Active lock-in options are a class of complex derivatives characterized by pronounced path dependence and optimal decision making features, and they possess significant application value in the design of structured financial products and risk management. This paper investigates the pricing of active lock-in call options under a stochastic volatility framework. The lock-in decision is formulated as an optimal stopping problem and is further reformulated as a partial differential equation with obstacle constraints. By introducing a linear complementarity problem formulation, the structural properties of the option value function and the optimal lock-in boundary are systematically characterized. From a numerical perspective, an IMEX time discretization scheme is employed to transform the continuous problem into a sequence of time-layered discrete complementarity systems. These systems are efficiently solved using the projected successive over relaxation (PSOR) algorithm. Numerical experiments are conducted to analyze the structural features and economic interpretations of the value function and the associated free boundary surface.
