Participation and unbiased pricing in CDS settlement mechanisms

Transcription

1 Partcpaton and unbased prcng n CDS settlement mechansms Ahmad Pevand February 2017 Abstract The centralzed market for the settlement of credt default swaps (CDS), whch governs more than $10 trllon s worth of outstandng CDS contracts, has been crtczed for msprcng the defaulted bonds that underle the contracts. I take a mechansm desgn approach to the settlement of CDS contracts. I fully characterze robust settlement mechansms that delver unbased prces for the underlyng assets and show that all robust settlement mechansms are payoff equvalent to a posted prce mechansm. I explot my analyss to propose a modfcaton to the exstng CDS market and the settlement procedure to mprove the effcency of the mechansm. Because forced partcpaton n the settlement mechansm s not possble, my approach requres the development of a new noton of the core of games of ncomplete nformaton. Ths new noton can be appled to mechansm desgn envronments n whch sde trades are allowed or when jonng the mechansm s a cooperatve decson. I am ndebted to Jeffrey Ely and Rakesh Vohra for our valuable conversatons and ther encouragement to wrte ths paper. I have also receved helpful comments from Ajay Subramanan. Georga State Unversty, Robnson College of Busness. Emal: apevand@gsu.edu. 1

2 1 Introducton A credt default swap (CDS) s an nsurance contract aganst the rsk of default. 1 After the default of a bond ssuer, the correspondng CDS contracts are settled through a centralzed mechansm. Ths mechansm produces a prce for the defaulted bond that s used to measure the amount of loss due to the default. The current mechansm n use underprces the asset n most cases and s also senstve to the number of CDS postons, whch results n effcency losses due to undernsurance (Gupta and Sundaram (2013) and Chernove et al. (2013)). I show that all robust mechansms that delver unbased prces n expectaton and all robust mechansms that are not senstve to the agent s number of postons, are pay-off equvalent to a posted prce mechansm. Snce a posted prce mechansm s mpossble or very dffcult to mplement, the results of the paper show that there s no practcal aucton procedure that would delver unbased prces. I explot my analyss to propose a modfcaton to the exstng CDS market and the settlement procedure to mprove the effcency of the mechansm. I show the proposed mechansm delvers partal partcpaton and unbased prcng. In addton, I model the partcpaton of players n ths mechansm and, thereby, develop a novel noton of the core of games of ncomplete nformaton. The new noton s necesstated by the fact that the decson to jon the blockng mechansm precedes partcpaton n the mechansm. Therefore, ths noton can be used to model partcpaton to centralzed markets n whch sde trades or sde matches pror to, or concurrent wth, the centralzed mechansm are allowed. Examples nclude dark pools 2 and some centralzed job markets, such as the Natonal Resdency Matchng Program (NRMP). Each CDS contract corresponds to a reference entty s bond. The correspondng CDS contracts can be settled va physcal settlement or cash settlement. In the case of cash settlement, the protecton seller pays the face value mnus the value of the defaulted bond to the protecton buyer. In the case of physcal settlement, the protecton buyer hands the defaulted bond to the protecton seller and receves the face value of the bond. Although physcal settlement has the advantages of not requrng a prce and the protecton buyer s full nsurance, t s often mpossble to physcally settle all contracts. 1 More broadly, a CDS s an nsurance contract aganst the rsk of a credt event. 2 Dark pools are prvate platforms to trade securtes. 2

