In this paper, we consider optimal pairs trading strategies in terms of static optimality and dynamic optimality under mean–variance criterion. The spread of the entity pairs is assumed to be mean-reverting and follows an Ornstein–Uhlenbeck process. A constrained optimal control problem is considered, and the Lagrange multiplier technique is adopted to transform the primal problem into a family of linear-quadratic optimal control problems that can be solved by the classical dynamic programming principle. Both solutions for static and dynamic optimal pairs trading problems are derived and discussed. We show that the “static and dynamic optimality” is a viable approach to the time-inconsistent control problem. Furthermore, numerical experiments are presented to demonstrate the performance of the optimal pairs trading strategies.