Applications of Optimal Stopping to Continiuos Time Markov Chains

Specialeforsvar ved Emil Fischer Kristensen

Titel: Applications of Optimal Stopping to Continiuos Time Markov Chains

Abstract: The theory of optimal stopping seeks to maximize an expected payoff through choosing a time to take a given action. Problems of this kind are frequently encountered in the area of mathematical finance and the theory can even be incorporated in certain aspects of life insurance mathematics. In this thesis we will present applications of optimal stopping theory to both the American put option and the surrender option in life insurance. The models we use to describe these contracts are based on continuous time Markov chains, hence a thourough examination of optimal stopping theory under continuous time Markov chains is required. The majority of material on optimal stopping is based on diffusion processes like the geometric Brownian motion, and therefore we need to adjust the theory to the case of Markov chains. The main result of this thesis is the early exercise premium representation under time continuous Markov chains. This result allows us to seperate an early exercise premium from the value function of an optimal stopping problem. In cooporation with the free-boundary problem associated to the optimal stopping problem we are then able to formulate a free- boundary equation, that is solved by the optimal boundary

 
Vejleder:  Jesper Lund Pedersen
Censor:    Jesper Olesen, Danica Pension