# Time dependent parameters in Hull-White model

Hull-White: $$d r = [\theta(t) - ar]d t + \sigma d W_t.$$ There is a statement in John Hull's book:

The advantage of making $a$ or $\sigma$, or both, functions of time is that the models can be fitted more precisely to the prices of instruments that trade actively in the market.
The disadvantage is that the volatility structure becomes nonstationary. The volatility term structure given by the model in the future is liable to be quite different from that existing in the market today.

$\theta(t)$ can already match the initial curve in the market, what's the meaning of fitted more precisely to the prices?

$\sigma(t)$ can match today's implied vol of all maturities, it should be better than $\sigma.$ How to understand nonstationary?

On your first question, the fact that you fit to the yield curve $-$ which is what the standard Hull-White model with time-dependent $\theta(t)$ allows to do $-$ does not mean that you are fitting the prices of more complex products such as options. For that you need to make $a$ and/or $\sigma$ also time-dependent.
On your second question, it simply means that you no longer have a flat, constant value for $\sigma$ but a much complex structure which depends on time $-$ I don't think there is anything more to that statement.
• @AFK I know, but therefore if volatility is constant it is stationary, as do $r$ increments "w.r.t. to volatility" $-$ although increments are not stationary due to time-dependence of $\theta$. What could be the meaning otherwise? – Daneel Olivaw Apr 11 '18 at 22:23