Hull-White Model

Overview

The Hull‑White model is a one‑factor short‑rate model used to price interest‑rate derivatives. It assumes the instantaneous short rate follows a mean‑reverting process with normally distributed shocks. The model is widely used because it can be calibrated to fit the current term structure of interest rates while remaining mathematically tractable.

Key features

• Single‑factor model: dynamics driven by one source of randomness (a single Brownian motion).
• Mean reversion: short rates tend to move back toward a time‑dependent long‑run level.
• Normal shocks: rate changes are normally distributed, allowing for (low‑probability) negative rates.
• Calibratable drift: a time‑dependent drift term enables exact fit to the observed initial yield curve.
• Used to price: bond options, interest rate caps/floors, swaptions, mortgage‑backed securities and other interest‑rate derivatives.

Model dynamics (standard form)

A common specification of the Hull‑White one‑factor model is the stochastic differential equation

dr(t) = [θ(t) − a r(t)] dt + σ dW(t)

where:
* r(t) is the instantaneous short rate,
* a > 0 is the mean‑reversion speed,
* σ ≥ 0 is the volatility,
* θ(t) is a deterministic function chosen to fit the initial term structure,
* W(t) is a standard Brownian motion.

Because θ(t) is time‑dependent, the model can reproduce today’s entire yield curve rather than relying on a single rate observation.

Relationship to other models

• Extends Vasicek: Hull‑White generalizes the Vasicek model by allowing a time‑dependent drift θ(t).
• Contrast with CIR: CIR also models mean reversion but ensures nonnegative rates by using square‑root diffusion; Hull‑White permits negative rates due to normal shocks.
• Compared to HJM and BGM: HJM models the evolution of forward rates (infinite‑dimensional framework) and BGM focuses on observable forward LIBOR rates; Hull‑White centers on the instantaneous short rate and is lower dimensional and easier to implement.

Advantages

• Analytical tractability: closed‑form solutions are available for many instruments (e.g., bond prices, bond options, swaptions under certain assumptions).
• Flexible calibration: θ(t) lets practitioners fit the current yield curve exactly.
• Simplicity: one‑factor setup keeps implementation and calibration relatively straightforward.

Limitations and special considerations

• Negative rates: because shocks are normally distributed, the model allows negative short‑rate realizations (possible in low‑rate environments).
• Single factor: cannot capture all sources of term‑structure dynamics (e.g., complex volatility structures) that multi‑factor or HJM models can.
• Calibration over time: while θ(t) fits the initial curve, recalibration may be needed as market conditions change.

Origins

The model was introduced by John C. Hull and Alan D. White in 1990. Their formulation made a practical, calibratable extension to existing mean‑reverting short‑rate models and has since become a standard tool in interest‑rate modeling and derivative pricing.

Summary

The Hull‑White model offers a practical balance between tractability and flexibility for pricing interest‑rate derivatives. Its ability to fit the current yield curve and provide analytical pricing formulas makes it popular, but users must account for its allowance of negative rates and its single‑factor limitations.