Under an assumption of continuity on the price process, and under an assumption that a continuum of calls on the underlying are traded, the work of Neuberger and Dupire gives that the price for the variance swap is equal to twice the price of a log contract. This price is model-free.

But what if we are not prepared to assume continuity? Then, given call prices a range of possible prices is consistent with no-arbitrage. In this talk we try to characterise this range.