Bonds Futures: Delta? No Gamma!

Abstract

Bond futures are liquid but complex instruments. Here they are analysed in a one-factor Gaussian HJM model. The in-the-model delta and out-of-the-model delta and gamma are studied. An explicit formula is provided for in-the-model delta. The out-of-the-model delta and gamma are equivalent to partial derivatives with respect to discount factors. In particular cases the derivative can not be obtained by standard techniques. The same situations lead to cases where the gammas (second order partial derivatives) do not exist.

Keywords: Bond future, delivery option, delta, gamma, HJM Gaussian model, in-the-model, out-of-the-model

JEL Classification: G13, E43