DVA from issued debt will be called DVA2 in line with the convention adopted by John Hull and Alan White (2012b).
3.1.1 Close-out and CVA
When a default occurs the non-defaulting party must lodge a claim of money owed with the bankruptcy court as part of bankruptcy proceedings. If the derivative portfolio has been traded under an ISDA agreement then the value of the claim is governed by the legal language enshrined in the version of the ISDA master agreement that has been signed, either 1992 (ISDA, 1992) or 2002 (ISDA, 2002), or the more recent ISDA close-out protocol (ISDA, 2009c). These different ISDA documents use different language to describe what can be taken into consideration when calculating the close-out value and the mechanics by which the calculation is done. Several leading quantitative analysts have highlighted issues with potential ambiguity of language in these documents leading to calls for further clarification from ISDA (Brigo and Morini, 2011; Carver, 2011).
The precise nature of the close-out value has a significant impact on the exposure at default and hence on the CVA itself. The following possible variations of close-out have been identified:
• Risk-free close out. The close-out value is the portfolio value assuming no counterparty credit risk at all. This is the simplest case as it corresponds to the sum of the underlying derivative values netted according to the netting agreements in place between the two parties. One could argue this case is the one likely to be implemented in practice, given the simplicity.
• Risky close out. The close-out value of the portfolio is its value immediately prior to default including all valuation adjustments. The valuation adjustments made could include the CVA, DVA and FVA, immediately prior to default, depending on which type of adjustments and models are used by the non-defaulting party. Risky close-out is chosen as an example by Burgard and Kjaer (2011b); Burgard and Kjaer (2011a); Burgard and Kjaer (2013) in their papers on FVA, while Wei (2011) has examined the impact of risky-close out on a standard interest rate swap. If the total valuation adjustment was negative on the value of the portfolio of trades with the defaulted counterparty, risky close-out would lower the claim value and the ultimate recovery amount. Hence it is not clear whether the non-defaulting party would seek to include a negative adjustment on a rational basis. Nevertheless, anecdotal evidence suggests that dealers are keen on using a risky close-out assumption.
• Replacement cost: DVA-only. If a bilateral CVA model is used the one possible close-out would ignore the CVA prior to default but include the DVA term. This would be done on the basis that any replacement trade would naturally include the non-defaulting party’s DVA as the counterparty replacing the transaction would inevitably charge for their CVA which would always equal this DVA. Brigo and Morini (2011) and Gregory and German (2012) examine the impact of this close-out condition. Gregory and German highlight that this potentially removes some of the difficulties surrounding the use of DVA at the expense of additional complexity in the CVA calculation, while Brigo and Morini point to the lower overall recovery if replacement cost is used. The actual wording of the 2009 ISDA close-out protocol is “quotations (either firm or indicative) for replacement transactions supplied by one or more third parties that may take into account the creditworthiness of the Determining Party at the time of the quotation” (ISDA, 2009c). This suggests that the correct procedure would be to include DVA. Given that DVA would increase the book value of the trade this would also be entirely rational as it would increase the claim value and hence the potential recovery value, In practice whether this would be successful is not clear. It is also worth noting that any actual replacement trade used by the non-defaulting party to close out the market risk position would be transacted with interbank counterparties under a CSA. If the original trade was unsecured then the CVA and DVA of this actual replacement trade would be much smaller than that of the original trade and the close-out provision would not necessarily represent the actual replacement cost of the original trade.
• Replacement cost: Funding costs. The ISDA close-out protocol also includes the statement that “the Determining Party may include costs of funding, to the extent costs of funding are not and would not be a component of the other information being utilised” (ISDA, 2009c). Hence another possible adjustment would be to include funding costs, possibly alongside DVA, depending on the type of model used.
For the purposes of this chapter, I will consider only risk-free close-out as the other options add considerably to the complexity of the CVA (and FVA) calculation. However, I will return to the topic of close-out when considering models of FVA in Chapter 9.
3.2 Unilateral CVA Model
In this section I introduce the unilateral CVA model that assumes only the counterparty can default. Two different approaches are taken to derive the CVA, an expectation-led approach and a replication-based approach. In the context of pure CVA it could be argued that the expectation approach is simpler; however, I will use replication repeatedly throughout the book and so it is useful to also present this method of derivation.
3.2.1 Unilateral CVA by Expectation
Consider a portfolio of trades between a bank B and a counterparty C traded on an unsecured basis. The portfolio of trades generates a series of cash flows, C(t, T), between time t and the maturity time T in the absence of a default. The cash flows described by C(t, T) may be contingent in the sense that they may depend on option exercises but are not dependent on the default of the counterparty in any way. I assume that only the counterparty can default and that this occurs at time τ. Under these conditions the fair value of the portfolio is given by
The value of the cash flows prior to the default time is given by
where
where (Vτ)+ and (Vτ)− are the positive and negative parts of the portfolio value respectively and R is the recovery rate or fraction. Note that the portfolio value at close-out, Vτ, is the risk-free close-out discussed above. Hence, using equations (3.5), (3.4) and (3.6), we obtain an expression for the portfolio value at time t when the counterparty is subject to default risk,
(3.7)
Using the linearity of expectations and the tower property this reduces to
(3.8)
This expression can be simplified by using the fact that Vt = (Vt)+ + (Vt)−, giving
(3.9)
Using the fact that
