Monte Carlo Simulation of the LFM

Black's caplet formula is analytically exact in the LFM, but most derivatives that reference multiple LIBOR rates — swaptions, Bermudan swaptions, range accruals, CMS spreads — require numerical pricing. Monte Carlo simulation under the LFM is the workhorse for these products. The simulation discretises the BGM SDE system for all forward rates simultaneously, using correlated Brownian shocks generated from the correlation matrix $\rho$. The key practical issues are: (1) choosing the right numeraire measure to control variance; (2) the log-Euler (predictor-corrector) discretisation that avoids negative rates and reduces discretisation bias; (3) generating correlated shocks via Cholesky factoring; and (4) testing the simulation against known analytical benchmarks (caplets, European swaptions via the Rebonato approximation). This lesson works through each step with a concrete Python implementation.