Decomposing Smiles: A Time Change Approach

We develop a novel time-change approach to study the shape of implied volatility smiles. The method is applicable to common semimartingale models, including jump-diffusion, rough volatility and infinite activity models. We approximate the at-the-money skew and curvature with an improved moment-based formula. The moments are further explicitly computed under a time change framework. The limiting skew and curvature for several models are considered. We also test the accuracy of the short-term approximation results on models via numerical methods and on empirical data. Finally, we apply the method to the calibration problem.