Partial Payoff Swap Model
Partial payoff swap pays periodically, the payoff from a particular European style put option on the spread between respective ten and two-year CMS rates.
Pricing Partial Payoff Swap
Partial payoff swap pays periodically, the payoff from a particular European style put option on the spread between respective ten and two-year CMS rates. Moreover, this payoff is algebraically equivalent to the sum of the spread above and the payoff from a related European style put option.
Let
denote a swap rate for a swap specified by
ten year maturity,
6-month JPY Libor paid semi-annually, in arrears,
a fixed rate paid semi-annually,
represent a swap rate for a swap specified by
two year maturity,
6-month JPY Libor paid semi-annually, in arrears,
a fixed rate paid semi-annually.
Here one party must pay, semi-annually,
,
at time , where
is a 1,000,000,000 JPY notional amount,
is an accrual period,
In addition the party receives period payments based on JPY Libor.
Let be a strike level. Recall that one party must pay, periodically,
.
Moreover,
. (3.1)
We note that the price is
,
which can be viewed as the payoff from a European style put option specified by
strike,
,
underlying security,
.
The remaining term,, is valued.
Let
denote a reset time,
be the corresponding payment time.
We assume that the forward swap rate process, , satisfies under the
-forward probability measure an SDE, of the form
, (3.1.1)
where
is a constant volatility parameter,
is a standard Brownian motion.
Here is a timing and convexity adjusted, forward swap rate; the forward swap rate, convexity and timing adjustments are respectively computed.
Note that the forward swap rate process above may be assumed to satisfy an SDE of the form (3.1.1) under a corresponding forward swap measure; moreover, the forward swap rate will then not be log-normally distributed under the -forward probability measure.
Let
denote a reset time,
be the corresponding payment time,
represent the price at time
of a zero coupon bond that matures at
.
Then
where
· and
respectively denote expectation under the
and
-forward probability measure,
· is a forward JPY Libor rate that sets at
for the accrual period
.
Assume that, under the -forward measure,
are independent, standard Brownian motions. Moreover, assume that
where is a constant volatility parameter. Furthermore let
From the above, under the -forward probability measure,
,
and
are independent, standard Brownian motions.
Assume that, under -forward measure,
and
are independent, standard Brownian motions. Observe that
Let
,
and assume that
.
Let ; then
where
From the above, under -forward probability measure,
and
are independent, standard Brownian motions. Let
· denote a forward swap rate, which sets at time
with respect to an underlying 10 year swap,
· denote a forward swap rate, which sets at time
with respect to an underlying 2 year swap.
Assume that, under -forward probability measure,
Then, under -forward measure,
References:
https://finpricing.com/lib/EqBarrier.html
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