The efficient use of Type Ia supernovae (SNIa) for cosmological studies requires knowledge ofany parameter that can affect their luminosity in either systematic or statistical ways. Observationalsamples of SNIa commonly use the metallicity of the host galaxy, Zhost, as an estimatorof the supernova progenitor metallicity, ZIa, that is one of the primary factors affecting SNIamagnitude. Here, we present a theoretical study of the relationship between ZIa and Zhost. Wefollow the chemical evolution of homogeneous galaxy models together with the evolution ofthe supernova rates in order to evaluate the metallicity distribution function, MDF( Z), i.e.the probability that the logarithm of the metallicity of a SNIa exploding now differs in lessthan Z from that of its host. We analyse several model galaxies aimed to represent fromactive 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 ourapproach: IMF, star-formation law, stellar lifetime, stellar yields, and SNIa delay-time distribution(DTD). Our results show a remarkable degree of agreement between the mean ¯ZIa ina galaxy and its Zhost when they both are measured as the CNO abundance, especially if theDTD peaks at small time delays, while the average Fe abundance of host and SNIa may differup to 0.4-0.6 dex in passive galaxies. The dispersion of ZIa in active galaxy models is quitesmall, meaning that Zhost is a quite good estimator of the supernovametallicity. Passive galaxiespresent a larger dispersion, which is more pronounced in low mass galaxies.We present aprocedure to generate random SNIa metallicities, given the host metallicity. We also discussthe use of differentmetallicity indicators: Fe vs. O, and gas-phasemetallicity vs. stellar metallicity.Finally, the results of the application of our formalism to a galactic catalogue (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 largeuncertainties in the metallicities derived from the catalogue, the gross trends of ¯ZIa vs. Zhostobtained from VESPA for different galaxy types are roughly consistent with our theoreticalestimates.