Valuation of asset and volatility derivatives using decoupled time-changed L\'evy processes

In this paper we propose a general derivative pricing framework which employs decoupled time-changed (DTC) L\'evy processes to model the underlying asset of contingent claims. A DTC L\'evy process is a generalized time-changed L\'evy process whose continuous and pure jump parts are allowed to follow separate random time scalings; we devise the martingale structure for a DTC L\'evy-driven asset and revisit many popular models which fall under this framework. Postulating different time changes for the underlying L\'evy decomposition allows to introduce asset price models consistent with the assumption of a correlated pair of continuous and jump market activities; we study one illustrative DTC model having this property by assuming that the instantaneous activity rates follow the the so-called Wishart process. The theory developed is applied to the problem of pricing claims depending not only on the price or the volatility of an underlying asset, but also to more sophisticated derivatives that pay-off on the joint performance of these two financial variables, like the target volatility option (TVO). We solve the pricing problem through a Fourier-inversion method; numerical computations validating our technique are provided.