Abstract: The aim of this paper is to analyze which risk factors and which distribution features models for the joint evolution of stock log-returns and their volatility should target in order to provide a robust calibration, repricing and hedging performance. The setup is based on time changed Lévy process with both diffusive and jump components, and leverage effects originated by both factors. The analysis is carried out considering S&P500 options. Joint work with Grégory Rayée (ULB).