LIBOR Market Model

Short-rate models and the general HJM framework are built on an instantaneous rate — a rate over an infinitesimally short instant — which is a mathematically convenient fiction that no bank actually quotes or trades. What banks quote and trade is a discrete forward LIBOR (or now SOFR) rate over a real period, like three months. The LIBOR Market Model's insight is to stop modeling the fictional instantaneous rate and instead directly model the dynamics of these real, observable forward rates, each assumed lognormal — exactly the assumption already baked into the Black formula that the whole cap and floor market uses to quote prices. The payoff is that the model is automatically calibrated to match market caplet prices almost by construction, something short-rate models have to work much harder to achieve.

The model

Each forward rate $F_i(t)$, covering the period from $T_i$ to $T_{i+1}$, follows

$$dF_i(t) = \mu_i(t)\, F_i(t)\, dt + \sigma_i(t)\, F_i(t)\, dW_i(t),$$

where $\sigma_i(t)$ is that specific forward rate's own volatility (the same number the market quotes as its Black implied vol) and $\mu_i(t)$ is a drift that depends on which probability measure you're working under and is generally not zero except under that rate's own natural "forward measure." In plain English: each forward rate moves proportionally to its own level (lognormal, so it can't go negative under this specification) with a volatility that's directly the number quoted in the caplet market — no translation needed between "what the model uses" and "what the market quotes."

Worked example 1 — pricing a caplet is now trivial

Under its own forward measure, $F_i(t)$ has zero drift, so pricing caplet $i$ collapses to plugging $F_i(0)$, the strike $K$, and $\sigma_i$ straight into the Black formula — the exact same calculation as in the caps-and-floors worked examples. With $F_i(0) = 5.20\%$, $K = 5.00\%$, $\sigma_i = 20\%$, $t=1$: this reproduces the same $d_1 = 0.296$, $d_2 = 0.096$ computation as before, giving a caplet value around $12,143 on a $10,000,000, 3-month notional — the LIBOR Market Model doesn't invent a new caplet price, it's defined so that it reproduces the market's own Black-formula price exactly for every individual caplet.

Worked example 2 — where the real work is: correlation

The hard part isn't any single caplet, it's a swaption, which depends on several forward rates together, and those rates are correlated, not independent. Suppose two adjacent 3-month forwards have volatilities 20% and 19%, and a correlation $\rho = 0.85$ between their random shocks. The approximate variance of a 6-month rate built by combining them scales with $\sigma_1^2 + \sigma_2^2 + 2\rho\sigma_1\sigma_2 = 0.04 + 0.0361 + 2(0.85)(0.20)(0.19) = 0.04 + 0.0361 + 0.0646 = 0.1407$, giving an approximate combined volatility of $\sqrt{0.1407} \approx 37.5\%$ — noticeably less than the $20\% + 19\% = 39\%$ you'd get by naively adding vols, because imperfect correlation ($\rho < 1$) provides some diversification even between two adjacent forward rates on the same curve.

Each simulated path above is lognormal, never crossing zero — exactly the property the LIBOR Market Model imposes on every one of its many forward rates simultaneously, all correlated with each other.

What this means in practice

The LIBOR Market Model is the industry-standard engine for pricing anything whose payoff depends on multiple forward rates jointly — Bermudan swaptions, CMS spread options, and other exotic rate products — precisely because it's built directly on the quantities the vanilla cap and swaption markets already price, so calibrating it to match those markets is comparatively direct.