XVA. Green Andrew

seen in the context of multiple yield curve models. These models were introduced to account for tenor basis or the large discrepancies observed between instruments with different payment frequencies, observable through the prices of basis swaps. Cross-currency basis was already well established in yield curve frameworks and pre-crisis yield curve models were already contructed in such a fashion as to reprice both single currency and cross-currency swaps. These cross-currency models implicitly used 3M US Dollar LIBOR as the primary discount curve and all other currencies were marked with currency basis with respect to US Dollars. The main justification for the use of US Dollar discounting was its role as global reserve currency. A number of papers were published on multiple curve discount models including Henrard (2007); Henrard (2009), Ameritrano and Bianchetti (2009), Chibane and Sheldon (2009), Fujii, Shimada and Takahashi (2010b) and Bianchetti (2010). A number of authors then extended the multiple curve frameworks into models designed to value exotic interest rate derivative products while accounting for both tenor and cross-currency basis and models of this type were presented by Mercurio (2010b), Fujii, Shimada and Takahashi (2009) and by Kenyon (2010) in the short rate modelling framework.

      A parallel development to this was the realisation that the use of 3M xIBOR curves as the primary discounting curve was incorrect for derivative portfolios secured by collateral under CSA agreements. Market practitioners realised that the correct discount curve was in fact the OIS curve, at least for CSAs which accepted collateral in a single currency and paid the overnight unsecured rate of interest on posted collateral on a daily basis. This was driven by a realisation that the OIS curve was considered risk free while the rapidly increased divergence between the OIS rate and 3M xIBOR rates clearly demonstrated that xIBOR was perceived as far from risk free by market participants (see for example Hull and White, 2013). Piterbarg clearly demonstrated that the interest rate paid on collateral was the correct discount rate under a perfect single currency CSA agreement (Piterbarg, 2010) and subsequently in the multi-currency case (Piterbarg, 2012).10

      Unsecured derivatives were seen as just another discount curve, with valuations either remaining at 3M xIBOR discounted values or moved to a discount curve equivalent to the bank cost of funds if this was higher than 3M xIBOR. It quickly became apparent that such models led to double counting of benefit from DVA and funding for those institutions that used bilateral CVA models. The primary driver for both funding and DVA was the market perception of credit worthiness; in the context of funding this was seen through the yield on bank funding instruments and their spread over instruments considered risk free such as high quality government bonds and through the bank CDS curve in the case of DVA. This led Burgard and Kjaer (2011b) and Burgard and Kjaer (2011a) to produce a self-consistent framework the included CVA, DVA and FVA through a similar PDE approach to that used by Piterbarg in the context of collateralisation. Morini and Prampolini (2011) also developed a model including both DVA and FVA from a probabilistic approach. Kenyon and Stamm (2012) developed a portfolio level model for FVA, while Pallavicini, Perini and Brigo (2012) produced a portfolio level model that incorporates cash flows from collateral as well as individual trades using a probabilistic approach.

      In the 25th anniversary edition of Risk Magazine, John Hull and Alan White (2012b) wrote an article arguing that FVA should not be applied to derivatives. The response from market practioners was an immediate vigorous counterargument that funding costs should be priced into derivatives, beginning with Laughton and Vaisbrot (2012) in the next issue of Risk Magazine and followed by Castagna (2012) and Morini (2012). Hull and White published a further two papers as the debate continued (2012c; 2014b). Kenyon and Green (2014c) and Kenyon and Green (2014b) continued the debate with Hull and White (2014c) in the context of the implications of regulatory associated costs.

      Hull and White (2012b) based their argument on eight key points:

      1. Discounting at the risk-free rate is a consequence of risk-neutral valuation.

      2. Hedging involves buying and selling zero cost instruments and so hedging does not affect valuations.

      3. The Fischer-Hirshleifer Separation Principle (Hirshleifer, 1958)/Modigliani-Millar Theorem (Modigliani and Miller, 1958) imply that pricing and funding should be kept separate.

      4. Banks invest in Treasury instruments and other low-yielding securities without charging funding costs.

      5. FVA is equal to the change in DVA from the fair value option on the bank’s own issued debt.

      6. FVA is a form of anti-economic valuation adjustment.

      7. Proponents of FVA do not require the derivatives desk to earn a bank’s weighted average cost of capital.

      8. The FVA adjusted price is a Private Valuation.

      Laughton and Vaisbrot (2012) countered that Hull and White’s arguments were based on complete markets where all risks can be hedged, while in practice markets are incomplete and this introduces subjectivity into valuations; hence the law of one price no longer holds. Furthermore, Laughton and Vaisbrot argue that the Black-Scholes-Merton model relies upon both no arbitrage and the ability to borrow and lend at the risk-free rate in unlimited size, while in reality there is no deep liquid two-way market in borrowing and lending cash and that apparent arbitrage opportunities are visible in the market because of the practical difficulties in conducting arbitrage. Furthermore Laughton and Vaisbrot suggest that models should be practically useful to traders and that as a result the cost of borrowing is an exogenous factor that is unaffected by a single trade and that no value should be attributed to profit or loss on own default through DVA as it is impossible to monetise.

      Antonio Castagna (2012) argued against points 1, 2, 3, 4 and 5 in the list above. Discounting using the risk-free rate may not be appropriate, argues Castagna, as it does not cover the cost of the replication strategy. Hedging does not always involve buying and selling zero cost instruments as the funding rate has to be paid if money is borrowed to purchase an asset. Castagna argues that the Modigliani-Millar theorem does not apply to derivatives. The argument proposed by Hull and White (2012b) that banks invest in low yielding instruments is simply false as these assets are mostly funded through the repo market. Finally, Castagna suggests that FVA cannot be offset by gains or losses through DVA from the fair value of own debt option as any such gains cannot be realised in the event of default.

      Massimo Morini (2012) argues that it makes sense for a lender to charge funding costs as it will be left with a carry loss in the event it does not default itself, even if it does not make sense for the borrower. Morini suggests that FVA might be a benefit for shareholders on the basis that the shareholders of a limited liability company are effectively holding a call option on the value of the company. Like Castagna, Morini argues against the use of the Modigliani-Millar theorem but on the basis that Hull and White assume that the market response to the choice of projects undertaken by the company is linear as expressed through funding costs. Morini demonstrates that even in the case of a simple Black-Cox model the market response is nonlinear. Finally, Morini agrees with Hull and White's assertion that risk-free discounting is appropriate in derivative pricing.

      What does this debate actually mean in practice? The answer to at least some degree is that quants protest too much when theoretical arguments are challenged by the market. Ultimately mathematics is a tool used in finance to, for example, value products, risk manage them, produce economic models, etc. All models have assumptions and if the market changes and the assumption is no longer valid the models have to change to remain useful. This is not the first time that models have “failed” to some degree. Skew and smile on vanilla options have long been present showing deviations from the Black-Scholes model. This demonstrates that the market believes the distribution of the underlying asset is not log-normal as assumed by the Black-Scholes model. During the credit crisis the Gaussian Copula models used to value CDOs were unable to match market prices. However, it is clear that FVA presents a broader challenge to quantitative finance than previous issues such as volatility smile. The fundamental assumptions underlying much of quantitative finance theory since Black-Scholes are challenged by FVA. Valuation is undergoing a paradigm shift away from these standard assumptions towards models that encompass more realism and away from the simplifying assumptions of the Black-Scholes framework.

      Kenyon and Green (2014c) and Kenyon and Green (2014b) demonstrated

10

OIS discounting models are discussed in Chapter 9.