Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 29 / 6 / a

If the discounted zero-coupon bond price is a martingale, its expectation is $Z_{0,T}=e^{-A(0,T)}$. The Gaussian moment-generating function applied to the representation above gives $\mathbb{E}{Z}_{s,T}=e^{-A(s,T)+v(s,T)/2}$, soDifferentiate in maturity $T\ge s$. The two mean derivatives are ${\mu }_{s,T}$ and ${\mu }_{0,T}$. Symmetry of the covariance in its last two arguments makes the two boundary terms in the derivative of the double integral equal. HenceWe obtain the Gaussian forward-rate covariance drift restrictionwhich proves that the martingale condition implies the mean restriction. Only maturity differentiation is used; a time derivative of the covariance is unnecessary.