Nonparametric Calibration of Inhomogeneous Lévy Processes using Fourier Techniques

In this report, inhomogeneous Lévy processes are studied in a discrete observational model based on derivatives of the process. First, homogeneous Lévy models are defined and an already known nonparametric method, using Fourier techniques and call and put option prices, for estimating the parameters of the model is described based on Belomestny and Reiẞ (2006a). Previous research suggests that there is a need for an extension of this concept since option prices with different maturities produce significantly different results. After all, the assumption that the parameters of the model are the same for any time window is not realistic and better results could be achieved once this premise is rejected. That is why inhomogeneous Lévy processes are introduced and studied in this report. The estimation method for the homogeneous model from Belomestny and Reiẞ (2006a) is extended to fit into the inhomogeneous framework. Next, asymptotic normality of the estimators is proven for these processes in this setting and confidence intervals are constructed using the finite sample variance method. Asymptotic normality has already been shown and confidence intervals have been constructed in the homogeneous framework in the continuous observational model by Söhl (2014). Finally, data is simulated from an inhomogeneous Merton model to test the performance of the method and options from the S&P 500 index are used as a real-world application