Derivative Securities – Fall 2007 – Section 10
Notes by Robert V. Kohn, extended and improved by Steve Allen.
Courant Institute of Mathematical Sciences.
Options on interestbased instruments:
pricing of bond options, caps, floors,
and swaptions.
This section provides an introduction to valuation of options on interest
rate products. We focus on two approaches: (i) Black’s model, and (ii) trees. Briefly: we’re
taking the same methods developed earlier this semester for options on a stock or forward
price, and applying them to interestbased instruments.
The material discussed here can be found in chapters 26, 27, and 28 of Hull; I’ll also take
some examples from the book
Implementing Derivatives Models
by Clewlow and Strickland,
Wiley, 1998. (Steve Allen’s version of these notes includes a few pages about how a Monte
Carlo scheme based on the HeathJarrowMorton theory works. I don’t attempt that here.)
************************
Black’s model.
Recall the formulas derived in Section 5 for the value of a put or call on
a forward price:
c
[
F
0
, T
;
K
]
=
e
−
rT
[
F
0
N
(
d
1
)

KN
(
d
2
)]
p
[
F
0
, T
;
K
]
=
e
−
rT
[
KN
(

d
2
)
 F
0
N
(

d
1
)]
where
d
1
=
1
σ
√
T
bracketleftBig
log(
F
0
/K
) +
1
2
σ
2
T
bracketrightBig
d
2
=
1
σ
√
T
bracketleftBig
log(
F
0
/K
)

1
2
σ
2
T
bracketrightBig
=
d
1

σ
√
T.
Black’s model values interestbased instruments using almost the same formulas, suitably
interpreted.
One important difference:
since the interest rate is no longer constant, we
replace the discount factor
e
−
rT
by
B
(0
, T
).
The essence of Black’s model is this: consider an option with maturity
T
, whose payoff
φ
(
V
T
) is determined by the value
V
T
of some interestrelated instrument (a discount rate,
a term rate, etc). For example, in the case of a call
φ
(
V
T
) = (
V
T

K
)
+
. Black’s model
stipulates that
(a) the value of the option today is its discounted expected payoff.
No surprise there – it’s the same principle we’ve been using all this time for valuing options
on stocks. If the payoff occurs at time
T
then the discount factor is
B
(0
, T
) so statement
(a) means
option value =
B
(0
, T
)
E
∗
[
φ
(
V
T
)]
.
We write
E
∗
rather than
E
RN
because in the stochastic interest rate setting this is
not
the
riskneutral expectation; we’ll explain why
E
∗
is different from the riskneutral expectation
later on. For the moment however, we concentrate on making Black’s model computable.
For this purpose we simply specify that (under the distribution associated with
E
∗
)
1
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(b) the value of the underlying instrument at maturity,
V
T
, is lognormal; in other words,
V
T
has the form
e
X
where
X
is Gaussian.
(c) the mean
E
∗
[
V
T
] is the forward price of
V
(for contracts written at time 0, with
delivery date
T
).
We have not specified the variance of
X
= log
V
T
; it must be given as data. It is customary
to specify the “volatility of the forward price”
σ
, with the convention that
log
V
T
has standard deviation
σ
√
T
.
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 Fall '11
 Bayou
 Finance, Derivatives, Derivative, Interest Rates, Interest, Options, Mathematical finance

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