3 Frst, n most cases, the number of outstandng CDS contracts s more than the number of bonds. 3 Second, even f protecton buyers could purchase the defaulted bonds for physcal settlement, dong so would artfcally rase the prce of the defaulted bond. For these reasons, an alternatve way of settlng the contracts by cash transfer has emerged. The challenge for cash settlement s to dentfy a value for the defaulted bond. To determne the quantty of contracts to be settled physcally or by cash transfer, as well as a prce for cash settlement, the Internatonal Swaps and Dervatves Assocaton (ISDA) ntroduced a two-stage mechansm. In the frst stage, only agents wth CDS contracts partcpate. In ths stage, the mechansm determnes the number of defaulted bonds to be bought or sold n the second stage of the mechansm and a prce cap or floor. In the second stage, a unform prce aucton determnes a prce for the defaulted bond. As of 2009, all CDS contracts are pegged to the value of the defaulted bond determned by ths mechansm, unless both the protecton buyer and protecton seller choose to opt out. 4 In addton, agents may sell ther CDS contracts to other agents pror to partcpaton n the settlement mechansm. The mechansm used by the ISDA has been the subject of crtcsm. Chernove et al. (2013) have observed that the defaulted bonds n ths mechansm are underprced by an average of 6%. Due to the wnner s curse n the second stage of the current mechansm, msprcng s nevtable. Ths msprcng mples that the protecton buyer cannot fully nsure aganst the rsk of default by the ssuer of the bond and that there s uncertanty n regard to the future payoff of a defaulted bond. Also, the outcome of the mechansm s dependent form the number of bonds that are traded at the second stage of aucton, Chernove et al. (2013). Ths creates extra uncertanty for CDS traders, snce when agents trade CDSs the number of physcal bonds and the total number of CDSs at the tme of CDS settlement are unknown. The goal of any desgn should be to settle contracts wth unbased prces. A set- 3 As stated n Summe and Mengle (2011), at the tme of Delph Corporaton s bankruptcy, t was estmated that there were $28 bllon n CDSs outstandng but only $2 bllon n defaulted bonds. If short sellng were facltated n ths market, n a physcal settlement, the protecton buyer would short sell the defaulted bond rather than hand t to the protecton seller. Because defaulted bonds are traded over the counter, short sellng the defaulted bonds s dffcult or even mpossble. 4 The ISDA argues that requrng all partes to a CDS to be bound by the results of the mechansm ensures certanty, consstency, enhanced transparency, and lqudty; see 3

4 tlement mechansm s unbased f the cash settlement prce s equal to the value of the defaulted bond or f an agent s payoff s equal to the payoff from physcal settlement of all contracts. I take a mechansm desgn approach and look for a settlement mechansm that s unbased. Moreover, the mechansm must satsfy three mportant propertes: 1. Ex-post ncentve compatblty: The mechansm s ncentve compatble for all possble agents belefs. 2. Weak budget balance: The desgner does not have to ncur a cost to execute the settlement mechansm. 3. Robustness wth respect to agents partcpaton decsons: The settlement delvers unbased prces regardless of the agent s partcpaton choce. The revelaton prncple mples that, wthout loss of generalty, one can restrct attenton to drect settlement mechansms that are ncentve compatble (Myerson (1981)). The frst property s a robustness property aganst agents belefs (see Bergemann and Morrs (2005)). Because there s cash transfer n the mechansm, the second property s also standard. The thrd property s also mportant, as agents cannot be compelled to partcpate n the ISDA settlement mechansm. If both partes of a CDS contract agree, they can choose to settle some of ther contracts outsde of the settlement mechansm. Also, an agent may sell some of hs contracts to another agent pror to partcpatng n the settlement clearnghouse. I defne partcpaton choce to descrbe the agents decson about how they partcpate n the central mechansm as well as how they settle contracts outsde of the mechansm. Partcpaton choces should satsfy two mportant crtera. Frst, when a par of agents make a decson about ther partcpaton, t should beneft both of them. Second, these partcpaton decsons should be self-confrmng. Ths means that, when a par of agents engage n sde settlement or CDS trade, each agent update hs belef about other agent s type. Gven the updated belefs, agents choose to ext when there s a beneft to dong so. The robustness property noted above requres the mechansm to be unbased n all possble agent partcpaton choces. The man theoretcal nnovaton of ths paper s to model how agents choose ther level of partcpaton n the mechansm. 4

