3.5 Quantifying CCR
3.5.3 Portfolio Decomposition
The approaches described above, while they aid in understanding and would suitably be in a position to utilize existing risk and pricing systems, lack the crucial explicit representation of the replicating portfolio. In short they are not flexible enough to enable the understanding of CVA as a standalone component of the price of the contingent claim. The analysis of CVA is important for the understanding of both its effect on the nature of the contract and the prescription of hedges.
Cheburini [23], considers linear instruments, more specifically forwards, and the pricing of counterparty risk embedded on them. The author concludes that the incorporation of counterparty risk may lead to the existence of gamma exposures that would render the linear instrument non-linear. Analysing this further, the author notes that the contracts are also sensitive to volatility. This is not surprising as the consideration of counterparty risk points to the existence of default optionality and options are known for their sensitivity to volatility. Some of the material below can also be found from Gregory [46], Duffie [29], Cheburini [23], Brigo and Masseti [16], Duffie and Singleton [30].
Unilateral CVA
Unilateral CVA assumes the existence of two counterparties in a derivative transac- tion which are normally described as the investor and the counterparty. The investor
3.5. QUANTIFYING CCR 38
is assumed default free and the counterparty is assumed to be default prone. At a default time τ there are three possible states that are of interest:
1. The value to the counterparty is negative, which implies that it is positive to the investor, i.e., Vt > 0. This means that the investor receives RtVt of the value of the contract and incurs an immediate loss of (1−Rt)Vt.
2. The value to the counterparty is positive, which implies that it is negative towards the investor, i.e., Vt<0. The investor would then pay the amount Vt to the counterparty and both parties walk away.
3. The value of the contract to the counterparty and to the investor is zero; no party is thus required to pay anything.
The states above lead to the general formula for pricing unilateral counterparty credit risk[46, 16].
Proposition 3.2 (General Unilateral Counterparty Credit Risk Pricing Formula).
Let the value of a contingent claim at time t with maturity T between two coun- terparties A and C, with A assumed to be default free and C assumed to be default prone, be denoted byVb(t, T). Let the value of an equivalent contingent claim between
the two counterparties with the assumption that C is also default free be denoted by
V(t, T). The values Vb(t, T) and V(t, T) are from A’s point of view, then we have
that b V(t, T) =V(t, T)−E˜P IτC≤T(1−R C τC)D(t, τC)V(τC, T) +| F t (3.18) =Vt−UCVA(t).
Proof. The above points (1-3) can be mathematically written as follows,
b V(t, T) =E˜Pt[D(t, τC)RtCV(τC, T)++ D(t, τC)V(τC, T) − ) +IτC>TV(t, T) +IτC≤T(V(t, τC)], (3.19) and also, V(τC, T)−=V(τC, T)−V(τC, T)+. (3.20)
3.5. QUANTIFYING CCR 39
Substituting (3.20) into the terms in (3.19) we obtain: b V(t, T) =EPt˜[IτC>TV(t, T) +IτC≤T(V(t, τC) + D(t, τC)RCtV(τC, T)++ D(t, τC)V(τC, T)−D(t, τC)V(τC, T)+)] =EPt˜[IτC>TV(t, T) +IτC≤T(V(t, τC) + D(t, τC)V(τC, T)) + D(t, τC)IτC≤T(R C t V(τC, T)+−V(τC, T)+)] =V(t, T)−EP˜ t IτC≤TD(t, τC)(1−R C t )V(τC, T)+ (3.21) =V(t, T)−UCVA(t). (3.22)
What is immediately obvious is that the unilateral CVA att which we denoted by UCVA(t) is positive and is an option with zero strike on the value of the contract and is also dependent on default. The positive nature of UCVA implies that the value of the derivative with counterparty risk taken into consideration is less than the value of an identical derivative that is considered default free. The hedges that are suggested by the UCVA are options. In the case of a standard interest rate swap it would be a series of swaptions that would form the hedging portfolio, while with a credit default swap it would be CDS options, traded on the counterparty. The use of such intricate products shows how hedging CVA can be a complex task.
Bilateral CVA
The bilateral CVA extends the ideas underlying the unilateral CVA. The difference is that the investor and the counterparty both have a positive probability of defaulting. We denote their default times as τA and τC respectively, with the first default time τ =τA∧τC. The maturity of the contract is, as usual, T. We denote the default free value of the contract at time tby V(t, T) and the corresponding counterparty default prone contract byVb(t, T). All values are viewed from the investor’s point of view. The following cases need to be considered:
1. There is no default during the life of the contract, i.e., τ > T. All parties are able to honor their contractual obligations and thus we must have that
b
V(t, T) =V(t, T).
