Pricing futures contracts
Now let us proceed to the pricing of futures contracts. As we did with forward contracts, we consider the case of a generic underlying asset priced at $100. A futures contract calls for delivery of the underlying asset in one year at a price of $108. Let us see if $108 is the appropriate price for this futures contract.
Suppose we buy the asset for $100 and sell the futures contract. We hold the position until expiration. For right now, we assume no costs are involved in holding the asset. We do, however, lose interest on the $100 tied up in the asset for one year. We assume that this opportunity cost is at the risk-free interest rate of 5 percent.
Recall that no money changes hands at the start of a futures contract. Moreover, we can reasonably ignore the rather small margin deposit that would be required. In addition, margin deposits can generally be met by putting up interest-earning securities, so there is really no opportunity cost. As discussed in the previous posts, we also will assume away the daily settlement procedure; in other words, the value of the futures contract paid out at expiration is the final futures price minus the original futures price. Because the final futures price converges to the spot price, the final payout is the spot price minus the original futures price.
So at the contract expiration, we are short the futures and must deliver the asset, which we own. We do so and receive the original futures price for it. So we receive $108 for an asset purchased a year ago at $100. At a 5 percent interest rate, we lose only $5 in interest, so our return in excess of the opportunity cost is 3 percent risk free. This risk-free return in excess of the risk-free rate is clearly attractive and would induce traders to buy the asset and sell the futures. This arbitrage activity would drive the futures price down until it reaches $105.
If the futures price falls below $105, say to $102, the opposite arbitrage would occur. The arbitrageur would buy the futures, but either we would need to be able to borrow the asset and sell it short, or investors who own the asset would have to be willing to sell it and buy the futures. They would receive the asset price of $100 and invest it at 5 percent interest. Then at expiration, those investors would get the asset back upon taking delivery, paying $102. This transaction would net a clear and risk-free profit of $3, consisting of interest of $5 minus a $2 loss from selling the asset at $100 and buying it back at $102. Again, through the buying of the futures and shorting of the asset, the forces of arbitrage would cause prices to realign to $105.
Some difficulties occur with selling short certain assets. Although the financial markets make short selling relatively easy, some commodities are not easy to sell short. In such a case, it is still possible for arbitrage to occur. If investors who already own the asset sell it and buy the futures, they can reap similar gains at no risk. Because our interest is in financial instruments, we shall ignore these commodity market issues and assume that short selling can be easily executed.
If the market price is not equal to the price given by the model, it is important to note that regardless of the asset price at expiration, the above arbitrage guarantees a risk-free profit. That profit is known at the time the parties enter the transaction.
The transactions we have described are identical to those using forward contracts. We did note with forward contracts, however, that one can enter into an off-market forward contract, having one party pay cash to another to settle any difference resulting from the contract not trading at its arbitrage-free value up front. In the futures market, this type of arrangement is not permitted; all contracts are entered into without any cash payments up front.