For a family of functions G, we define the G-variation, which generalizes power variation; G-variation swaps, which pay the G-variation of the returns on an underlying share price F; and share-weighted G-variation swaps, which pay the integral of F with respect to G-variation. For instance, the case G(x)=x^2 reduces these notions to, respectively, quadratic variation, variance swaps, and gamma swaps.

We prove that a multiple of a log contract prices a G-variation swap, and a multiple of an F log F contract prices a share-weighted G-variation swap, under arbitrary exponential Levy dynamics, stochastically time-changed by an arbitrary continuous clock having arbitrary correlation with the Levy driver, under integrability conditions. We solve for the multipliers, which depend only on the Levy process, not on the clock.

In the case of quadratic G and continuity of the underlying paths, each valuation multiplier is 2, recovering the standard no-jump variance and gamma swap pricing results. In the presence of jump risk, however, we show that the valuation multiplier differs from 2, in a way that relates (positively or negatively, depending on the contract) to the Levy measure's skewness.

This work, joint with Peter Carr, extends Carr-Lee-Wu's treatment of variance swaps, by generalizing from quadratic variation to G-variation; and by encompassing not only unweighted but also share-weighted payoffs.