# Implication of forward-rate dynamics when the short-rate follows a normal process

In the section 3.2.3 of the second edition of "Interest Rate Models - Theory and Practice" by Brigo and Mercurio, the forward-rate dynamics implied by the CIR model is derived as follow:

The CIR short-rate dynamics under the risk-neutral measure:

$$d r(t)=k(\theta-r(t)) d t+\sigma \sqrt{r(t)} d W^Q(t)$$

The forward-rate dynamics under the forward measure: $$d F(t ; T, S)=\sigma \frac{A(t, T)}{A(t, S)}(B(t, S)-B(t, T)) \exp \{-(B(t, T)-B(t, S)) r(t)\} \sqrt{r(t)} d W^{S}(t) = \sigma\left(F(t ; T, S)+\frac{1}{\gamma(T, S)}\right) \sqrt{(B(t, S)-B(t, T)) \ln \left[(\gamma(T, S) F(t ; T, S)+1) \frac{A(t, S)}{A(t, T)}\right]} d W^{S}(t)$$

[Question 1]

When the short-rate follows a normal process for example like the following:

$$d r(t)=k[\theta(t)-\alpha(t)r(t)] d t+\sigma(t) d W^Q(t)$$,

and if I follow the same derivation as done for the CIR model, am I correct to assume that the forward-rate dynamics would look like a shifted-lognormal process like this?:

$$d F(t ; T, S)=\sigma(t) (B(t, S)-B(t, T))\left(F(t ; T, S)+\frac{1}{\gamma(T, S)}\right) d W^{S}(t)$$

[Question 2]

Then, if I further assume that $$B(t,T)=T-t$$, the forward-rate dynamics would become

$$d F(t ; T, S)=\sigma(t) \gamma(T, S) \left(F(t ; T, S)+\frac{1}{\gamma(T, S)}\right) d W^{S}(t)$$.

Would this mean that when $$F(t ; T, S)$$ is near zero, it behaves like a normal process with the short-rate volatility?

I.e. $$d F(t ; T, S)=\sigma(t) d W^{S}(t)$$ ?