2. Suppose that the counterparty defaults, i.e., τ =τC ≤T. If the value of the contract to the counterparty is positive, i.e., negative towards the investor, then V(τ, T) < 0. The investor will pay the market value of the contract to the counterparty. If, however, the value to the investor is positive, which means thatV(τ, T)>0, then the investor will receive the recovery value equal toRCτV(τ, T) and immediately incurs a loss of (1−RCτ)V(τ, T).
3.5. QUANTIFYING CCR 40
3. Suppose that the investor defaults, i.e., τ =τA≤T. Then if the value of the contract to the counterparty is positive, i.e., negative towards the investor,then V(τ, T)<0. The counterparty will only receive RAτV(τA, T) and immediately incurs a loss of (1−RAτ)V(τA, T). If however, the value of the contract is positive to the investor, then the investor will receive V(τA, T) in full.
4. If at default the contract is valued at zero, then both parties walk away without having to pay anything.
Proposition 3.3(The General Formula for Bilateral Credit Value Adjustment). Let
b
V(t, T) be the value of a contingent claim between two counterparties, an investor (A) and a counterparty (C). Both counterparties are assumed to be default prone with their default times denoted by τA and τC, such that τA6=τC, for all events in Ω. Let τ = τA∧τC denote the first default time. Let V(t, T) be the value of an identical contingent claim assuming all parties to be default free. Both Vb(t, T) and V(t, T) are calculated from the point of view of A. Then we have that,
b V(T, t) =V(t, T)−E˜P[ Iτ≤TD(t, τ)((Iτ=τC(1−R C τC)V(τC, T) +) −Iτ≤T(Iτ=τA(1−R A τA)(−V(τA, T)) +))| F t]. (3.23)
Proof. Looking at the cases above it is clear that,
b
V(t, T) =E˜P[Iτ >TV(t, T) +Iτ≤TD(t, τ)(V(t, τ) +Iτ=τC(R
C t V(τ, T)+ +V(τC, T)−) +Iτ=τA(R A t V(τA, T) − +V(τ, T)+))| Ft]. (3.24) Looking at the case whereτ ≤T and also noting the following results:
V(τA, T)+=V(τA, T)−V(τA, T)− (3.25) V(τC, T)−=V(τC, T)−V(τC, T)+, (3.26) we can show that
V(τC, T)−+RCV(τC, T)+=V(τC, T)−V(τC, T)++RCV(τC, T)+
=V(τC, T)−(1−RC)V(τC, T)+ (3.27) and also
V(τA, T)++RAV(τA, T)−=V(τA, T)−V(τA, T)−+RAV(τA, T)− =V(τA, T)−(1−RA)V(τA, T)−
3.5. QUANTIFYING CCR 41
Since
V(t, T) =EP˜[Iτ >TV(t, T) +Iτ≤T(Iτ=τAD(t, τA)V(τA, T)
+Iτ=τCD(t, τC)V(τC, T) +V(t, τ))| Ft], (3.29)
we can substitute (3.27) and (3.28) into (3.25) and also using (3.29) we can group the terms that are obtained through the substitutions and obtain
b V(T, t) =V(t, T)−E˜P[Iτ≤TD(t, τ)((Iτ=τC(1−R C τC)V(τC, T) +) −Iτ=τA(1−RAτ A)(−V(τA, T)) +)| F t]. (3.30)
Re-arranging (3.30) above we arrive at b V(T, t) =V(t, T)−EP˜[ Iτ≤T(Iτ=τCD(t, τ)(1−R C τC)V(τC, T) +)| F t] +EP˜[Iτ≤T(Iτ=τAD(t, τ)(1−R A τA)(−V(τA, T)) +| F t], (3.31) which is normally written as [46, 16],
b
V(T, t) =V(t, T)−UCVA(t) + DVA(t). (3.32)
The important result from (3.31) above is that the UCVA is, as already noted, a series of options to enter into the underlying derivative (possibly with a new coun- terparty at default) and such options need to be financed by the UCVA, which explains its negative contribution. There is another term which contributes posi- tively, known as the debt valuation adjustment denoted by DVA above. The DVA indicates that the value of the derivative could actually be increased by the consid- eration of both counterparties default risk. This happens due to the fact that on occurrence of a default event, the defaulting entityicould be viewed as benefiting a value of (1−Rτi)V(τ, T), as it will not pay it. This further implies that the worsening credit quality of a counterparty actually boost profit margins for that counterparty, a result that is counterintuitive as it requires that the hedges be locked in prior to default. This leaves the paradoxical question: How does an entity hedge against its own default?
