The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 727 Wednesday, February 4, 2004 1:12 PM

Credit Risk Modeling and Credit Default Swaps 727

If pA equals pB then pAB = pA and default of either asset results in

default of the other. In this instance the correlation is at its maximum of

100%.

As correlations go negative, a point arrives at which there is zero

probability of both assets defaulting together. Graphically, there is no

intersection between the two circles, as shown in Exhibit 22.10, and we

have pAB = 0. The correlation becomes

• pApB
  ρ = ----------------------------------------
  1 – pA 1 – pB

A negative correlation of –100% can only occur if pA = 1 – pB—that is,

for every default of asset A, asset B survives and vice versa.

The price of the first-to-default basket is simply the area of the two

nonoverlapping circles

Ωρ = ρ = pA + pB

This is when the default basket is most expensive.

We have seen above the price of a basket in the limits of low, high,

and zero correlation. Given that Ω = pA + pB – pAB , we can write the

price of a basket in terms of the default correlation as

Ω = p^22

A + pB – pApB – ρ pA – pA pB – pB

EXHIBIT 22.10 Negative Default Correlation Case

As the default correlation becomes negative, the two circles separate implying that

the joint default probability has fallen to zero.