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# Hull-White

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## Overview (cf. Wikipedia)

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

$dr(t) = (\theta(t) - \alpha(t) r(t))\,dt + \sigma(t)\, dW(t)\,\!$

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:

θ a constant - known as the Vasicek model
θ depending on t - known as the Hull-White model
θ and α 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 θ(t)/α) and positive if the current value is small. That is, the stochastic process is a 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 caplets and swaptions readily tradeable in the market.

When $\alpha, \theta$ and $\sigma$ are constant, Ito's lemma can be used to prove that

$r(t) = e^{-\alpha t}r(0) + \frac{\theta}{\alpha} \left(1- e^{-\alpha t}\right) + \sigma e^{-\alpha t}\int_0^t e^{\alpha u}\,dW(u)\,\!$

which has a distribution

$r(t) \sim N(e^{-\alpha t} r(0) + \frac{\theta}{\alpha} \left(1- e^{-\alpha t}\right), \frac{\sigma^2}{2\alpha} \left(1-e^{-2\alpha t}\right)).$