The inclusion of DVA in the fair-value of derivative transactions has now become standard accounting practice in most parts of the world. Furthermore, some sophisticated banks are including an FVA (Funding Valuation Adjustment), but since DVA can be interpreted as a funding benefit the oft-debated issue regarding a possible double-counting of funding benefits arises, with little consensus as to its resolution. One possibility is to price the derivative by replication, guaranteeing a consistent inclusion of costs and benefits. However, as has recently been noted, DVA is (at least partially) unhedgeable, having no exact market hedge. Furthermore, current frameworks shed little light on the controversial question, raised by Hull (2012), of whether the effect a derivative has on the riskiness of an institution's debt should be taken into account when calculating FVA. In this paper we propose a solution to these two problems by identifying an instrument, a fictitious CDS written on the hedging counterparty which is implicitly contained in any given derivatives transaction. This allows us to show that the hedger's unhedged jump-to-default risk has, despite not being actively managed, a well defined value associated to a funding benefit. Carrying out the replication including such a CDS, we obtain a price for the derivative consisting of its collateralized equivalent, a contingent CVA, a contingent DVA, and an FVA, coupled to the price via the hedger's short-term bond-CDS basis. The resulting funding cost is non-zero, but substantially smaller than what is obtained in alternative approaches due to the effect the derivative has on the recovery of the hedger's liabilities. Also, price agreement is possible for two sophisticated counterparties entering a deal if their bond-CDS bases obey a certain relationship, similar to what was first obtained by Morini and Prampolini (2010).
