View source for Hull-White

== Overview (cf. Wikipedia)==

The Hull-White model is a short-rate model. In general, it has dynamics

:<math>dr(t) = (\theta(t) - \alpha(t) r(t))\,dt + \sigma(t)\, dW(t)\,\!</math>

There is a degree of ambiguity amongst practitioners about exactly which parameters are time-dependent. The most commonly accepted choices are, starting with the most popular one:

:&theta; a constant - known as '''the [[Vasicek model]]'''
:&theta; depending on t - known as '''the Hull-White model'''
:&theta; and &alpha; both time-dependent - known as  '''the extended [[Vasicek model]]'''

For the rest of this article we assume that only theta has t-dependence.
Neglecting the stochastic term for a moment, notice that a change in r is negative if r is currently "large" (greater than &theta;(''t'')/&alpha;) and positive if the current value is small. That is, the stochastic process is a [[mean reversion|mean-reverting]] [[Ornstein-Uhlenbeck process]].

&theta; is calculated from the initial [[yield curve]] describing the current term structure of interest rates. Typically, &alpha; is left as a user input (for example, it may be estimated from historical data). &sigma; is determined via [[calibration]] to a set of [[caplet]]s and [[swaption]]s readily tradeable in the market.

When <math>\alpha, \theta </math> and <math>\sigma</math> are constant, [[Ito'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 a distribution
:<math>r(t) \sim N(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)).</math>

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