Path-dependent PDEs for volatility derivatives

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.