Modeling Mortgage Defaults with a CIR Rate and Poisson Jumps
Summary
The document considers simulating defaults in a pool of mortgage loans whose principal declines through scheduled payments and nonrandom prepayments. The author proposes modeling the conditional default rate with a nonnegative, mean-reverting square-root diffusion, augmented by Poisson jumps. The diffusion is intended to represent ordinary default-rate variation, while jumps stand in for crisis periods and clustered defaults. The author emphasizes that this rate is not itself portfolio loss, since losses depend on the remaining principal balance.
The central questions are how to simulate the proposed process in discrete time and whether it is sensible for mortgage defaults. The document supplies no answer, calibration, or simulation evidence, so it does not establish a preferred discretization or validate the model. It also leaves important modeling choices open, including how jumps affect default rates and how defaults interact with the shrinking pool. Treat the process as a candidate specification rather than a tested approach.
Key ideas
- The proposed default-rate model combines a mean-reverting square-root diffusion with Poisson jumps.
- The diffusion represents ordinary variation, while jumps are intended to capture crisis-driven default clustering.
- A conditional default rate is distinct from portfolio losses, which depend on outstanding principal.
- The document asks about simulation and model suitability but provides no implementation guidance or validation.
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Full text
# Simulating a square root process with jumps for mortgage defaults
# 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 thisShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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