In this thesis, we address the problem of hedging American options. Unlike European options, which can be replicated, American options generally cannot be replicated due to the possibility of early exercise. Therefore, an alternative method for hedging our position must be found. Using supermartingales, the Snell envelope and Doob's decomposition, we prove a theorem that allows for the determination of the fair initial price, the hedging strategy, and the optimal stopping time. Finally, through an example using dynamic programming, we construct a self-financing portfolio that effectively protects against risk in incomplete markets. This means it minimizes the probability that we will be unable to meet our obligations in a given scenario.
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