1 of 5
NBA673
Introduction to Derivatives I
Tibor Jánosi
Spring 2006 (1
st
half)
Homework 2: Arbitrage, Forwards, Futures
Solutions
Problem 1:
(a)
987153
.
365
100
0475
.
1
1
=
⋅
+
=
B(t, T)
59
.
45
$
987153
.
45
$
)
,
(
)
(
=
=
=
T
t
B
t
S
F(t, T)
(b)
At maturity, the value of the original forward contract will be
S(T)
$50.25, while the value of
the new forward contract will be $45.59
S(T)
.
The value of the net position will be
V(T
)=$4.66.
This example helps us understand the true meaning of “offsetting.”
The second forward
contract offsets the first one in the sense that it eliminates future uncertainty about payoffs; in
effect (in this case) it “freezes in” the losses.
The upside is that one is no longer exposed to the
possibility that the stock price will fall further, thus increasing losses even more.
Of course, the
stock price might also rise, case in which the offsetting transaction might not appear to be such
a great idea in
hindsight
(but hindsight is always 20/20).
(c)
We can determine the present value of the offsetting position by discounting the future (known)
net cash flow on the two contracts.
60
.
4
$
987153
.
66
.
4
$
)
,
(
)
(

=
⋅

=
⋅
=
T
t
B
T
V
V(t)
Problem 2:
The formula linking simple discount rates and the prices of zeros is
360
1
)
,
0
(
T
i
T
B
d
⋅

=
.
From here,
we obtain immediately the prices for US zerocoupon bonds:
0 = “3 months ago”
t
= “now”
T = “expiration”
“100 days”
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Maturity (Days)
US Discount Rate
Price of US Zeros
30
5.46
.995450
90
5.77
.985575
180
5.96
.970200
We write the equation giving the forward price in generic form (but we use subscript Y to denote yen
denominated zerocoupon bonds, as opposed to F, for a generic foreign zerocoupon
bond):
)
S(
,T)
(
B
,T)
F(
,T)
B(
Y
0
0
0
0
⋅
=
⋅
. From here, we immediately get the prices of yen
denominated zerocoupon bonds.
Using the same 360day year, we then compute the implied simple
discount rate for Japanese zeros.
For this, we use the formula that expresses the simple discount rate as
a function of the corresponding zerocoupon bond:
[ ]
T
T
J
i
j
360
)
,
0
(
1
⋅

=
(we have used
J(0,T)
to
denote the price of Japanese zeros, and
i
j
to denote the Japanese simple discount rate, but this is still
the usual formula).
Maturity
(Days)
Price of US Zeros
Price of Japanese
Zeros
Japanese
Discount Rate
30
.995450
.998886
1.34
90
.985575
.995780
1.69
180
.970200
.991350
1.73
Problem 3
:
This is a simple application of the formula given in class:
]
[
)
0
(
)
,
0
(
)
,
0
(
0
contract
the
of
life
the
over
dividends
PV
S
T
B
T
F

=
⋅
.
We can immediately write the following:
9741
.
2
9967
.
50
.
1
37
.
63
9512
.
)
,
0
(
⋅

⋅

=
⋅
T
F
From this equation we get that the forward price is
00
.
63
$
)
,
0
(
=
T
F
.
Problem 4:
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 Spring '06
 JANOSI,TIBOR
 Net Present Value, Forward price

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