American options give the holder the right to exercise the option at or before the maturity time. This characteristic of American options makes it somewhat difficult to value numerically. However, the pricing of the American option is one of the most important and interesting topics in financial engineering. The numerical pricing of American option is vital to securities market participants, because of the practical usage of American option and the importance in the trading market.
Based on the option pricing theory, we will apply Monte Carlo method and binomial method to price American put option, and investigate comparatively the performance of these numerical methods. It is well-known that, there is no closed form solution for American options, except for American call options without paying dividends, thus numerical pricing is essential and important for American option. The Black-Scholes (B-S) option pricing theory will be a guide for our pricing.
The martingale theory for option pricing will be used to analyze the American options pricing. A perpetual mixed American option is constructed. By martingale theory and dynamic programming, we get closed form solution for this option. Also a perpetual mixed option with knock-out barrier is constructed, and we analyze and get the closed form solution for this option.
