  We focus on extending existing short-rate models, enabling control of the
generated implied volatility while preserving analyticity. We achieve this goal
by applying the Randomized Affine Diffusion (RAnD) method to the class of
short-rate processes under the Heath-Jarrow-Morton framework. Under
arbitrage-free conditions, the model parameters can be exogenously stochastic,
thus facilitating additional degrees of freedom that enhance the calibration
procedure. We show that with the randomized short-rate models, the shapes of
implied volatility can be controlled and significantly improve the quality of
the model calibration, even for standard 1D variants. In particular, we
illustrate that randomization applied to the Hull-White model leads to dynamics
of the local volatility type, with the prices for standard volatility-sensitive
derivatives explicitly available. The randomized Hull-White (rHW) model offers
an almost perfect calibration fit to the swaption implied volatilities.
