Option Pricing con volatilità stocastica: analisi ed implementazione del modello di Heston

The present work aims at evaluating options using the Heston model. This model is presented both from a theoretical and a practical point of view. Initially, we review the salient mathematical steps that lead to its formulation. In particular, the hedging portfolio of the underlying asset price and variance is constructed in order to derive the exercise probability of the option. Afterward, some important aspects related to the implementation of the model are analyzed and the main numerical approaches are discussed: the integration methods and the approaches based on the Fast Fourier Transform (FFT) and the Fractional Fast Fourier Transform (FRFT). We then illustrate the minimization methods of the objective function useful to perform the calibration of the model and, finally, we show the results of an empirical analysis carried out on real data.