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.
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
In the vasicek model, the short rate follows the following dynamic:
$dr_{t} = a(b-r_{t})dt + \sigma dW_{t}$
With :
$a$ constant positive, which represents the return force
$b$ constant positive, which represents the long-term rate
$\sigma$ constant positive, which represents the volatility
The parameters of the model then become $\vartheta$, $a$ and $\sigma$
Under vasicek, the short rate follows an Ornstein-Uhlenbeck process (O-U), and the corresponding solution is:
$$r_t = r_0 e^{-at} + b(1-e^{-at}) + \sigma e^{-at} \int_0^t e^{au} dW_u$$
This Ornstein-Uhlenbeck (O-U) process has been established, following a Gaussian law of parameters its expectation and variances, an Ornstein-Uhlenbeck process is in fact both Gaussian and Markovian.
Let be $X$ a random variable according to a normal distribution. Then $e^X$ follows a log-normal law and therefore:
$\mathbb{E}(e^X) = e^{\mathbb{E}(X) + \frac{1}{2} \mathbb{V}(X)}$