Chapter 33: Modeling Forward Rates

The Heath–Jarrow–Morton (HJM) Framework (1992)

Instead of modeling the short rate, the HJM framework models the behavior of the entire forward rate curve.

The Model

The process for the instantaneous forward rate $f(t,T)$:

$df(t,T)=\alpha (t,T)dt+\sigma (t,T)dz$

where $\sigma (t,T)$ is the volatility of the forward rate. The key result:

$\alpha (t,T)=\sigma (t,T)\cdot {\int }_t^T\sigma (t,\tau )d\tau$

The drift is determined by the volatility structure — a no-arbitrage condition. Once the volatility structure is specified, the drift follows automatically.

Implementation

• Forward rates are discretized into a finite number of periods
• The volatility structure $\sigma (t,T)$ determines the tree geometry
• Non-Markov: the evolution depends on the whole path of the process

Advantages

• Consistent with any initial term structure
• Can incorporate a rich volatility structure
• Most general framework

Disadvantages

• Generally non-recombining trees (computationally expensive)
• Non-Markov property makes most implementations path-dependent
• Many volatility structures lead to non-Markov models

The BGM (Brace–Gatarek–Musiela) / LIBOR Market Model (1997)

Also called the LIBOR Market Model (LMM) — models observable forward LIBOR rates (now being adapted to SOFR).

Model Setup

Each forward rate $F_k(t)$ for the period $[T_k,T_{k+1}]$ follows:

$\frac{d{F}_k(t)}{F_k(t)}={\mu }_k(t)dt+{\sigma }_k(t)d{z}_k$

where ${\sigma }_k(t)$ is the volatility of forward rate $k$ and $d{z}_k$ are correlated Wiener processes. Under the forward measure for $T_{k+1}$, the drift ${\mu }_k(t)$ is zero (the forward rate is a martingale).

Key Features

• Forward rates are lognormal (always positive)
• Market observables: the model works with actual rates quoted in the market (caplet volatilities)
• Calibration: volatilities ${\sigma }_k$ are directly calibrated to cap/floor prices
• Correlations between forward rates (caplets to swaptions) are captured by the correlation matrix of $d{z}_k$

Calibration to Swaptions

The BGM model can be calibrated to both caplet and swaption volatilities. The swaption volatility is a function of forward rate volatilities and their correlations:

${\sigma }_{\text{swaption}}^2={\sum }_{i,j}{w}_i{w}_j{\rho }_{ij}{\sigma }_i{\sigma }_j$

where $w_i$ are weights derived from the swap rate's sensitivity to each forward rate.

Advantages over Short-Rate Models

• Directly uses market-quoted volatilities
• More intuitive: traders think in terms of forward rate volatilities
• Natural for pricing caps, floors, and swaptions

Implementation

• Monte Carlo simulation is the primary implementation method
• Each forward rate is simulated step by step
• Drift adjustments are needed when moving between forward measures

Agency Mortgage-Backed Securities (MBS)

The BGM model is particularly useful for valuing mortgage-backed securities because:

• MBS cash flows depend on future interest rate paths (prepayment behavior)
• Prepayment modeling (behavioral component) is path-dependent
• The model captures the correlation structure needed for MBS valuation

IOs and POs

• IO (Interest Only): Receives only the interest portion of mortgage payments
• PO (Principal Only): Receives only the principal portion
• These react differently to prepayment rates: IOs lose value when prepayments increase (less interest collected); POs gain value (principal received earlier)

Valuing IOs and POs requires modeling the entire path of interest rates to capture prepayment behavior correctly — a natural application for the BGM model with Monte Carlo simulation.