Dangers of Bilateral Counterparty Risk: the fundamental impact of closeout conventions
aa r X i v : . [ q -f i n . P R ] N ov Dangers of Bilateral Counterparty Risk: the fundamentalimpact of closeout conventions
Damiano Brigo
Dept. of MathematicsKing’s College, London
Massimo Morini ∗ Banca IMI, Intesa-SanPaoloand Bocconi University, Milan
First version March 13, 2010. This version November 1, 2018.
Abstract
We analyze the practical consequences of the bilateral counterparty risk adjustment.We point out that past literature assumes that, at the moment of the first default, arisk-free closeout amount will be used. We argue that the legal (ISDA) documentationsuggests in many points that a substitution closeout should be used. This would take intoaccount the risk of default of the survived party. We show how the bilateral counterpartyrisk adjustment changes strongly when a substitution closeout amount is considered. Wemodel the two extreme cases of default independence and co-monotonicity, which highlightpros and cons of both risk free and substitution closeout formulations, and allow us tointerpret the outcomes as dramatic consequences on default contagion. Finally, we analyzethe situation when collateral is present.
AMS Classification Codes : 62H20, 91B70
JEL Classification Codes : G12, G13 keywords : Bilateral Counterparty Risk, Credit Valuation Adjustment, Debit Valuation Ad-justment, Closeout, Default Contagion, Bond Pricing, Default Correlation, Co-monotonic De-faults, Collateral Modeling.
In this paper we analyze the practical consequences of the bilateral counterparty risk adjust-ment. We point out that past literature assumes that, at the moment of default, a risk-freecloseout amount will be used. The closeout amount is the net present value of the residualdeal which is computed when one party defaults, and that is used for default settlement. Arisk-free closeout amount is a net present value that assumes that the surviving counterpartyis default-free.We argue here that the legal (ISDA) documentation on the settlement of a default doesnot confirm this assumption. Documentation suggests in many points that a ‘substitution ∗ Corresponding author. This paper expresses the views of its authors and does not represent the institutionswhere the authors are working or have worked in the past. Such institutions, including Banca IMI, are notresponsible for any use which may be made of this paper contents. We thank Giorgio Facchinetti, MarcoBianchetti, Luigi Cefis, Martin Baxter, Andrea Bugin, Vladimir Chorny, Josh Danziger, Igor Smirnov and otherparticipants to the ICBI 2010 Global Derivatives and Risk Management Conference for helpful discussion. Theremaining errors are our own. The authors would also like to give special thanks to Andrea Pallavicini andAndrea Prampolini for thoroughly and deeply discussing the research issues considered in this paper. . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk Risk-free vs substitution closeout: Practical consequences
We analyze the practical consequences of the two different ways of computing the closeoutamount. We show on a simple derivative that the standard formula for bilateral counterpartyrisk adjustment, assuming risk-free closeout, is not consistent with the market practice onsimple uncollateralized defaultable claims such as bonds or loans or options. In fact it causesthe initial price of a deal to depend crucially on the risk of default of one party that has nofuture obligations in the deal. In case of a simple ‘bond’ or ‘option’ deal, it would make theprice to depend on the risk of default of the ‘lender’, namely the party which has not futurepayments to make in the deal, such as the bond buyer or the option buyer. This feature is notconsistent with current market practices and is counterintuitive. We show that this featurecan be avoided by using the formula with substitution closeout that we introduce in this paper.In this sense, the substitution closeout formula that we introduce appears as the right choiceto achieve consistency with market practice for uncollateralized deals.However, the substitution closeout formula is not immune to problems. The main oneregards the consistency with collateral computations for collateralized deals. The collateral iscalculated computing a risk-free net present value. If the closeout is not risk-free how couldthe collateral amount and the closeout amount match at default in such a way that no lossesare suffered by any party, as one would expect for collateralized deals? In the following wepropose an explanation of how this can be the case even for a substitution closeout, but theissue would appear to be trivially solved under a risk-free closeout.We also observe that, if we consider the above simple uncollateralized ’bond’ deals underscenarios of perfect default dependency between the two counterparties, the risk-free closeoutCVA is either the same as the substitution closeout CVA (this happens when the ’lender’ haslower credit spreads, so that he never defaults first and we are in practice in a unilateral CVA)or it has a behaviour that could be considered not to be illogic, contrary to the substitutioncloseout. This second case happens when the ’lender’ has higher credit spreads; in this casethe risk-free closeout says that any bond should be treated as risk-free; this could be justifiedby the fact that the lender always defaults first so he will never be impacted by the defaultof the borrower. The situation changes when we consider counterparties with independentdefault risk, as we detail below.
