The density functional theory and full-potential linearized augmented plane wave (FLAPW) method are used to study the anisotropic optical properties of superconductor MgB2. The optical conductivity spectra，the reflectivity spectra and electron energy loss spectra have been calculated. The calculated optical conductivity spectra along xx-axis display quite different features in comparison with that along the zz-axis，while the reflectivity spectra coincide well with the corresponding electron energy loss spectra，especially with respect to the characteristic peaks. In the optical conductivity spectra，the first intra-band absorption peak along xx-axis appears at 20000cm-1 while along zz-axis it appears at a high frequency of about 40000cm-1. In order to simulate the temperature effect in these spectra，the Lorentz expansion (δ=0.10eV) is added to the optical matrix elements in the calculation for 0K. In comparison with existing experimental data，a minor discrepancy of 1000cm-1 (～0.124eV) has been found in the calculated optical spectra.