The goal of the this project is to study two problems linked to American options. Unlike European options, they allow their buyer to exercise his rights at any time prior to the maturity of the option. In order to both price and hedge such financial products, it is necessary for the seller to be able to understand and compute which instants are optimal for the buyer to potentially exercise the aforementioned rights.
We first study the so-called "Perpetual" American options, which have the desirable feature that their pricing and hedging admit explicit solutions, before going back to the more standard case in finite horizon which we will treat numerically.