BGM Dynamics and the LFM under Multiple Measures

Under its own $T_k$-forward measure $Q^{T_k}$, the forward LIBOR rate $F_k$ is driftless and lognormal — this is the content of the last lesson and gives Black's caplet formula exactly. But caplets are single-rate payoffs. Pricing a swaption requires the joint dynamics of $F_{k}, F_{k+1}, \ldots, F_{k+n}$ simultaneously, which means working under a single common measure where all rates are alive at once. When you shift $F_k$ from its natural $Q^{T_k}$ measure to any other measure — the spot LIBOR measure, the terminal measure, or the $Q^{T_j}$ measure with $j \neq k$ — the rate acquires a non-zero drift. This drift is a sum of pairwise covariance terms between $F_k$ and the adjacent rates. The resulting system of SDEs under the spot LIBOR measure is the BGM model (Brace-Gatarek-Musiela 1997). Solving for these drifts via the change-of-numeraire toolkit is the central technical step of the LFM; the result is a tractable system for Monte Carlo simulation.