Forms of contagion implied by the two closeouts in different situations
Another important point in comparing the two closeout formulations is understanding whathappens upon default of one counterparty. In particular we focus on the case that shows themost striking differences with the unilateral case: the default of the counterparty with nofuture payments in the deal, the ‘lender’ in a synthetic loan transaction. Let us start with aquite relevant problem of the risk free closeout, having possibly destabilizing consequences.When one of our counterparties defaults, we not only have losses if we are creditors of the . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk
Structure of the paper
In Section 2 we present the formulas introduced by previous literature for counterparty riskadjustments. In Section 3 we analyze these formulas to clarify what they assume about thecomputation of the closeout amount, and we propose a closeout amount different from therisk-free one, namely the substitution closeout. Then in Sections 4 and 5 we analyze ISDAdocumentation. In Section 6 we perform the quantitative analysis of the problem. We firstapply the alternative formulas to a very simple payoff, revealing that only the substitutioncloseout appears consistent with market practice for bonds and loans, while the risk-free close-out introduces even at initial time 0 a dependence on the risk of default of the party with nofuture obligation. The pattern of this unusual dependence is detailed numerically. Then wequantify the losses or the benefit, respectively, that a borrower would suffer in case of defaultof the lender when the substitution or risk-free closeouts are applied in the respective cases ofindependence and total default dependence. We have decided to consider only these limit cases- perfect dependence and independence - because they allow us to make claims which are asmodel-independent as possible, avoiding the use of ’black-box’ copula models and associatedconcepts such as ’default correlation’.In Section 7 we show that when the substitution closeout is used the combination of adefaultable deal and a collateral agreement may lead naturally to a default-free deal, as itshould be. Finally in the Appendix we show a number of mathematical properties, such assymmetry, of the formula with substitution closeout that we introduce in this work. . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk We consider two parties in a derivative transaction: A (investor) and B (counterparty). Wecall τ X , R X and L X = 1 − R X , respectively, the default time, the recovery and the loss givendefault of party X , with X ∈ { A, B } . The risk-free discount factor is D ( t, T ) = e − R Tt r ( s ) ds ,where r ( t ) is the risk-free short-rate. We define Π A ( t, T ) to be the discounted cashflows ofthe derivative from t to T seen from the point of view of A , with Π B ( t, T ) = − Π A ( t, T ). Thenet present value of the derivative at t is V A ( t ) := E t [Π A ( t, T )] , where E t indicates the risk-neutral expectation based on market information up to time t .Notice that this, in general, includes default monitoring, i.e. the filtration at time t includes { τ X > t } . We denote by Q t the risk neutral probability measure conditional on the sameinformation at time t .The subscript A indicates that this value is seen from the point of view of A , the superscript0 indicates that we are considering both parties as default-free. Obviously, V B ( t ) = − V A ( t ).The early literature on counterparty risk adjustment, see for example Brigo and Masetti(2005), introduced ‘unilateral risk of default’. Here only the default of counterparty B isconsidered, while the investor A is treated as default free. Under this assumption, the adjustednet present value to A is V BA ( t ) = E t (cid:8) { τ B >T } Π A ( t, T ) (cid:9) +(1) + E t n { τ B ≤ T } h Π A (cid:0) t, τ B (cid:1) + D (cid:0) t, τ B (cid:1) (cid:16) R B (cid:0) V A (cid:0) τ B (cid:1)(cid:1) + − (cid:0) − V A (cid:0) τ B (cid:1)(cid:1) + (cid:17)io = V A ( t ) − E t h L B { τ B ≤ T } D (cid:0) t, τ B (cid:1) (cid:0) V A (cid:0) τ B (cid:1)(cid:1) + i =: V A ( t ) − CVA A ( t ) . The superscript B indicates that this value allows for the risk of default of B . Notice thatwe always assume both parties to be alive at t . The approach is easily extended to the casewhen B is treated as default-free, but the default of investor A is taken into account. Now theadjusted net present value to A is V AA ( t ) = E t (cid:8) { τ A >T } Π A ( t, T ) (cid:9) +(2) + E t n { τ A ≤ T } h Π A (cid:0) t, τ A (cid:1) + D (cid:0) t, τ A (cid:1) (cid:16)(cid:0) V A (cid:0) τ A (cid:1)(cid:1) + − R A (cid:0) − V A (cid:0) τ A (cid:1)(cid:1) + (cid:17)io = V A ( t ) + E t h L A { τ A ≤ T } D (cid:0) t, τ A (cid:1) · (cid:0) − V A (cid:0) τ A (cid:1)(cid:1) + i =: V A ( t ) + DVA A ( t ) . The extension to the most realistic case when both A and B can default is less trivial. Thisis called ’bilateral risk of default’ and it is introduced for interest rate swaps in Bielecki andRutkowski (2001), Picoult (2005) (where a simplified and approximated use of the indicators isadopted), Gregory (2009), Brigo and Capponi (2008), and Brigo Pallavicini and Papatheodorou(2009). In these previous works the net present value adjusted by the default probabilities ofboth parties is given by V A ( t ) = E t { Π A ( t, T ) } (3) + E t n A h Π A (cid:0) t, τ A (cid:1) + D (cid:0) t, τ A (cid:1) (cid:16)(cid:0) V A (cid:0) τ A (cid:1)(cid:1) + − R A (cid:0) − V A (cid:0) τ A (cid:1)(cid:1) + (cid:17)io + E t n B h Π A (cid:0) t, τ B (cid:1) + D (cid:0) t, τ B (cid:1) (cid:16) R B (cid:0) V A (cid:0) τ B (cid:1)(cid:1) + − (cid:0) − V A (cid:0) τ B (cid:1)(cid:1) + (cid:17)io , . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk = 1 { T < min( τ A ,τ B ) } A = 1 { τ A ≤ min( T,τ B ) } B = 1 − A − = 1 { τ B <τ A } { τ B ≤ T } . Notice that V B ( t ) = − V A ( t ) ,thus this formula enjoys the symmetry property that one would expect. In the next sectionwe analyze these formulas to understand what they implicitly assume about the settlement ofa default event, and discuss the realism of such assumptions, and their practical consequences. Knowing which default happens first is of fundamental importance for determining the actualpayout. Thus it is convenient to rewrite formula (3) to make the order of the default eventsexplicit.We define 1 to be the first entity to default, and 2 to be the second one so that(4) τ = min (cid:0) τ A , τ B (cid:1) , τ = max (cid:0) τ A , τ B (cid:1) . With these definitions the pricing formula (3) simplifies to V A ( t ) = E t { Π A ( t, T ) } (5) + E t n (1 B − A ) h Π (cid:0) t, τ (cid:1) + D (cid:0) t, τ (cid:1) (cid:16) R (cid:0) V (cid:0) τ (cid:1)(cid:1) + − (cid:0) − V (cid:0) τ (cid:1)(cid:1) + (cid:17)io . Let us compare this bilateral pricing formula with the price V BA ( t ) given in (1). There thedistinction between τ and τ is meaningless, since only the counterparty B can default, sothat τ = τ A = + ∞ ,τ = τ B =: τ Thus, V BA ( t ) = E t (cid:8) { τ>T } Π A ( t, T ) (cid:9) (6) + E t n { τ ≤ T } h Π A ( t, τ ) + D ( t, τ ) (cid:16) R (cid:0) V A ( τ ) (cid:1) + − (cid:0) − V A ( τ ) (cid:1) + (cid:17)io . Now we introduce the main theme of this paper. All formulas for counterparty risk adjustmentare based on precise assumptions on what happens when there is a default event. Let us startfrom the unilateral case (6). When the default of the counterparty happens before maturity(look at the part of the formula following the indicator of event { τ ≤ T } ), the total payout ismade of two parts: the cashflows received before default, Π A ( t, τ ), and the present value ofthe payout at default time τ . At τ the residual deal is marked-to-market. The mark-to-marketof the residual deal at an early termination time is called closeout amount in the jargon ofISDA documentation. Here it is given by V A ( τ ) = − V B ( τ ) , . