Dynamics of short rate in HJM

Question

According to a simplified HJM framework, we have:

Forward Rate: $f(t,T)=\sigma W_t +f(0,T) +\int_0^t{\alpha(s,T)}ds$, where $W_t$ is brownian motion.

Dynamics of forward rate: $d_tf(t,T)=\sigma dW_t+\alpha(t,T)dt$

Short Rate: $r_t = \sigma W_t + f(0,t) +\int_0^t{\alpha(s,t)}ds$

and $r_t = f(t,t)$

I tried to find the dynamics of the short rate ($dr_t$) by applying Ito's formula, but I could only obtain $dr_t= d_tf(t,t) + \frac{\partial f}{\partial t}(0,t)$.

My steps:

Let $r_t = X_t + f(0,t)$ where $X_t = \sigma W_t +\int_0^t{\alpha(s,t)}ds$ such that $dX_t = \sigma dW_t + \alpha(t,t) dt$. Applying Ito's formula, $dr_t= d_tf(t,t) + \frac{\partial f}{\partial t}(0,t)$.

However the text states that the dynamics is actually: $dr_t= d_tf(t,t) + \frac{\partial f}{\partial T}(t,t)$.

May I know why there is this discrepancy?

Answer 1

You just have to be careful with the notation because $t$ can appear in two arguments.

The forward rate is

$$f(t,T)=f(0,T) + \int_{0}^t \alpha(s,T)ds+ \sigma W_t.$$

The short rate is

$$r_t=f(t,t)=f(0,t) + \int_{0}^t \alpha(s,t)ds+ \sigma W_t = F(t,X_t),$$

where $F_X = 1$, $F_{XX} = 0$ and

$$X_t = \int_{0}^t \alpha(s,t)ds+ \sigma W_t,\\dX_t = df(t,T)|_{T=t}= \alpha(t,t)dt+\sigma dW_t.$$

Using Ito's lemma

$$dr_t = F_tdt+ F_XdX_t+ \frac1{2}F_{XX}(dX_t)^2$$

Then

$$dr_t=\left[\frac{\partial f }{\partial t}(0,t)+\int_{0}^t \frac{\partial }{\partial t}\alpha(s,t)ds\right]dt+dX_t \\=\frac{\partial f }{\partial T}(t,t)+df(t,T)|_{T=t}$$