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the discount function was obtained on any date through log-linear interpolation.

      Valuing interest rate swaps was straightforward and the discounting curve was driven by xIBOR rates. Cross-currency swaps were more involved and a spread or basis between currencies was required to revalue cross-currency swaps to par. Nevertheless single currency derivative books were typically valued using the single currency discount curve so that cross-currency and single currency books were valued inconsistently. Tenor basis existed in the sense that anyone who wanted to swap a 3M xIBOR rate for a 6M xIBOR rate was charged a spread but these spreads were small and reflected operational costs of the transaction. Tenor basis was essentially ignored for valuation purposes.

      In 1996 CVA was not calculated and funding costs were not explicitly considered. Counterparty credit risk was managed through traditional credit limits set by the credit risk function within a bank. The Basel I framework had already been introduced but capital management was considered a back office function.

      By the time the credit crisis began in 2007, many banks used a multi-currency discounting framework where USD was usually considered the primary curve with no currency basis and all other currencies had separate projection and discounting curves constructed in such a way as to reprice to par all of the single currency instruments used to construct the curve and the cross-currency swaps. CVA was commonly calculated and charged by tier one banks, with some institutions using unilateral models and some bilateral models. FAS 157 (2006) introduced a CVA(DVA) reporting requirement with wording that implies a bilateral model should be used.

      1.2.2 The Post-Crisis World…

      As the crisis hit, the market reacted rapidly and tenor basis spreads and the spread between three-month LIBOR and OIS widened rapidly. The USD three-month LIBOR and OIS reached a peak of 365bp just after the collapse of Lehman Brothers in 2008, having averaged 10bp prior to August 2007 (Sengupta and Tam, 2008). This prompted banks to switch to using multiple projection curves for rates with different tenors (see for example Kenyon and Stamm, 2012). It was also realised that the discount curve, normally believed to be based on three-month rates was not the appropriate choice for discounting trades transacted under CSA agreements. CSAs typically pay a rate of interest on posted collateral equal to the overnight rate in the collateral currency and hence it rapidly became market practice to use OIS-based curves to discount collateralised cash flows (Piterbarg, 2010; Piterbarg, 2012).

      While CVA had been marked by many tier one banks prior to the crisis, a number of banks switched from the use of unilateral to bilateral models during the crisis. Bank CDS spreads widened significantly leading to dramatic increases in DVA accounting benefit where it was marked. Competitive pressure on pricing also pushed banks to include DVA in derivative pricing as many corporate customers had CDS spreads that were significantly tighter than the bank counterparties they were trading with. Accounting standards moved to recommend bilateral CVA models because of the symmetrical valuation. Some banks chose to actively manage DVA, while others chose to warehouse some or all of the risk.

      Unsecured funding costs became a focus as bank funding spreads widened significantly as their CDS spreads did and a number of models were produced (see for example Burgard and Kjaer, 2011b; Burgard and Kjaer, 2011a; Morini and Prampolini, 2011; Pallavicini, Perini and Brigo, 2012)). FVA remains controversial as will be discussed in detail in section 1.4; however, most practitioners accept that it must be included in pricing and accounting practice with most major banks now taking FVA reserves.

      Post-Crisis, regulatory capital is a scarce and expensive resource that must be carefully managed by banks. Capital management is no longer a back office function and now resides firmly in the front office as a core activity for tier one banks. The cost of regulatory capital has to be priced into every new transaction to determine if a trade is expected to be profitable. Capital modelling is now based in the much more complex Basel II.5 and Basel III regimes (and their implementation by regional regulatory bodies).

In 2015 pricing a “vanilla” interest rate swap involves multiple projection and discount curves for the baseline valuation and a large-scale Monte Carlo simulation at counterparty level to calculate CVA, FVA and KVA; it is a long way from the single yield curve discount models of the mid-1990s (see Table 1.1). Indeed, as will become apparent in this book, single trades can no longer be valued in isolation and trade valuation is an exercise in allocation of portfolio level numbers down to individual trades.

Table 1.1 Components of derivative prices before and after the financial crisis of 2007–2009.

      1.3 Trade Economics in Derivative Pricing

      1.3.1 The Components of a Price

      Table 1.1 illustrates the components of pricing before and after the financial crisis of 2007–2009. Not all of these components apply to all trades and to understand the terms under which a derivative is transacted it is useful to divide counterparties into three types: unsecured, CSA and CCP. Of course, in reality the range of counterparty arrangements is a continuum between unsecured and an idealised CSA with full instantaneous transfer of collateral but it is useful to separate them for the purpose of this discussion. Each of these three cases will have different pricing components.

      Unsecured Pricing

      • Risk-neutral valuation

      • Hedging/management costs

      • CVA

      • FVA

      • KVA

      • Profit (Tax/TVA).

      For trades that are unsecured the components of the price begin with the baseline risk-neutral valuation, although there remains industry debate about the appropriate choice of discount curve for this, with both OIS and xIBOR-based discount curves used in the market. Hedging and trading desk management costs should be charged and this includes effects such as bid-offer, lifetime re-hedging costs and cost of supporting infrastructure such as system maintenance and staff. A profit margin will also typically be charged. Given there is no collateral to mitigate the exposure to the counterparty, CVA will be calculated based on the full expected exposure profile. If a bilateral CVA model is used there will also be a DVA benefit term. The same expected exposure will give rise to FVA. The lifetime cost of maintaining regulatory capital, KVA, will also be included, although this might be a hurdle rate or minimum return level rather than a cash amount.

      CSA Pricing

      • Risk-neutral valuation: CSA-based (OIS) discounting

      • Hedging/Management costs

      • Residual CVA (including impact of collateralisation)

      • Residual FVA (including impact of collateralisation)

      • COLVA/Collateral effects

      • KVA

      • Bilateral Initial Margin MVA

      • Profit.

      For trades covered by a CSA the baseline risk-neutral valuation will be discounted using a curve appropriate to the terms of the CSA. Given that the CSA is imperfect in the sense that the collateral transferred to support the mark-to-market of the trade is done on a discrete periodic basis rather than a continuous basis, a residual counterparty exposure will remain. This residual exposure leads to residual CVA and residual FVA. There may also be a COLVA adjustment to account for collateral effects in pricing that cannot be captured by a discounting approach such as collateral optionality. Capital must be held against collateralised portfolios and this gives rise to KVA, although the presence of collateral significantly reduces the amount of capital that must be held through the counterparty credit risk and CVA capital terms. The leverage ratio comes into importance here, however, as while collateral reduces the CCR and CVA terms, it has a restricted impact on the leverage ratio. Market risk capital will be held unless there are other market risk offsetting trades. Hedging and trading desk management costs should again be charged as should the profit margin. Under BCBS 226 (2012e) and BCBS 242 (2013e), trades supported by CSA agreements will also require

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