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk B , which is the defaultingparty, and negative to A , which has not defaulted, A will pay this amount entirely to theliquidators of the counterparty. If the closeout amount is instead positive to A and negative to B , the liquidators of the latter will pay to A only a recovery fraction of the closeout amount.These provisions lead to the payout at default of (6), given by(7) (cid:16) R (cid:0) V A ( τ ) (cid:1) − (cid:0) − V A ( τ ) (cid:1) + (cid:17) . Notice that, as indicated by the superscript 0, here the closeout amount V A ( τ ) is computedtreating the residual deal as a default-free deal. The reason for that is obvious. There are twoparties A and B , and party A is supposed default-free so it will never default, while party B has already defaulted and this is taken into account by the fact that, in case V B ( τ ) < B will pay only a recovery fraction of the default-free closeout amount. This default-free closeoutis also a substitution closeout, in the sense that, if A wanted to substitute the defaulted dealwith another one where the counterparty is default-free, the counterparty would ask A to pay V A ( τ ), a risk-free closeout since both parties are risk-free.Now let us look at the pricing formula (5) for bilateral risk of default. The payout atdefault is now given by (1 B − A ) h R (cid:0) V (cid:0) τ (cid:1)(cid:1) + − (cid:0) − V (cid:0) τ (cid:1)(cid:1) + i Here both A and B can default, and what matters is who defaults first. If the counterparty B defaults first, then 1 B = 1, 1 A = 0, τ = τ B and we have the same payout as in the unilateralcase of (7). If the investor A defaults first, the payout is reversed: when the closeout amountis positive to the defaulted investor, this amount is received fully, while if it is negative only arecovery fraction will be paid to the counterparty, leading to h(cid:0) V A (cid:0) τ (cid:1)(cid:1) + − R (cid:0) − V A (cid:0) τ (cid:1)(cid:1) + i . Notice that, like in the unilateral case, also with bilateral risk of default the closeout amount is computed treating the residual deal as default-free (8) V (cid:0) τ (cid:1) = − V (cid:0) τ (cid:1) . We are again excluding the possibility of default of either party. Is this assumption as obviouslyjustified here as it was in the unilateral case? Not quite. Only the assumption of ignoringthe risk of default of 1 is justified obviously. In fact 1 has defaulted, and this is accounted forcorrectly by computing a closeout amount where there is no possibility of another default of1, and then, in case this amount is negative to 1, assuming that 1 will pay only a recoveryfraction of it. But the other party 2 has not defaulted, and now it is not true that it will neverdefault in the future. There is a non-negligible probability that it defaults before the maturity T of the residual deal. Thus it is unclear why the mark-to-market of a residual deal whereone of the two parties has not yet defaulted and can default in the future should be treatedas default free. Here the risk-free closeout amount (8) is not a substitution closeout. In factif the survived party 2 wanted to substitute the defaulted deal with another one where themarket counterparty is default free, the counterparty would ask 2 to pay not the opposite of(8) but(9) V (cid:0) τ (cid:1) = − V (cid:0) τ (cid:1) , because the market counterparty cannot ignore the default risk of party 2 from τ to thematurity T of the residual deal. Thus the amount (9) will be called in the following substitutioncloseout amount . It is given by (1) when 1 = B and by (2) when 1 = A . . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk V A ( t ) = E t { Π A ( t, T ) } (10) + E t n A h Π A (cid:0) t, τ A (cid:1) + D (cid:0) t, τ A (cid:1) (cid:16)(cid:0) V BA (cid:0) τ A (cid:1)(cid:1) + − R A (cid:0) − V BA (cid:0) τ A (cid:1)(cid:1) + (cid:17)io + E t n B h Π A (cid:0) t, τ B (cid:1) + D (cid:0) t, τ B (cid:1) (cid:16) R B (cid:0) V AA (cid:0) τ B (cid:1)(cid:1) + − (cid:0) − V AA (cid:0) τ B (cid:1)(cid:1) + (cid:17)io . In the rest of the paper we analyze this formula under two points of view. First, we want tounderstand if it is more appropriate than formula (3) used in the previous literature. For thisanalysis we consider1) the ISDA documentation on derivatives, to understand the legal prescriptions and thefinancial rationale that should be followed in computing the closeout amount2) which one between the substitution closeout amount assumed by (10) and the risk-freeone assumed by (3) is simpler to be applied in case of a real default.3) the financial effects of using (10) rather than (3), to see which one minimizes the defaultcontagion in case of default, which one is more consistent with market standards on consoli-dated financial products and which one fits better with market practices to minimize defaultrisk such as CSA collateral agreements.One observation is in order about this plan of analysis . The reader may think that thefirst step to take is to ask experienced practitioners which approach is applied in practice forthe computation of closeout. We have done this (and we thank for that in particular the par-ticipants to Global Derivatives 2010 in Paris) and surprisingly enough we have found differentopinions. This variety of opinions can be related to the fact that the ISDA documentationhad given a (relatively) open definition of closeout amount, and more importantly to the factthat full awareness of the importance of CVA and DVA has arisen just after
Lehman’s default,the last important default that may work as a benchmark. At that time “Libor discounting”was mainly used for closeout, and this is somehow in-between a risk-free discounting and adiscounting taking full consideration of the risk of default of the remaining party.Then, in the appendix, we analyze the new formula (10) under a mathematical point ofview. We first show that it enjoys the symmetry property ˆ V A ( t ) = − ˆ V B ( t ) as well. Then weshow that is equivalent toˆ V A ( t ) = E t { Π A ( t, T ) } + E t n h D (cid:0) t, τ A (cid:1) (cid:16) − E τ A h L B { τ B ≤ T } D (cid:0) τ A , τ B (cid:1) · (cid:0) V A (cid:0) τ B (cid:1)(cid:1) + i + (cid:0) − R A (cid:1) (cid:16) − V A (cid:0) τ A (cid:1) + E τ A h L B { τ B ≤ T } D (cid:0) τ A , τ B (cid:1) · (cid:0) V A (cid:0) τ B (cid:1)(cid:1) + i(cid:17) + (cid:19)(cid:21)(cid:27) + E t n h D (cid:0) t, τ B (cid:1) (cid:16) E τ B h L A { τ A ≤ T } D (cid:0) τ B , τ A (cid:1) · (cid:0) − V A (cid:0) τ A (cid:1)(cid:1) + i − (cid:0) − R B (cid:1) (cid:16) V A (cid:0) τ B (cid:1) + E τ B h L A { τ A ≤ T } D (cid:0) τ B , τ A (cid:1) · (cid:0) − V A (cid:0) τ A (cid:1)(cid:1) + i(cid:17) + (cid:19)(cid:21)(cid:27) , and that, using the definitions in (4) and recalling that V B ( t ) = − V A ( t ), it can be simplifiedin ˆ V A ( t ) = E t { Π A ( t, T ) } + E t n (1 B − A ) h Π (cid:0) t, τ (cid:1) + D (cid:0) t, τ (cid:1) (cid:16) R (cid:0) V (cid:0) τ (cid:1)(cid:1) + − (cid:0) V (cid:0) τ (cid:1)(cid:1) + (cid:17)io . . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk The document that should set a standard for the computation of the close-out amount in caseof a default event is the ISDA (2009) Close-out Amount Protocol. Nowhere in this documentone finds a precise formula for the computation of the close-out amount, however one can findthere practical principles that can shed some light on the issue we are considering. We readat page 13 that ”If the Early Termination Date results from an Event of Default”, this earlytermination will be settled by the transfer of ”the Close-out Amount or Close-out Amounts(whether positive or negative) determined by the Non-defaulting Party”. The non-defaultingparty that determines the closeout amount is party 2 in our notation. Then at page 15 we havethe following prescription: ”In determining a Close-out Amount, the Determining Party mayconsider any relevant information, including, without limitation, one or more of the followingtypes of information: (i) quotations (either firm or indicative) for replacement transactionssupplied by one or more third parties that may take into account the creditworthiness of theDetermining Party at the time the quotation is provided”. This is in contrast with the default-free closeout amount V (cid:0) τ (cid:1) prescribed by the classic formula, and seems instead consistentwith the substitution-cost closeout amount V (cid:0) τ (cid:1) prescribed by the formula given in thispaper, that includes the risk of default of the survived party 2. The ISDA documentation isnot so strict to make this a binding prescription - the document speaks of a determining partythat may take into account its own creditworthiness. Thus the risk-free closeout amount isnot excluded.Various other points in the documentation confirm that a substitution closeout is morelikely. One of the clearest is in the Market Review of OTC Derivative Bilateral Collateral-ization Practices, published by ISDA on March 1, 2010, that says: ”Upon default close-out,valuations will in many circumstances reflect the replacement cost of transactions calculatedat the terminating party’s bid or offer side of the market, and will often take into account thecredit-worthiness of the terminating party” .The same document, however, points out that this substitution closeout risks being aproblem for collateralized derivatives, since it seems at odds with the computation of collateralamount: “However, it should be noted that Exposure is calculated at mid-market levels so asnot to penalize one party or the other. As a result of this, the amount of collateral heldto secure Exposure may be more or less than the termination payment determined upon aclose-out”. We analyze this issue in detail in Section 7. The above analysis shows a financial rationale in the ISDA documents in favor of a substitutioncloseout, but it also shows that the same documents leave room for other solutions. There isone point in strong favor of a risk-free closeout: the simplicity of computation, since it does notrequire an assessment of the risk of default of the survived party. In the settlement of a default,if a risk-free closeout is used, then all contracts with the same payoff would have the samecloseout value, irrespectively of the counterparty, which is an important simplification. Thisrisk-free amount would not be difficult to compute in the market because it would correspondto the mark-to-market of a deal equal to the residual deal but for the fact of being collateralized.Of course some parties could find it unfair. In the end the defaulted deal was not collateralized,and at inception it had different prices for different parties because of the different default risksof the parties. Some may not accept now a unique ’collateralized’ closeout amount. In spite ofthis, the homogeneity of the risk-free closeout amount is a strong point in favor of this solution,maybe just as a simpler approximation to the substitution closeout, if the error turns out tobe not too big. . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk
We perform a mathematical and numerical analysis of the consequences of the two approaches.We consider a contract where party B enters at time 0 into the commitment to pay a unitof money at time T to party A . This claim has exactly the same payoff as the prototypicalzero-coupon bond or loan of the textbooks on finance. Party B is the borrower or bond issuer,and A is the lender or bond holder. This payoff is convenient for our purposes because thereis a consolidated standard on how to price it. We will see which one between risk-free andsubstitution liquidation is consistent with the practice developed in the loan and bond markets.The comparison with the bond market is particularly interesting for a further reason. Whenbilateral counterparty risk was first introduced, there was some discussion in the market aboutone consequence of it: when a bank includes its own risk of default in the pricing of a deal,it can actually book a profit when there is an increase of its credit spreads. This is a ratherbizarre fact. However supporters of this approach pointed out that this already happens forbanks with reference to bond issuances. Banks have the so-called fair value option , namelythe possibility to account for issued bonds in their balance-sheets at mark-to-market. Whenthis is done, the bond liabilities decrease in value when credit spreads increase, and the bankcan mark a profit. This consistency with the treatment of bonds has contributed to makethe DVA more accepted in the market. Below we will show some practical effects of bilateralcounterparty risk adjustments under risk-free closeout or substitution closeout. By observingthese effects on a bond-like payoff, we are using the payoff that has already been the mainreference for understanding the appropriateness of counterparty risk adjustments. In the following we take deterministic interest rates.
First we evaluate this ’derivative bond’ deal using unilateral formulas, namely considering onlythe default of the borrower counterparty B (here the default of A is neglected from the payoutirrespectively of the fact that it may default or not). We apply (1) to this payoff, getting V BA ( t ) = E t n { τ B >T } e − R Tt r ( s ) ds o + E t (cid:26) { τ B ≤ T } (cid:20) e − R τBt r ( s ) ds R B (cid:16) e − R TτB r ( s ) ds (cid:17) + (cid:21)(cid:27) (11) = e − R Tt r ( s ) ds E t (cid:2) { τ B >T } (cid:3) + R B e − R Tt r ( s ) ds E t (cid:2) { τ B ≤ T } (cid:3) This is the standard formula for the pricing of a defaultable bond or loan. We have V BA (0) = e − R T r ( s ) ds Q (cid:0) τ B > T (cid:1) + R B e − R T r ( s ) ds Q (cid:0) τ B ≤ T (cid:1) , which says that the price of a defaultable bond equals the price of a default-free bond multipliedby the survival probability of the issuer, plus a recovery part received when the issuer defaults. . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk V AA ( t ), the value when only thedefault of the lender A is taken into account. We have(12) V AA ( t ) = E t n { τ A >T } e − R Tt r ( s ) ds o + E t (cid:26) { τ A ≤ T } e − R τAt r ( s ) ds (cid:16) e − R TτA r ( s ) ds (cid:17) + (cid:27) = e − R Tt r ( s ) ds . We have obtained the price of a risk-free bond, thus the formula says that in a loan or bondwhat matters is the risk of default of the borrower. If we consider only risk of default forthe lender, and not default-risk of the borrower, we get just the price of a default-free loan orbond. We have no influence of the risk of default of a party, the lender, that in this contracthas no future obligations. Both formula (11) and (12) are in line with market practice.
