When pricing derivatives using Monte Carlo methods, we take outset in the risk neutral pricing formula which states that we need to calculate the expected value of the discounted cashflows. To do this, we simulate the market variables which - depending on the model and the product of consideration - might require that we actually simulate some state variables and then transform these into the market variables.
We can do all sorts of tricks to improve the accuracy / speed of the simulation such as using methods of variance reduction, quasi-MC methods with "different random variables", running the simulations in parallel, etc.
However, we might also use a change of measure: To get the correct price under a new measure, we will have to change the dynamics of the state / market variables and the numeraire accordingly. One example where such change of measure is benificial is for interest rate derivatives where we can avoid simulating the joint probability of the numeraire and the payoff function, $g$, by changning from the risk-neutral measure, $Q$, to the $T$-forward measure $Q^T$, that is
$$V_0 = B(0) \cdot E^Q_0 \left[\frac{1}{B(T)} \cdot g(r_T)\right] = P(0,T)\cdot E^T_0 \left[g(r_T)\right],$$
where $B$ denotes the bank account, $P$ denotes the zero coupon bonds. In a short rate model (such as Vasicek) this would allow us to easily use an exact simulation scheme without first finding the joint distribution of the $B$ and $g(r_T)$ (which both depend on $r$) or using a discretization scheme.
My question is: How do we decide which measure to price a claim under?
Does it entirely depend on the model and the product? Or does there exist any ways of deciding this a prior whichout performing numerical tests?
