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We consider the class of affine LIBOR models with multiple curves, which isan analytically tractable class of discrete tenor models that easilyaccommodates positive or negative interest rates and positive spreads. Byintroducing an interpolating function, we extend the affine LIBOR models to acontinuous tenor and derive expressions for the instantaneous forward rate andthe short rate. We show that the continuous tenor model is arbitrage-free, thatthe analytical tractability is retained under the spot martingale measure, andthat under mild conditions an interpolating function can be found such that theextended model fits any initial forward curve. This allows us to compute valueadjustments (i.e. XVAs) consistently, by solving the corresponding`pre-default' BSDE. As an application, we compute the price and valueadjustments for a basis swap, and study the model risk associated to differentinterpolating functions.

Continuous tenor extension of affine LIBOR models with multiple curves and applications to XVA