Now we price the deal considering the default risk of both parties, and assuming first a sub-stitution closeout . We apply formula (10) introduced in this paper, putting (11) and (12) intothis formula. We getˆ V A ( t ) = E t n e − R Tt r ( s ) ds o + E t (cid:26) A (cid:20) e − R τAt r ( s ) ds (cid:16) e − R TτA r ( s ) ds E τ A (cid:2) { τ B >T } (cid:3) + R B e − R TτA r ( s ) ds E τ A (cid:2) { τ B ≤ T } (cid:3)(cid:17) + (cid:21)(cid:27) + E t (cid:26) B (cid:20) e − R τBt r ( s ) ds R B (cid:16) e − R TτB r ( s ) ds (cid:17) + (cid:21)(cid:27) = E t n e − R Tt r ( s ) ds o + E t n A h e − R Tt r ( s ) ds E τ A (cid:2) { τ B >T } (cid:3) + R B e − R Tt r ( s ) ds E τ A (cid:2) { τ B ≤ T } (cid:3)io + E t n B h R B e − R Tt r ( s ) ds io Notice that 1 A = 1 { τ A ≤ min( T,τ B ) } is τ A -measurable, namely it is known at τ A , so that the second one of the three terms of ˆ V A ( t )above can be rewritten as E t n A h e − R Tt r ( s ) ds E τ A (cid:2) { τ B >T } (cid:3) + R B e − R Tt r ( s ) ds E τ A (cid:2) { τ B ≤ T } (cid:3)io = e − R Tt r ( s ) ds E t (cid:2) E τ A (cid:2) A { τ B >T } (cid:3)(cid:3) + R B e − R Tt r ( s ) ds E t (cid:2) E τ A (cid:2) A { τ B ≤ T } (cid:3)(cid:3) = e − R Tt r ( s ) ds E t (cid:2) A { τ B >T } (cid:3) + R B e − R Tt r ( s ) ds E t (cid:2) A { τ B ≤ T } (cid:3) where in the last passage we have used the law of iterated expectations. Now in ˆ V A ( t ) we canfactor together the terms that are multiplied by R B and those that are not, gettingˆ V A ( t ) = e − R Tt r ( s ) ds E t [1 ] + e − R Tt r ( s ) ds E t (cid:2) A { τ B >T } (cid:3) + R B e − R Tt r ( s ) ds E t (cid:2) A { τ B ≤ T } (cid:3) + R B e − R Tt r ( s ) ds E t [1 B ]= e − R Tt r ( s ) ds E t (cid:2) + 1 A { τ B >T } (cid:3) + R B e − R Tt r ( s ) ds E t (cid:2) A { τ B ≤ T } + 1 B (cid:3) Now we concentrate on the indicators. It is easy to see that1 A { τ B >T } = 1 { τ A ≤ min( T,τ B ) } { τ B >T } = 1 { τ A ≤ T } { τ B >T } , A { τ B ≤ T } = 1 { τ A ≤ min( T,τ B ) } { τ B ≤ T } = 1 { τ A ≤ τ B } { τ B ≤ T } = 1 { T < min( τ A ,τ B ) } = 1 { T <τ A } { T <τ B } , . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk V A ( t ) = e − R Tt r ( s ) ds E t (cid:2) { T <τ A } { T <τ B } + 1 { τ A ≤ T } { τ B >T } (cid:3) + R B e − R Tt r ( s ) ds E t (cid:2) { τ A ≤ τ B } { τ B ≤ T } + 1 { τ B <τ A } { τ B ≤ T } (cid:3) . Since 1 { T <τ A } + 1 { τ A ≤ T } = 1 { τ A ≤ τ B } + 1 { τ B <τ A } = 1, we haveˆ V A ( t ) = e − R Tt r ( s ) ds E t (cid:2) { τ B >T } (cid:3) + R B e − R Tt r ( s ) ds E t (cid:2) { τ B ≤ T } (cid:3) (13) = e − R Tt r ( s ) ds (cid:2) Q t (cid:0) τ B > T (cid:1) + R B Q t (cid:0) τ B ≤ T (cid:1)(cid:3) . In spite of the somewhat lengthy computations, we have found a very simple and reasonableresult. We have ˆ V A ( t ) = V BA ( t ), and we have that the risk of default of the bond-holder in abond, or the lender in a loan, does not influence the value of the contract. Remark 1.
