In this work, the analytical pricing of maximum and minimum options under the integrated scenario of interdependent stochastic volatility and jump, and the stochastic interest rate, were investigated by means of the composite Mellin transform approach. The analytic expressions of Mellin transform functions of the price of the maximum put option, minimum call option, and the exchange option were derived by different partial differential-integral equations (PDIEs). Meanwhile, the explicit price of other maximum and minimum options were obtained by means of the payoff decomposition technique and the parity relation of max-min and exchange options. In addition, the convergence of solutions of PDIEs was further demonstrated by transform techniques and decomposition skills. The simulation of the price process of two underlying assets was given to present the effectiveness and uniqueness of the proposed model. Finally, numerical analysis was implemented to examine the accuracy of the PDIE method and the validity of key parameters.