A L evy HJM Multiple-Curve Model with Application to CVA Computation

Abstract

We consider the problem of valuation of interest rate derivatives in the post-crisis setup. We develop a multiple-curve model, set in the HJM framework and driven by a Levy process. We proceed with joint calibration to caps and swaptions of different tenors, the calibration to caps guaranteeing that the model correctly captures volatility smile e ffects (in strike) and the calibration to at-the-money swaptions ensuring an appropriate term structure of the volatility in the model. To account for counter-party risk and funding issues, we use the calibrated multiple-curve model as an underlying model for CVA computation. We follow a reduced-form methodology through which the problem of pricing the counter-party risk and funding costs can be reduced to a pre-default Markovian BSDE, or an equivalent semi-linear PDE. As an illustration we study the case of a basis swap, for which we compute the counter-party risk and funding adjustments.