Title: Optimal Multiple Stopping Approach to Mean Reversion Trading
Description: This thesis studies the optimal timing of trades under mean-reverting price dynamics subject to fixed transaction costs. We first formulate an optimal double stopping problem whereby a speculative investor can choose when to enter and subsequently exit the market. The investor's value functions and optimal timing strategies are derived when prices are driven by an Ornstein-Uhlenbeck (OU), exponential OU, or Cox-Ingersoll-Ross (CIR) process. Moreover, we analyze a related optimal switching problem that involves an infinite sequence of trades. In addition to solving for the value functions and optimal switching strategies, we identify the conditions under which the double stopping and switching problems admit the same optimal entry and/or exit timing strategies. A number of extensions are also considered, such as incorporating a stop-loss constraint, or a minimum holding period under the OU model. A typical solution approach for optimal stopping problems is to study the associated free boundary problems or variational inequalities (VIs).

For the double optimal stopping problem, we apply a probabilistic methodology and rigorously derive the optimal price intervals for market entry and exit. A key step of our approach involves a transformation, which in turn allows us to characterize the value function as the smallest concave majorant of the reward function in the transformed coordinate. In contrast to the variational inequality approach, this approach directly constructs the value function as well as the optimal entry and exit regions, without a priori conjecturing a candidate value function or timing strategy. Having solved the optimal double stopping problem, we then apply our results to deduce a similar solution structure for the optimal switching problem. We also verify that our value functions solve the associated VIs. Among our results, we find that under OU or CIR price dynamics, the optimal stopping problems admit the typical buy-low-sell-high strategies. However, when the prices are driven by an exponential OU process, the investor generally enters when the price is low, but may find it optimal to wait if the current price is sufficiently close to zero.

In other words, the continuation (waiting) region for entry is disconnected. A similar phenomenon is observed in the OU model with stop-loss constraint. Indeed, the entry region is again characterized by a bounded price interval that lies strictly above the stop-loss level. As for the exit timing, a higher stop-loss level always implies a lower optimal take-profit level. In all three models, numerical results are provided to illustrate the dependence of timing strategies on model parameters.