We point out that in the above formula for ˆ V A ( t ) there can be dependence from therisk of default of the lender through the terms Q t (cid:0) τ B > T (cid:1) and Q t (cid:0) τ B ≤ T (cid:1) when the formulais evaluated at a future time t > . In fact in some models terms such as Q t (cid:0) τ B > T (cid:1) dodepend on the the risk of default of A . For example, when using a copula model, at times t > the fact that a correlated counterparty has defaulted or not by t changes the default probabilityof a counterparty still alive, since dependency is set on unobservable latent variables (theexponential triggers), about which one can get information by observing if correlated companieshave defaulted or not. In other models such as the structural first passage model of Black andCox (1976) or the multivariate exponential Marshall Olkin (1967) model this does not happen.We point out that however also within models, like copulas, where terms such as Q t (cid:0) τ B > T (cid:1) do depend on the the risk of default of A , pricing is always made assuming t = 0 , where even inthese models there is independence from the the risk of default of A . The situation is differentinstead when in a copula we make a forward valuation where we need to assume t > , as wedo in the following when we asses the value of a deal just before and just after a default. Therewe show how conditioning on information about A modifies Q t (cid:0) τ B > T (cid:1) . We can conclude that, although bilateral risk of default matters in general for bilateralcontracts, as confirmed by the fact that ˆ V A (0) is in general much more complex that V BA (0),with substitution closeout we have that when the contract has no future obligations for aparty A the risk of default of A does not influence the price. Only the risk of default of thebond-issuer or borrower matters for valuation. This result is also in line with market practice. Now we apply instead the formula (3) that assumes a risk-free closeout . Since V A ( τ ) = e − R Tτ r ( s ) ds , we have V A ( t ) = E t n e − R Tt r ( s ) ds o + E t (cid:26) A (cid:20) e − R τAt r ( s ) ds (cid:16) e − R TτA r ( s ) ds (cid:17) + (cid:21)(cid:27) + E t (cid:26) B (cid:20) e − R τBt r ( s ) ds R B (cid:16) e − R TτB r ( s ) ds (cid:17) + (cid:21)(cid:27) = e − R Tt r ( s ) ds E t [1 + 1 A ] + e − R Tt r ( s ) ds R B E t [1 B ] . The reader may think that the risk of default of the lender could in practice influence the price of a bond orloan through its effect on the cost of funding for the lender. See Morini and Prampolini (2010) for a discussionon this. However, when liquidity costs are note considered, like in this paper, this effect does not exist and,like in classic bond pricing, one expects the risk of default of the holder not to influence the price of a bond. . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk V A ( t ) = e − R Tt r ( s ) ds Q t (cid:2) T < min (cid:0) τ A , τ B (cid:1) ∪ τ A ≤ min (cid:0) T, τ B (cid:1)(cid:3) + e − R Tt r ( s ) ds R B Q t (cid:2) τ B < τ A ∩ τ B ≤ T (cid:3) . Playing with indicators one can also obtain the alternative expressions V A ( t ) = e − R Tt r ( s ) ds (cid:0) Q t [ τ B > min( τ A , T )] + R B Q t [ τ B < min( τ A , T )] (cid:1) (15) = e − R Tt r ( s ) ds (cid:0) Q t [ τ B > T ] + Q t [ τ A < τ B < T ] + R B Q t [ τ B < min( τ A , T )] (cid:1) Reaching this result for V A applied to a ‘derivative bond’ has been far easier than reaching theanalogous one for ˆ V A , but the result looks more complex. In fact it introduces a dependenceon the risk of default of the lender even at time 0, and on the exact individuation of the firstdefault, that was not there in ˆ V A . This is in contrast with the market practice in the bond orloan markets. In particular, let us compare V A ( t ) with ˆ V A ( t ). Since1 + 1 A = 1 { τ B >T } { τ A >T } + 1 { τ B ≥ τ A } { τ A ≤ T } ≥ { τ B >T } = 1 { τ B >T } { τ A >T } + 1 { τ B >T } { τ A ≤ T } and R B ≤
1, we have V A ( t ) ≥ ˆ V A ( t ) = V BA ( t ) . Thus a risk-free liquidation increases the value of a ’derivative bond’ to the bond holdercompared to the value that a bond has in the market practice. Symmetrically, the value isreduced to the bond issuer, and this reduction is an increasing function of the default risk ofthe bond holder. This confirms that a substitution closeout guarantees the borrower, the partywhich has payment to do in the future, making its risk of default the only relevant variablethat determines the value of a bond or loan. This is a crucial feature for the stability of a debtmarket, but it has implications that can appear paradoxical. An example is in the following.
Let us consider again the case of a risk free zero coupon bond, and assume the total default de-pendence case, as represented by two co-monotonic default times. We assume co-monotonicityas τ A = ψ ( τ B )for a deterministic and strictly increasing function ψ . Let us further limit ourselves to situ-ations where the lender is riskier than the borrower in terms of probability of default. Thisis represented by assuming that ψ ( x ) < x , so that τ A < τ B in all scenarios. This is clearlya very extreme case; some example may happen if B is somehow a subsidiary of A , althoughin a real-world case a default time will never be exactly a deterministic function of anotherdefault time (unless we consider the case of simultaneous defaults, that in fact is what usuallyholds for subsidiaries). Some models based on a physical explanation of default dependency,such as structural models or the multivariate exponential Marshall Olkin (1967) model we usein the following, do not even admit comonotonicity outside the case of simultaneous default.However this is a scenario admitted by other common models, like a bivariate default intensitymodel where default intensities of A and B satisfy λ A > λ B and where the two default timeexponential triggers are connected by the co-monotonic copula, or equivalently by a Gaussiancopula with correlation 1. Here default of a company triggers the default of a second one ina fully predictable way, but the default of the second one may happen any number of years . