# Hull-White Extension of Vasicek Model

I am reading the book Interest Rate Models by Brigo and Mercurio and try to understand the Hull White Model Extended Vasicek Model. They start off by defining the instantaneous short-rate process under the risk-neutral Measure by

\begin{align} dr(t)=[\theta (t) -a(t)r(t)]dt + \sigma dW(t) \end{align}

with $\theta$, $a,$ $\sigma$ being deterministic functions of time. I dont fully understand why those dynamics describe the risk neutral one. Doesnt it mean that the drift $\theta (t) -a(t)r(t)$ is the riskless return and if yes why?

Thanks for any help.

$\theta(t) - a(t) r(t)$ is the risk neutral drift. The Hull & White models posits the dynamics $dr(t) = (\theta(t) - a(t) r(t)) dt + \sigma dW(t)$ under the risk neutral measure $P$ and then calibrates $\theta(t)$ so that the risk neutral condition $$E^P\left[e^{-\int_0^T r(u) du} \right]=\text{discount}(T)$$ is satisfied and $P$ is indeed the risk neutral measure. Calibration of $\theta(t)$ is exact and explicit.
• I see, that makes sense. So then you first solve the SDE to get r(t), substitute it into the risk neutral condtion above and solve for $\theta(t)$? For discount(T), zero coupon bond prices are used? Thanks! – Mh Aztec Mar 26 '18 at 14:51
• yes to both questions. There is a closed form formula to compute $\theta(t)$ see for instance quant.stackexchange.com/questions/38739/… – Antoine Conze Mar 26 '18 at 14:56