Before revision:  We determine necessary and sufficient model-free conditions for the price of a continuously monitored weighted variance swap to be consistent with absence of arbitrage. We discuss in detail the types of arbitrage that may arise when the determined conditions are not satisfied.
After revision:  We determine necessary and sufficient model-free conditions for the price of a continuously monitored weighted variance swap along with super- and sub- replicating strategies which enforce them. We find that market quotes for variance swaps are surprisingly close to the model-free lower bounds we determine. We solve the problem by transforming it into an analogous question for a European option with a convex payoff. The lower bound becomes a problem in semi-infinite linear programming which we solve in detail. The upper bound is explicit. We work in a model-independent and probability-free setup.