e are considering the right to enter into a given CDS contract at some time
in the future (maturity of the option). If the reference name of CDS defaults
before the expiration of the option then the option vanishes. Hence, if the
spread widens then the option appreciates greatly but also is more likely to
vanish. Let
denote the market value of the CDS with coupon
as observed at time
.
The option matures at
.
The payoff at the time
is
.
The value of the option at the time
is
(
)
We perform the change of measure
where we introduced the expectation
with respect to the numeraire
that makes such separation possible (see the
(
Common_application_of_change_of_measure
)).
The existence of such numeraire was discussed before, see
(
Risky annuity
).
At this point we recall from (
Risky annuity
)
that the
itself is a risk neutral expectation of cashflows discounted with
.
It easy to see that such cashflows combine with
yielding
