The shape memory alloy NiTi is particularly popular for medical applications and implants due to its excellent biocompatibility and corrosion resistance. Detailed knowledge of its properties and interactions with other elements is crucial to ensure a safe and optimal application of the material. The focus of this work is the characterization of NiTi hydrides using the methods of density functional theory and the VASP code. Based on experimental observations, structural models have been created for different hydrogen concentrations and distributions. The formation energy has been calculated and compared between the models to find the most stable hydride structure. Furthermore, optimal cell parameters for a supercell which is stabilized in the B2 phase have been calculated using a manual fit and automated structural optimization. For some possible hydride structures, the elastic constants have been evaluated to assess mechanical stability. Diffraction patterns have been simulated and compared to experiments. Through the comparison of the different models, a few potentially mechanically and energetically stable structures could be identified. Furthermore, a phase transformation (change in cell shape) as a function of hydrostatic pressure (volume) has been predicted. Although pressure induced phase transformations have been reported for pure NiTi before, this analysis suggests that NiTiH exhibits a more complex behavior.