I could need some advice on extensions of the CIR model.
The standard CIR reads
$dr(t)=\kappa(\theta-r(t))dt + \sigma \sqrt{r(t)} dW(t)$.
A possible extension, if we would like the short-rate to also include negative values, could be a displaced version, so that $r(t)+\alpha$, where $\alpha>0$, follows a CIR model.
Further, to fit the initial term structure one could also consider the CIR++ (can be seen in Brigo et al) which is that
$r(t)=x(t)+\phi(t)$,
where $x$ is CIR and $\phi(t)$ is deterministic and chosen to fit the initial term structure.
My question is if it would make sense to consider a displaced CIR++, that is that $r(t)+\alpha=x(t)+\phi(t)$. My immediate thought is that the $\alpha$ does not provide any additional value for the model, and that the $\phi$-function already makes it possible for the short-rate to be negative?
