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Chapter 24
Interest Rate Models
Question 24.1.
a)
F
=
P (
0
,
2
) /P (
0
,
1
)
=
.
8495
/.
9259
=
.
91749.
b)
Using Black’s Formula,
BSCall (.
8495
,.
9009
×
.
9259
1
,
0
,
1
,
0
)
=
$0
.
0418
.
(1)
c)
Using put call parity for futures options,
p
=
c
+
KP (
0
,
1
)
−
FP (
0
,
1
)
=
.
0418
+
.
8341
−
.
8495
=
$0
.
0264
(2)
d)
Since 1
+
K
R
=
1
.
1, the caplet is worth 1.1 one year put options on the two year bond
with strike price 1
/
1
.
1
=
0
.
9009 which is the same strike as before. Hence the caplet is worth
1
.
11
×
.
0264
=
.
0293.
Question 24.2.
a)
The two year forward price is
F
=
P (
0
,
3
) /P (
0
,
2
)
=
.
7722
/.
8495
=
.
90901.
b)
Since
0
,
2
)
=
P (
0
,
3
)
the ±rst input into the formula will be .7722. The present value
of the strike price is
.
9
P (
0
,
2
)
=
.
9
×
.
8495
=
.
76455. We can use this as the strike with no interest
rate; we could also use a strike of .9 with an interest rate equal to the 2 year yield. Either way the
option is worth
7722
76455
105
,
0
,
2
,
0
)
=
$0
.
0494
(3)
c)
Using put call parity for futures options,
p
=
c
+
0
,
2
)
−
0
,
2
)
=
.
0494
+
.
76455
−
.
7722
=
$0
.
4175
.
(4)
d)
The caplet is worth 1.11 two year put options with strike 1
/
1
.
=
.
9009. The no interest
formula will use
(.
9009
)(.
8495
)
=
.
7653 as the strike. The caplet has a value of
1
.
BSPut (.
7722
7653
105
,
0
,
2
,
0
)
=
$0
.
0468
.
(5)
304
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Question 24.3.
We must sum three caplets. The one year option has a value of
.
0248, the two year option has a
value
.
0404, and the three year option has a value of
.
0483. The three caplets have a combined
value of
1
.
115
(.
0248
+
.
0404
+
.
0483
)
=
$0
.
1266
.
(6)
Question 24.4.
A Fat yield curve implies the two bond prices are
P
1
=
e
−
.
08
(
3
)
=
.
78663 and
P
2
=
e
−
.
08
(
6
)
=
.
61878. If we have purchased the three year bond, the duration hedge is a position of
N
=−
1
2
P
1
P
2
1
2
e
3
(.
08
)
.
63562
(7)
in the six year bond. Notice the total cost of this strategy is
V
8%
=
.
78663
−
.
63562
(.
61878
)
=
.
39332
(8)
which implies we will owe
.
39332
e
.
08
/
365
=
.
39341 in one day. If yields rise to 8.25%, our portfolio
will have a value
V
8
.
25%
=
e
−
.
0825
(
3
−
1
/
365
)
−
.
63562
e
−
.
0825
(
6
−
1
/
365
)
=
.
39338
.
(9)
If yields fall to 7.75%, the value will be
V
7
.
75%
=
e
−
.
0775
(
3
−
1
/
365
)
−
.
63562
e
−
.
0775
(
6
−
1
/
365
)
=
.
39338
.
(10)
Either way we lose
.
00003. This is a binomial version of the impossibility of a no arbitrage Fat
(stochastic) yield curve.
Question 24.5.
±or this question, let
P
1
be the price of the 4 year, 5% coupon bond and let
P
2
be the price of the
8 year, 7% coupon bond. Note we use a continuous yield and assume the coupon rates simple. We
also use a continuous yield version of duration, i.e.
D
∂P/∂y
P
.
a)
The bond prices are
P
1
=
4
X
i
=
1
.
05
e
−
.
06
(i)
+
e
−
.
06
(
4
)
=
.
95916
.
(11)
305
Part 5 Advanced Pricing Theory
and
P
2
=
8
X
i
=
1
.
07
e
−
.
06
(i)
+
e
−
.
06
(
8
)
=
1
.
0503
.
(12)
The (modifed) durations are
D
1
=
∑
4
i
=
1
.
05
ie
−
.
06
(i)
+
4
e
−
.
06
(
4
)
.
95916
=
3
.
7167
.
(13)
and
D
2
=
∑
8
i
=
1
.
07
−
.
06
(i)
+
8
e
−
.
06
(
8
)
1
.
0503
=
6
.
4332
.
(14)
b)
IF we buy one 4year bond the duration hedge involves a position oF
N
=−
D
1
D
2
P
1
P
2
.
5276
(15)
oF the 8year bond. This has a total cost oF
.
9516
−
.
5276
(
1
.
0503
)
=
.
39746. The next day we will
owe
.
39746
e
.
06
/
365
=
.
39753. IF yields rise in the next instant to 6
.
25 then bond prices will be
P
1
=
4
X
i
=
1
.
05
e
−
.
0625
(i
−
1
/
365
)
+
e
−
.
0625
(
4
−
1
/
365
)
=
.
95045
(16)
and
P
2
=
8
X
i
=
1
.
07
e
−
.
0625
(i
−
1
/
365
)
+
e
−
.
0625
(
8
−
1
/
365
)
=
1
.
0338
.
(17)
The duration hedge will have a value oF
.
95045
−
.
5276
(
1
.
0338
)
=
.
40502
<.
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This note was uploaded on 12/06/2011 for the course ECON 101 taught by Professor Adam during the Spring '06 term at Neumann.
 Spring '06
 Adam

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