Randomization of Short-Rate Models, Analytic Pricing and Flexibility in Controlling Implied Volatilities

Abstract: 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.