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The Journal of Credit Risk (65–92) Volume 7/Number 1, Spring 2011

A generalized credit value adjustment

Mats KjaerQuantitative Analytics, Barclays Capital, 5 The North Colonnade,Canary Wharf, London E14 4BB, UK; email: [email protected]

In this paper we prove three results about the valuation of over-the-counter deriva-tive portfolios under counterparty risk. First, we derive a generalized credit valueadjustment (CVA) under the assumption that both parties can default. Second,we show how this CVA can be hedged in a simple bilateral credit model usingsingle-name credit default swaps and vanilla options on the underlying portfolio.Third, we prove the conditions under which adding a CVA to the counterpartydefault risk-free mark-to-market value of a portfolio is equivalent to discountingthe portfolio cashflows with the risky curve of the counterparty. The generalizedCVA derived in this paper contains the standard bilateral CVA as well as non-standard CVAs such as the so-called extinguisher and set-off CVAs as specialcases.

1 INTRODUCTION

The importance of correctly pricing and hedging counterparty risk in over-the-counter(OTC) transactions became very clear during the turbulent market events of the sum-mer and autumn of 2008. However, valuing securities under the assumption that theparties of the transaction could default has a longer history than that. Perhaps thefirst example is the pricing of corporate bonds, where the price discount comparedwith the equivalent government bond1 can be seen as a compensation for credit risk.2

The introduction of OTC contracts like swaps and options in the 1970s triggered thedevelopment of derivative valuation techniques for the case when one of the partiesmay default. An early paper on the topic by Johnson and Stulz (1987) shows how

The author thanks Vladimir Piterbarg, Christoph Burgard and Tom Hulme of the QuantitativeAnalytics Group at Barclays Capital for proposing improvements to the manuscript and for fruitfuldiscussions regarding the role of the funding curve in the context of counterparty risk. This paperrepresents the views of the author alone and not the views of Barclays Capital or Barclays Bank Plc.1 We assume that governments cannot default since they can print money in order to pay off theirdebts, which is the case at least for the US.2 Differences in liquidity and tax treatment between government and corporate bonds also contributeto the price difference. Corporate bonds may also require a higher amount of capital to be held againstthem.

65

66 M. Kjaer

the valuation of vanilla options changes when the option writer may default. Theyrefer to these options as vulnerable options, and use a Merton-type structural creditmodel. Jarrow and Turnbull (1995) extend this work by using reduced-form creditmodels and by considering vulnerable options on risky corporate zero-coupon bonds,where both the option writer and the zero-coupon bond issuer may default. Sorensenand Bollier (1994) discuss the valuation of vulnerable swaps, where both parties candefault, and show how the credit risk can be hedged using swaptions and credit defaultswaps (CDSs).

An OTC derivative is a legally binding contract between the two parties and, inorder to simplify the paperwork, the legal contract is usually derived from the 2002International Swaps and Derivatives Association (ISDA) Master Agreement.3 Thisdocument stipulates that the main mechanism used to reduce counterparty risk isnetting, that is, trades with positive value are offset against trades with negative valuewhen determining the claim submitted to the bankruptcy administrators. In additionto netting, the ISDA Master Agreement offers two more optional mechanisms forreducing counterparty risk. The first one is a so-called credit support annex, whichdetails the posting of collateral. The second mechanism is an additional terminationevent, which allows a party to terminate a portfolio if the credit rating of the otherparty falls below a predetermined threshold. This paper will not discuss credit supportannexes or additional termination events.

As discussed in Redon (2006), the Basel II agreement requires financial institu-tions to monitor their exposure given default to their counterparties. Moreover, newfair-value accounting rules require that the reported mark-to-market values of OTCportfolios include a credit value adjustment (CVA) to reflect counterparty risk.

Because of the presence of netting in the 2002 ISDA Master Agreement, the resultsof Johnson and Stulz (1987) and Jarrow and Turnbull (1995) cited above are notapplicable, since they only deal with single options (and hence extend to nonnettedportfolios). Papers, book chapters and books that develop methodologies for valuationof derivative portfolios under counterparty risk include, but are not limited to, Brigoand Mercurio (2007), Brigo and Chourdakis (2008), Li and Tang (2007), Gregory(2009), Alavian et al (2008), Redon (2006) and Pykhtin and Zhu (2007). The mainresult of the references cited above is a generic formula for an additive CVA thatholds regardless of the model used for the asset price and credit. More specifically,the total value of a portfolio including counterparty risk is given by the portfoliovalue, calculated assuming all transaction parties are default free (referred to as the“counterparty risk-free value” in this paper), minus a CVA. This result has influencedthe way that many financial institutions organize their trading and hedging. The nor-mal desks (fixed income, commodities, foreign exchange, etc) value and hedge the

3 URL: www.isda.com.

The Journal of Credit Risk Volume 7/Number 1, Spring 2011

A generalized credit value adjustment 67

counterparty-credit risk-free value, whereas pricing and hedging of the CVA part isperformed by a separate CVA desk.

Until the start of the credit crisis in the summer of 2007, the credit spreads of mostfinancial institutions were so small compared with those of the counterparties that mostmodels that were used to calculate the CVA assumed that the financial institution couldnot default. The resulting CVA is referred to as the unilateral CVA. As the spreadswidened, however, the need for a so-called bilateral CVA,4 which takes into accountthe fact that both parties can default, arose. A derivation of the standard bilateralCVA formula can be found in many of the references cited above (see, for exampleBrigo and Capponi (2008)). Gregory (2009) also derives this model but criticizes it forimplicitly assuming that the parties can short their own credit. In Section 5 we discussthis topic further and demonstrate how a party may short its own credit in practice aspart of a hedging strategy. This result is very similar to the one derived in Burgard andKjaer (2010), although that paper uses delta hedging and partial differential equationtechniques, whereas this paper uses a semistatic replication. Burgard and Kjaer (2010)also derive a bilateral CVA formula in the presence of positive funding spreads.

Other recent papers on counterparty risk include Blanchett-Scalliet and Patras(2008), Brigo and Capponi (2008), Brigo and Chourdakis (2008), Brigo andPallavicini (2007), Crépey et al (2009), Jarrow and Yu (2001), Leung and Kwok(2005) and Lipton and Sepp (2009). These papers primarily study counterparty riskfor CDSs.

The derivations of the unilateral and bilateral CVA formulas both assume thatthe cashflows at default to be the ones specified in the ISDA Master Agreement.Sometimes, however, parties agree other terms, such as “extinguishers”, where thesurviving party never has to pay the defaulting party, or “set-offs”, which allow thesurviving party to pay any obligations with bonds issued by the defaulting partyat face value. Extinguishers and set-off contracts may be unilateral or bilateral andresult in different CVA formulas compared with the regular unilateral or bilateralCVAs implied by the ISDA Master Agreement.

