Variance swaps on time-changed Lévy processes

We prove that a multiple of a log contract prices a variance swap, under arbitrary exponential Lévy dynamics, stochastically time-changed by an arbitrary continuous clock having arbitrary correlation with the driving Lévy process, subject to integrability conditions. We solve for the multiplier, which depends only on the Lévy process, not on the clock. In the case of an arbitrary continuous underlying returns process, the multiplier is 2, which recovers the standard no-jump variance swap pricing formula. In the presence of negatively-skewed jump risk, however, we prove that the multiplier exceeds 2, which agrees with calibrations of time-changed Lévy processes to equity options data. Moreover we show that discrete sampling increases variance swap values, under an independence condition; so if the commonly-quoted 2 multiple undervalues the continuously-sampled variance, then it undervalues even more the discretely-sampled variance. Our valuations admit enforcement, in some cases, by hedging strategies which perfectly replicate variance swaps by holding log contracts and trading the underlying. JEL Classification G13