Pricing 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 as a special case of our framework. In the presence of negativelyskewed jump risk, however, we prove that the multiplier exceeds 2, which agrees with calibrations of time-changed Lévy processes to equity options data. Finally 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 furthermore the discretely-sampled variance.