Optimal Stopping Time and Pricing of Exotic Options

The purpose of this paper is to develop a general framework to analyze the optimal stopping problem relevant to the exotic option. In general, the pricing of the American style path-dependent option is equivalent to solving an optimal stopping problem. We turn the optimal stopping problem relevant to the American style path-dependent option into a constrained nonlinear programming in a infinite dimensional Banach space, and prove that the existence of the optimal stopping time of the optimal stopping problem relevant to a class exotic option (Asian option, Lookback option).