The Thesis consists in two fundamental chapters: First, I analyze the presence of jumps in high-frequency financial series, with an extensive application to most liquid American equities. I develop a statistical test that formally treats the multiple testing issue and show, in substance, that discontinuities are much rarer events than previously thought. Second, I introduce a novel, scalable numerical pricing framework for American options under multi-factor models. I make use of adaptive sparse grids to alleviate the curse of dimensionality. My methodology offers a new direction for research in the pricing of derivative contracts with early-exercise features under models actually capable of fitting statistical properties of financial series.