Empirical semi-groups and calibration

We give a model free calibration algorithm based on a finite element type approach. Although no probabilistic model is a priori needed it turns out that the setting of our problem automatically produces a Markov dynamics for the underlying stock. Markovianity is our only a priori assumption. We test our algorithm using syntethic data produced by Dupire's model with several types of local volatility. In order to test the flexibility of our algorithm. We also consider data produced by Heston's model which is a stochastic volatility model which is no more Markovian and does not enter in our frame.