Variation and share-weighted variation swaps on time-changed Lévy processes

Abstract

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 FlogF contract prices a share-weighted G-variation swap, under arbitrary exponential Lévy dynamics, stochastically time-changed by an arbitrary continuous clock having arbitrary correlation with the Lévy driver, under integrability conditions.

We solve for the multipliers, which depend only on the Lévy 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 specified G) to the Lévy measure’s skewness.

In three directions this work extends Carr–Lee–Wu, which priced only variance swaps. First, we generalize from quadratic variation to G-variation; second, we solve for not only unweighted but also share-weighted payoffs; and third, we apply these tools to analyze and minimize the risk in a family of hedging strategies for G-variation.