Before revision:  In particular we find that prices of European call/puts are not enough for the upper bound price of the vanilla variance swap to be finite. We show that given an extra piece of information, namely the price of an additional asset, a finite bound can be explicitly determined .
After revision:  In particular we use and extend F\"ollmer's pathwise stochastic calculus. Appropriate notions of arbitrage and admissibility are introduced. This allows us to establish the usual hedging relation between the variance swap and the 'log contract' and similar connections for weighted variance swaps. Our results take form of a FTAP: we show that the absence of (weak) arbitrage is equivalent to the existence of a classical model which reproduces the observed prices via risk-neutral expectations of discounted payoffs .