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Funding, Collateral and Hedging: uncovering the mechanics and the subtleties of funding valuation adjustments

  • Andrea Pallavicini
  • Daniele Perini
  • Damiano Brigo

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.

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Paper provided by in its series Papers with number 1210.3811.

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Date of creation: Oct 2012
Date of revision: Dec 2012
Handle: RePEc:arx:papers:1210.3811
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  1. Morten L. Bech & Antoine Martin & James McAndrews, 2012. "Settlement liquidity and monetary policy implementation—lessons from the financial crisis," Economic Policy Review, Federal Reserve Bank of New York, issue Mar, pages 3-20.
  2. Markus K. Brunnermeier & Lasse Heje Pedersen, 2007. "Market liquidity and funding liquidity," LSE Research Online Documents on Economics 24478, London School of Economics and Political Science, LSE Library.
  3. Damiano Brigo & Kyriakos Chourdakis, 2009. "Counterparty Risk For Credit Default Swaps: Impact Of Spread Volatility And Default Correlation," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(07), pages 1007-1026.
  4. Andrea Pallavicini & Daniele Perini & Damiano Brigo, 2011. "Funding Valuation Adjustment: a consistent framework including CVA, DVA, collateral,netting rules and re-hypothecation," Papers 1112.1521,, revised Dec 2011.
  5. Fries, Christian P., 2010. "Discounting Revisited. Valuations under Funding Costs, Counterparty Risk and Collateralization," MPRA Paper 23082, University Library of Munich, Germany, revised 30 May 2010.
  6. Damiano Brigo & Andrea Pallavicini & Vasileios Papatheodorou, 2011. "Arbitrage-Free Valuation Of Bilateral Counterparty Risk For Interest-Rate Products: Impact Of Volatilities And Correlations," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(06), pages 773-802.
  7. Damiano Brigo & Cristin Buescu & Massimo Morini, 2011. "Impact of the first to default time on Bilateral CVA," Papers 1106.3496,
  8. Damiano Brigo & Andrea Pallavicini & Vasileios Papatheodorou, 2009. "Bilateral counterparty risk valuation for interest-rate products: impact of volatilities and correlations," Papers 0911.3331,, revised Feb 2010.
  9. Damiano Brigo & Massimo Morini, 2010. "Dangers of Bilateral Counterparty Risk: the fundamental impact of closeout conventions," Papers 1011.3355,
  10. Damiano Brigo, 2011. "Counterparty Risk FAQ: Credit VaR, PFE, CVA, DVA, Closeout, Netting, Collateral, Re-hypothecation, WWR, Basel, Funding, CCDS and Margin Lending," Papers 1111.1331,, revised Jun 2012.
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