Interest Rate Model Families

Selecting an interest rate model is like choosing an engine for different aircraft—short-rate models power simple products, HJM handles complex term structure dynamics, and market models excel for benchmark rates. Each family has distinct calibration requirements and use cases.

Model Family Overview

Short-Rate Models

Hull-White (One-Factor)

Dynamics: dr = [θ(t) - κr] dt + σ dW

Parameters:

• κ (kappa) ≈ 0.02-0.10: Mean reversion speed
• σ (sigma) ≈ 0.005-0.02: Volatility of short rate
• θ(t): Time-dependent drift (calibrated to curve)

Calibration inputs:

• Current yield curve (determines θ(t))
• Cap/floor volatilities (determines σ, κ)

Advantages:

• Analytical solutions for bonds and options
• Fast calibration and pricing
• Naturally mean-reverting

Limitations:

• Rates can go negative (may or may not be realistic)
• Single factor limits curve dynamics
• Poor for swaption smiles

Two-Factor Extensions

Adding a second factor improves curve dynamics:

• G2++ model: Two correlated short-rate factors
• Captures yield curve steepening/flattening
• More calibration parameters but better fit

HJM Framework

Dynamics: df(t,T) = α(t,T) dt + σ(t,T) dW(t)

Where f(t,T) is the instantaneous forward rate at time t for maturity T.

Key insight: Drift α(t,T) is determined by volatility σ(t,T) under no-arbitrage: α(t,T) = σ(t,T) × ∫[t to T] σ(t,s) ds

Parameters: Volatility function σ(t,T) fully specifies the model.

Calibration inputs:

• Yield curve
• Cap/swaption volatility surface
• Correlation structure (multi-factor)

Advantages:

• Flexible term structure dynamics
• Any volatility structure can be accommodated
• Theoretical foundation for market models

Limitations:

• Non-Markovian in general (path-dependent)
• Implementation complexity
• Calibration can be challenging

Market Models (BGM/LMM)

Dynamics (LIBOR Market Model): dL_i(t) / L_i(t) = μ_i dt + σ_i(t) dW_i

Where L_i is the forward LIBOR for period [T_i, T_{i+1}].

Parameters:

• σ_i(t): Volatility of each forward rate (≈ 15-30% typical)
• ρ_ij: Correlation between forward rates
• Often parametric (e.g., exponentially decaying vol)

Calibration inputs:

• Caplet volatilities → individual σ_i
• Swaption volatilities → correlations ρ_ij
• Skew data → shift/SABR extensions

Advantages:

• Direct modeling of market-quoted rates
• Natural for Bermudan swaption pricing
• Smile extensions available (shifted, SABR-LMM)

Limitations:

• High-dimensional (one factor per tenor)
• Monte Carlo required for most products
• Calibration is numerically intensive

Comparison Matrix

Model Selection Triggers

Use this checklist to select the appropriate model:

• Use Hull-White 1F when: Pricing caps, floors, or simple European swaptions; need fast calibration and pricing; smile is not critical
• Use G2++ when: Curve dynamics matter (e.g., curve steepening trades); moderate complexity acceptable; still want analytical tractability
• Use HJM when: Path-dependent rate derivatives (e.g., ratchet caps); custom volatility structures needed; willing to invest in implementation
• Use LMM when: Pricing Bermudan swaptions; need to match swaption volatility surface; production environment with Monte Carlo infrastructure

Operational Complexity

For real-time trading desks, simpler models dominate. For risk management and end-of-day valuations, accuracy justifies LMM complexity.

Calibration Tenor Range Example

LMM calibration instruments:

Calibrate to this grid, interpolate between points, and validate out-of-sample swaptions.

Next Steps

For calibration and validation procedures, see Model Calibration and Validation.

To understand swaps and the products these models price, review Interest Rate Swaps: Fixed vs. Floating.