It is well known that, under a continuity assumption on the price of a stock S, the realized variance of S for maturity T can be replicated by a portfolio of calls and puts maturing at T. This paper assumes that call prices on S maturing at T are known for all strikes but makes no continuity assumptions on S. We derive semiexplicit expressions for the supremum lower bound Vinf on the hedged payoff, at maturity T, of a long position in the realized variance of S. Equivalently, Vinf is the supremum strike K such that an investor with a long position in a variance swap with strike K can ensure a nonnegative payoff at T. We study examples with constant implied volatilities and with a volatility skew. In our examples, Vinf is close to the fair variance strike obtained under the continuity assumption.