Derivation — G2++ Bond Price and Forward-Rate Volatility-Correlation Structure

The G2++ bond price formula rests on a single central calculation: showing that $I(t,T) = \int_t^T [x(u)+y(u)]\,du$ is Gaussian conditional on $\mathcal{F}_t$, with explicitly computable mean and variance. This note carries out that proof via the moment-generating function of Itô stochastic integrals, derives the mean $M(t,T)$ and variance $V(t,T)$ in closed form, and then extracts the most consequential implication: the forward-rate volatility structure $\sigma_f(t,T)$ and instantaneous correlation between forward rates at different tenors. Both $\sigma_f$ and the correlation are needed to understand what the G2++ model can and cannot fit in the swaption vol matrix, and when the model produces a humped instantaneous-forward vol.