The G2++ Two-Additive-Factor Gaussian Model: Dynamics and Bond Prices

The G2++ model — two correlated zero-mean OU processes plus a deterministic shift that fits the initial yield curve exactly — is the benchmark tractable two-factor model for rates. What makes it more than just "two Hull-Whites added together" is the correlation $\rho$ between the factors: this single parameter allows the model to fit the swaption vol matrix's slope and level simultaneously, generating imperfect correlation between rates at different tenors that no one-factor model can achieve. At the same time, both factors are Gaussian, so the integral $\int_t^T r(s)\,ds$ remains Gaussian, and the bond price retains a closed-form affine structure $P(t,T) = A(t,T)e^{-B^x x(t)-B^y y(t)}$. This note derives the full model dynamics, the bond price formula, and the calibration condition for $\phi(t)$.