# Risk neutral measure of short rate model

As we all know, all affine term-structure models are members of HJM model. Under HJM model, there is a unique risk-neutral measure in both forward-rate process and bond evolving process. Hence, the model is complete. However, there is no unique risk-neutral measure in short rate models like Vasicek, CIR model (the measure is adjusted by the parameter lambda). Thus, the model is incomplete.

The question is: How to justify the existence and absence of a unique risk-neutral measure in forward rate models (HJM) and short-rate models (Vasicek) respectively? Are there any contradictions?

For any given process for the short rate $\{r_t,, t >0\}$, the price at time $t$ of a zero-coupon bond with maturity $T$, where $t\le T$, is given by \begin{align*} P(t, T) = E\left(e^{-\int_t^T r_s ds}\,\big|\, \mathcal{F}_t\right). \end{align*} Since, for $t\le T$, \begin{align*} \frac{P(t, T)}{e^{\int_0^tr_s ds}} = E\left(e^{-\int_0^T r_s ds}\,\big|\, \mathcal{F}_t\right) \end{align*} is a martingale under the risk-neutral measure, we can assume that the dynamics for $r_t$ is already defined in the risk-neutral measure.
For the forward rate $f(t, T)$, we note that $r_t = f(t, t)$ and \begin{align*} P(t, T) = e^{-\int_t^T f(t, u)du}. \tag{1} \end{align*} We assume that $f(t, T)$ follows, under the risk-neutral measure, the HJM model, that is, \begin{align*} df(t, T) = \alpha(t, T) dt + \sigma(t, T) dW_t, \end{align*} where $\{W_t, \, t \ge 0\}$ is a standard Brownian motion. From $(1)$, \begin{align*} d\ln P(t, T) &= f(t, t) dt -\int_t^T df(t, u) du\\ &=r_t dt - \left(\int_t^T \alpha(t, u) du\right)dt - \left(\int_t^T \sigma(t, u) du\right)dW_t. \end{align*} Then \begin{align*} \frac{dP(t, T)}{P(t, T)} &= \frac{1}{P(t, T)}d\left(e^{\ln P(t, T)} \right)\\ &=\frac{1}{P(t, T)}\left(e^{\ln P(t, T)} d\ln P(t, T) + \frac{1}{2}e^{\ln P(t, T)} d\langle \ln P, \ln P\rangle_t\right)\\ &=\left(r_t - \int_t^T \alpha(t, u) du +\frac{1}{2}\left(\int_t^T \sigma(t, u) du\right)^2 \right)dt - \left(\int_t^T \sigma(t, u) du\right)dW_t. \end{align*} Note that, under the risk-neutral measure, the drift term of $dP(t, T)$ is $r_t$. That is, \begin{align*} \int_t^T \alpha(t, u) du = \frac{1}{2}\left(\int_t^T \sigma(t, u) du\right)^2. \end{align*} Consequently, \begin{align*} \alpha(t, T) = \sigma(t, T)\int_t^T \sigma(t, u) du. \end{align*}