The three-factor model without a linear return generating process

From a theoretical point of view, the Fama and French three-factor model requires the following implicit assumptions: (i) the excess return of an asset is correlated with market, size, and book-to-market factors, and (ii) the return generating process is linear. In this note, we demonstrate that the linearity assumption of the return generating process can be relaxed. This suggests that assumption (i) alone is sufficient for the three-factor model.