  In this paper we propose a novel pricing-hedging framework for volatility
derivatives which simultaneously takes into account rough volatility and
volatility jumps. Our model directly targets the instantaneous variance of a
risky asset and consists of a generalized fractional Ornstein-Uhlenbeck process
driven by a L\'{e}vy subordinator and an independent sinusoidal-composite
L\'{e}vy process. The former component captures short-term dependence in the
instantaneous volatility, while the latter is introduced expressly for
rectifying the activity level of the average forward variance. Such a framework
ensures that the characteristic function of average forward variance is
obtainable in semi-closed form, without having to invoke any geometric-mean
approximations. To analyze swaps and European-style options on average forward
volatility, we introduce a general class of power-type derivatives on the
average forward variance, which also provide flexible nonlinear leverage
exposure. Pricing-hedging formulae are based on a modified numerical Fourier
transform technique. A comparative empirical study is conducted on two
independent recent data sets on VIX options, before and during the COVID-19
pandemic, to demonstrate that the proposed framework is highly amenable to
efficient model calibration under various choices of kernels.
