The Valuation Of American Option And Martingale Method

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. However, because of the practical importance of American options, their efficient and accurate pricing is vital for option market participants.This paper simply summarizes a method about valuation of American options-martingale method and the result turns out that it is rather effective in valuation of some American options. In addition, barrier option is also introduced and researched in the article, and as a new option, it has played more and more important role in financial exchange market. However, although the introduction of barrier reduces the price of options, it also leads to the diffucult of valuation. On the basis of the research of the former, the author obtains the analytical solution of American call-options in the presense of a "down-and-out" barrier through the transformation of a variational inequality.