# Minimum variance hedge ratio price difference vs. log-returns

So from my understanding Hull (2012) f.e. shows that the optimal hedge ratio minimizes the variance of the returns. But what happens to the variance of the prices? Is the Minimum variance hedge portfolio also the one, which has the lowest variance at the price level? For example, if I compute the optimal hedge ratio for the returns and then I simulate the spot and futures path with Monte-Carlo and use the optimal hedge ratio to construct the hedge portfolio, will it in the end also be the one with the lowest variance? I was wondering because when going from the returns (which are by assumption normally distributed) to the price level (log-normal) the distribution changes so this might not hold anymore.