Funding Valuation Adjustment: a consistent framework including CVA, DVA, collateral,netting rules and re-hypothecation

In this paper we describe how to include funding and margining costs into a risk-neutral pricing framework for counterparty credit risk. We consider realistic settings and we include in our models the common market practices suggested by the ISDA documentation without assuming restrictive constraints on margining procedures and close-out netting rules. In particular, we allow for asymmetric collateral and funding rates, and exogenous liquidity policies and hedging strategies. Re-hypothecation liquidity risk and close-out amount evaluation issues are also covered. We define a comprehensive pricing framework which allows us to derive earlier results on funding or counterparty risk. Some relevant examples illustrate the non trivial settings needed to derive known facts about discounting curves by starting from a general framework and without resorting to ad hoc hypotheses. Our main result is a bilateral collateralized counterparty valuation adjusted pricing equation, which allows to price a deal while taking into account credit and debt valuation adjustments along with margining and funding costs in a coherent way. We find that the equation has a recursive form, making the introduction of an additive funding valuation adjustment difficult. Yet, we can cast the pricing equation into a set of iterative relationships which can be solved by means of standard least-square Monte Carlo techniques.