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What appears to be a minor extension of the European contract results
in considerable mathematical difficulties: so much in fact, that an
explicit and exact pricing formula cannot anymore be found analytically.
Fortunately, the variational calculus introduced in sect.5.3.1
to implement the
finite element method
provides a solid theoretical framework that can be extended to obtain a
numerical solution with only minor changes to account for the difference
in the European and American exercise styles.
6.2 Methods for American options: finite elements (FEM)