Arbitrage Bounds for Weighted Variance Swap Prices

Consider a frictionless market trading a finite number of co-maturing European call and put options written on a risky asset plus an instrument with path-dependent payoff known as a weighted variance swap, e.g. a vanilla variance swap or a corridor variance swap. The question we ask is: Do the traded prices admit an arbitrage opportunity? 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. 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.