I'm trying to derive the valuation equation under a general stochastic volatility model. What one can read in the litterature is the following reasonning:
One consider a replicating self-financing portfolio $V$ with $\delta$ underlying and $\delta_1$ units of an other derivative $V_1$. One writes Itō on one hand, and the self-financing equation on the other hand, and then one identifies the terms in front of the two brownian motions and in front of $dt$. The first two identifications give $\delta$ and $\delta_1$, and the last identification gives us a PDE in $V$ and $V_1$.
Then what is commonly done is to write it with a left hand side depending on $V$ only, and a write hand side depending on $V_1$ only. So you get $f(V) = f(V_1)$. We could have chosen $V_2$ instead of $V_1$ so one gets $f(V) = f(V_1) = f(V_2)$. Thus $f(W)$ does not depends on the derivative $W$ one chooses, and is called the market price of the volatility risk. What I cannot understand in this reasonning is why $V$ does not depend on the derivatives $V_1$ you choose to hedge the volatility risk in your portfolio. As far as I see it, one should write $V(V_1)$ instead of $V$. Then one has $f(V(V_1)) = f(V_1)$ and $f(V(V_2)) = f(V_2)$ so one gets no unique market price of the volatility risk.
Does anyone know why does the price of a derivative does not depend on the derivative you choose to hedge against the volatility risk?
Thank you in advance!