Some optimal stopping problems with applications in mathematical finance

Abstract

In this thesis, we mainly concern three topics arising in optimal stopping theory: an employee stock option (ESO) valuation problem, American put option with an up-and-out barrier and a finite time horizon optimal prediction problem involving running maximum processes. The first two topics fall into the category of optimal stopping problems for one-dimensional regular and reflected diffusions. The ESO valuation problem studies a perpetual employee stock option, which com- bines the feature of early exercise and an additional feature that it also gives the holder an additional choice of a certain amount of cash besides the stock option. From a con- trol viewpoint, we can mathematically formulate it as an optimal stopping problem. We shall consider this optimal stopping problem under one-dimensional regular diffu- sions as well as reflected diffusions and provide complete characterization for the value function and optimal exercise strategy. For risk management and hedging purpose, it is often important for financial institutions to be fully aware of the model behaviour, that is to understand, the impacts of model parameters on the value function and optimal exercise strategy. Typically, this is often the main responsibility for the Model Valida- tion Quant Group in an investment bank. To tackle this issue, we develop two unified approaches to study the impact of internal and external parameters, respectively. As a result, several internal and external comparison principles are obtained, which provide mathematical description for model behaviour. The valuation of American put option with an up-and-out barrier was first consid- ered by Karatzas and Wang under the Black-Scholes pricing framework. We further generalize their results for general one-dimensional regular diffusions. Furthermore, we also obtain internal and external comparison principles for the model. The last topic of this thesis is devoted to a finite time horizon optimal prediction problem involving running maximum processes. It aims to find the best trading strat- egy, by which the distance under certain measurement between the selling price and global maximum price over a given investment time horizon is minimized.