This work introduces the first 2-way fluid-structure interaction (FSI) computational model to study the effect of aortic annulus eccentricity on the performance and thrombogenic risk of cardiac bioprostheses. The model predicts that increasing eccentricities yield lower geometric orifice areas (GOAs) and higher normalized transvalvular pressure gradients (TPGs) for healthy cardiac outputs during systole, agreeing with in vitro experiments. Regions with peak values of residence time and shear rate are observed to grow with eccentricity in the sinus of Valsalva, indicating an elevated risk of thrombus formation for eccentric configurations. In addition, the computational model is used to analyze the effect of varying leaflet rigidity on both performance, thrombogenic and calcification risks with applications to tissue-engineered prostheses, observing an increase in systolic and diastolic TPGs, and decrease in systolic GOA, which translates to decreased valve performance for more rigid leaflets. An increased thrombogenic risk is detected for the most rigid valves. Peak solid stresses are also analyzed, and observed to increase with rigidity, elevating risk of valve calcification and structural failure. The immersed FSI method was implemented in a high-performance computing multi-physics simulation software, and validated against a well known FSI benchmark. The aortic valve bioprosthesis model is qualitatively contrasted against experimental data, showing good agreement in closed and open states. To the authors' knowledge this is the first computational FSI model to study the effect of eccentricity or leaflet rigidity on thrombogenic biomarkers, providing a novel tool to aid device manufacturers and clinical practitioners.