Counterparty Risk on Interest Rate Derivatives in a Multiple Curve Setup

We study the valuation and hedging of CSA interest rate derivatives. By CSA interest rate derivatives, we mean a portfolio of OTC interest rate derivatives between two defaultable counterparties, connected by the means of a netting agreement regarding the counterparty risk related cash flows between the two parties. CSA cash flows comprise the collateral relative to this netted set of contracts, and the close-out cash flows in case of default of either party. The first step consists in studying the so-called counterparty clean valuation of the portfolio, namely the valuation in a hypothetical situation where the parties would be risk-free, yet accounting for the post-crisis discrepancy between the risk-free discount curve, and the LIBOR fixing curve. Toward this view we resort to a defaultable HJM methodology, in which this discrepancy is accounted for by the possibility of a stylized default of the LIBOR contributing banks, which would be priced by the market. Specific short rate examples are given in the form of an extended CIR and a Lévy Hull–White model for the risk-free short rate and the LIBOR short credit spread. In the second step, the counterparty clean value process of the portfolio is used as an underlier to an option, which prices the correction in value (CVA) to the portfolio due to the counterparty risk, in accordance with the corresponding CSA. The post-crisis multiple curve issue (including the above discrepancy) implies that the CVA should also account for the costs of funding a position in the portfolio and in its collateral, and of setting-up a related hedge. We model funding costs in the form of liquidity bases. We then develop a reduced-form, pre-default, backward stochastic differential equations (BSDE) approach to the problem of pricing and hedging interest rate counterparty risk. The problem of pricing and hedging interest rate counterparty risk, can thus reduced to low-dimensional Markovian pre-default CVA BSDEs, or equivalent semi-linear PDEs.