Markovian and multi-curve friendly parametrisation of a HJM model used in valuation adjustment of interest rate derivatives

We consider feasible Heath-Jarrow-Morton framework specifications that are easily implementable in XVA engines when pricing linear and non-linear interest rate derivatives in a multi-curve environment. Our particular focus is on relatively less liquid markets (Polish PLN) and the calibration problems arising from that fact. We first develop the necessary tool-kit for multi- -curve construction and XVA integration and then show and discuss various specifications of the HJM model with regard to their practical usage. We demonstrate the importance of the Cheyette subclass and derive the dynamics of instantaneous forward rates in generic forms of different models. We performed calibrations of several one-factor models of that form and found that even with a relatively simple specification, i.e. Hull-White with two summands, we may achieve satisfactory results in terms of the quality of the calibration and calculation time.