Consider a stochastic volatility model. As there are two sources of risk and one asset only, this is an incomplete market. One can complete the market by considering a derivative V1 used to hedge the volatility risk. My question is: Do derivatives prices depends on the derivative V1 one chooses to complete the market? And if no, why?
Of course. The whole point of pricing derivatives is that it is an interpolation exercise. Saying that the market is incomplete means there are not enough constraints to determine the price of all derivatives. It's a bit as if you had a polynomial of degree 2 but you gave only 2 points rather than 3.
So yes, the price of your V1 will be vital because if you need to use it to hedge your position then for sure you need to know how much it costs...