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

Abstract

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.

Keywords:

• Futures
• Futures options
• Basis risk
• Modified Brownian bridge process

Notes

†Yan (2002) defined the basis as the difference between the log futures price and the log spot price.

†When for , which leads to φ (0, T, U) = 0, the FOBR model will degenerate into the Black model under a stochastic interest rate.

†Generally, MAE represents the expected average error, while RMSE punishes larger errors because they receive greater weights when squared; thus, RMSE is more appropriate for risk-averse investors.

‡The empirical analysis is basically an in-sample or goodness-of-fit test, and the FOBR model, with at least seven parameters for fitting, may perform better than a three-parameter Black model. Further research could compare the FOBR model with more general option models.