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Abstract
This dissertation contains two parts: the first chapter is concerned with identifying the highs and lows of prices forstock trading. The underlying asset prices fluctuate in a mean reversion fashion.
The purpose is to maximize the overall profit in the long run. Ideally, we want to sell high and buy low. However, it is extremely difficult to identify when is low and when is high in practice. Under the mean reversion model, we follow a dynamic programming approach and determine these key thresholds to optimize our profit. In the second chapter, we discuss an optimal pairs trading rule. A pairs position consists of a long position in one stock and a short position in the other. The problem is to find stopping times to open and then close the pairs position to maximize expected reward functions. We consider the optimal pairs trading rule with one round trip. The underlying stock prices follow a general geometric Brownian motion with regime switching. The optimal policy is characterized by threshold curves obtained by solving the associated HJB equations (quasi-variational inequalities). Moreover, numerical examples are provided to illustrate optimal policies.