In this paper we extend the results given in the papers and books cited above inthree directions. First, we derive a generalized CVA formula, from which the unilateraland bilateral CVA, extinguisher CVA and set-off CVA follow as corollaries. The firstbenefit of this result is a clearer link between the contractual cashflows occurring givena default and the resulting CVA formula. The second benefit is that many differenttypes of CVA can be computed with one formula. Second, we extend the work ofSorensen and Bollier (1994) and propose a discrete hedging strategy for the CVA in asimple credit model where the two default times are independent of each other and all

4 The portion of the bilateral CVA reflecting an institution’s own default risk is sometimes referredto as debt value adjustment (DVA).

Research Paper www.thejournalofcreditrisk.com

68 M. Kjaer

the asset prices. The replicating portfolio contains options on the portfolio as well assingle-name CDSs on the two parties, and we prove that the hedge error goes to zeroas the hedging frequency goes to infinity. As part of the hedging section, we show thatthe strategy always generates enough cash at the right times to allow a party to buyand sell protection on itself and the counterparty. We believe that this hedge strategywill be helpful when risk-managing counterparty risk in practice. The third resultshows under what circumstances subtracting a CVA from the counterparty risk-freevalue is equivalent to replacing discounting the cashflows with the risky rate of thecounterparty. An implication of this result is that it is easy to compute the CVA ofhighly exotic derivatives, as long as the cashflows are always nonnegative.

This paper is organized as follows. In Section 2 we review the existing CVA modelsfrom the papers cited above and set some notation. Section 3 presents the modelassumptions behind the generalized CVA derived in Section 4. Having derived thegeneralized CVA, Section 5 shows how it can be hedged in a simple credit model.The equivalence between CVA and discounting with the risky rate of the counterpartyis discussed in Section 6 before giving some numerical examples in Section 7. Weconclude in Section 8. Lengthy proofs are found in Appendix A and Appendix B.

2 NOTATION AND EXISTING CVA MODELS

In this section we will introduce some notation and review the standard unilateral andbilateral CVA formulas derived in many of the papers, book chapters and books citedin Section 1: see, for example, Brigo and Capponi (2008) or Gregory (2009).

Let .˝;F ; P;Ft / be a filtered probability space satisfying the usual conditionsas defined in Protter (1990). Here Ft D �.Gt [Ht /, where Gt contains all marketinformation (including credit spreads) up to time t and Ht carries information aboutthe default times. The probability measure P is the equivalent martingale pricingmeasure corresponding to the numeraire process N.t/ > 0 and expectations withrespect to this measure are denoted by E.

The two parties are labeled B (“the issuer”) and C (“the counterparty”) and theirdefault times are denoted by �B and �C, respectively, and their recovery rates by RB

and RC, respectively. Moreover, we let the first default time be given by � � �B ^ �C

and assume that the last cashflow of the OTC portfolio occurs at time T > 0. Finally,we let V.t/ be the value of the portfolio to issuer B at time t > 0, assuming thatneither of the parties can default. Similarly, OV .t/ is the value of the portfolio if theparties can default.

All the derivations of the standard bilateral CVA cited in Section 1 implicitly orexplicitly make the following assumptions on the cashflows given default.

(1) The surviving party pays the defaulting party in full if owing money (ie, V.�/ <0 from the perspective of the surviving party).



(2) The defaulting party pays the surviving party as much as it can if it owes money(ie, V.�/ > 0 from the perspective of the surviving party). This is modeled aspaying a known recovery rate times the money it owes.5

Under the assumptions above, and using the notation V C.t/ � max.V .t/; 0/ andV C.t/ � min.V .t/; 0/, many of the papers cited in Section 1 (see, for example,Gregory (2009)) show that V.t/ and OV .t/ are linked through the relation:

V.t/ D OV .t/C .t/ (2.1)

where the regular bilateral CVA .t/ is given by:

.t/ D .1 �RB/E

�V �.�/1f�D�B;�6T g

N.t/

N.�/

ˇ̌̌ˇ Ft

�

C .1 �RC/E

�V C.�/1f�D�C;�6T g

N.t/

N.�/

ˇ̌̌ˇ Ft

�(2.2)

By also assuming that party B cannot default in Equation (2.2), we obtain the regularunilateral CVA �.t/ as:

�.t/ D .1 �RC/E

�V C.�/1f�C6T g

N.t/

N.�/

ˇ̌̌ˇ F t

�(2.3)

The CVAs in Equations (2.2) and (2.3) are entirely general in that they hold for anycombination of credit and market models. In order to actually compute the CVA,however, we need to specify models for the distribution of �B, �C and V.t/. Animportant feature of a model is its capability to capture wrong-way and right-wayrisk, a concept discussed in detail in Loosely speaking, it means that the nature of thedependence between the portfolio value and the default time is such that default occurswhen exposures are higher than expected (wrong-way risk) or lower than expected(right-way risk). The simplest and most common credit model, which we shall referto hereafter as the fully independent CVA model, independent CVA model, assumesthat the default times �B and �C are the first jump times times of two independent timeinhomogeneous Poisson processes with deterministic hazard rates �B and and �C,respectively. These default times are also assumed to be independent of the portfoliovalue V.t/, so it follows that this model cannot capture wrong-way and right-wayrisk. Under this assumption the bilateral CVA (2.2) becomes:

.t/ D .1 �RB/

Z T

t

E.t; u/fB.t; u/GC.t; u/ du

C .1 �RC/

Z T

t

E.t; u/fC.t; u/GB.t; u/ du (2.4)

5 This recovery rate is called the swap recovery rate and may be different from the bond recoveryrate of the corporate bonds used to build the credit curve.


70 M. Kjaer

where:

E.t; u/ D E

�V �.u/

N.t/

N.u/

ˇ̌̌ˇ Ft

�is the negative expected exposure

E.t; u/ D E

�V C.u/

N.t/

N.u/

ˇ̌̌ˇ Ft

�is the positive expected exposure

GB.t; u/ D exp

��

Z u

t

�B.s/ ds

�is the conditional probability of party B

GC.t; u/ D exp

��

Z u

t

�C.s/ ds

�is the conditional probability of party C

fB.t; u/ D �B.u/GB.t; u/ is the conditional density of �B

fC.t; u/ D �C.u/GC.t; u/ is the conditional density of �C

Similarly, the unilateral CVA (2.3) becomes:

�.t/ D .1 �RC/

Z T

t

E.t; u/fC.t; u/ du (2.5)

In practice, we may only have the positive and negative expected exposures availableat the discrete set of fixed6 exposure dates fTngNnD0withT0 D 0,Tm D t andTN D T .If this is the case, the integrals (2.4) and (2.5) are often discretized as:

N .t/ D .1 �RB/

N�1XnDm

Em;n

pm;n;nC1qm;nC1

C .1 �RC/

N�1XnDm

Em;npm;n;nC1 qm;nC1 (2.6)

�N .t/ D .1 �RC/

N�1XnDm

Em;npm;n;nC1 (2.7)

where Ei;n D E.Ti ; Tn/, Ei;n D

E.Ti ; Tn/ and:

pi;j;k D GB.Ti ; Tj / �GB.Ti ; Tk/

pi;j;k D GC.Ti ; Tj / �GC.Ti ; Tk/ q i;k D GB.Ti ; Tk/

qi;k D GC.Ti ; Tk/

6 Using fixed exposure dates will make the CVAs calculated on different dates more comparable.The hedging strategy in Section 5 is also easier to formulate within a fixed exposure date framework.



