Forwards vs futures

It can be shown --Hull Appendix 3B-- that when interest rates are constant and the same for all maturities, then the futures and forward prices are the same. If the interest rates are stochastic, this relationship does not hold. Whether the forward price is lower than the futures price or higher will depend on the correlation of the underlying asset with the interest rates. This situation arises from the daily settlement procedure that takes place in the futures market. Remember that there is no secondary market for the forward contracts.

Example 7 (Spot-interest correlation)   Suppose that the interest rates and the underlying asset are negatively correlated. That is to say that on average, when the interest rates fall the price of the underlying asset increases, something that is true in the stock markets. Consider an investor that holds a long futures position. When the asset price increases, because of the marking-the-market procedure, the investor is making an immediate gain --the basis increases. This extra gain will be invested at an interest rate which is lower than average, due to the negative correlation. In a similar fashion, when the price of the underlying falls, the immediate loss will have to be financed at a rate which is above the average. Forwards are not subject to daily settlements, and therefore not affected by the spot-interest correlation. This makes forward contracts more attractive; in an efficient market when the spot-interest correlation is negative we expect forward prices to be higher than the futures ones.

Obviously the inverse will also hold, that is to say when the spot-interest correlation is positive we expect forward prices to be lower than the futures ones.

These differences have only a theoretical value, in practice these differences are ignored. Usually the maturity of futures contracts is quite short, and the spot-interest correlation is not that high in absolute terms to imply significant differences. Therefore handbooks and practitioners make the assumption that futures and forwards have the same price, even when interest rates are uncertain. Of course one has to be careful when dealing with longer maturity futures, since then the differences might become quite significant.

In the remaining of these notes we will use the same notation $F\left( t,\tau \right)$ for both forwards and futures, recognizing the pitfalls.