MAFS 5030
Quantitative Modeling of Derivatives Securities
Solution to Homework One
Course Instructor: Prof. Y.K. Kwok
1. There will be two coupons before delivery: one in 6 months and one just prior to
delivery. Using the present value formula:
$94
.
6 =
F
+ $4
e
0
.
1
+
$4
e
0
.
05
,
where
F
is the forward price, $4 is the coupon amount and interest is compounded
continuously at the rate of 10%. We obtain
F
= $94
.
6
e
0
.
1

$4
e
0
.
05

$4 = $96
.
34
.
Alternatively, we can interpret coupons as negative cost of carry and apply the
formula:
F
=
S

D
B
(
τ
)
,
where
D
is the sum of present value of the coupons. We then have
F
=
$94
.
6

$4
e

0
.
05

$4
e

0
.
1
e

0
.
1
= $96
.
34
.
2. Consider the fixedleg and floatingleg of a swap of unit notional.
Suppose the
current time is indexed by 0 (i.e.
t
= 0), and
t
= 1 means one year away from now.
Fixedleg
At
t
=
1
4
, receives
10%
2
= 0
.
05 of interest.
At
t
=
3
4
, receives
10%
2
= 0
.
05 of interest.
Floatingleg
At
t
=
1
4
, receives
L
1
2
parenleftbigg

1
4
parenrightbiggparenleftbigg
3
4

1
4
parenrightbigg
=
1
2
L
1
2
parenleftbigg

1
4
parenrightbigg
.
Here,
L
1
2
parenleftbigg

1
4
parenrightbigg
denotes halfyear LIBOR reset at an earlier time
t
=

1
4
.
At
t
=
3
4
, receives
L
1
2
parenleftbigg
1
4
parenrightbiggparenleftbigg
3
4

1
4
parenrightbigg
=
1
2
L
1
2
parenleftbigg
1
4
parenrightbigg
.
1
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Discount bond prices observed at
t
= 0:
B
parenleftbigg
0
,
1
4
parenrightbigg
= 0
.
972
, B
parenleftbigg
0
,
3
4
parenrightbigg
= 0
.
918. The
floating rate bond that receives 1 +
1
2
L
1
2
parenleftbigg

1
4
parenrightbigg
at time
1
4
is now priced at 0
.
992.
This gives the present value of
1
2
L
1
2
parenleftbigg

1
4
parenrightbigg
= 0
.
992

B
parenleftbigg
0
,
1
4
parenrightbigg
= 0
.
992

0
.
972 = 0
.
02
.
Also, the implied present value of
1
2
L
1
2
parenleftbigg
1
4
parenrightbigg
=
B
parenleftbigg
0
,
1
4
parenrightbigg

B
parenleftbigg
0
,
3
4
parenrightbigg
= 0
.
972

0
.
918 = 0
.
054
.
Therefore, the present value of the floatingleg payments is $0
.
02 + $0
.
054 = $0
.
074
per unit notional. The present value of the fixedleg payments is
0
.
05
bracketleftbigg
B
parenleftbigg
0
,
1
4
parenrightbigg
+
B
parenleftbigg
0
,
3
4
parenrightbiggbracketrightbigg
= ($0
.
05)(0
.
972 + 0
.
918) = $0
.
0945
per unit notional.
The value of the swap to the fixedrate payer with notional one million
=
$1
,
000
,
000
×
(0
.
074

0
.
0945)
=

$20
,
500
.
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 Spring '11
 A
 Derivative, Option style, early exercise

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