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Pricing a Collateralized Derivative Trade with a Funding Value Adjustment

Chadd Hunzinger, Coenraad Labuschagne

2015
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Journal of Risk and Financial Management
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... von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Abstract: The 2008 credit crisis changed the manner in which derivative trades are conducted. One of these changes is the posting of collateral in a trade to mitigate the counterparty credit risk. Another is the realization that banks are not risk-free and, as a result, cannot borrow at the risk-free rate any longer. The latter led banks to introduced the controversial adjustment to derivative prices, known as a funding value adjustment (FVA), which is interlinked with the posting of collateral. In this paper, we extend the Cox, Ross and Rubinstein (CRR) discrete-time model to include collateral and FVA. We prove that this derived model is a discrete analogue of Piterbarg's partial differential equation (PDE), which describes the price of a collateralized derivative. The fact that the two models coincide is also verified by numerical implementation of the results that we obtain. JEL classifications: C51, G12, C53, G01 J. Risk Financial Manag. 2015, 8 18 1. Introduction The 2008 credit crisis emphasized the importance of managing counterparty credit risk correctly. One of the ways to mitigate counterparty credit risk is by posting collateral within a derivative trade. Collateral is a borrower's pledge of specific assets to a lender, to secure repayment of a liability. For exchange traded derivatives, i.e., stock options, counterparty credit risk is reduced, because the two counterparties in the trade are required to post margins to the exchange. However, this does not fully eliminate counterparty credit risk. Central counterparty clearing houses (CCPs) are responsible for facilitating trades between the two counterparties. CCPs benefit both parties in a transaction, because they bear most of the credit risk; however, CCPs are not risk-free. As a result, counterparty credit risk is present in exchange traded derivatives. This paper focuses on over-the-counter (OTC) traded derivatives. Mitigation of counterparty credit risk in OTC derivatives requires collateral, but the posting of collateral is not mandatory. The posting of collateral in a derivative trade is regulated by a credit support annex (CSA). A CSA is a contract that documents collateral agreements between counterparties in trading OTC derivative securities. The trade is documented under a standard contract, called a master agreement, developed by the International Swaps and Derivatives Association (ISDA). Prior to the 2008 credit crisis, the London Interbank Offered Rate (LIBOR) was seen as an acceptable proxy for the risk-free rate. LIBOR is the rate at which banks could freely borrow and lend. The LIBOR rate includes a spread for the credit risk of the banks. An overnight interest rate swap (OIS) is a swap for which the overnight rate is exchanged for a fixed interest rate for a certain tenor (see Hull and White [1]). An overnight index swap references an overnight rate index, such as the Fed funds rate, as the underlying one for its floating leg, while the fixed leg would be set at an assumed rate. The difference between LIBOR and the OIS rate is known as the LIBOR-OIS spread. Prior to the 2008 credit crisis, this spread was only a few basis points; it was stable and not significant (see Gregory [2] (p. 286)). Hunzinger and Labuschagne [3] reports that the default of banks, such as Lehman Brothers and Bear Stearns, disproved the myth in the 2008 credit crisis that banks are risk-free. As a result, the LIBOR-OIS spread spiked to hundreds of basis points in the aftermath of the Lehman default in September 2008, and has remained significant ever since. The LIBOR-OIS spread reached 364.42 basis points (see Gregory [2]). These shifts made it apparent that LIBOR incorporates an adjustment for the credit risk of the banks; therefore, LIBOR is an imperfect proxy for the risk-free rate. The OIS rate appears to be the preferred choice as a proxy for the interest-free rate (see Hull and White [1]). The 2008 credit crisis drove home the realization that banks are not risk-free, and banks became unable to borrow at preferential rates. This resulted in banks charging a funding value adjustment (FVA) on transactions. FVA is the difference between the price of the derivative and the collateral posted in the trade, multiplied by the difference between growing this amount at the funding rate and the collateral rate. This amount is then discounted back to the initial time. In essence, FVA is a correction made to the risk-free price of an OTC derivative to account for the funding cost in a financial institution. The inclusion of FVA in pricing financial instruments is a controversial issue. Hull and White [4] argue against it. They argue that the funding costs and benefits realized in a trade violate the idea of risk-neutral pricing and should not be included in the pricing of the derivative. Inclusion of FVA into the

doi:10.3390/jrfm8010017
fatcat:jyvj77f5c5cw7gtdgan456fhiy