Hull-White Model
Also known as: Extended Vasicek, Generalized Vasicek
The Hull-White model (1990) is a one-factor short-rate model with time-dependent mean reversion and volatility, designed to fit the initial yield curve exactly. It generalizes the Vasicek model to allow better calibration to observed bond and derivative prices, and is widely used for pricing interest rate exotics and managing interest rate risk.
Key highlights
- Exact fit to yield curve: time-dependent theta(t) ensures the model reproduces current bond prices with no initial drift
- Closed-form bond prices: analytical formulas for zero-coupon bonds and bond options available, enabling fast pricing
- Calibration flexibility: both drift and volatility can be tailored to market data independently
- Mean reversion: natural bounds prevent rates from becoming unrealistic; stable long-term distributions
Intuition
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How it works
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When to use it
Use Hull-White for pricing interest rate derivatives when accurate calibration to the current yield curve is necessary. It is particularly valuable for pricing bonds, swaptions, and exotic interest rate products. The model is less suitable for very long-dated products where two-factor extensions may be needed.
Strengths & limitations
- Exact fit to yield curve: time-dependent theta(t) ensures the model reproduces current bond prices with no initial drift
- Closed-form bond prices: analytical formulas for zero-coupon bonds and bond options available, enabling fast pricing
- Calibration flexibility: both drift and volatility can be tailored to market data independently
- Mean reversion: natural bounds prevent rates from becoming unrealistic; stable long-term distributions
- Negative rates: allows negative interest rates without special handling, which is unphysical for some applications (though realistic post-2008)
- One-factor assumption: cannot capture decorrelated moves of short and long rates (level vs slope effects)
- Volatility assumption: constant volatility sigma may not match empirical changes in rate volatility over time
- Parameter sensitivity: small changes in calibration parameters can significantly alter long-dated option prices
Common pitfalls
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Applications
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Frequently asked
How does Hull-White differ from Vasicek?
Vasicek has constant parameters and cannot fit the initial yield curve (it has initial valuation bias). Hull-White generalizes Vasicek by making the drift time-dependent via theta(t), allowing perfect fit to the market curve. Both allow negative rates; both have mean reversion.
What does theta(t) do?
Theta(t) is the time-dependent drift function. It is chosen such that the model exactly reproduces observed zero-coupon bond prices. Intuitively, theta(t) compensates for the difference between the model's long-term mean and the market's expectation of future short rates implicit in the curve.
Can I use constant volatility in Hull-White?
Yes, constant sigma simplifies calibration and is common in practice. However, empirical evidence shows volatility changes over time. If you need to price options with different maturities, consider time-dependent sigma(t) or stepping stones (different sigma per tenor) to improve fit.
How do I build a Hull-White binomial tree?
Discretize the SDE, then at each node calculate the bond price analytically and adjust node spacing to ensure no arbitrage. The tree must satisfy the recombining condition and avoid negative rates or explosive paths. Many banks use Jarrow-Rudd or Black-Derman-Toy tree adjustments to handle this.
Sources
- 1.Hull, J., & White, A. (1990). Pricing interest-rate-derivative securities. Review of Financial Studies, 3(4), 573-592.
- 2.Brigo, D., & Mercurio, F. (2006). Interest Rate Models: Theory and Practice (2nd ed.). Springer-Verlag.
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ScholarGate. (2026, June 3). Hull-White Model. ScholarGate. https://scholargate.app/quantitative-finance/hull-white-model