Zhiming Lu: Optimal Stopping Theory with Applications to American Options and Game Options

Abstract.

The purpose of this thesis is to investigate the intricate world of financial derivatives, with a focus on mathematical modelling and analysis of option valuation and optimal decision-making. We will delve into the optimal value function and the optimal stop- ping time for both American options and Game options in the infinite time horizon. The basic idea we are using in this thesis is to try to connect the optimal stopping problem to the free boundary value problem. In order to achieve this we will introduce stochastic integrals and stochastic differential equations(SDE). Finally, we use the results we derived to do some simulations and make some explanations.