Advanced Interest Rate Models (HJM and LMM) › Summary

This chapter developed two foundational frameworks for advanced interest rate modeling. The Heath-Jarrow-Morton approach models the evolution of the entire instantaneous forward rate curve, with absence of arbitrage completely determining the forward rate drift from its volatility structure. This drift condition, $\alpha(t,T) = \sigma(t,T) \int_t^T \sigma(t,s) \, ds$, ensures that discounted bond prices are martingales under the risk-neutral measure.

The LIBOR Market Model takes a more practical approach by directly modeling discrete forward rates observable in the market. Under each forward rate's natural measure, the rate follows driftless lognormal dynamics, leading directly to Black's formula for caplet pricing. Simulating multiple rates requires working under a common measure, introducing drift terms that depend on the correlation structure.

Key implementation concepts include:

• Forward measure numeraire: The bond $P(t,T)$ is numeraire under the $T$-forward measure, making the corresponding forward rate a martingale
• Black's formula: Provides closed-form caplet and floorlet prices under lognormal forward rate assumptions
• Correlation parameterization: Exponential decay structures capture the intuition that adjacent forward rates are more correlated than distant ones
• Two-stage calibration: First fit cap volatilities, then calibrate correlations to swaption prices

The frameworks balance flexibility with tractability. While they can match observed market prices more accurately than short-rate models, they require careful attention to numerical implementation, particularly for Monte Carlo simulation. Modern implementations often extend these basic frameworks with stochastic volatility or local volatility modifications to capture smile dynamics observed in the market.