 Gaussian bond-option formula

For a maturity-$T$ call on a unit maturity-$U$ zero-coupon bond, where $t<T<U$, deterministic forward-rate volatilities make the bond-price ratio lognormal under the T-forward measure. Its integrated variance is $V={\int }_t^T\Vert \Sigma (s,U)-\Sigma (s,T){\Vert }^{2}ds$. For $V>0$, put $d_1=[\log (P(t,U)/(KP(t,T)))+V/2]/\sqrt{V}$ and $d_2=d_1-\sqrt{V}$. Integrating the lognormal distribution above the strike gives the displayed price. At $V=0$, the price is $(P(t,U)-KP(t,T){)}^{+}$. The forward measure is essential: the raw bond price has a stochastic discount rate under the money-market measure.