= Gaussian bond-option formula
{c}
{title2=$C_t=P(t,U)\Phi(d_1)-KP(t,T)\Phi(d_2)$}
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 $V=\int_t^T\|\Sigma(s,U)-\Sigma(s,T)\|^2ds$. 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.
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