Assuming there are no arbitrage opportunities, the price of a derivative depends on the prices of other derivatives available in the market. If you introduce a derivative without giving it a price, or by giving it a price that is already theoretically determined by your initial setup, then you do not affect the prices of other derivatives.

To take a simpler example, suppose we have two sources of risk, two (biased) coin tosses $\omega_1$ and $\omega_2$ and the only tradable asset is $S=\$1_{H(\omega_1)} 1_{H(\omega_2)}$ i.e. we get 1 dollar if both coin tosses are heads and 0 otherwise.

Now we can complete the market by introducing two options having payoffs
$$
V_1=h S +\$1_{H(\omega_2)}
$$
$$
V_2=\$1_{H(\omega_1)}
$$
where $h$ is a huge number,

Assume there are no arbitrage opportunities and the price of the option $V_1$ is $\$1$. Then $S=\$1$ must be very unlikely and so the original asset $S$ is now essentially worthless. But if we don't assume the price of any introduced option to be given then introducing the option has no effect.