TABLE 1 Cashflow scenarios at default.

Scenario V.�/ > 0 V.�/ < 0

� D �B, � < T ˛V.�/ ˇV.�/

� D �C, � < T �V.�/ ıV .�/

� D �B D �C, � < T "V.�/ �V .�/

with Ti 6 Tj 6 Tk . The advantage of this discretization of the integral (2.4) oversome other scheme (eg, Simpson, Gauss–Legendere) is that the total probability massis preserved for any number of exposure dates N , and not only when N !1.

It is also possible to assume that �B and �C are joined by a copula but are still ofV.t/. Alternatively, each default time could be modeled as the first jump of a Coxprocess whose hazard rate is correlated with the hazard rate of the other party as wellas with V.t/, but the Poisson jumps are independent of each other and V.t/. If theportfolio contains CDSs, it may be desirable to create a stronger dependence betweenthe default times and the portfolio value. Brigo and Chourdakis (2008) propose usingstochastic hazard rates joined by a copula model in this case.

The remainder of the paper will deal with generalizing Equation (2.2) and we willnot assume any particular credit model, except for in Sections 5 and 7, where we willuse the fully independent CVA model described above.

3 ASSUMPTIONS ON CONTRACTUAL CASHFLOWS GIVENDEFAULT

Having fixed the notation in Section 2 we now state the generalized default-timecashflow scenarios in Assumption 3.1 below.

Assumption 3.1 (Cash flows given default) Let˛,ˇ,� , ı, ", and � be real numbersin Œ0; 1�. Then we assume that the cashflows from counterparty C to issuer B at thedefault time � D min.�B; �C/ are given by Table 1.

In Assumption 3.1 we allow for simultaneous defaults as in Gregory (2009). Inpractice, a simultaneous default reflects the fact that a bankruptcy may take months tosettle, during which it is possible for the surviving party to default as well. The defaultof the first party may even contribute to the default of the second party, a phenomenonknown as default contagion. When this happens, the party that defaulted first wouldreceive less than the full amount if it was owed money.

In Section 4 we will show that different values for the parameters ˛ to � result indifferent CVA types.


72 M. Kjaer

4 THE GENERALIZED CREDIT VALUE ADJUSTMENT FORMULA

We start this section by introducing some notation. The contracts of the portfolio areassumed to generate a sequence of random cashflows f.tm; Ym/gMmD1 satisfying thefollowing conditions:

(1) tm is an Ft stopping time;

(2) Ym 2 Ftm and Ym 2 L1.˝;F ; P /;

(3) tm and Ym do not depend on any credit information about parties B and C.

The last condition is necessary for the standard result OV .t/ D V.t/ � .t/ to hold,and excludes all Bermudan and American options as well as all physically settledEuropean options. The intuition behind this is that an optimal exercise decision whenmaximizing OV .t/ rather than V.t/ will depend on the credit of the two parties.

Next we define the random variable C.t; T / for 0 6 t 6 T as:

C.t; T / D

MXmD1

Ym

N.tm/1ft6tm<T g (4.1)

which can be interpreted as the sum of numeraire-adjusted cashflows in Œt; T /. Thisrandom variable satisfies C.t; t/ D 0 and:

C.t; T1/ D C.t; T0/C C.T0; T1/ (4.2)

for t 6 T0 6 T1, which ensures that there is no double counting of cashflows.Moreover, it follows thatC.t; T / 2 L1.˝;F ; P /, so, by standard derivatives pricingtheory, the counterparty-credit risk-free value V.t/ of the portfolio at some time t isgiven by:

V.t/ D N.t/EŒC.t; T / j Ft � (4.3)

With the different default scenarios in Assumption 3.1 in mind, we define the fourdisjoint default scenarios SA, SB, SC and SD as:

SA D f�B > T g \ f�C > T g

SB D f�B < �Cg \ f�B 6 T gSC D f�C < �Bg \ f�C 6 T gSD D f�C D �Bg \ f�C 6 T g

which implies that:1 D 1SA C 1SB C 1SC C 1SD (4.4)

In order to prove our main result we need the following lemma.



Lemma 4.1 LetC.t; T / denote the sum of numeraire-adjusted cashflows as definedin (4.1). Then, for k D B;C;D, it holds that:

EŒC.�k; T /1SkN.t/ j Ft � D E

�V.�k/1Sk

N.t/

N.�k/

ˇ̌̌ˇ Ft

�

where �D D � .

Proof See Appendix A. �

Proposition 4.2 (Generalized CVA) Let the cashflows from the portfolio underthe default scenarios SA–SD be the ones given in Assumption 3.1. Then the totalportfolio value OV .t/ at time t > 0 satisfies:

OV .t/ D V.t/ � .t/

where the generalized CVA .t/ is given by:

.t/ D E

�f.1 � ˛/1SB C .1 � �/1SC C .1 � "/1SDgV

C.�/N.t/

N.�/

ˇ̌̌ˇ Ft

�

C E

�f.1 � ˇ/1SB C .1 � ı/1SC C .1 � �/1SDgV

�.�/N.t/

N.�/

ˇ̌̌ˇ Ft

�

Proof See Appendix B. �

If the probability of a simultaneous default is zero, then .t/ simplifies to:

.t/ D E

�f.1 � ˛/1SB C .1 � �/1SCgV

C.�/N.t/

N.�/

ˇ̌̌ˇ Ft

�

C E

�f.1 � ˇ/1SB C .1 � ı/1SCgV

�.�/N.t/

N.�/

ˇ̌̌ˇ Ft

�(4.5)

If party B cannot default, we obtain the generalized unilateral CVA of Corollary4.3 below.

Corollary 4.3 (Generalized unilateral CVA) The generalized unilateral CVA�.t/is given by:

�.t/ D E

�f.1 � �/V C.�/C .1 � ı/V �.�/g1f�C6T g

N.t/

N.�/

ˇ̌̌ˇ Ft

�


74 M. Kjaer

Proof The assumption that party B cannot default is equivalent to 1SB D ;, 1SC D

1f�C6T g and 1SD D ;. Inserting these identities into Proposition 4.2 then gives theresult. �

We conclude this section by showing that some common variations of CVA, suchas the standard bilateral CVA, are special cases of the CVA that was presented inProposition 4.2.

