In difference to European options American options are significantly harder to price because the possibility of early exercise leads to a free boundary problem. In our studies we first look at literature regarding this boundary problem and try to find the critical boundary close to expiry. This leads to terms for the behaviour of the critical boundary close to expiry. We try to summarize the findings and extract the most plausible solution. Returning to the problem of finding the option value we address Employee stock options. In comparison to traditional American options we have to account for non-hedgeability, the vesting period and dilution. Traditionally ESOs have been valued with the intrinsic value method, but we try to value them with an enhanced FASB 123 model. With models proposed by Hull and others we try to find an approximate solution for the option value and the expected life. Via experimentation and plotting our findings we study the inner workings of those methods. By comparing the binomial and the trinomial models we try to find an accurate and fast method - with the goal of establishing a practical one - for option valuation.
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