Bond-price dynamics in the Vasicek model

Question

Hello I am studying about interest rate modeling

There is one good source about Vasicek (link: https://web.mst.edu/~bohner/fim-10/fim-chap4.pdf). However there is one equation that I try but unable to replicate which is:

$dP(t,T) = r(t)P(t,t)dt - \sigma B(t,T)P(t,T)dW(t)$

This equation on 2nd page (or page 18th according to document paging). It locates about 1/3 page top down. Anyone understand how we get this one? What border me is why there is $B(t,T)$ appear. I tried but unable to obtain that result.

Besides, the side question is why in interest rate stochastics process it is always express under risk neutral $\mathbb{Q}$ why a traditional stock price S is often expressed in $\mathbb{P}$

Thank you so much

Answer 1

You know the bond price formula takes this form:

$P \left( t, T \right)= A \left( t, T \right) e^{ -r_{t} B \left(t, T \right) }$

Now apply Ito's lemma, so you will get after some manipulation:

$\frac{dP}{P}= \left(\frac{1}{A} \frac {\partial A}{\partial t} -r \frac {\partial B}{\partial t} - \kappa \theta B + \kappa r B+ \frac{1}{2} B^2 {\sigma}^2\right) dt - \sigma B d w_{t}$

The expected return under the risk neutral measure must be r, so you can set the drift term equal to r, and you get your equation.

Don't think it is necessary to represent risk neutral measure by Q, but Q has become sort of conventions. Probably originated from the fact that people use P to represent probability, and then use the next letter Q, to denote the next measure one want to talk about.