Let RswapB , Rbond

B , RswapC R

swapC and Rbond

C denote the swap and bond recovery ratesfor B and C, respectively. Then the generalized CVA (4.5) can be used to constructthe following:

The standard bilateral CVA. This CVA corresponds to setting ˛ D 1, ˇ D RswapB ,

� D RswapC and ı D 1 in the CVA formula (4.5).

The set-off CVA. This CVA is obtained by setting ˛ D 1, ˇ D RswapB , � D Rswap

C andı D Rbond

C , which means that if party C defaults first and is owed money, then partyB can pay with the defaulting party’s bonds (counted at par) instead of cash.

The one-sided extinguisher CVA. This CVA is obtained by setting˛ D 1,ˇ D RswapB ,

� D RswapC and � D R

swapC and ı D 0, which means that if party C defaults first

and is owed money,

The two-sided extinguisher CVA. This CVA is obtained by setting˛ D 0,ˇ D RswapB ,

� D RswapC ı D 0, which means that the surviving party does not have to pay

anything if owing the defaulting party money.

From Proposition 4.2 it follows that if ˛ D ˇ, � D ı and " D �, then:

.t/ D E

�f.1 � ˛/1SB C .1 � �/1SC C .1 � "/1SDgV.�/

N.t/

N.�/

ˇ̌̌ˇ Ft

�

which implies that the CVA is linear in the portfolio in a particular sense. Morespecifically, if a trade is added to an existing portfolio, we would normally haveto recalculate the CVA of the entire new portfolio, which may be computationallydemanding. In this special case, however, we only need to add the CVA of the newtrade to the CVA of the old portfolio. The condition of the coefficients above is satisfiedby total extinguishers, where the portfolio is terminated on default and none of theparties gets any cashflows (˛ D ˇ D � D ı D " D � D 0).



We conclude this section by noting that, in the full independence model presentedin Section 2, the generalized CVA in Proposition 4.2 becomes:

.t/ D

Z T

t

f.1 � ˛/E.t; u/C .1 � ˇ/ E.t; u/gfB.t; u/GC.t; u/ du

C

Z T

t

f.1 � �/E.t; u/C .1 � ı/ E.t; u/gfC.t; u/GB.t; u/ du (4.6)

The discretized version N of (2.4) is given by:

N .t/ D

N�1XnDm

f.1 � ˛/Em;n C .1 � ˇ/ Em;ng

pm;n;nC1qm;nC1

C

N�1XnDm

f.1 � �/Em;n C .1 � ı/ Em;ngpm;n;nC1

qm;nC1 (4.7)

where again we assumed that t D Tm.Comparing the generalized CVA (4.6) with the regular bilateral CVA (2.4) (or the

corresponding formulas for N ) shows that, in the full independence model, we cancompute a generalized CVA with the same formula as the regular bilateral CVA (2.4),provided that we modify the positive and negative exposures and use zero recoveryrates. This approach is practical since very little extra work is needed to implementthe generalized CVA formula once the regular bilateral CVA is implemented.

5 HEDGING

In this section we show how we can hedge the generalized CVA (4.6) in the fullyindependent credit model described in Section 2 using single-name CDSs on partiesB and C, together with call and put options on the underlying portfolio. If these optionsdo not trade in the market, we assume that they can be synthetically replicated. Thestrategy requires some assumptions about hedging transactions that are consideredadmissible that may not hold in practice. These assumptions are not necessary tohedge the unilateral equivalent of (4.6).

All results in this section assume that issuer B hedges his/her counterparty riskbut, by symmetry, all results hold for counterparty C as well. Moreover, we assumethe existence of a default-risk-free third party A, which acts as counterparty for thehedging transactions. The latter part of this section is dedicated to showing that, givena set of assumptions about the rules for collateral, the hedge contains enough assets forissuer B to post as collateral to justify the selling of put options and CDS protectionon itself to party A. In practice, the issuer might want to replicate the short CDSsynthetically in order to satisfy the regulators.


76 M. Kjaer

To derive the hedging strategy we start by rearranging (4.7) as:

N .t/ D

N�1XnDm

f.1 � ˛/ pm;n;nC1qm;nC1 C .1 � �/pm;n;nC1

qm;nC1gEm;n

C

N�1XnDm

f.1 � ˇ/ pm;n;nC1qm;nC1 C .1 � ı/pm;n;nC1

qm;nC1g

Em;n

�

N�1XnDm

�m;nEm;n C

N�1XnDm

�m;n

Em;n (5.1)

This way of presenting the CVA N shows that it is equal to a portfolio of long callsand short puts on the underlying portfolio, which is a generalization of the work bySorensen and Bollier (1994), who show that the CVA of a swap is a default probabilityweighted portfolio of swaptions.

A default of a party triggers one of the cashflow scenarios specified in Assump-tion 3.1 due to (partial) repayment of the underlying portfolio value. However, thereis also a second cashflow coming from liquidating the remaining puts and calls of theCVA hedge, so in general, the CDS notional needed for full protection will not equalthe exposure. Before defining our hedging strategy ˘N .t/ in Definition 5.1 below,we define the forward starting CVA N .t; TmC1/ for t 2 .Tm; TmC1� as:

N .t; TmC1/ D

N�1XnDmC1

�mC1;nE.t; Tn/C

N�1XnDmC1

�mC1;n

E.t; Tn/ (5.2)

so, by definition N .Tm; Tm/ D N .Tm/. Next we define our hedging strategy inDefinition 5.1.

Definition 5.1 (The hedging strategy ˘N ) Let˘N .t/ denote the value of a port-folio containing the following assets for t 2 .Tm; TmC1�, 0 6 m < N :

(1) long calls on V with maturities fTngN�1nDmC1 and notionals f�mC1;ngN�1nDmC1;

(2) short puts on V with maturities fTngN�1nDmC1 and notionals f �mC1;ng

N�1nDmC1;

(3) short CDS protection on issuer B for the period .Tm; TmC1� with notional:

.1 � ˛/Em;m C .1 � ˇ/ Em;m � N .T

Cm ; TmC1/

(4) long CDS protection on counterparty C for the period .Tm; TmC1�with notional:

.1 � �/Em;m C .1 � ı/ Em;m � N .T

Cm ; TmC1/



It is important to note that the assumption of deterministic hazard rates means thatwe only have to hedge the default risk and that there is no need to hedge credit-spread movements. By Definition 5.1, it follows that ˘N .t/ D N .t/ only whent 2 fTng

N�1nD0 and we do not know if ˘N .t/ is self-financing or not, so the question

is whether ˘N .t/ is a good hedge of the CVA (4.6) or not. Proposition 5.2 gives ananswer to this question.

Proposition 5.2 The strategy ˘N .t/ introduced in Definition 5.1 satisfies:

limN!1

˘N .t/ D .t/

for all t 6 T . Moreover, ˘N is asymptotically self-financing as N !1.

