New answers tagged modelling
A simple shifting trick is to put $r(t)-f$ instead of $r(t)$ under the square root in your expression. Then $f$ is the new, possibly negative, interest rate floor. If, for example, $f=-100bp$ then the process is defined for all $r(t)>-100bp$. Same for Sabr, where instead of square root you have another exponent.
You are right. In the CIR++, $\alpha$ parameter is absorbed into $\phi$. With the CIR++, $\phi(t)$ will allow you to have to have negative rates. You will calibrate your $\phi$ to fit the discount factors. The shifted idea is the one used to handle negative rates problem in caplet, swaption...
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