CIR calibration

Question

I'm using a CIR short rate model to forecast interest rate paths. I've been thinking and also searching online about different ways of estimating its parameters (a, b and sigma). While there are a number of relatively intuitive ways with their pros and cons (e.g. reshape into a linear regression and do OLS or MLE), I cannot find anywhere how to actually calibrate them and fit them. Is there a way to measure the goodness of fit? From my calculations, the R-squared from the linear regression is meaningless. In what way can you test that your fitted values are similar to your observed values?

To clarify, this question is not about term structures. I'm only interested to see how well it fits to the observed historical time series values before I run simulated paths.

Thanks for any advice!

Answer 1

CIR can be used to simulate paths, although forecasting with a model of that class is a bit unintuitive. Why? The results are highly dependent on the stochastic parameters of the equation. So, let's say you obtained a calibrated model and simulated it 5 times. You would (most likely) get very different results with the same model but with different random numbers. You could average the numbers, but then all you're getting is a mean-reversion result based on historic parameters that are calibrated on a random walk (w/ drift). Overall, the parameters are really just an extension of a random walk and won't describe time series data well.

Therefore, the economic value of a CIR simulation is limited if forecasting. But, using the paths to price and model a security or derivative is indeed a very useful tool and you can likely see why.