Consider the one-factor Hull-White model
$$ \mathrm{d}r(t) = (\theta(t)-\kappa r(t))\mathrm{d}t + \sigma\mathrm{d}W(t) $$
When one calibrates the model to market data one chooses
$$ \theta(t) = \frac{\partial f^M}{\partial T}(0,t) + \kappa f^M(0,t) + \frac{\sigma^2}{2\kappa}\left(1-\mathrm{e}^{-2\kappa t}\right) $$
where $f^M(0,T) = -\frac{\partial}{\partial T}\log(P^M(0,T))$ with the observed bond term structure $P^M(0,T)$ at the time of calibration.
I have several questions regarding this calibration:
How do I come up with this formula for $\theta(t)$? I always read that this aligns the model with the market's zero curve. How can you derive that this formula in fact establishes the desired consistency?
How do I come up with $P^M(0,T)$ and $f^M(0,T)$? Am I right that the following approach is taken?
- First the zero curve $y^M(t)$ is bootstrapped using coupon bearing instruments.
- The bootstrapped curve is just given at a finite number of points given by the maturities of the considered instruments. We interpolate these points (e.g. via spline interpolation) to obtain the function $y^M(t)$ on a continuous domain.
- Now we obtain $P^M(0,T)$ by setting $P^M(0,T) = \exp(-y^M(T)T)$.
- In a final step we compute the derivative $f^M(0,T):=-\frac{\partial}{\partial T}\log (P^M(0,T)) = -\frac{\partial}{\partial T}\left(y^M(T)T\right) = -\left(\frac{\partial}{\partial T}y^M(T)\right)T - y^M(T)$