In recent decades, options and other financial derivatives have become a cornerstone not only of the financial industry but also of the global economy at large. Their widespread use has driven innovation in risk management, investment strategies, and market efficiency. This thesis covers the mathematical foundations of financial derivatives, with a particular emphasis on option pricing models. It contains a discussion of selected types of options, the mathematical frameworks used for their valuation, and reviews the current state of the art in numerical methods. Traditional numerical methods for option pricing are often tailored to specific contract types, limited either by conceptual constraints or computational cost. To address these limitations, I introduce a novel machine learning based approach for option pricing: the SWIM-PDE solver. This method tackles the valuation problem by solving the associated partial differential equation using an ansatz based on separation of variables. The spatial component is represented by Sample Where It Matters (SWIM) networks, enabling a flexible and adaptive approximation of the solution. Inserting the ansatz into the equation transforms the PDE into an ODE with respect to the time component, which can be solved with an ODE solver. The theoretical part of this work outlines the structure and working principles of the SWIM-PDE solver, while the numerical experiments demonstrate its performance in comparison to conventional solvers. Results show that the method achieves competitive accuracy and, importantly, generalizes well across various option types without requiring substantial modifications. These findings highlight the strong potential of the SWIM-PDE solver as a powerful and versatile tool for deriva- tive valuation. The full implementation of the SWIM-PDE solver, along with all numerical experiments, is available on GitHub     «

In recent decades, options and other financial derivatives have become a cornerstone not only of the financial industry but also of the global economy at large. Their widespread use has driven innovation in risk management, investment strategies, and market efficiency. This thesis covers the mathematical foundations of financial derivatives, with a particular emphasis on option pricing models. It contains a discussion of selected types of options, the mathematical frameworks used for their valua...     »