Discounting Revisited. Valuations under Funding Costs, Counterparty Risk and Collateralization.

Fries, Christian P. (2010): Discounting Revisited. Valuations under Funding Costs, Counterparty Risk and Collateralization.

Abstract

Looking at the valuation of a swap when funding costs and counterparty risk are neglected (i.e., when there is a unique risk free discounting curve), it is natural to ask "What is the discounting curve of a swap in the presence of funding costs, counterparty risk and/or collateralization".

In this note we try to give an answer to this question. The answer depends on who you are and in general it is "There is no such thing as a unique discounting curve (for swaps)." Our approach is somewhat "axiomatic", i.e., we try to make only very few basic assumptions. We shed some light on use of own credit risk in mark-to-market valuations, giving that the mark-to-market value of a portfolio increases when the owner's credibility decreases.

We present two different valuations. The first is a mark-to-market valuation which determines the liquidation value of a product. It does, buy construction, exclude any funding cost. The second is a portfolio valuation which determines the replication value of a product including funding costs.

We will also consider counterparty risk. If funding costs are presents, i.e., if we value a portfolio by a replication strategy then counterparty risk and funding are tied together:

- In addition to the default risk with respect to our exposure we have to consider the loss of a potential funding benefit, i.e., the impact of default on funding.

- Buying protection against default has to be funded itself and we account for that.

The valuation naturally attributes for wrong-way-risk (i.e., the correlation between counterparty default and counterparty exposure).

• Item Type: MPRA Paper
• Language: English
• Keywords: Discounting, Valuation; Counterparty Risk; Collateral; Netting; CVA; DVA; Bond; Swap; Credit Risk; Own Credit Risk; Wrong Way Risk; Liquidity Risk
• Subjects: G - Financial Economics > G1 - General Financial Markets > G12 - Asset Pricing ; Trading Volume ; Bond Interest Rates; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C15 - Statistical Simulation Methods: General; G - Financial Economics > G1 - General Financial Markets > G13 - Contingent Pricing ; Futures Pricing

References

Brigo, Damiano; Pallavicini, Andrea; Papatheodorou, Vasileios: Bilateral counterparty risk valuation for interest-rate products: impact of volatilities and correlations. 2010.

Ebmeyer, Dirk: Essays on Incomplete Financial Markets. Doctoral Thesis. University of Bielefeld, Bielefeld.

Fries, Christian P.: Mathematical Finance. Theory, Modeling, Implementation. John Wiley & Sons, 2007. ISBN 0-470-04722-4}.

Mercurio, Fabio: LIBOR Market Models with Stochastic Basis, March 2010.

Morini, Massimo: Solving the puzzle in the interest rate market. 2009.

Morini, Massimo; Prampolini, Andrea: Risky funding: a unified framework for counterparty and liquidity charges. 2010.

Piterbarg, Vladimir: Funding beyond discounting: collateral agreements and derivatives pricing. Risk magazine. February 2010.

Whittall, Christopher: The price is wrong. Risk magazine. March 2010.