Question

6. Forward and Futures Prices (16 points) Suppose the S&R index is 800, and that the dividend yield is 0. You are an arbitrageur with a continuously compounded borrowing rate of 5.5% and a continuously compounded lending rate of 5%. Supposing that there are no transaction fees, show that a cash-and-carry arbitrage is not profitable if the 1-year forward price is less than 845.23, and that a reverse cash-and-carry arbitrage is not profitable if the 1-year forward price is greater than 841.02.

Answer 1

In cash and carry arbitrage, the arbitrageur takes a long position in the asset i.e. buys the index and sells the forward on the index. To take the long position, the arbitrageur borrows the money.

Lets call the arbitrageur Mr X for simplicity

Mr X borrows 800 at 5.5% continuously compounding rate.

Amount due after 1 year = 800 x e^{(5.5% x 1)} = 845.23

Thus after 1 year, Mr X has to pay 845.23. If forward price was less than or equal to 845.23, Mr X will not make any profit.

In reverse cash and carry arbitrage, Mr X will initiate a short position in S&R index and will buy the forward on the index

Proceeds from taking short position in index = 800

Lending at 5%, after 1 year Mr X receives = 800 x e^{(5% x 1)} = 841.02

Now, after 1 year, Mr X will need to square off his position in forward contract by paying the agreed price 1 year back. If this price is greater than or equal to 841.02, Mr X will not make any profit as the amount he received by taking a short position in index and lending will be offset by squaring off the forward contract.