Proof The first claim follows easily by the fact that if the positive and negativeexpected exposures E.t; u/ and

E.t; u/ defined in Section 2 are reasonably regular

functions of u, then N .t/ ! .t/ as N ! 1. The claim now follows from theearlier observation that ˘N .Tm/ D N .Tm/ for all fTmgN�1mD0.

To prove the asymptotic self-financing property, we first compute the revenue of theportfolio rebalancing transactions at timeTm. Negative revenues should be interpretedas costs and negative long CDS notionals should be interpreted as the notional onwhich protection has been sold.

(1) Exercise the call and put options expiring atTm. This generates a revenue I1.m/given by:

I1.m/ D �m;mEm;mC �m;m

Em;m

(2) Increase the remaining long-call-option notionals from �m;n to �mC1;n and theremaining short put option notionals from

�m;n to

�mC1;n. The notionals satisfy:

�m;n D qm;mC1 qm;mC1�mC1;n and

�m;n D qm;mC1

qm;mC1

�mC1;n

so the revenue I2.m/ from this transaction is given by:

I2.m/ D .qm;mC1 qm;mC1 � 1/ N .T

Cm ; TmC1/

(3) Buy CDS protection against the default of counterparty C with notional equalto the cashflow given default according to Assumption 3.1 minus the cashflowfrom liquidating the remaining calls and puts. This results in a revenue I3.m/given by:

I3.m/ D �f.1 � �/Em;m C .1 � ı/ Em;m � N .T

Cm ; TmC1/gpm;m;mC1


78 M. Kjaer

(4) Buy CDS protection against own default with notional equal to the cashflowgiven default according to Assumption 3.1 minus the cashflow from liquidatingthe remaining calls and puts. This results in a revenue I4.m/ given by:

I4.m/ D �f.1 � ˛/Em;m C .1 � ˇ/ Em;m � N .T

Cm ; TmC1/g

pm;m;mC1

Adding the cost and revenues above gives that the total profit I.m/ D I1.m/ C

I2.m/ � I3.m/ � I4.m/ is given by:

I.m/ D �pm;m;mC1 pm;m;mC1f..1 � ˛/C .1 � �//Em;m

C ..1 � ˇ/C .1 � ı// Em;mg

C pm;m;mC1 pm;m;mC1 N .T

Cm ; TmC1/

where we have used that qm;mC1 D 1 � pm;m;mC1 and qm;mC1 D 1�

pm;m;mC1.

Before proceeding, we note that the hedging error I.m/ over the period .Tm; TmC1�equals the total cashflow triggered if both parties default in the period .Tm; TmC1�times the probability of this happening. The fact that I.m/ ¤ 0means that the strategyis not self-financing, but we are going to show that the sum of the hedging errorsconverges to zero as N !1, since the probability of a joint default in .Tm; TmC1�is proportional to jTmC1 � Tmj2 when jTmC1 � Tmj is “small”.

The total hedging gain (or loss):

IN �

N�1XmD0

I.m/

is an FT -measurable random variable satisfying:

jIN j 6 max06m<N

pm;m;mC1

N�1XmD0

pm;m;mC1j N .TCm ; TmC1/j

C max06m<N

pm;m;mC1f.1 � ˛/C .1 � �/g

N�1XmD0

pm;m;mC1Em;m

C max06m<N

pm;m;mC1f.1 � ˇ/C .1 � ı/g

N�1XmD0

pm;m;mC1j

Em;mj

When N !1 we have that:

max06m<N

pm;m;mC1 ! 0

max06m<N

pm;m;mC1 ! 0



N�1XmD0

pm;m;mC1 N .TCm ; TmC1/!

Z T

t

�C.u/j .u/j du

N�1XmD0

pm;m;mC1Em;m !

Z T

t

�C.u/E.u; u/ du

N�1XmD0

pm;m;mC1j

Em;mj ! �

Z T

t

�B.u/ E.u; u/ du

where we have used that pm;m;mC1 � �C.Tm/.TmC1 � Tm/ when jTmC1 � Tmjis “small”. This in turn implies that jIN j ! 0 almost surely, since the integralsabove are finite almost surely because the underlying portfolio generates finitelymany integrable cashflows. �

In the above strategy issuer B sells put options and may also sell CDS protectionagainst the default of B and/or C. Any sold protection against the default of C doesnot require any collateral, since the third party A will only lose a payout if bothissuer B and counterparty C default in the same time interval .Tm; TmC1�. WhenT D jTmC1 � Tmj is small, then the probability of this event is proportional to.T /2, so we could argue that this probability is so small that it can be ignored.Repeating the arguments of the proof of Proposition 5.2 shows that the total cost ofthis protection converges to zero as N !1.

The strategy ˘N in Definition 5.1 requires issuer B to sell put options and CDSprotection on itself to the risk-free party A. Both these transactions only make senseif issuer B posts collateral with party A to eliminate any losses for A if issuer Bdefaults. In order for the issuer to be able to cover these collateral requirements usingthe assets available within the hedging strategy ˘N , we need to make the followingassumptions:

� The issuer B hedges the default-risk-free portfolio value V and is allowed topost this hedge as collateral with party A at its mark-to-market value V (eg,when V < 0) when hedging the CVA. Moreover, the hedge of V is funded atthe risk-free rate.

� All long call options can be posted as collateral with party A at their mark-to-market value.

� The issuer B can sell CDSs on itself to partyA provided that it posts the notional(given in Definition 5.1) as collateral with party A.


80 M. Kjaer

The amount X to be posted as collateral is obtained by adding the value of the putoptions and the short CDS notional on B, so:

X D �

N�1XnDMC1

�mC1;n

Em;n � .1 � ˛/Em;m � .1 � ˇ/

Em;m C N .T

Cm ; TmC1/

D

N�1XnDMC1

�mC1;nEm;n � .1 � ˛/Em;m � .1 � ˇ/ Em;m

Let Y denote the value of the assets that issuer B has available to post as collateral.Above, we assume that the long call options can be used by issuer B as collateral attheir mark-to-market value, so:

Y D

N�1XnDMC1

�mC1;nEm;n

which implies that:

Y �X D .1 � ˛/Em;m C .1 � ˇ/ Em;m

and it follows that if Em;m < 0, then Em;m D 0, which in turn implies that the call

options alone will not provide sufficient collateral. However, we assume above thatissuer B hedges the derivative portfolio and can post the hedge as collateral withparty A. Here

Em;m < 0 means that the account holding the replicated option value

contains the positive amount � Em;m. Of this amount, a proportion ˇ of this can be

used to pay counterparty C according to the cashflow Assumption 3.1, whereas theproportion .1�ˇ/ can be posted as collateral with the third party A. In reality, ˇ oftenrepresents a recovery rate, which is not known or specified in any ISDA agreement.Since issuer B has no contractual obligations to return any specified proportion of theportfolio value to C, he or she can post any proportion of the replicated portfolio thatis required as collateral with the third party A. Note that issuer B only posts collateralwith party A in order to be able to hedge the CVA, but does not post any collateralwith counterparty C as part of a credit support annex.

