About The Problem Of Pricing American Options

Option as a financial derivative products, option pricing is a modern financial scienceresearch topics, the development of options for the financial markets’ prosperity has madegroundbreaking contributions.1973, university of Chicago’s Black F and Scholes M usedpartial differential equations deduced the famous Black-Scholes model, and solved astandard European option price formula. As American option within the agreed time hasthe advance of implementation, so the pricing is much more complicated than Europeanoption. Since the option price with the underlying stock price and execution time haveclosely related, which made breakthrough contribution for the research of the optionpricing theory. Then paper mainly from the stock price fluctuations in the impact ofAmerican option price and the execution time impact of the American option price have adiscussed.The paper main content is following:To ensure the income in the case of non-negative, according to the option pricedetermined to continue to hold regional or cease to hold regional. When the underlyingstock’s transaction price determined, continues or ceases to be determined based on thestock price. Assuming parameter c in the option price range. Solving c with the underlyingstock price and strike price relationship. Minimize the cost of the consumer J. Assume thatthe parameter space is finite dimensional, derived the finite dimensional approximation ofthe parameter space, the parity relation between call option and put option was obtained.The results also were expand to the case of stock paided continuous dividend.Compare the difference of American options and European options is that Americanoption execution time is uncertainly. The paper combine martingale stops with Brownianmotion calculate the probability of expected value. Then combined with standardEuropean option price formula, obtained permanent American put and call price formulamixed. Studied the default time tends to infinity analytical expression. The results can befurther extended to the underlying stocks continuous pay dividends of American optionprice.