HullWhite

Hull-White refers to a family of interest rate models, named after John Hull and Alan White, introduced in 1990 as a time-dependent extension of the Vasicek model. The one-factor version is a short-rate model where the instantaneous interest rate follows a mean-reverting Gaussian process, designed to fit the current term structure of interest rates.

In the one-factor Hull-White model, the short rate r_t evolves under the risk-neutral measure as

dr_t = [theta(t) - a r_t] dt + sigma dW_t,

where a > 0 is the mean-reversion speed, sigma > 0 is the volatility, W_t is a Brownian motion, and theta(t) is a time-dependent drift chosen to fit the initial yield curve. Theta(t) can be expressed in terms of the initial forward rate curve f(0,t) as

theta(t) = ∂f(0,t)/∂t + a f(0,t) + (sigma^2 / (2a)) (1 - e^{-2 a t}).

Bond prices take the form B(t,T) = A(t,T) exp(-B(t,T) r_t), with B(t,T) = (1 - e^{-a (T-t)})/a, and A(t,T) determined by a, sigma, and the initial term structure. This structure yields closed-form expressions for zero-coupon bonds and provides semi-closed forms or efficient numerical methods for many interest-rate derivatives, such as caps, floors, and swaptions, often using Jamshidian’s trick or simple Monte Carlo simulations.

A two-factor extension exists, introducing a second Brownian motion to enhance the model’s flexibility in fitting the initial term structure and cap volatilities. Hull-White remains popular in quantitative finance for its analytical tractability, arbitrage-free dynamics, and compatibility with observed yield curves.