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Futures Pricing

In the previous section, we established the relationship between spot (S) and futures (F) prices as S = F d(0,T), where d(0,T) is the discount rate over the period of the futures contract. If it is observed that S < F d(0,T) for which the current spot price is below the present value of the futures price there will be an incentive to short the futures contract, buy and store the asset, and then deliver it at time T for F. If, on the other hand, S > F d(0,T), there is an incentive to short the asset, invest the proceeds at d(0,T) and go long the futures, taking delivery of the asset at time T for a price F. Either way, there is an arbitrage profit to be made, and since these opportunities will be driven from the market, we conclude that the relationship between spot and futures prices must be fully consistent with the existing term structure of interest rates. Let's now take a closer look at this logic.

Suppose you borrow an amount S, buy one unit of the underlying asset on the spot market at price S, and take a short position in the futures market (to deliver this asset at price F). The total cost of this portfolio is zero, that is, the portfolio is [–S/d(0,T), F]. At time T, you deliver the asset for F and repay the loan amount S/d(0,T). This is an equilibrium result. This relationship says that the current spot price is equal to the discounted present value of the futures price. If there were storage costs involved, then F must compensate for those as well as ...