The problem of calibrating time-inhomogeneous exponential Lévy models with finite jump activity based on market prices of plain vanilla options is studied. Belomestny and Reiß introduced an estimation procedure for calibration in the homogeneous case with one maturity. The open-ended question that will be addressed is if we can extend this model to use all listed plain vanilla options with multiple maturities. This opens a way to use all available data to create a time-inhomogeneous model with time-dependent Lévy triplets between subsequent intervals based on the maturities.

We establish via an adapted Lévy-Khintchine representation an estimation procedure on the explicit inversion of the option price formulas in the spectral domain with a cut-off scheme for the regularisation of high frequencies. The estimation procedure will be shown to be well-defined and the parameters to be normally distributed asymptotically.

Practical implications imply that using the asymptotic variances of the normal distributions leads to insufficient confidence intervals for non-asymptotical cases. We, therefore, construct confidence intervals using an approximated finite sample variance.

Monte Carlo simulations are implemented in the computational software R to evaluate the stability and accuracy of the estimation procedure. Furthermore, the finite sample confidence intervals are assessed with coverage probabilities.

In the end, the estimation procedure is evaluated by calibrating to market data of plain vanilla S\&P500 options, which contain numerous maturities. The estimated parameters with confidence intervals between maturities support time-dependency and the constructed time-inhomogeneous exponential L{\'e}vy models appear favorable in contrast to the time-homogeneous model.

A model is designed for solving gravitational profiles of self-gravitating and rotating planets via the use of Poisson's equation for total gravity, i.e., the sum of the gravitational and rotational potential. Poisson's equation is a partial differential equation that is solved with the usage of Green's functions. This is a major advantage because Earth's surface is an equipotential, which leads to a Dirichlett boundary condition. We apply this methodology to a spherical Earth, where the Green's function is closed. It is demonstrated that, when this Green’s functions based methodology is applied to a homogeneous density, a linear spherical density and the PREM density profiles (where all density profiles are spherically symmetric), then discontinuities occur in the accelerationof gravitation across the surface of the Earth, in the range of 0.01 %- 0.13 %. Such an effect is a symptom of incompatibility with the laws of physics for geostaticalequilibrium, and it is certainly significant as compared to the numerical accuracy of the solution. Moreover, the discrepancy is dependent on geographical latitude, as was to be expected, because it somehow reflects the centrifugal effect. The conclusion is made that this method is indeed capable of detecting, and even quantifying, the incompatibility of spherically symmetric mass distributions with the fundamental laws of physics for self-gravitating and rotating planets. This opens a way -and this is one of the main results- to computing corrections to spherically symmetric mass distributions, such as the PREM model. Based on this method, a further way forward to this end is suggested.