e saw in the previous section that the present value of a CDO tranche is
expressed in terms of quantities of the form
for a variety of functions
.
Hence, we would like to compute distribution of the quantity
for all payment dates
.
We introduce discretization of the notional. Let
be a small notional amount. We introduce the hut operation:
and the quantity
We perform the separation trick for
as described in the section
(
Credit correlation
).
Hence, we will be assuming that the credit events of obligors are
(conditionally) independent. We denote
the defaulted notional of portfolio of
obligors.
We start induction in
with consideration of a portfolio consisting of only one
obligor.
Since
is the total event we
continue
We apply the formula (
Bayes
formula
)
We introduce the notation
Prob
and use the delta symbol
(in C++ notation) then we
conclude
where the
is the fractional notional of the first obligor. The
comes from the observation that if
then the first (and only) obligor defaulted. Hence,
.
We proceed with the general induction steps using the same
tools.
Note that the probabilities
Prob
were considered in the section
(
CDS
). These probabilities come
from calibration to observed market quotes for CDSs.