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk V A ( t ) = e − R Tt r ( s ) ds whereas the substitution closeout formula yieldsˆ V A ( t ) = e − R Tt r ( s ) ds Q t ( τ B > T ) . We are in a situation where whenever there is default, A defaults always first. As a consequence, A will never be impacted by B ’s default, and one could expect then the price of the bond to A not to depend on the default risk of B . While this happens with the risk-free closeout, thisdoes not happen with the substitution closeout, that maintains dependence on default risk of B . We have to say, however, that although in this stylized example it would make sense forthe bond price not to depend on the default probability of B , this does not happen in themarket, where the bond price remains ˆ V A ( t ) regardless of default dependence issues, makingthe price of bonds the same for all buyers. To quantify the size of the above difference, and to analyze numerically the practical effect ofeither assumption on closeout, we need to have a model for the default times of our two names.Consistently with the purpose of keeping complexity as low as possible since we are dealingwith very fundamental issues, we will use the simplest bivariate extension of the standardsingle name credit model. Like in the single name market credit model, we assume that thedefault time of the name X , X ∈ { A, B } , is exponentially distributed. We take a flat defaultintensity λ X , so that the survival probability is Q ( τ X > T ) = e − λ X T , and for consistency also r ( s ) is taken flat so that the default-free bond is P T = e − rT . As baseline hypotheses, we take the two extreme scenarios, namely one scenario where thevariables τ A and τ B are independent, and a second one where they are co-monotonic. We startwith the independent case. It is well known that in this case min (cid:0) τ A , τ B (cid:1) is also exponentiallydistributed of parameter λ A + λ B , since Q (cid:0) min (cid:0) τ A , τ B (cid:1) > T (cid:1) = Q (cid:0) τ A > T (cid:1) Q (cid:0) τ B > T (cid:1) = e − ( λ A + λ B ) T It is easy to compute the terms needed to apply Formula (14). First, one has Q (cid:0) τ A < min( τ B , T ) (cid:1) = λ A λ A + λ B (cid:16) − e − ( λ A + λ B ) T (cid:17) , as one can easily show by solving the integral Q (cid:0) { τ B > τ A } ∩ { T > τ A } (cid:1) = Z T Q ( τ B > t ) Q ( τ A ∈ dt ) = Z T e − λ B t λ A e − λ A t dt . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk Q (cid:0) τ A < τ B (cid:1) = λ A λ A + λ B which is obtained as a limit case of the earlier expression when T ↑ ∞ . We have all we needfor computing V A ( t ) = e − rT (cid:8) Q (cid:2) T < min (cid:0) τ A , τ B (cid:1)(cid:3) + Q (cid:2) τ A < min (cid:0) T, τ B (cid:1)(cid:3) + R B Q (cid:2) τ B < min (cid:0) τ A , T (cid:1)(cid:3)(cid:9) == e − rT (cid:26) e − ( λ A + λ B ) T + λ A λ A + λ B (cid:16) − e − ( λ A + λ B ) T (cid:17) + R B λ B λ A + λ B (cid:16) − e − ( λ A + λ B ) T (cid:17)(cid:27) (16)Notice that in our model the probability of events such as τ A = T or τ A = τ B is zero.In the comonotonic default case, we assume that τ A = ξ/λ A , τ B = ξ/λ B , with ξ standard exponential random variable. It follows that(17) τ A = λ B λ A τ B In this case it will be easy to compute the terms appearing in (14). We begin with the casewhere λ B > λ A . Now τ B always happens first, so we have(18) V A ( t ) = e − R Tt r ( s ) ds (cid:0) Q t [ τ B > T ] + R B Q t [ τ B < T ] (cid:1) = ˆ V A ( t ) . Hence in this case risk free closeout and substitution closeout agree. Furthermore, here it doesnot make sense to ask what happens at the default time of the lender, because the borrowerdefaults first and closeout will always happen before the default of the lender.The interesting case where the two formulations disagree also in the co-monotonic case iswhen λ B < λ A , and hence τ B > τ A . In this case one can see by looking at (15) and (13) that V A ( t ) = e − R Tt r ( s ) ds , which makes sense, given that default of B will never happen to a solvent A , while the substi-tution closeout formula remains (13),ˆ V A ( t ) = e − R Tt r ( s ) ds (cid:2) Q t (cid:0) τ B > T (cid:1) + R B Q t (cid:0) τ B ≤ T (cid:1)(cid:3) , although we have to take into account Remark 1 if t > With this simple model we can test numerically the behaviour of the formula (3) or (16) withrisk-free closeout. Set the risk-free rate at r = 3%, and consider a bond with maturity 5 years.The price of the bond varies with the default risk of the borrower, as usual, and here also withthe default risk of the lender, due to the risk-free closeout. In Figure 1 we show the price of thebond for intensities λ Lender , λ Borrower going from zero to 100%. We consider R Borrower = 0so that the level of the intensity approximately coincides with the market CDS spread on the5 year maturity. . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk a company which has a net creditor position in aderivative and suddenly defaults, can have a strong financial gain stemming from default itself ,whose benefit will go to its liquidators.Symmetrically, and this is the most worrying part, the counterparty that did not defaultand that is a net debtor of the defaulted company will have to book a sudden loss due to defaultof the creditor . Notice that this is at odds with standard financial wisdom. It is natural thata creditor of company is damaged by the default of the company, but here we have somethingmore: when (3) is used, there is a damage not only to creditors, but also to debtors . This isworrying since it implies that with a risk-free closeout the default contagion spreads also tonet debtors, not only to net creditors.Let us observe a numerical example. We start from the above r = 3% and a bond withmaturity 5 years, for a 1bn notional. Now we take R Borrower = 20% and two risky parties.We suppose the borrower has a very low credit quality, as expressed by λ Borrower = 0 .
2, thatmeans Q (cid:0) τ Borrower ≤ y (cid:1) = 63 . . The lender has a much higher credit quality, as expressed by λ Lender = 0 .
04. This means alower probability of default, that however is not negligible, being Q (cid:0) τ Lender ≤ y (cid:1) = 18 . P T = 860 . mn Using the formula with risk-free closeout, we get that a risky bond, within the two partiesabove, has price V Lender = 359 . mn to be compared with the price coming form the formula with substitution closeoutˆ V Lender = 316 . mn. These figures can be very easily computed via (3) or more directly with (16). The exampleconfirms a difference in the valuation given by the two formulas. The higher value of V Lender depends on the probability of default of the lender.The difference of the two valuations is not negligible but not dramatic. More relevant isthe difference of what happens in case of a default under the two assumptions on closeout. Wehave the following risk-adjusted probabilities on the happening of a default event Q (cid:2) τ Borrower < min (cid:0) y, τ Lender (cid:1)(cid:3) = 58% , Q (cid:2) y < min (cid:0) τ Lender , τ
Borrower (cid:1)(cid:3) = 30% , Q (cid:2) τ Lender < min (cid:0) y, τ Borrower (cid:1)(cid:3) = 12% . The two formulas agree on what happens in case of no default or in case of default of theborrower first. These are the most likely scenarios, totalling 88% probability. But with anon-negligible probability, 12%, the lender can default first. Let us analyze in detail whathappens in this case. Suppose the exact day when default happens is τ Lender = 2 . years. Just before default, at 2 . V Borrower (cid:0) τ Lender − d (cid:1) = − . . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk V Borrower (cid:0) τ Lender − d (cid:1) = − . A and B the borrower, that when τ A ≤ min (cid:0) T, τ B (cid:1) and V B (cid:0) τ A (cid:1) ≤ V B (cid:0) τ A (cid:1) = V B (cid:0) τ A (cid:1) .Thus the book value of the bond becomes simply the value of a risk free bond, V B (cid:0) τ A + 1 d (cid:1) = − . . − . . V Borrower (cid:0) τ Lender + 1 d (cid:1) = − . X is a lender when V X ( t ) > T > t >
0, while he is a borrower if V X ( t ) < T > t >
