Correlated time-changed L\'evy Processes

Carr and Wu (2004), henceforth CW, developed a framework that encompasses almost all of the continuous-time models proposed in the option pricing literature. Their main result hinges on the stopping time property of the time changes, but all of the models CW proposed for the time changes do not satisfy this assumption. In this paper, when the time changes are adapted, but not necessarily stopping times, we provide analogous results to CW. We show that our approach can be applied to all models in CW.