Editing Hull–White model (section)

==Analysis of the one-factor model==
For the rest of this article we assume only <math>\theta </math> has ''t''-dependence.
Neglecting the stochastic term for a moment, notice that for <math>\alpha > 0</math> the change in ''r'' is negative if ''r'' is currently "large" (greater than <math>\theta(t)/\alpha)</math> and positive if the current value is small. That is, the stochastic process is a [[Regression toward the mean|mean-reverting]] [[Ornstein–Uhlenbeck process]].

θ is calculated from the initial [[yield curve]] describing the current term structure of interest rates. Typically α is left as a user input (for example it may be estimated from historical data). σ is determined via [[calibration]] to a set of [[Interest rate cap and floor|caplets]] and [[swaption]]s readily tradeable in the market.

When <math>\alpha</math>, <math>\theta</math>, and <math>\sigma</math> are constant, [[Itô's lemma]] can be used to prove that

:<math> r(t) = e^{-\alpha t}r(0) +  \frac{\theta}{\alpha} \left(1- e^{-\alpha t}\right) + \sigma e^{-\alpha t}\int_0^t e^{\alpha u}\,dW(u),</math>

which has distribution
:<math>r(t) \sim \mathcal{N}\left(e^{-\alpha t} r(0) +  \frac{\theta}{\alpha} \left(1- e^{-\alpha t}\right), \frac{\sigma^2}{2\alpha} \left(1-e^{-2\alpha t}\right)\right),</math>

where <math>\mathcal{N}( \mu ,\sigma^2 )</math> is the [[normal distribution]] with mean <math>\mu</math> and variance <math>\sigma^2</math>.

When <math>\theta(t)</math> is time-dependent,
:<math> r(t) = e^{-\alpha t}r(0) +  \int_{0}^{t}e^{\alpha(s-t)}\theta(s)ds + \sigma e^{-\alpha t}\int_0^t e^{\alpha u}\,dW(u),</math>

which has distribution
:<math>r(t) \sim \mathcal{N}\left(e^{-\alpha t} r(0) +  \int_{0}^{t}e^{\alpha(s-t)}\theta(s)ds, \frac{\sigma^2}{2\alpha} \left(1-e^{-2\alpha t}\right)\right).</math>