Study On The Pricing Of The Capped Call Option

The option is one of the most important financial derivatives. Since the stock options traded in American organized Exchange in 1973 for first time , the option market has developed rapidly. Due to the market demand, financial institutes have designed many financial derivatives which we call exotic options, whose payoff functions are more complex than those of the standard European and American call or put options. Unceasing development of the new financial derivatives as well as the pricing study of them is an important field of financial research.Since Fisher Black and Myron Scholes proposed the famous option pricing formula in 1973,the Black-Scholes frame became the mainstream in the option pricing research field. Based on the fundamental assumption of the Black-Scholes frame, the pricing of the capped call option is discussed in this paper comprehensively and systematicly.Chapter one of this paper deals with the research background and the literature summary, both domestic and abroad.Chapter two studies the European capped call option.First we obtain the pricing formula to the option under the Black-Scholes circumstance, then the model is extended to get the precise pricing of the option when the parameters are time-dependment as well as the transaction costs are considered. Finally the characters of the option price and risk management are studied.Chapter 3 is devoted to the American capped call options, in which two pricing models are given, the free boundary model and the variational inequality model. As to the free boundary model, we give a kind of numerical algorithm method which is called layer slice method, and as to the variational inequality model, we solve it by two finite difference methods, the explicit difference and the implicit difference respectively. This paper pays more attention to the explicit difference, not only gives its numberical method in detail, but also prooves its convergence for both the solution and the free boundary. Some characters of the American capped call options are also obtained from its numerical solutions.