Chapter 6
The Wide World of Futures Contracts
Question 6.1
The current exchange rate is 0.02
€
/¥, which implies 50¥/
€
. The euro continuously compounded interest
rate is 0.04, the yen continuously compounded interest rate is 0.01. Time to expiration is 0.5 years. Using
the foreign exchange forward formula, (6.2), we find:
(0.04
0.01)
0.5
(0.01 0.04)
0.5
euro/yen forward
0.02
0.02
1.015113
0.020302
yen/euro forward
50
50
0.98511
49.2556
e
e
−
×
−
×
=
=
×
=
=
=
×
=
Note that when we use euro/yen, our “home” currency is the euro; hence, when applying the formula, we
are European. It is the price we (as Europeans) have to pay for the “foreign” currency (the yen). Hence
we use our home interest rate (i.e.,
4%)
r
=
in the forward exchange formula and we use the yen interest
rate as the foreign interest rate (i.e.,
1%).
y
r
=
Similarly, when we use yen/euro, our “home” currency is the yen; hence when applying the formula, we
are Japanese. It is the price we (as Japanese) have to pay for the “foreign” currency (the euro). Hence we
use our home interest rate (i.e.,
1%)
r
=
in the forward exchange formula and we use the euro interest
rate as the foreign interest rate (i.e.,
4%).
y
r
=
Question 6.2
The current spot exchange rate is $0.008/ ¥, the oneyear continuously compounded dollar interest rate is
5%, and the oneyear continuously compounded yen interest rate is 1%. This means that we can calculate
the noarbitrage price of a oneyear $/yen forward to be:
(0.05 0.01)
dollar/yen forward
0.008
0.008
1.0408108
0.0083265
e
−
=
=
×
=
We can see that the observed forward exchange rate of 0.0084 $/ ¥ is too expensive, relative to the fair
forward price. We therefore go short the forward contract and synthetically create a long forward position
(buy yen):
Today
At expiration of
the contract
Description
in $
in yen
in $
in yen
Sell $/ ¥ forward
0
—
$0.0084
−
1
Buy yen for 0.0079204 dollar
−
0.0079204
−
0.99005
—
—
Lend 0.99005 yen
—
−
0.99005
—
1
Borrow 0.0079204 dollar
+
0.0079204
—
−
0.0083265
—
Total
0
0
0.0000735
0
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McDonald •
Fundamentals of Derivatives Markets
Note that, when we enter into the short forward position, we short 1 yen. This is why we buy 0.99005 yen
as
.01
0.99005.
e
−
=
We could have done an arbitrage by buying 1 yen and shorting 1.01005
.01
(i.e.
)
e
yen
forward.
With the above arbitrage strategy, we’ve earned 0.0000735 dollars, without any exchange risk or initial
investment involved. We have exploited the arbitrage opportunity and would like to increase the scale of
this strategy as much as possible (i.e., buy as many yen on the spot market and sell them forward).
With a forward exchange rate of 0.0083, the observed price is too cheap. We will buy the forward and
synthetically create a short forward position.
Today
At expiration of
the contract
Description
in $
in yen
in $
in yen
Buy $/ ¥ forward
0
—
−
0.0083$
+
1
Sell yen for 0.0079204 dollar
+
0.0079204
−
0.99005
—
—
Borrow 0.99005 yen
—
+
0.99005
—
−
1
Lend 0.0079204 dollar
−
0.0079204
—
+
0.0083265
—
Total
0
0
0.0000265
0
Therefore, we made an arbitrage profit of 0.0000265 dollars.
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 Fall '08
 Staff
 Exchange Rate, Interest, Interest Rate, Rational pricing

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