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Valuation of American Options

The valuation of American options is a difficult problem. The basic reason is that the asset price at which early exercise is optimal isn’t known in advance and has to be found as part of the solution of the problem. In mathematical terms, a partial differential equation known as Black–Scholes equation has to be solved with a moving boundary condition. This is known in general as a moving boundary problem. Analytic solutions of this kind of problems can be found only in very special cases (e.g. for the American call on an asset paying a single discrete dividend during the lifetime of the option). However, because of the practical importance of American options, their efficient and accurate pricing is vital for option market participants. Finite difference methods can be used to solve the differential equation numerically, but in order to obtain an accurate solution, considerable computational effort is necessary. Therefore other, more efficient methods have been developed. In this thesis a comparison of numerical and approximative methods to solve this problem for equity (representing options on assets with discrete known payments) and FX options (representing options on assets with a continuous dividend yield or holding costs) is presented. The numerical methods are based on a binomial tree. A finite difference method the solution of which is considered exact is used as a benchmark all the other methods are compared to. The result is an assessment when these methods can be successfully applied.