he
credit default swap contract is designed to provide protection from a credit
event associated with a risky bond. The coupon leg of the CDS pays coupon
while there is no credit event. The protection leg of the credit default swap
pays only if there is a credit event before maturity of the CDS. At the time
of the credit event the protection buyer (=coupon payer) receives par from the
protection seller (=coupon receiver) and delivers the bond to the protection
seller. On default the coupon leg pays the accrued interest.
Let
be the coupon payment dates,
be the effective date of the swap,
be
the coupon,
the default time (first and the only one),
be recovery rate at default and
is the MMA interest rate. The value of the coupon leg at time
is
The value of the protection leg
is
We proceed to evaluate each of the components. We apply the formula
(
Common_application_of_change_of_measure
)
and results of the section
(
T-forward probability
measure
):
where we introduced the
notation
We introduce a fine mesh
on the interval
and use the previously developed techniques (see section
(
Distribution of Poisson
process
section
)):
The other leg of the contract is valued in the similar manner:
where we introduced the new function
modelling the expectation of the recovery rate. We collect all the pieces
together:
Conventionally, it is assumed that the
is flat:
.
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