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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 $dt$?

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  • $\begingroup$ I do not think you got the short rate process $r_t$ right, with the $\sqrt{r_t}$ term. The bond price solution is certainly not for the short rate process you wrote down. $\endgroup$ – Hans May 3 '16 at 19:06
  • $\begingroup$ does it have anything to do with the fact that the $rd_t$ is given as risk-neutral dynamics? $\endgroup$ – Michal May 3 '16 at 19:31
  • $\begingroup$ No. Risk neutral measure would not affect the $dW_t$ term. There must be a typo. The bond price solution for the short rate process you have written is a lot more complicated than what you have written here. Besides, where does the $\sigma$ in your bond price formula come from while it is not in the short rate process? $\endgroup$ – Hans May 3 '16 at 19:45
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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.

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  • $\begingroup$ I thought about typos too, but I was not sure if I was missing something. (1) is not even CIR model because sigma is missing and the drift is different (CIR $dr_t = (b - \beta r_t)dt + \sigma \sqrt{r_t} dW_t \quad $ ). Assuming (1) is $dr_t = \kappa (\theta - r_t)dt + \sigma dW_t$ the final equation should translate into $$dr_t(\tau)= \frac{B(t,\tau)}{\tau}(\kappa (\theta - r_t)dt + \sigma dW_t)) \quad \quad (8)$$ $\endgroup$ – Michal May 4 '16 at 13:55
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    $\begingroup$ @Michal. Yes, but note that (1) is indeed CIR (with $\sigma=1$). It is a matter of convention. You could write CIR model $dr_t = (b-\beta r_t)dt + \sigma \sqrt{r_t} dW_t$ as $dr_t = \beta(b/\beta- r_t)dt + \sigma \sqrt{r_t} dW_t$ and then let $\beta = \kappa$, $b/\beta = \theta$ and $\sigma=1$ to fall-back on the (wrong) dynamics mentioned in (1). $\endgroup$ – Quantuple May 4 '16 at 14:03

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