Futures and futures options with basis risk: theoretical and empirical perspectives

Under a no-arbitrage assumption, the futures price converges to the spot price at the maturity of the futures contract, where the basis equals zero. Assuming that the basis process follows a modified Brownian bridge process with a zero basis at maturity, we derive the closed-form solutions of futures and futures options with the basis risk under the stochastic interest rate. We make a comparison of the Black model under a stochastic interest rate and our model in an empirical test using the daily data of S&P 500 futures call options. The overall mean errors in terms of index points and percentage are -4.771 and -27.83%, respectively, for the Black model and 0.757 and 1.30%, respectively, for our model. This evidence supports the occurrence of basis risk in S&P 500 futures call options.