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The efficient use of Type Ia supernovae (SNIa) for cosmological studies requires knowledge of any parameter that can affect their luminosity in either systematic or statistical ways. Observational samples of SNIa commonly use the metallicity of the host galaxy,  Z  host , as an estimator of the supernova progenitor metallicity,  Z  Ia , that is one of the primary factors affecting SNIa magnitude. Here, we present a theoretical study of the relationship between  Z  Ia  and  Z  host . We follow the chemical evolution of homogeneous galaxy models together with the evolution of the supernova rates in order to evaluate the metallicity distribution function,  MDF (Δ Z ) , i.e. the probability that the logarithm of the metallicity of an SNIa exploding now differs in less than  Δ Z   from that of its host. We analyse several model galaxies aimed to represent from active to passive galaxies, including dwarf galaxies prone to experience supernova‐driven outflows. We analyse as well the sensitivity of the MDF to the most uncertain ingredients of our approach: initial mass function, star formation law, stellar lifetime, stellar yields and SNIa delay‐time distribution (DTD). Our results show a remarkable degree of agreement between the mean  in a galaxy and its  Z  host  when they both are measured as the CNO abundance, especially if the DTD peaks at small time delays, while the average Fe abundance of host and SNIa may differ up to 0.4–0.6 dex in passive galaxies. The dispersion of  Z  Ia  in active galaxy models is quite small, meaning that  Z  host  is quite a good estimator of the supernova metallicity. Passive galaxies present a larger dispersion, which is more pronounced in low‐mass galaxies. We present a procedure to generate random SNIa metallicities, given the host metallicity. We also discuss the use of different metallicity indicators: Fe versus O and gas‐phase metallicity versus stellar metallicity. Finally, the results of the application of our formalism to a galactic catalogue [VErsatile SPectral Analysis (VESPA)] suggest that SNIa come, in average, from small metallicity progenitors both at low redshifts (contrary to expectations) and in galaxies with high star formation activity. In spite of large uncertainties in the metallicities derived from the catalogue, the gross trends of  versus  Z  host  obtained from VESPA for different galaxy types are roughly consistent with our theoretical estimates.

Is the metallicity of their host galaxies a good measure of the metallicity of Type Ia supernovae?