seek to replace a large portfolio with a smaller number of trades with the same risk profile. Active management of the CVA and CCR terms is embedded into Basel III through the capital mitigation that is available from the use of single name and index CDS trades to hedge counterparty risk. In theory trades could be used to hedge the market risk sensitivities of the lifetime cost of capital. Such trades would be in place to generate retained profits at exactly the time additional capital would be needed. In reality, however, these transactions would likely attract additional capital themselves making hedging involve iteration/optimisation rather than simply constructing a trade to offset the risk directly.
One further point to note here is that the amount of capital required depends on the regulatory approvals that the institution has in place. For each of three main contributors to regulatory capital listed above, the calculation method depends on status, with more advanced institutions allowed to use internal modelling subject to appropriate approvals and oversight. The different approaches are illustrated in Figure 1.1.
Figure 1.1 The different calculation methodologies available for market risk, counterparty credit risk and CVA under the Basel III framework. The more complex methodologies requiring regulatory approval are lower down the figure.
It should be immediately clear that regulatory capital will not be the same for all market participants because the methodology in use in each institution is different. The cost of capital of each institution will also be different as each will set individual target returns. Even in the absence of book-specific effects, such as netting, the cost of capital embedded in the derivative price can be significantly different between different banks.
Regulatory uncertainty and regulatory divergence are also major issues. Further regulatory proposals have been made by the Basel Committee and more will be made in the future. For example, under the “ review of the trading book” fundamental (2012g; 2013c) major changes will be made to the market risk capital framework, with the standardised approach revised and changes to IMM including a switch to expected shortfall from VaR. The two non-IMM approaches to counterparty credit risk will be replaced in 2017 with the standardised approach (BCBS, 2014b). The cost of future regulation is unknown but it will certainly apply to long-dated derivatives that are transacted today.
1.4 Post-Crisis Derivative Valuation or How I Learned to Stop Worrying and Love FVA
1.4.1 The FVA Debate and the Assault on Black-Scholes-Merton
As noted earlier, bank CDS spreads were very narrow prior to the start of the credit crisis in 2007. AA or better rated banks were able to fund themselves at the rate implied by the LIBOR discounting curve or in some cases below. FVA adjustments were not made as they were not needed. After the default of Lehman brothers as credit spreads widened sharply so did funding costs. The spread between overnight rates and rates of longer tenors widened significantly with the US Dollar OIS-3M LIBOR spread reaching 365bp just after the collapse of Lehman Brothers in September 2008 (Sengupta and Tam, 2008). Unsecured funding costs became very significant for the first time in the history of derivative markets.
Initially unsecured funding was 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
