American options valuation using optimal stopping

Abstract

This dissertation aims at giving a theoretical study of the optimal stopping problem together with its application in the pricing of American options. The study initiates by discussing the two main approaches used in optimal stopping; the Martingale and the Markovian Approach. Considering the Martingale approach one can solve the optimal stopping problem either by backward induction or by the method of essential supremum. Although the former requires the time horizon to be finite, the latter can be extended to the infinite horizon case. The link between potential theory and Martingale theory allows one to characterize the value function in the Markovian approach as the smallest superharmonic function dominating the gain function. This superharmonic characterization will be studied in detail and will lead us to solve the optimal stopping problem by reducing it to a free-boundary problem. This free boundary problem is then used to solve the problem of pricing American options. In the infinite horizon case the optimal stopping problem is solved explicitly however in the case of finite horizon numerical techniques will then be addressed.