Analytic value function for optimal regime-switching pairs trading rules

We introduce a regime-switching Ornstein-Uhlenbeck (O-U) model to address an optimal investment problem. Our study gives a closed-form expression for a regime-switching pairs trading value function consisting of probability and expectation of the double boundary stopping time of the Markov-modulated O-U process. We derive analytic solutions for the homogenous and non-homogenous ODE systems with initial value conditions for probability and expectation of the double boundary stopping time, and translate the solutions with boundary value conditions into solutions with initial value conditions. Based on the smoothness and continuity of the value function, we can obtain the optimum of the value function with thresholds and guarantee the existence of optimal thresholds in a finite closed interval. Our numerical analysis illustrates the rationality of theoretical model and the shape of transition probability and expected stopping time, as well as discusses sensitivity analysis in both one-state and two-state regime-switching models. We find that the optimal expected return per unit time in the two-state regime-switching model is higher than that of one-state regime-switching model. Likewise, the regime-switching model's optimal thresholds are closer and more symmetric to the long-term mean.