We consider weighted variance contracts in which the realized variance is subjected to a spot-dependent weighting function, a notable example of which is the corridor variance swap. Such payouts admit a quasi-static hedge involving European-style options, with expiries at all dates up to the maturity of the contract. This paper proposes a method for overhedging weighted variance using only a finite number of maturities. Moreover, this approach is shown to have good convergence properties and allows one to treat dividends in a natural way. As an application, the method is used to relate corridor variance with the variance implicit in the definition of the HSI Volatility Index.