The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 720 Wednesday, February 4, 2004 1:12 PM

720 The Mathematics of Financial Modeling and Investment Management

either defaulting gives rise to the probability that only the second name

(but not the first) defaults:

T

∫ – dQ[^1 (^0 ,t)Q^2 (^0 ,t)]+ dQ^1 (^0 ,t)

0



= [ 1 – Q 1 ( 0 ,T)Q 2 ( 0 ,T)]– [ 1 – Q 1 ( 0 ,T)]

= Q 1 ( 0 ,T)[ 1 – Q 2 ( 0 ,T)] (22.15)

This probability is equal to the probability of survival of the first name

and default of the second name; thus, it is with this probability that the

payoff to the second name is paid. By the same token, the default prob-

ability of the first name is 1 – Q 1 (0,T), and it is with this probability

that the payoff regarding to the first name is paid.

In a basket model specified in equation (22.13), the final formula for

the price of an N bond basket under independence is

T N k k – 1

V = ∫∑P( 0 ,t)– d ∏Ql( 0 ,t)+ d ∏Ql( 0 ,t) [ 1 – Rk ()t ] (22.16)

0 k =^1 l =^1 l =^0

where Q 0 (t) = 1 and hence dQ 0 (t) = 0. Equation (22.16) assumes that

the last bond (i.e., bond N) has the highest priority in compensation,

that is, if the last bond jointly defaults with any other bond, the payoff

is determined by the last bond. The second to last bond has the next

highest priority in a sense that if it jointly defaults with any other bond

but the last, the payoff is determined by the second to last bond. This

priority prevails recursively to the first bond in the basket.

Investment banks that sell or underwrite default baskets are them-

selves subject to default risks. If a basket’s reference entities have a

higher credit quality than their underwriting investment bank, then it is

possible that the bank may default before any of the issuers. In this case,

the buyer of the default basket is subject to not only the default risk of

the issuers of the bonds in the basket, but also to that of the bank as

well—that is, the counterparty risk. If the counterparty defaults before

any of the issuers in the basket do, the buyer suffers a total loss of the

whole protection (and the spreads that had been paid up to that point in

time). We modify equation (22.16) to incorporate the counterparty risk

by adding a new asset with zero payoff to the equation:

TN + 1 k k – 1

V = ∫∑P( 0 ,t) – d ∏Qj( 0 ,t)+ d ∏Ql( 0 ,t) [ 1 – Rk ()t ] (22.17)

0 k =^1 l =^1 l =^0