American options are the most commonly traded options in the market. They are used to mitigate risk, speculate about the future, and are the key components of complex trading strategies. In this dissertation, we propose a new front-fixing finite element method for the valuation of American options. One of the attractive qualities of our front-fixing finite element method is that the early exercise boundaries and the option prices can be computed simultaneously with very high accuracy.;We study in detail our front-fixing finite element method for the valuation of American options on stocks, American options on zero-coupon bonds under a class of one-factor models of the short interest rate, and American options on stocks under a regime-switching model. In all three cases we establish stability, present numerical results, examine our method, and compare it with others.
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