This paper proposes a pairs trading model that incorporates time-varying volatility of the Constant Elasticity of Variance (CEV) type, extending traditional pairs trading frameworks. The authors formulate the problem as a stochastic control problem where a trader holds a portfolio of two co-integrated assets and seeks to maximize expected power utility from terminal wealth over a fixed horizon. The optimal trading strategies are characterized by solving a Hamilton-Jacobi-Bellman (HJB) partial differential equation using a Finite Difference method. Model parameters are estimated via the Generalized Method of Moments (GMM) on historical daily data. The results demonstrate that accounting for time-varying volatility leads to significantly different optimal strategies compared to constant volatility models, and empirical tests show improved performance, highlighting the importance of modeling volatility dynamics in pairs trading.