In my dissertation I develop three important essays in asset pricing. In Chapter 1, I document a dramatic increase in crash risk premia in individual stocks after the 2008/2009 financial crisis, indicating that investors are willing to pay high insurance to hedge against crashes in individual stocks. I theoretically explain this puzzling feature in an economy where investors face short-sale constraints. In Chapter 2 my coauthors and I develop a new efficient numerical methodology to price American options, the Fast Recursive Projection methodology. In Chapter 3 we use this new pricing technique to show that we can rationalize up to 25% of the apparent suboptimal non-exercise of American call options estimated by Pool, Stoll, and Whaley (2008), by allowing a more sophisticated stochastic process for the underlying stock and by taking into account the discrete dividend payments correctly.