Background: The reaction rates used in γ-process nucleosynthesis network calculations are mostly derived from theoretical, statistical model cross sections. Experimental data is scarce for charged particle reactions at astrophysical, low energies. Where experimental (α,γ) data exists, it is often strongly overestimated by Hauser-Feshbach statistical model calculations. Further experimental α-capture cross sections in the intermediate and heavy mass region are necessary to test theoretical models and to gain understanding of heavy element nucleosynthesis in the astrophysical γ process. Purpose: The aim of the present work is to measure the Sb121(α,γ)I125, Sb121(α,n)I124, and Sb123(α,n)I126 reaction cross sections. These measurements are important tests of astrophysical reaction rate predictions and extend the experimental database required for an improved understanding of p-isotope production. Method: The α-induced reactions on natural and enriched antimony targets were investigated using the activation technique. The (α,γ) cross sections of Sb121 were measured and are reported for the first time. To determine the cross section of the Sb121(α,γ)I125, Sb121(α,n)I124, and Sb123(α,n)I126 reactions, the yields of γ rays following the β decay of the reaction products were measured. For the measurement of the lowest cross sections, the characteristic x rays were counted with a low-energy photon spectrometer detector. Results: The cross section of the Sb121(α,γ)I125, Sb121(α,n)I124, and Sb123(α,n)I126 reactions were measured with high precision in an energy range between 9.74 and 15.48 MeV, close to the astrophysically relevant energy window. The results are compared with the predictions of statistical model calculations. The (α,n) data show that the α widths are predicted well for these reactions. The (α,γ) results are overestimated by the calculations but this is because of the applied neutron and γ widths. Conclusions: Relevant for the astrophysical reaction rate is the α width used in the calculations. While for other reactions the α widths seem to have been overestimated and their energy dependence was not described well in the measured energy range, this is not the case for the reactions studied here. The result is consistent with the proposal that additional reaction channels, such as Coulomb excitation, may have led to the discrepancies found in other reactions.