Two common criticisms in the literature and among practitioners regarding theusage of the bilateral CVA are that it is not possible for a party to sell CDSs on itselfand that there is limited value in acknowledging positive cashflows triggered by anown default. In this section we have shown how it is possible to sell put optionsand CDS protection by making these transactions fully collateralized, and that theself-financing hedging strategy provides enough assets to post as collateral. As aconsequence, any positive cashflows occurring because of a default of issuer B willbe seized by party A and will not benefit issuer B. Instead the benefit to B consists of



the ability to raise funds by selling put options and protection on itself without havingto pay any additional unilateral CVA charges to party A.

Of course, the assumptions above about collateral are rather idealistic, but since thebilateral CVA is commonly used, we believe that it is important to show the assump-tions that are needed to make it the initial cost of an admissible hedging strategy. Ifone believes that these assumptions are too idealistic, one might want to adjust thegeneralized CVA formula. For example, one could choose not to collateralize the soldput options, and instead pay an additional unilateral CVA to party A on these options.

The proof of Proposition 5.2 used the fact that �B and �C are assumed to be indepen-dent, which implies that the probability of a simultaneous default in a time intervalof length T is proportional to .T /2. One fortunate consequence of this is thatthe hedging can be done using single-name CDS only. The Marshall–Olkin copulaused in Gregory (2009), for example, has a probability of simultaneous default pro-portional to T , so any hedging strategy for this model would need one CDS thatpays out if B defaults conditional on C being alive, and another that pays if C defaultsconditional on B being alive. We leave it to future research to investigate the hedgingof CVA in these more general dependence models. This work is important since thereare currently strong systemic fears within the CDS market, making the independenceassumption rather unrealistic.

6 CREDIT VALUE ADJUSTMENT AND DISCOUNTING WITH THERISKY CURVE

The value OP .t; T / at time t of a zero coupon bond issued by party C is given by:

OP .t; T / D P.t; T /f1 � .1 �RC/P.�C 6 T /g

which is equal to P.t; T /C �.t/, where �.t/ is the regular unilateral CVA of Corol-lary 4.3. Alternatively, the risky zero coupon bond price can be seen as a unit cashflowdiscounted by the risky discounting curve of party C, also known as the funding curveof party C. This section will under what conditions discounting cashflows with therisky curve of the counterparty is equivalent to adding a CVA to the counterparty-risk-free value of the portfolio as described in Proposition 4.2. The answer is givenin Proposition 6.1 below.

Proposition 6.1 (Relation between CVA and discounting with the risky curve)Let the following three conditions be satisfied:

(1) only party C can default;

(2) all cashflows Ym are nonnegative;

(3) the default time �C is independent of the portfolio value V.t/ for all t > 0.


82 M. Kjaer

Then the credit risky value OV .t0/ is given by:

OV .t0/ D P.t0; T /f1 � .1 �RC/P.t0 < � 6 T /gEŒC.t; T / j Ft0 �

Proof By the nonnegativity of the cashflows Ym it is sufficient to prove the claimfor a single random cashflow Y occurring at time T .

Corollary 4.3, the corollary to Theorem 18 in Chapter 1 of Protter (1990) and thenonnegativity of cashflows yield:

OV .t0/ D V.t0/ � .1 � �/N.t/EŒVC.�/1f�6T g=N.�/ j Ft0 �

D V.t0/ � .1 � �/N.t/EŒV .�/1f�6T g=N.�/ j Ft0 �

D V.t0/ � .1 � �/N.t/EŒN.�/EŒY=N.T / j F� �1f�6T g=N.�/ j Ft0 �

The independence of � and Y yields that EŒY=N.T / j F� � D EŒY=N.T /�, so:

OV .t0/ D V.t0/ � .1 � �/N.t/EŒEŒY=N.T /�1f�6T g j Ft0 �

D V.t0/ � .1 � �/N.t/EŒY=N.T /�EŒ1f�6T g j Ft0 �

D V.t0/ � .1 � �/V .t0/P.t0 < � 6 T /D V.t0/f1 � .1 � �/P.t0 < � 6 T /gD N.t0/f1 � .1 � �/P.t0 < � 6 T /gEŒY=N.T / j Ft0 �

If we choose N.t/ D P.t; T / and � D RC, this becomes equivalent to discountingwith the risky bond. �

Negative cashflows should be discounted with the risk-free rate since they representthe exposure of party C to party B and party B is assumed default free. Burgard andKjaer (2010) derive a CVA formula for the case when party B is not default free andconsequently has to fund itself at a rate higher than the risk-free rate.

If the default time and the portfolio value are dependent, we have a situation referredto in Redon (2006) as wrong-way or right-way risk depending on the direction of thedependence. Wrongly applying Proposition 6.1 in this case and discounting withthe funding curve would ignore right-way and wrong-way risk and would henceoverestimate or underestimate the CVA.

One implication of this result is that it is easy to compute the CVA of highly exoticderivatives, as long as their cashflows are always nonnegative. This is because thereis no need to compute any positive and negative expected exposures, which typicallyrequires Monte Carlo simulation. In other words, it is easier to compute the CVAof an Asian foreign exchange barrier option, for example, than of a vanilla interestrate swap.



7 EXAMPLES

In this section we calculate the generalized unilateral and bilateral CVA for somevalues on the parameters ˛, ˇ, � , ı, " and �. For simplicity, we consider a portfoliocontaining the single commodity forward specified in Table 2 on the next page. Tosimplify matters further, we select a hybrid market and credit model with a constantand deterministic risk-free interest rate r > 0 so that the bank account B.t/ used asnumeraire evolves as:

dB.t/

B.t/D r dt

Moreover, we assume that the WTI forward price F.t; T / follows the dynamics:

dF.t; T /

F.t; T /D .�1e��.T�t/ C �2/ dW.t/

where �1 > 0, �2 > 0 and > 0 are deterministic constants and W.t/ is a Wienerprocess under the equivalent martingale measure induced by choosing B.t/ as thenumeraire. Finally, we let the default times �B and �C be the first jump times oftwo independent Poisson processes with hazard rates �B and �C, respectively. As inSection 2, we take �B and �C to be independent of the WTI forward price processF.t; T /.

The generalized CVA that we are going to compute is given in Equation (4.7), sowe need to calculate the positive and negative expected exposures. For the commodityforward in Table 2 on the next page, it is easy to show that E.t; u/ is a call option ona forward contract expiring at time u, which yields that:

E.t; u/ D e�r.T�t/fF.t; T /˚.d1/ �K˚.d2/g E.t; u/ D e�r.T�t/fF.t; T / �Kg �E.t; u/

where:

d1 Dlog.F.t; T /=K/C 1

2A2.t; u; T /

A.t; u; T /

d2 Dlog.F.t; T /=K/ � 1

2A2.t; u; T /

A.t; u; T /

A2.t; u; T / D�212.e�2�.T�u/ � e�2�.T�t//

C �22 .u � t /C 2�1�2

.e��.T�u/ � e��.T�t//

When computing (4.7) we use the market data parameters in Table 3 on the next pageand the resulting positive and negative expected exposures are shown in Figure 1 onthe next page and Figure 2 on page 85 for three different values of the initial WTIforward price.


