Funding, Collateral and Hedging: Uncovering the Mechanics and the Subtleties of Funding Valuation Adjustments

38 Pages Posted: 14 Oct 2012 Last revised: 12 Dec 2012

See all articles by Andrea Pallavicini

Andrea Pallavicini

Intesa Sanpaolo

Daniele Perini


Damiano Brigo

Imperial College London - Department of Mathematics

Date Written: December 3, 2012


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.

Keywords: Funding Cost, Cost of Funding, Bilateral Counterparty Risk, Credit Valuation Adjustment, Debit Valuation Adjustment, Collateral Modeling, Margining Cost, Close-Out, Re-hypothecation, Default Correlation, Gap Risk, Central Counterparty

JEL Classification: G12, G13

Suggested Citation

Pallavicini, Andrea and Perini, Daniele and Brigo, Damiano, Funding, Collateral and Hedging: Uncovering the Mechanics and the Subtleties of Funding Valuation Adjustments (December 3, 2012). Available at SSRN: or

Andrea Pallavicini (Contact Author)

Intesa Sanpaolo ( email )

Largo Mattioli 3
Milan, MI 20121

Daniele Perini

Mediobanca ( email )

Piazzetta Enrico Cuccia, 1
Milano, MI 20121

Damiano Brigo

Imperial College London - Department of Mathematics ( email )

South Kensington Campus
London SW7 2AZ, SW7 2AZ
United Kingdom


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