On the Hull-White model with volatility smile for Valuation Adjustments

Affine Diffusion (AD) dynamics are frequently used for Valuation Adjustments ( $ {\rm xVA} $ xVA) calculations due to their analytic tractability. However, these models cannot capture the market-implied skew and smile, which are relevant when computing $ {\rm xVA} $ xVA metrics. Hence, additional degrees of freedom are required to capture these market features. In this paper, we address this through an SDE with state-dependent coefficients. The SDE is consistent with the convex combination of a finite number of different AD dynamics. We combine Hull-White one-factor models where one model parameter is varied. We use the Randomized AD (RAnD) technique to parameterize the combination of dynamics. We refer to our SDE with state-dependent coefficients and the RAnD parametrization of the original models as the rHW model. The rHW model allows for efficient semi-analytic calibration to European swaptions through the analytic tractability of the Hull-White dynamics. We use a regression-based Monte-Carlo simulation to calculate exposures. In this setting, we demonstrate the significant effect of skew and smile on (potential future) exposures and $ {\rm xVA} $ xVAs of linear and early-exercise interest rate derivatives.