Options#2 - 1
OPTIONS #2: INTEREST RATE DERIVATIVES: THE
STANDARD MARKET MODELS
(Hull Ch. 28, 7
th
Ed.)
Extensions of Black’s Model (Delayed Payoff)
As we’ll see below, there are option contracts in which the
decision to exercise occurs at date
T
, but the payoff
doesn’t actually occur until a later date,
T
*
>
T
.
In this
case we discount the payoff from time
T
*
, but the
probabilities in Black’s formula all depend on date
T
.
Options#2 - 2
Let
r
*
be the zero-coupon yield for maturity
T
*
.
Then,
equations (28.1) and (28.2) 7
th
Ed. [(26.1) and (26.2) 6
th
Ed.] become
c
=
*
*
T
r
e
±
[
FN
(
d
1
)
±
XN
(
d
2
)],
=
b
(0,
T
*
)[
FN
(
d
1
)
±
XN
(
d
2
)]
p
=
*
*
T
r
e
±
[
XN
(
±
d
2
)
±
FN
(
±
d
1
)],
=
b
(0,
T
*
)[
XN
(
±
d
2
)
±
FN
(
±
d
1
)],
where
d
1
=
T
T
X
F
V
V
2
2
1
)
/
ln(
²
,
Options#2 - 3
and
d
2
=
T
T
X
F
V
V
2
2
1
)
/
ln(
±
=
d
1
±
V
T
.
and
b
(0,
T
*
) is the price of a zero coupon bond that pays
$1 at time
T
*
.
Interest Rate Caps (Hull § 26.3, 6
th
Ed., §28.2, 7
th
Ed.)
Suppose you have a floating-rate loan based on six-month
LIBOR.
If the principal is $100 million, and if
Options#2 - 4
payments are made semi-annually, then you would
normally pay
0.5
u
100
u
R
,
where
R
is the six-month LIBOR (with semi-annual
compounding) from the
beginning
of the six-month
period.
Now suppose a Financial Institution (FI) has offered you a
cap
on your payments of 10% per year (compounded
semi-annually).
In that case, your payment,
PMT
, is

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