Pricing Futures and Futures Options with Basis Risk

In this article, assuming the stochastic behavior of basis as a modified Brownian bridge process, we obtain closed-form solutions of futures and futures options generalizing the Black (1976). The arrangement permits the formulas of futures and futures options to be functions of spot price, volatility of spot return, initial basis, basis volatility, as well as the correlation coefficient between basis and spot return. In the meantime, it also ensures the basis to be zero at maturity of futures contract.