A Levy 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 effects (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.

Keywords: interest rate derivative, multiple-curve term structure model, Levy process, credit valuation adjustment (CVA), funding

JEL Classification: G12, E43