2.4 Forward and Futures 2 SOLUTIONS
Here is the arbitrage argument of why this has to be the market
price. This is basically the same argument that you have seen in
class. Suppose you enter into a contract to sell silver in 9-months
at price F . The fair price F must be equal to the cost of replicating
portfolio, or otherwise there is an arbitrage opportunity. We will
discuss the case that there is a mispricing and a strategy to take
advantage of that in part b. Here is how the replicating portfolio
will be constructed.
To replicate the future contract you would borrow money today
($13.50 to buy an ounce of silver), buy silver and store that for
9 months. Note that your initial net cash flow is zero. You will
have to also pay storage fees once every quarter. By the end of
9 months, you will owe 13.50
¡
1 +
0.05
4
¢
3
= 14.013 due to your
initial borrowing and, in addition, you will owe
0.1
4
¡
1 +
0.05
4
¢
3
+
0.1
4
¡
1 +
0.05
4
¢
2
+
0.1
4
¡
1 +
0.05
4
¢
1
= 0.077 due to the storage fees that
you have had to pay. On the other hand, you own an ounce of
silver which you can sell at agreed price of F . In order to have
the net cash flow zero at maturity, F must be equal to the total
money you owe. Putting all these together you get that F must
be equal to the value given above in order to have no arbitrage.
(b) If the actual futures price is below $14.09 you would like to take
advantage of this mispricing by buying the futures contract, i.e.
entering into the contract to buy Silver in 9 months. This is also
sometimes referred to as ”taking a long position in the futures
contract”. You can then use the reverse of the above replication
argument to synthesize a short position in the future to offset your
long position. Alternatively, if the price is above, you would enter
in a short position in the futures contract, i.e. agree to sell Silver
in 9-months at a level above $14.09, and use the above replication
strategy to replicate a long position in the future. The main idea
here is that the cost of the replicating strategy is $14.09. Hence if
you can buy the future at lower price or sell at higher, you should
do that and use the replication strategy to offset your position.
This way you will eliminate all your risk and make risk-free or
arbitrage money. The argument works exactly the same way, with
signs reversed, in b oth case. We consider both cases below.
For concreteness, let’s assume you can sell the contract at $14.30,
which is higher that the fair price you calculated in part a. To
take advantage of this, you would do the following:
i. Enter into a contract to sell Silver in 9 months at $14.30
ii. Borrow $13.50 now and buy an ounce of silver. You will have
to borrow this money for 9 months.
49
c
° 2008, Andrew W. Lo and Jiang Wang