I just wanted to clear somethings up when it comes to pricing Mortgage backed securities using Monte-Carlo methods. I understand that interest rate paths have to be modelled in order to come up with prepayment models which are then used to figure out the cashflows. What I am confused about is what interest rate is being modelled i.e what is the term of the simulated interest rate is it the one month rate? The one year rate? etc.
Wouldn't it make sense to produce a simulation of the one month rate for a fixed number of time periods, say the life of the mortgage. Then at each time period the cash-flows can be discounted month by month until we get to time 0. And then repeat this process n number of times to get the average present value.
For a very simplistic example we simulate the 1 month rate r starting at time t as, for 3 months to get [$(0, r1), (2, r2), (3, r3)$] then using our prepayment model we generate cashflows for each month [$(1, CF1), (2, CF2), (3, CF3)$] then we can discount each cash flow using our simulated rates to get PV at time 0 $PV = (((CF3d(2,3) + CF2) *d(1,2) + CF1) *d(0,1))$ where $(dt_1,t_2)$ is the discount factor from time $t_1$ to time $t_2$. repeating this whole process n number of times.
Would this be a valid method or have I completely missed the mark? Any help is appreciated thanks!