0. Payoffs that generate this situationare very common. The simplest example is an option contract. The option buyer is the lender,the option writer is the borrower.Additionally, some time after inception all deals, even the most complex derivatives, canhave mark-to-market far away from zero, so that for one party V X ( t ) ≪
0. Such partybecomes a net borrower and in case of default of the other party would suffer, following arisk-free closeout, losses similar to those outlined above. . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk We now take an example where the two default times have constant intensities and are co-monotonic, and we will see, conversely, that this time the substitution closeout has a dramaticeffect for the creditors and liquidators of the defaulted company.We assume λ Borrower = 3 . λ Lender = 4%, r = 3%, T = 5 y and R Borrower = 0. Thissetting means that
Lender always defaults first, since following formula (17) τ Borrower = λ Lender λ Borrower τ Lender = 1 . τ Lender
We still assume a notional of 1bn. The initial value to the borrower of the bond under bilateralcounterparty risk for the risk free closeout is − e − . · bn = − . mn, namely the risk-free value, consistently with the example we showed in Section 6.1.5. Thevalue of the bond under substitution closeout is − e − (0 . · bn = − . mn Now let us see what happens when the lender A defaults at 2.5y. We need to remember thatwe are in a situation where A defaults always before B, and where we know exactly that, if Adefaults at 2.5y, B defaults in 2.77y. This is clearly a purely toy case, but it approximates apossibly realistic one: the case of a company B that, as an effect of the default of A, sees itsown risk of default increase dramatically, going to the verge of default.In such a case the risk free closeout gives us the same value both immediately before andafter default, namely, to the lender A: e − . · . bn = 927 . mn with no discontinuity, whereas the substitution closeout implies a jump from e − . · . Q (cid:0) τ Borrower > y | τ Lender > . y (cid:1) bn = e − . · . Q (cid:18) τ Lender > . y | τ Lender > . y (cid:19) bn = e − . · . e − . ( . ¯1 − . ) = 856 . bn to e − . · . Q (cid:0) τ Borrower > y | τ Lender = 2 . y (cid:1) bn = 0 . Due to the fact that the model implies that the borrower will default at 2.77y, which isbefore the maturity, the substitution closeout, that takes into account the default risk of thecounterparty, sees a null value for the bond.In other terms, at the default of the lender, the creditors and liquidators of the lendersee all their claims towards the borrower lose their value. This confirms that the substitutioncloseout reduces the claims of the creditors towards the debtors of the defaulted entity, andthis effect is stronger the stronger is the default dependence between the defaulted entity andits borrowers. This may be the case, for example, when the defaulted entity is a company witha relevant systemic impact.
There is one final thing to assess, and it is the behaviour of the two forms of closeout forcollateralized deals. A defaultable deal should satisfy the following property: the combinationof the defaultable deal and a collateral agreement must eliminate losses at default. . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk X the counterparty which isa net borrower provides the net lender with an amount of liquidity equal to the risk-free netpresent value of X . This amount of collateral generates interest and is regularly updated toremain equal to the net present value of X , and can be claimed back by the borrower only atmaturity of the deal if the borrower has not defaulted earlier. If instead the borrower defaultsearlier, the collateral must be used to offset the default loss.We analyze in detail what happens at default considering a very simple payoff where X = 1and must be paid at T , namely the simple ’derivative’ bond considered above. The borrowermust give at time 0 to the lender an amount of liquidity equal to the risk-free net present valueof the future payoff, C (0) = e − R T r ( s ) ds . Then the borrower must update regularly this quantity, in such a way that the collateralremains equal at any time t to the net present value of the future payoff, C ( t ) = e − R Tt r ( s ) ds . The collateral is updated daily. We approximate this daily settlement with a continuoussettlement. In this case, in order to keep the collateral C ( t ) at e − R Tt r ( s ) ds , the borrower mustpay to the lender continuously an amount(19) r ( t ) C ( t ) dt, in fact it is trivial to show that if dC ( t ) = r ( t ) C ( t ) dt,C (0) = e − R T r ( s ) ds , then C ( t ) = e − R Tt r ( s ) ds , as desired. On the other hand, the lender that keeps the collateral C ( t ) must give back tothe borrower the interest generated continuously by C ( t ). This transforms into a continuouspayment(20) r ( t ) C ( t ) dt. made by the lender to the borrower.We immediately notice that the settlement payment (19) made by the borrower cancelsout with the interest payment (20) made by the lender. Thus for a simple payoff the collat-eral agreement becomes a sort of self-financing strategy, namely it does not involve any netcontinuous exchange of money. The borrower B pays e − R T r ( s ) ds at time 0. The collateralremains with the lender and earns interest in time at rate r ( t ), this interest r ( t ) C ( t ) dt is notreturned to the borrower but added to the value of the collateral, that in this way is always C ( t ) = e − R Tt r ( s ) ds . If we consider the risk-free closeout, the requirement of a match between the closeout and thevalue of the collateral at the first default τ among those of the two counterparties appearstrivially obtained. The risk-free closeout amount is always V A ( τ ) = e − R Tτ r ( s ) ds . . Brigo and M. Morini. Impact of Closeout Conventions on Bilateral Counterparty Risk τ is always C ( τ ) = e − R Tτ r ( s ) ds . If we just say that the collateral is an amount of money to be used to offset the default closeout,it turns out that the lender always has the right amount of money to offset the default closeout.
However, if we take a less naive view we remember that collateral is not just an amount ofmoney but a contract with specific features. This contract has a value at default and it isthis value, not the amount of money held by the lender, that needs to match the closeoutamount. If this happens, there will be no transfer on money at default of a collateralized partyand no losses for any party, exactly as we expect for a collateralized deal. We show in thefollowing that this can be obtained rather easily with a substitution closeout . If we think ofthe practical working of collateral, we can treat collateral as an amount of money given to thelender from the borrower coupled with a claim of the borrower towards the lender to have thiscollateral back. If there is no default of the borrower before maturity, the borrower has a claimto have the collateral back at termination of the deal, namely a payoff C ( T ) at maturity T upon no default: 1 { τ B >T } C ( T ) = 1 { τ B >T } at T. If instead the default of the borrower happens before maturity, the borrower has a claim toreceive the collateral back at its own default time τ B { τ B
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