American option has the additional feature of early exercise, potentially it has higher value than European options and more hard to be priced. Meanwhile,since there’s no explicit option formula, we often price option by numerial methed. According to the different dividend payment, we disscuss the American option pricing under stochastic market option model respectively based on dividend and treatment fees.Firstly, if the dividend is continuously paid, we can take its effect into risk free rate and the treatment fee is propotional to stock price. Clearly, we can derive the patial different equation of American option, which is very difficult to solve analiticaly, but can be solved by finite difference method.Secondly, we analyze the option pricing of fixed dividend by split the stock price into two parts, risk part and riskless part. So does the option price. Then the risk part price got by finite difference method, which plus the riskless part is the option price of prefixed dividend.Lastly, if the dividend is propotional to stock price, the paid moment increase the probility of early exercise. We discuss the inequality in detail, which existents the early exercise happens before the paid moment. The stock price in inequality is time related, so we cannot solve this inequality by finite difference method. Under this circumustence, we apply ment carol method to many paths of stock price process, then calculate the option price by the least square method.
