I want to find the price of Zero coupon bond given a short rate model.
I think about Merton, Vasiceck, CIR, Ho & Lee models.
1) Given a simulation of $r_t$ how can I calculate $ P(t,T) = \mathbb{E}^Q\left[\left. \exp{\left(-\int_t^T r_s\, ds\right) } \right| \mathcal{F}_t \right] $ ?
Using the simulations i think it would be easy to calculate the integral. But how to calculate the integral knowing $\mathcal{F}_t$ ? Am I supposed to find an expression of $r_s$ depending on $r_t$ ?
2) How to deal with the risk neutral probability here ?
3) Would this approach still be ok with a time dependant model ? (Hull White) Would this approach still be good with multiple factor model ? (Logstaff Schwartz)