84 M. Kjaer

TABLE 2 Term sheet for the commodity forward used in this section.

Commodity West Texas Intermediate (WTI) crude oilAmount 1 barrelMaturity T October 20, 2016Strike K US$85.00

TABLE 3 Market data parameters for the November 2016 WTI contract used in this section(all parameters are annualized).

Valuation date t August 26, 2009F.t; T / Varies per moneyness scenario

˛ 0.60�1 0.25�2 0.80

r 4%�B 2%�C 2%

FIGURE 1 Positive expected exposures.

25

20

15

10

5

0

F = US$70

F = US$85

F = US$100

July 6, 2009 July 6, 2011 July 5, 2013 July 5, 2015

US

$

Positive expected exposures for the commodity forward specified in Table 2 with market data from Table 3 and initialforward prices F.t; T / D US$70, US$85 and US$100.



FIGURE 2 Negative expected exposures.

F = US$70

F = US$85

F = US$100

US

$

−25

−20

−15

−10

−5

0

July 6, 2009 July 6, 2011 July 5, 2013 July 5, 2015

Negative expected exposures for the commodity forward specified in Table 2 on the facing page with market datafrom Table 3 on the facing page and initial forward prices F.t; T / D US$70, US$85 and US$100.

Next, we compute the generalized CVA (4.7) for some different combinations of ˛,ˇ, � and ı and three initial values of the WTI forward contract and display the resultsin Table 4 on the next page. The results show that the CVA varies widely dependingon the type of agreement that is in place between the two parties. In particular, thebilateral extinguisher A agreement is also highly sensitive to the initial forward price.

Finally, we compute the hedging strategy discussed in Section 5 for the forward inTable 2 on the facing page as a function of the initial WTI forward price in the caseof a regular bilateral CVA with RB D RC D 0. The results are shown in Figure 3 onthe next page.

The results in Figure 3 on the next page show that the issuer always sells protectionagainst its own default and buys protection against counterparty default. We also seethe importance of including the cashflow from liquidating the remaining CVA hedgewhen calculating the CDS notionals. If we were to naively let the CDS notionals on Band C equal the positive and negative exposures, respectively, then the CDS notionalon C would be zero for F 6 US$85 and the CDS notional on B would be zero forF > US$85. One consequence of this would be that if counterparty C defaults, thenthe value of the remaining call options would not be enough to close out the remainingshort put options.


86 M. Kjaer

TABLE 4 Generalized CVAs for the forward in Table 2 under three different market sce-narios.

CVA CVA CVACVA type ˛ ˇ � ı OTM ATM ITM

Bilateral regular 1.00 0.40 0.40 1.00 �0.80 0.00 0.80Unilateral regular N/A N/A 0.40 1.00 0.33 0.72 1.30Bilateral extinguisher A 0.00 0.00 0.00 0.00 �2.67 0.00 2.67Unilateral extinguisher A N/A N/A 0.00 0.00 �1.42 0.00 1.42Bilateral extinguisher B 1.00 0.40 0.00 0.00 �2.43 �0.66 0.93Bilateral extinguisher C 1.00 0.40 0.00 0.40 �1.70 �0.22 1.20Unilateral extinguisher C N/A N/A 0.00 0.40 �0.63 0.48 1.72Bilateral set-off 1.00 0.40 0.40 0.40 �1.90 0.66 0.40Unilateral set-off N/A N/A 0.40 0.40 �0.85 0.00 0.85

Here the CVA N .t/ given in (4.7) has been evaluated using weekly time steps. Abbreviations: OTM stands for“on-the-money”, ATM stands for “at-the-money” and ITM stands for “in-the-money”. Here “unilateral” is equivalentto setting �B D 0. The OTM scenario corresponds to F.t; T / D US$70, ATM to F.t; T / D US$85 and ITM toF.t; T / D US$100.

FIGURE 3 Long CDS notional.

Counterparty CIssuer B

Long

CD

S n

otio

nal (

US

$)

−11

−6

−1

4

9

68 73 78 83 88 93 98

WTI forward price (US$)

Initial CDS (seen from the perspective of issuer B) notional on issuer B and counterparty C required to hedge aregular bilateral CVA on the forward in Table 2 on page 84 when following the strategy given in Definition 5.1. Herewe have taken RB D RC D 0 and the market data is given in Table 3 on page 84.



8 CONCLUSION

In this paper we first proved a generalized CVA formula from which we then derivedsome commonly used special cases such as the regular unilateral CVA, the regularbilateral CVA, the set-off CVA and the one-sided and two-sided extinguisher CVAs.This result allows for the unified calculation of different types of CVAs. In a numericalexample we computed the CVA on an OTC WTI oil forward contract, and we saw thatthe CVA varies widely depending on the type of agreement that is in place betweenthe two parties.

Next we proposed a hedging strategy of this generalized CVA in a model withdeterministic credit spreads. The starting point was a discrete-time hedging strategy,which was shown to be exact and self-financing in the limit of continuous time hedg-ing. We also demonstrated how the strategy contains enough assets to use as collateralin order to enable the issuer to sell put options and CDSs against own default providedthat we make some rather idealistic assumptions about the rules for collateral. We alsocomputed the hedge for a bought commodity forward and saw how important it is tohave CDS protection in place against counterparty default, even if there is no currentexposure.

We continued by demonstrating that, under certain circumstances, counterparty riskmay be accounted for simply by discounting with the risky curve of the counterparty.This makes it a lot easier to compute the CVA of portfolios of exotic trades providedthat their cashflows are all nonnegative.

All the results so far have relied on the fact that the hedging of the counterpartyrisk-free value V can be funded at the risk-free interest rate. Positive funding spreadsand counterparty risk are just two sides of the same coin, so in practice, given that bothparties B and C can default, it is more likely that this funding occurs at a funding ratehigher than the risk-free rate. Given that the funding spreads of financial institutionsare currently historically high, adding funding costs to the model would be a naturalextension of this paper. Also, the addition of a collateral agreement between B and Cwould be another possible extension.

APPENDIX A: PROOF OF LEMMA 4.1

Without loss of generality, we prove Lemma 4.1 for k D B only. In the proof we willmake extensive use of the corollary to Theorem 18 in Chapter 1 of Protter (1990). Itstates that if Y 2 L1.˝;F ; P / and S and T are stopping times, then:

EŒEŒY j FS � j FT � D EŒEŒY j FT � j FS �

D EŒY j FS^T � (A.1)


88 M. Kjaer

with S ^ T � min.S; T / and FS , FT and FS^T being the corresponding stoppingtime � -algebras.

