# What is actually going on in Monte-Carlo simulation for Mortgage backed securities?

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!

In my understanding, the mortgage prepayment option, at any point in time, is a function of the value of the mortgage from that point in time forward. This value, in turn, is a function of the future evolution of the interest rates and any optimal decision taken by the mortgagor along that path and all paths that evolve from any future 'branch'.

So in short, at every modeled decision time point, you need to come up with all paths (rates and mortgagor decisions) that evolve from there.

From that point, you can of course use your approach for discounting (to today) by iterative use of the year-to-year discount factors simulated along your path.

There is a lot of prepayment models for MBS, mostly every big bank has its own proprietary model.

But the prepayment model can take into account many variables than only interest rate. A probability of prepaymet also depends on geographical location of a debtor (for example there is a lower probability that a mortgage would be prepaid in New York than in Iowa because properties are pretty more expensive in NY and a mortgage is higher). Also social status of a debtor or point in financial cycle can be of importance (there is a lower probability a mortgage would be prepaid in recession than in times of growth because a debtor has lower income or a bank is unwilling to refinance during a crisis). Moreover, so-called MBS vintage (i.e. a year a mortgage was engaged) can be important as the year influence rate a mortgage was provided for.

• I don’t think this answer is relevant. PSA is a static model and has nothing to do with Monte Carlo simulation to price prepayment optionally using a rate sensitive, dynamic model. – Bond wiz Mar 29 '20 at 16:01
• @Bondwiz: Yes, you are right. I removed link to PSA. I just wanted to point out that interest rate, although probably the most important, is not only variable in prepayment model. – Martin Vesely Mar 29 '20 at 21:31
• Yes but rates are generally the only thing simulated in a Monte Carlo framework, which is what the initial question was about.I don’t see how everything else you mentioned is related to the question. – Bond wiz Mar 30 '20 at 21:26
• You can also model a decision of a debtor to prepay, it is also described by a probability distribution (probably conditioned by interest rates). – Martin Vesely Mar 30 '20 at 21:40