The linear elasticity of graft copolymer melts in the lamellar phase was examined by the self-consistent field theory solved in real space. The extensional and shear moduli, which are used to derive the Young's modulus, are found to be dependent on the architecture parameters of graft copolymers (the number of branches and the distribution of junction points). Compared with the shear modulus, the extensional modulus makes a greater contribution to the Young's modulus. The graft copolymers with the larger branch number exhibit the better mechanical properties. For the physical origin of the improvement of mechanical properties, the contribution of internal energy is the main drive, while the contribution of entropy to the moduli is negative or neglected. The distribution of junction points was also found to play a role in determining the elastic properties. These findings gained through the theoretical calculations may provide useful information for designing graft copolymers with enhanced properties.