# Simulating a square root process with jumps for mortgage defaults

I am trying to simulate the paydown of a large pool of mortgage loans. For each monthly period, I am reducing principal by the scheduled principal payment (approximated by the WAC of the underlying loans) and prepayments governed by a CPR that is not random in the model. I want to incorporate random defaults and am quite inexperienced with simulating stochastic processes so please let me know if I can explain in more detail.

One idea that I had was to simulate a process like a CIR square root process that incorporates a Poisson jump processes. The intuition behind this would be that there is a typical small, mean reverting default process in normal times, that cannot be negative. The jump would represent crisis times, and approximate the correlated nature of defaults in the mortgages.

I have found quite a bit of resources on simulating the CIR process in discrete time, but never one with jumps. Is that because there is something implicitly dumb about doing this? The process would look something like this, where $$X$$ is the conditional default rate:

$$dX_t = (\alpha - \beta)X_t dt + \sigma \sqrt{X_t} dZ + JdN_t$$, where $$J$$ is a constant jump size and $$N_t$$ is a Poisson process with constant intensity.

Note that, $$X$$ is not the losses on the portfolio. If there was a notional value of 1 mortgages to begin with, and after sometime only 0.5 was left, the same CDR level would cause a lower loss in the latter.

So my question really is, what is the best way to simulate from this process in discrete time. Is the answer just to trivially simulate both processes separately for each step since the Poisson process is independent of the CIR process? Is this just a totally wack way to be modeling mortgage defaults in the first place? I am interested in any an all feed back. Thanks for your help in advance, I am very lost with this

• You should be able to simulate this, no problem. Calibrating this, on the other hand, is a bit more burdensome. What you are looking for is a "doubly stochastic process", e.g. equation 3.11 in here web.stanford.edu/~duffie/pisa.pdf . In a nutshell: Yes, you can trivially simulate the jump (yes/no + jump size), and add the result on top of the CIR increment to arrive at a new default intensity. Then, draw a default based on this total intensity. – Kermittfrog Feb 11 at 8:06