Vasicek model problem

Question

I am analyzing a problem where the below is given

Vasicek model with risk-neutral dynamics $$dr_t = \kappa (\theta - r_t)dt + \sqrt{r_t} dW_t \quad \quad (1) $$

bond prices $$P(t,T)=e^{A(t,T)-B(t,T)r_t} \quad \quad (2)$$, where $$B(t,T)= \frac{1- e^{-\kappa (t-T)}}{\kappa} \quad \quad (3)$$ $$A(t,T)= (B(t,T)-(T-t))(\theta-\frac{\sigma^2}{2 \kappa^2})-(\frac{\sigma^2 B(t,T)^2}{4 \kappa}) \quad \quad (4)$$

Using Ito formula I am deriving the $r_t(\tau)$ (continuously compounded spot rate with constant maturity $\tau$) where $\tau$ is constant and $r_t(\tau)=r(t,t+\tau)$.

$$r(t,t+\tau)=\frac{-log(P(t,t+\tau))}{\tau} \quad \quad (5)$$ substituting A and B yields $$r(t,t+\tau)=\frac{-A(\tau)}{\tau}+ \frac{r_t}{\tau} B(\tau) \quad \quad (6)$$ applying Ito formula where $\quad f'(r_t)=\frac{B(t,\tau)}{\tau} \quad$ and $\quad f''(r_t)=0$ $$dr_t(\tau) = f'(r_t)dr_t + \frac{1}{2} f''(r_t) d<r>_t \ = \ \frac{B(t,\tau)}{\tau} dr_t \quad \quad (7) $$

substituting for the $dr_t$ gives me the equation $$dr_t(\tau)= \frac{B(t,\tau)}{\tau}(\kappa (\theta - r_t)dt + \sqrt{r_t} dW_t)) \quad \quad (8)$$

The final answer I got is different from the answere suggested by solution manual $dr_t(\tau)= \frac{B(t,\tau)}{\tau}(\kappa (\theta - r_t)dt + \frac{B(t,\tau)}{\tau} dt \quad$ which is confusing.


My questions
Vasicek model is given in this problem in a different form from the one usually seen in the books $dr_t = \kappa (\theta - r_t)dt + \sigma dW_t $ is this a spoiler here? how should this be analyzed?

second
in the final equation does $\sqrt{r_t} dW_t$ translates into $dt$?

Answer 1

For starters, the short rate model you mention in equation (1) is Cox-Ingersoll-Ross while the bond price in equations (2)-(4) correspond to the Vacisek model. So there is a problem somewhere, I would go for a typo in (1).

Second, what you wrote seems fine to me, so there must definitely be yet another typo in your solution manual. Note that if there is no $dW_t$ term in the SDE for the rate $r(t,t+\tau)$ like it seems to be stated in your manual then this quantity would be predictible, which defeats the purpose of establishing a stochastic interest rate model in the first place.