Forward Contract - Rational Pricing

Rational Pricing

If is the spot price of an asset at time, and is the continuously compounded rate, then the forward price at a future time must satisfy .

To prove this, suppose not. Then we have two possible cases.

Case 1: Suppose that . Then an investor can execute the following trades at time :

1. go to the bank and get a loan with amount at the continuously compounded rate r;
2. with this money from the bank, buy one unit of stock for ;
3. enter into one short forward contract costing 0. A short forward contract means that the investor owes the counterparty the stock at time .

The initial cost of the trades at the initial time sum to zero.

At time the investor can reverse the trades that were executed at time . Specifically, and mirroring the trades 1., 2. and 3. the investor

1. ' repays the loan to the bank. The inflow to the investor is ;
2. ' settles the short forward contract by selling the stock for . The cash inflow to the investor is now because the buyer receives from the investor.

The sum of the inflows in 1.' and 2.' equals, which by hypothesis, is positive. This is an arbitrage profit. Consequently, and assuming that the non-arbitrage condition holds, we have a contradiction. This is called a cash and carry arbitrage because you "carry" the stock until maturity.

Case 2: Suppose that . Then an investor can do the reverse of what he has done above in case 1. But if you look at the convenience yield page, you will see that if there are finite stocks/inventory, the reverse cash and carry arbitrage is not always possible. It would depend on the elasticity of demand for forward contracts and such like.