Instantaneous and Terminal Correlation of Forward Rates

Caplet prices depend on the marginal distribution of each $F_k$ under $Q^{T_k}$ — correlation plays no role. Swaption prices depend on the joint distribution of $F_{\alpha+1},\ldots,F_\beta$ at time $T_\alpha$, and correlation is essential. The LFM models instantaneous correlation via the matrix $\rho_{k,j} = d\langle Z_k, Z_j\rangle_t/dt$, but the quantity that matters for pricing is the terminal correlation $\text{Corr}(F_k(T), F_j(T))$ for a future time $T$. These are not the same: terminal correlation depends on both instantaneous correlation and the instantaneous volatility structure. A model can have high instantaneous correlation $\rho_{k,j} \approx 1$ but low terminal correlation if the vol functions are structured to be nearly orthogonal in time. This lesson derives the relationship, explains the Rebonato parametrisation for the instantaneous correlation matrix, discusses historical calibration vs swaption calibration, and introduces rank reduction for computational efficiency.