

We represent the outcomes of the two defaultable assets A and B

using a Venn diagram as shown in Exhibit 22.7. The left circle corre-

sponds to all scenarios in which asset A defaults before time T. Its area

is therefore equal to pA, the probability of default of asset A. Similarly,

the area within the circle labeled B corresponds to the probability of

default of asset B and equals pB. The area of the shaded overlap corre-

(^44) For an extension of the above two-company analysis to multiple companies, see
Chen and Sopranzetti, “The Valuation of Default-Triggered Credit Derivatives.”