MC Simulation for Hull-White Model

Recipe 1
Fit kappa and sigma by calibrating to swaptions/caps (swaption and cap prices can be expressed as Hull-White bond options, which in turn can be expressed as function of H-W bond price). Then calculate theta(t) by taking partial derivative of instantaneous forward rate f(0,t). Then do Euler MC simulation:

$r(t)=r(s)+(\theta (t) - \kappa r)dt + \sigma dW$

Recipe 2
This alternative method doesn't require taking partial on f(0,t). After fitting kappa and sigma,
then do the following MC simulation (it is by integrating the SDE):

$r(t)=r(s)e^{-\kappa (t-s)} + \alpha(t) - \alpha(s)e^{-\kappa(t-s)}+B$

where B is a Gaussian distributed white noise with variance

 $ \frac {\sigma^2}{2 \kappa} [1-e^{-2 \kappa (t-s)}]$

and

$\alpha(t) = f(0,t) + \frac {\sigma^2}{2\kappa^2}(1-e^{-\kappa t})^2$

Bottom line is, theta-fitting is about calibrating to the spot curve. But Recipe 2 does just that without invoking theta, because the instantaneous forward rate contains the same information.

Ref: Brigo pp.73

Also:
One can simulate the stochastic short rate process under the T-forward measure instead of the risk-neutral measure. The drift of the SDE will be different from that above, and would depend on T.

Advantage:
Since under the T-forward measure we discount the payoff with zero coupon bond, and the zero coupon bond price is completely determined by the short rate at a certain moment, we don't have to simulate too many points on a path, but we do need to do that for risk-neutral measure pricing in order to approximate the money-market account well.

Disadvantage:
P(t,T) bond is the natural numeraire to use for discounting, but what if we are pricing an instrument with multiple cash flows?