Pricing American options without expiry date

Abstract

The history of options trading started prior to 1973. Many different types of options are regularly traded throughout the world. Options on stocks have been traded in Hong Kong since September 1995. Because of the early exercise opportunity, American-type options are more flexible and popular than European-type options. Although many researchers have contributed to deriving pricing formulas for European options, however there are no closed-form formulas for the prices of American options in most cases. The main difficulty is that it is a free boundary value problem. To price an American option, it is important to determine the optimal exercise boundary (and the optimal stopping time). For a perpetual American option, the optimal exercise boundary turns out to be constant through time. The word "perpetual" means that the option has no expiry date. This thesis discusses the martingale approach to pricing perpetual American-type options. A main tool in our approach is the principle of smooth pasting. For simplicity, options in one-stock case are considered first. These options include the perpetual American put option, call option and the perpetual maximum option on one stock. Then we extend our analysis to two-stock case. The perpetual maximum option on two stocks, the perpetual uncapped Margrabe option, the perpetual capped Margrabe options and the perpetual dynamic fund protection are discussed.