The analytical tractability of affine (short rate) models, such as the Vasi\v{c}ek and the Cox-Ingersoll-Ross models, has made them a popular choice for modelling the dynamics of interest rates. However, in order to account properly for the dynamics of real data, these models need to exhibit time-dependent, or even stochastic, parameters. This in turn breaks their tractability, and modelling and simulating becomes an arduous task. We introduce a new class of Heath-Jarrow-Morton (HJM) models, that both fit the dynamics of real market data and remain tractable. We call these models consistent recalibration (CRC) models. These CRC models appear as limits of concatenations of forward rate increments, each belonging to a Hull-White extended affine factor model with possibly different parameters. That is, we construct HJM models from "tangent" affine models. We develop a theory for a continuous path version of such models, and discuss their numerical implementations within the Vasi\v{c}ek and Cox-Ingersoll-Ross frameworks.
