Forward-Rate Dynamics from a Normally Distributed Short Rate

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


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

## Answer by Arshdeep (score 2, accepted)

https://quant.stackexchange.com/a/55038

For Q1, Indeed the ratio of 2 zero coupon bonds associated with the forward is an exact lognormal process (Just apply Ito's lemma to the ratio, as you already know the dynamics of the 0 coupon bonds. You can disregard the drift term as the forward rate is a martingale in the bond forward measure.).

The forward rate is then obtained by just adding a scalar, so the dynamics of the forward rate you have written follow from there.

For Q2, you are right. Note that this would hold approximately for short dated forwards (the fwd rate is much smaller than the inverse of the day count fraction) as well.