  We regard options on VIX and Realised Variance as solutions to path-dependent
partial differential equations (PDEs) in a continuous stochastic volatility
model. The modeling assumption specifies that the instantaneous variance is a
$C^3$ function of a multidimensional Gaussian Volterra process; this includes a
large class of models suggested for the purpose of VIX option pricing, either
rough, or not, or mixed. We unveil the path-dependence of those volatility
derivatives and, under a regularity hypothesis on the payoff function, we prove
the well-posedness of the associated PDE. The latter is of heat type, because
of the Gaussian assumption, and the terminal condition is also path-dependent.
Furthermore, formulae for the greeks are provided, the implied volatility is
shown to satisfy a quasi-linear path-dependent PDE and, in Markovian models,
finite-dimensional pricing PDEs are obtained for VIX options.
