Azriel Levy
Introduction
everal authors have shown that the theoretical relationship between forward and fu-S tures prices depends primarily on the assumptions regarding the stochastic process of interest rates (Cox, et. al. (1981); Richard and Sundaresan (1981); Jarrow and Oldfield (1981); and French (1982)). The difference between the two prices arises from the marking-to-market feature of futures prices. In particular, it can be shown that the price of a forward contract is related to the interest rate on a pure discount bond that matures at the same time as the contract; whereas the futures price is related to the return of rolling over one day bonds until the contract matures (Cox, et. al. (1981)). It is concluded that if interest rates are nonstochastic; then, to avoid arbitrage opportunities, the forward price must be equal to the futures price (French (1982)).
This note argues that the restriction that interest rates are nonstochastic is not a necessary condition for the two prices to be equal. To obtain a riskless hedge through a futures contract it is sufficient to be able to forecast, with certainty, each trading day the next trading day's yield on a bond that matures at the delivery date of the futures contract. However, to obtain the riskless hedge, one need not know with certainty all interest rates until the maturity of the contract. If the futures contract can create a riskless hedge; then, to avoid arbitrage (one price law), its price must be equal to the forward price.
This theoretical result, which can be tested empirically, has practical implications. While it is impossible, in practice, to forecast with complete certainty the next trading day's interest rates, the variance of the forecast is much smaller than the variance of forecasting all interest rates from the purchase (sale) of the contract to the delivery date. Thus, it is not surprising that many empirical studies have found forward and futures prices to be equal.