Caplet Pricing in the LFM: Black's Formula Recovered

The LIBOR Market Model was built with one explicit goal: making Black's caplet formula exact. This lesson shows exactly how that works. Under the $T_k$-forward measure $Q^{T_k}$, the forward LIBOR rate $F_k$ is lognormal and driftless — its distribution at fixing time $T_{k-1}$ is known in closed form. A caplet payoff is a call option on this lognormal random variable, so the expectation evaluates to the Black formula with no approximation. The lesson also derives the relationship between the Black implied vol $v_k$ and the LFM's instantaneous vol function $\sigma_k(t)$, and discusses how caps — sums of caplets — are priced and what the cap vol quotes mean.

1. 1. The Caplet Payoff
2. 1. Pricing Under the $T_k$-Forward Measure
3. 1. Black's Caplet Formula
4. 1. The Implied Black Vol and Its Relationship to $\sigma_k$
5. 1. Cap Pricing and the Sum of Caplets
6. 1. Floor, Cap-Floor Parity, and Collars