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In this work, we obtain the symplectic partitioned Runge-Kutta (SPRK) scheme for the optimal control problem of stochastic differential equations (SDEs). In order to discretize the optimal control problem, there are two basic approaches: discretize-then-optimize and optimize-then-discretize. We mainly focus on SPRK scheme for the optimal control problem of SDEs by following the discretize-then-optimize approach. After we present Hamiltonian formulations for the stochastic optimal control problem, we discretize the cost functional and the state equation with the help of Runge-Kutta schemes. To obtain the optimality system, we state the discrete Lagrangian. Then, we get the stochastic adjoint pair. (Our main contribution is to obtain an implicit Runge-Kutta scheme for the adjoint pair. As applications, we choose some problems from finance. We compare the numerical results with the exact solutions.