A note on contracts on quadratic variation

Given a Black stochastic volatility model for a future F, and a function g, we show that the price of 1/2 integral(T)(0) g(t, F(t))F-2(t) sigma(2)(t)dt can be represented by portfolios of put and call options. This generalizes the classical representation result for the variance swap. Further, in a local volatility model, we give an example based on Dupire\u27s formula which shows how the theorem can be used to design variance related contracts with desirable characteristics