In this thesis we investigate Asian oating strike options. We particu-larly focus on options with early exercise - American options. This typeof options are very lucrative to the end-users of commodities or ener-gies who are tend to be exposed to the average prices over time. Asianoptions are also very popular with corporations, who have ongoing cur-rency exposures. The main idea of the pricing is to examine the freeboundary position on which the value of the option is depending. Wefocus on developing a ecient numerical algorithm for this boundary.In the rst Chapter we give an informative description of the nancialderivatives including Asian options. The second Chapter is devoted tothe analytical derivation of the corresponding partial dierential equa-tion coming from the original Black - Scholes equation. The problemis simplied using transformation methods and dimension reduction. Inthe third and fourth Chapter we describe important numerical methodsand discretize the problem. We use the rst order Lie splitting and thesecond order Strang splitting. Finally, in the fth Chapter we makenumerical experiments with the free boundary and compare the resultwith other known methods.