Elastic properties of the f.c.c. phase of hard spheres are determined by Monte Carlo simulations of the box fluctuations in the constant pressure ensemble with variable box shape (NpT). It is shown that the extrapolated data differ by only a f ew percent from those obtained by using systems as small as consisted of N = 108 spheres. The present results are also compared with literature results indicating systematic disagreement with some of them. For this reason, an independent method of direct determination of elastic properties by the free energy differentiation with respect to deformation in the fixed box ensemble (NVT) is also used. Very good agreement is observed for the results of the latter method and the NpT method when they are extrapolated to the infinitely large system limit. Moreover, the results obtained in the present paper fulfill the self-consistency test between the density dependence of the pressure and the bulk modulus much better than the literature data mentioned.