Smiles & smirks: Volatility and leverage by jumps

We propose a novel flexible framework for the joint evolution of stock log-returns and their volatility based on time changed Lévy processes. The novelty of the approach stems from the generality of the jump structure we endow our model with, and the ability of the model to generate leverage effects out of the pure jump component. We derive the characteristic function and the forward characteristic function of the log-returns, which allow for the efficient pricing of vanilla and forward-start-like option contracts by Fourier transform methods. The proposed framework achieves robust calibration performance properties especially in the case of pure jump specifications. The results offered in this paper could have potentially interesting implications in terms of design of models and hedging strategies, and their development.