# Intuitive explanation for theta Hull-White

I am having a hard time coming up with an intuitive explanation for the long term mean $$\theta$$ in the Hull-White model:

$$\mathrm{d}r_t=[\theta(t)-\alpha r_t]\mathrm{d}t+ \sigma_t \mathrm{d}W_t$$

So far I have the following expression for $$\theta(t)$$:

$$\theta(t)=\frac{\partial f_M(0,t)}{\partial t}+\alpha f_M(0,t)+\frac{\sigma^2(t)}{2 \alpha}\left(1-e^{-2 \alpha t}\right)$$

where $$f_M$$ is the forward curve calculated from zero-coupon prices, $$\alpha$$ the mean reversion speed, and $$\sigma(t)$$ the volatility of the rate at that pillar. I understand that $$\theta(t)$$ is the curve that the present term structure will revert to, considering the present shape of the forward curve (the partial derivative term), the level of the forward rate, and the variance of $$r_t$$. Is this the right interpretation?

• Yes, there is a tendency for the instantaneous rate $r_t$ to be pulled towards the value $\theta(t)$. This rather artificial mechanism is how the model attempts to fit the desired term structure. Dec 18 '20 at 19:54