Hull–White one-factor model

1. 01

   Multiply by the integrating factor

   Apply e^{at} so the mean-reversion term becomes an exact differential.

   $$d(e^{at}r_t)=e^{at}\theta(t)dt+\sigma e^{at}dW_t$$
2. 02

   Integrate from t to T

   Solve for r_T as decayed current state plus deterministic drift integral and a Gaussian stochastic integral.

   $$r_T=e^{-a\tau}r_t+\int_t^T e^{-a(T-u)}\theta(u)du+\sigma\int_t^T e^{-a(T-u)}dW_u$$
3. 03

   Compute the variance

   Itô isometry converts the stochastic-integral variance into an ordinary integral.

   $$\sigma^2\int_t^T e^{-2a(T-u)}du=\sigma^2(1-e^{-2a\tau})/(2a)$$
4. 04

   Fit the initial term structure

   Choose θ(t), or an equivalent deterministic shift φ(t), so model P(0,T) equals the observed discount curve at every maturity.

5. 05

   Calibrate option dynamics

   Fit a and σ—or piecewise volatility—to a governed cap/swaption basket after the curve fit.

Hull–White separates exact initial-curve fit from a parsimonious one-factor Gaussian dynamics used for optionality and scenarios.