The bilateral CVA formula assumes that the value at default will be the risk free value of the derivative, whereas the ISDA recommends that the value be given by a substitute counterparty eager to be counterparty to the surviving party. This eager counterparty will in most instances consider the credit risk due to the surviving counterparty in its valuation which would mean that the value of the derivative at the particular default time will also need to be credit value adjusted. This inconsistency,
3.5. QUANTIFYING CCR 42
and the dangers of not considering the technical and legal details of a default event, is discussed in Brigo and Capponi [11]. The benefit of this approach is that it leaves the value of the contingent claim symmetrical and both parties are able to agree on its value. For this reason it has been used in accounting standards rather than for actual hedging purposes. We will, however, suggest ways to hedge this later in the dissertation.
A natural extension of the above formula is to consider relaxing the condition that no simultaneous defaults can happen5. Using the above reasoning, the following result can be derived.
Proposition 3.4 (General Bilateral CVA with Simultaneous Defaults Allowed).
Assume P˜(τA=τC)>0. The value of the contingent claim from A’s point of view,
assuming the possibility of default of both A and C is given by,
b V(T, t) =V(t, T)−EP˜[Iτ≤TD(t, τ)(I{τ=τC}∩{τC6=τA}((1−R C τC)V(τC, T) +) −I{τ=τA}∩{τA6=τC}((1−R A τA)(−V(τA, T)) +) + IτA=τC(V(τ, T) −RAτV(τ, T)−−RτCV(τ, T)+))| Ft]. (3.33)
Proof. Again we look at the cases outlined earlier in the bilateral section where we had assumed that the two entities can not default at the same time. On occurrence of a simultaneous default event at τ, ifV(τ, T)>0 then A will receiveRCτCV(τ, T) if however, V(τ, T)<0 then C will receive RA
τV(τ, T). Combining the above cases and the new one we obtain,
b V(t, T) =EP˜[Iτ >TV(t, T) +Iτ≤TD(t, τ)(V(t, τ) +I{τ=τC}∩{τC6=τA}(R C t V(τ, T)+ +V(τC, T)−) +I{τ=τA}∩{τA6=τC}(R A t V(τA, T)−+V(τ, T)+) +IτA=τC(R C τV(τ, T)++RAτV(τ, T) − ))| Ft]. (3.34)
Substituting (3.27) and (3.28) into (3.34) and re-arranging the resulting terms we obtain an expression in which we can use the fact that,
V(t, T) =EP˜[Iτ >TV(t, T) +Iτ≤T(I{τ=τA}∩{τA6=τC}D(t, τA)V(τA, T)
+I{τ=τC}∩{τC6=τA}D(t, τC)V(τC, T) +IτA=τCV(τ, T) +V(t, τ))| Ft],
(3.35)
5
3.5. QUANTIFYING CCR 43 to obtain b V(T, t) =V(t, T)−EP˜[Iτ≤TD(t, τ)(I{τ=τC}∩{τC6=τA}((1−R C τC)V(τC, T) +) −I{τ=τC}∩{τC6=τA}((1−R A τA)(−V(τA, T)) +) + Iτ=τA=τC(V(τ, T) −RAτV(τ, T)−−RτCV(τ, T)+))| Ft]. (3.36)
The result (3.36) can be written as b
V(T, t) =V(t, T)−UCVA(t) + DVA(t) + SDVA(t), (3.37)
where the new component SDVA(t) stands forsimultaneous default value adjustment.
Bilateral Collateralized CVA
The literature on credit value adjustments does not have much on the calculation of CVA in the presence of a credit support annex or more generally a collateral agreement. In two recent papers by Brigo et al [12] and Fujiiet al [41], the subject is considered. Fujiiet al derive solutions using Gateux derivatives for the case where collaterazation is imperfect and asymmetric6. Brigoet al derive a model free formula
for CVA taking into account the presence of a collateral agreement. In what follows we refer to their setting.
Let the collateral account be denoted byCt, which is a stochastic process adapted to the filtrationFt. Denote the exposures at a default timeτ byτ,A for the investor and τ,C for the counterparty. As already mentioned in the section on bilateral CVA, τ is the first default time. Denote the loss given default of the investor by LGDA= 1−RAt and the corresponding loss given default of the counterparty denoted LGDC = 1−RCt . Furthermore we generalize to the case where the collateral may be re-hypothecated7 by including a separate recovery rate for the collateral account for both investor and counterparty, these are denoted byRAt0 and RCt0 such that the corresponding losses given default of the collateral account are given by LGD0Aand LGD0C respectively. By following similar reasoning as in the case of the unilateral and bilateral CVA, it can be shown that the bilateral collateralized CVA (BC-CVA), when assuming that the exposures are symmetric in the sense that they are mid- market exposures, is given by
6
Only one of the counterparties is required to post collateral.
7The party that receives the collateral is allowed to use the collateral in whatever way they may
3.6. REMARK 44