Reading Gatheral's The volatility surface, page 7.
The model they are talking about is
$$\begin{align}dS_t&=\mu_tS_tdt+\sqrt{\nu_t}S_tdZ_1\\d\nu_t&=\alpha(S_t,\nu_t,t)dt+\eta\beta(S_t,\nu_t,t)\sqrt{v_t}dZ_2\\\left[Z_1,Z_2\right]&=\rho dt\end{align}$$
where $S_t$ is a stock price, and $\nu_t$ stochastic volatility.
There is an option on $S$ with price $V(S_t,\nu,t)$, and another asset $V_1$ depending on the volatility.
A portfolio $\Pi=V-\Delta V-\Delta_1 V_1$, with $\Delta$ and $\Delta_1$ chosen such that this evolves like the money market account
$$dB_t=r_tB_tdt$$
with $r_t$ assumed deterministic.
[skipping the long equations]
They get a big differential operator applied to $V$ and applied to $V_1$ most both equal to some function which they write in the form $-(\alpha-\phi\beta\sqrt{\nu})$, for some function $\phi$. So far so good. And they call $\phi$ the market risk of volatility risk.
With the choices of $\Delta$ and $\Delta_1$ made, they form a portfolio $\Pi_1=V-\Delta S$ from where
$$d\Pi_1-r\Pi_1=\beta\sqrt{\nu}\frac{\partial V}{\partial \nu}\left(\phi dt+dZ_2\right)$$
So far so good, too. Now they say "defining the risk neutral drift as"
$$\alpha'=\alpha-\beta\sqrt{\nu}\phi$$
the equation for $\nu$ becomes
$$d\nu_t=\alpha'dt+\beta\sqrt{\nu}dZ_2$$.
Here is what I don't understand. Why are they free to define the risk neutral drift? Is the model is complete and arbitrage free the risk neutral drift is no something one chooses, isn't it? It comes out of the unique risk free measure, isn't it?
I am a beginner in this things. I could be very wrong.
Alternatively, why is the risk-free drift equal to $\alpha-\beta\sqrt{\nu}\phi$?
My possible answer. But I am not sure. Probably my confusion is that I don't know well the definition of risk free.
Is it that risk free just means to get the portfolio, $\Pi_1$ in this case, to have drift $r$, as it happens with the stock in the Black-Scholes model? We see that
$$d\Pi_1=r\Pi_1dt+\beta\sqrt{\nu}\frac{\partial V}{\partial \nu}d Z_2'$$ with $Z_2':=\int_{0}^{t}\phi+Z_2$.