Before revision:  We consider a multidimensional model which is an extension of the usual Black-Scholes (BS) model, in the sense that the volatility is a progressively measurable function of the stock . For this case we show that the super-replication price is the cheapest cost of a trivial super-replication strategy.
After revision:  We consider a multidimensional continuous time model, in which the discounted stock price process satisfies the conditional full support property. We show that the super-replication price is the cheapest cost of a trivial super-replication strategy.