  The main result of this paper is a collateralized counterparty valuation
adjusted pricing equation, which allows to price a deal while taking into
account credit and debit valuation adjustments (CVA, DVA) along with margining
and funding costs, all in a consistent way. Funding risk breaks the bilateral
nature of the valuation formula. We find that the equation has a recursive
form, making the introduction of a purely additive funding valuation adjustment
(FVA) 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. As a consequence, we find that identifying funding costs and debit
valuation adjustments is not tenable in general, contrary to what has been
suggested in the literature in simple cases. The assumptions under which
funding costs vanish are a very special case of the more general theory. We
define a comprehensive framework that allows us to derive earlier results on
funding or counterparty risk as a special case, although our framework is more
than the sum of such special cases. We derive the general pricing equation by
resorting to a risk-neutral approach where the new types of risks are included
by modifying the payout cash flows. We consider realistic settings and include
in our models the common market practices suggested by 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. Finally, relevant examples of non-trivial settings illustrate how to
derive known facts about discounting curves from a robust general framework and
without resorting to ad hoc hypotheses.
