Derivation — BGM Drift under the Spot LIBOR Measure

The previous lesson stated the BGM drift recursion (BGM) under the spot LIBOR measure and the drift under a general $T_i$-forward measure. This note derives both from the change-of-numeraire toolkit applied to the product numeraire $B_d$. The derivation has three steps: (1) express $B_d$ as a product of bond-price ratios in terms of the forward rates; (2) compute the Radon-Nikodym density $dQ^d/dQ^{T_k}$; (3) use Girsanov's theorem to identify the drift correction. A three-rate numerical illustration confirms the formula is internally consistent — the convexity adjustment has the right sign and magnitude.