We consider a financial market model, where the dynamics of asset prices are given by an R-d-valued continuous semimartingale and the information flow is right-continuous. Using the dynamic programming approach we express the variance-optimal martingale measure in terms of the value process of a suitable optimization problem and show that this value process uniquely solves the corresponding semimartingale backward equation. We consider two extreme cases when this equation admits an explicit solution. In particular, we give necessary and sufficient conditions in order that the variance-optimal martingale measure coincides with the minimal martingale measure as well as with the martingale measure appearing in the second extreme case.