# Stochastic process for interest rates allowing negative values

The Cox-Ingersoll-Ross process for the short term interest rate r(t) does not allow r(t) to become negative, but short-term rates are negative in much of the developed world. To account for this, do you use a CIR process for a shadow rate r'(t) that equals r(t) + c, where c = 0.01 if you think short-term rates cannot get more negative than 1%? Has there been research on this?

Yes, people have looked into that alreaday, for instance here and here (for lognormal models). Brigo and Mercurio took the CIR short rate $$(x_t)$$ and added a deterministic shift $$\vartheta(t)$$ to it in order to obtain the short rate process $$(r_t)$$ via $$r_t=x_t+\vartheta(t)$$. The function $$\vartheta$$ serves to guarantee a perfect fit with observed discount factors and hence, can lead to negative short rate. In their book, you can read about their CIR extension (named CIR++) in Section 3.9 which addresses the issue of positivity explicitly in 3.9.3.