Applying (A.1) with S D t and T D � � �B^ �C, and recalling that almost surely,yields:

EŒN.t/C.�B; T /1SB j Ft � D EŒN.t/C.�B; T /1SB j Ft^� �

D EŒEŒN.t/C.�B; T /1SB j F� � j Ft � (A.2)

Again, using (A.1) in (A.2) with S D �B and T D �C gives:

EŒN.t/C.�B; T /1SB j Ft � D EŒEŒN.t/C.�B; T /1SB j F�B^�C � j Ft � (A.3)

D EŒEŒEŒN.t/C.�B; T /1SB j F�B � j F�C � j Ft � (A.4)

D N.t/EŒEŒEŒC.�B; T / j F�B �1SB j F�C � j Ft � (A.5)

because 1SB is known conditional on �B and �C.Next, we prove that the inner expectation of (A.5) satisfies:

EŒN.�B/C.�B; T / j F�B � D V.�B/ (A.6)

To do this, we first rewrite (4.3) as:

V.t/=N.t/ D EŒC.t; T / j Ft �

D

MXmD1

EŒYm=N.tm/ j Ft �1ft6tmg

D

MXmD1

Vm.t/1ft6tmg (A.7)

Moreover, if �mB � �B ^ tm, then:

EŒC.�B; T / j Ft � DMXmD1

EŒYm=N.tm/ j F�B �1ft6tmg

D

MXmD1

EŒYm=N.tm/ j F�mB �1f�B6tmg

D

MXmD1

EŒVm.tm/ j F�mB �1f�B6tmg (A.8)

by (A.7) and the fact that 1f�B6tmg D 0 if �mB ¤ �B. �mB 6 tm and Vm.t/ is a uniformlyintegrable martingale for t 6 tm, Doob’s optional sampling theorem (see Chapter 1



of Protter (1990) or Chapter 4 in Durrett (1995)) gives that:

EŒVm.tm/ j F�mB � D Vm.�mB / (A.9)

so combining (A.9) and (A.8) allows us to write:

EŒC.�B; T / j Ft � DMXmD1

Vm.�mB /1f�B6tmg

D

MXmD1

Vm.�B/1f�B6tmg

D V.�B/=N.�B/

Having proved (A.6), we continue by inserting it into (A.5), which yields:

EŒN.t/C.�B; T /1SB j Ft �

D EŒN.t/EŒEŒV .�B/=N.�B/ j F�B �1SB j F�C � j Ft � (A.10)

D EŒN.t/EŒEŒV .�B/=N.�B/1SB j F�B � j F�C � j Ft � (A.11)

D EŒN.t/EŒV .�B/=N.�B/1SB j F� � j Ft � (A.12)

D EŒN.t/V .�B/=N.�B/1SB j F�^t � (A.13)

D EŒN.t/V .�B/=N.�B/1SB j Ft � (A.14)

Here (A.12) and (A.13) follow by using (A.1) with S D �B, T D �C and S D � ,T D t , respectively. Finally, (A.14) is a consequence of the fact that � > t almostsurely.

APPENDIX B: PROOF OF PROPOSITION 4.2

By standard arbitrage-free pricing theory, OV .t/ is given by valuing the cashflowsgiven the default scenarios SA, SB, SC and SD, so:

OV .t/ D EŒC.t; T /N.t/1SA j Ft �

C E

��C.t; �B/N.t/C ˛V

C.�B/N.t/

N.�B/C ˇV �.�B/

N.t/

N.�B/

�1SB

ˇ̌̌ˇ Ft

�

C E

��C.t; �C/N.t/C �V

C.�C/N.t/

N.�C/C ıV �.�C/

N.t/

N.�C/

�1SC

ˇ̌̌ˇ Ft

�

C E

��C.t; �/N.t/C "V C.�/

N.t/

N.�/C �V �.�/

N.t/

N.�/

�1SD

ˇ̌̌ˇ Ft

�(B.1)


90 M. Kjaer

By (4.4) it follows that:

1SA D 1 � 1SB � 1SC � 1SD

so after substituting this into (B.1) we obtain:

OV .t/

D EŒC.t; T /N.t/ j Ft �

CE

��.C.t; �B/�C.t; T //N.t/C˛V

C.�B/N.t/

N.�B/CˇV �.�B/

N.t/

N.�B/

�1SB

ˇ̌̌ˇFt

�

CE

��.C.t; �C/�C.t; T /N.t/C�V

C.�C/N.t/

N.�C/C ıV �.�C/

N.t/

N.�C/

�1SC

ˇ̌̌ˇFt

�

CE

��C.t; �/�C.t; T /N.t/C "V C.�/

N.t/

N.�/C �V �.�/

N.t/

N.�/

�1SD

ˇ̌̌ˇFt

�(B.2)

Invoking the relation:

C.t; T / D C.t; �/C C.�; T /

then yields:

OV .t/ D EŒC.t; T /N.t/ j Ft �

� E

��C.�B; T /N.t/ � ˛V

C.�B/N.t/

N.�B/� ˇV �.�B/

N.t/

N.�B/

�1SB

ˇ̌̌ˇ Ft

�

� E

��C.�C; T /N.t/ � �V

C.�C/N.t/

N.�C/� ıV �.�C/

N.t/

N.�C/

�1SC

ˇ̌̌ˇ Ft

�

� E

��C.�; T /N.t/ � "V C.�/

N.t/

N.�/� �V �.�/

N.t/

N.�/

�1SD

ˇ̌̌ˇ Ft

�(B.3)

Applying (4.3) to the first line of (B.3) and using Lemma 4.1 in the three last lines of(B.3) then results in the equality:

OV .t/ D V.t/ � E

�fV.�B/ � ˛V

C.�B/ � ˇV�.�B/g

N.t/

N.�B/1SB

ˇ̌̌ˇ Ft

�

� E

�fV.�C/ � �V

C.�C/ � ıV�.�C/g

N.t/

N.�C/1SC

ˇ̌̌ˇ Ft

�

� E

�fV�D � "V

C.�/ � �V �.�/gN.t/

N.�/1SD

ˇ̌̌ˇ Ft

�(B.4)

� V.t/ � .t/ (B.5)



Since V.t/ D V.t/C C V.t/� and �k D � on Sk for k D A; B; C; D, .t/ mayfinally be rewritten as:

.t/ D E

�f.1 � ˛/V C.�/C .1 � ˇ/V �.�/g

N.t/

N.�/1SB

ˇ̌̌ˇ Ft

�

C E

�f.1 � �/V C.�/C .1 � ı/V �.�/g

N.t/

N.�/1SC

ˇ̌̌ˇ Ft

�

C E

�f.1 � "/V C.�/C .1 � �/V �.�/g

N.t/

N.�/1SD

ˇ̌̌ˇ Ft

�(B.6)

from which the desired result follows by rearranging the terms.

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