5 A mechansm that sets the cash settlement prce equal to the expected value of the defaulted bond, condtonal on the desgner nformaton, and sets a constant cash settlement quantty s called a posted prce mechansm. I show that all mechansms wth propertes 1 3 lsted above are payoff equvalent to a posted prce mechansm. The dffculty n desgnng a settlement mechansm, when partcpaton s voluntary, s that agents may manpulate the outcome of the mechansm through partcpaton. Because the desgner faces the budget constrant, he cannot pay the agents to partcpate n the mechansm. When a posted prce mechansm s employed, agents can no longer manpulate the settlement procedure through strategc choce of partcpaton. As dscussed, partcpaton n ths mechansm s voluntary. 5 Because CDS contracts are blateral between pars of agents, no settlement mechansm can enforce partcpaton. Choosng not to fully partcpate harms the transparency of the ISDA settlement procedure and drves lqudty away from the aucton. Ths s mportant because lqudty and transparency were among the man reasons offered for a central mechansm to settle CDSs n the frst place. I show that all unbased mechansms that nduce full partcpaton of all agents wth all of ther contracts are payoff equvalent to a posted prce mechansm. The current mechansm n use and the posted prce mechansm are robust wth respect to the network, n the sense that they treat two networks wth the same number of contract n the same way. A mechansm s robust wth respect to the network f the prce and quanttes are the same n two networks, for whch agents have a net equal number of contracts. The ratonale for ths property s that t lowers the systemc rsk and transacton costs. I extend the characterzaton result n two ways. Frst, I consder the case n whch agents cannot sell ther CDS contracts to other agents pror to the settlement mechansm. However, a par of CDS traders can choose ther level of partcpaton by settlng some of ther contracts outsde of the mechansm. Gven the same model of partcpaton as above, I agan show that the only mechansm that satsfes ex-post ncentve compatblty and robustness wth respect to agents decsons about partcpaton s a posted prce mechansm. Second, I generalze the noton of unbasedness. I consder 5 Some CDS contracts are between agents n dfferent countres, whch makes enforcng partcpaton even more dffcult. 5

6 settlement mechansms for whch, from an ex-ante pont of vew, the payoff of each agent s equal to hs payoff from cash settlement wth some known prce. Ths property ensures that the ex-ante payoff of agents s proportonal to ther net number of contracts. Ths s relevant here because CDS contracts are homogeneous. I show that the characterzaton results hold f one replaces unbasedness wth ths property. Taken together, my results show that the posed prce mechansm s the only robust settlement mechansm that delvers a prce that s not senstve to the number of postons. Therefore, there s no aucton procedure that would acheve the effcent outcome of unbased prcng. I explot my analyss to propose a modfcaton to the exstng CDS market and the settlement procedure to mprove the effcency of the mechansm. At the tme of contractng, the buyer and the seller of CDS decde whether the CDS contract s covered or naked. The holder of a covered CDS s requred to submt the underlyng defaulted bonds to the settlement mechansm n the frst round of the settlement procedure. If ths does not occur, the buyer does not receve the face value of the bond from the CDS seller. Wth ths modfcaton, I show all covered CDSs are settled physcally n the clearnghouse. Covered CDS buyers buy CDSs to hedge aganst the rsk of default of the underlyng bond. Therefore, based prcng of the CDSs does less harm to naked CDS compared to covered CDS buyers who buy the CDS contract to hedge aganst the rsk of default. I show that wth ths modfcaton, covered CDS traders fully partcpate n the settlement mechansm. 2 Related lterature My study contrbutes more broadly to the market desgn lterature and, more specfcally, to the nascent lterature on the valuaton and settlement of CDS contracts. Studes that examne the CDS contract settlement mechansm focus exclusvely on the propertes and modfcatons of the current mechansm n use. Gupta and Sundaram (2013) observe that there s a prce bas for auctons held n the perod. Smlarly, Helwege et al. (2009) compare the mechansm prce to the pre- and post-aucton prces of the defaulted bond n a sample of ten early auctons and fnd no msprcng n ther sample. Coudert and Gex (2010) study the settlement procedure for a number of cases. Ther emprcal study also reveals a prce bas n the aucton. Du and Zhu 6

7 (2015) develop a theoretcal model to explan why the current aucton msprces the defaulted bond and propose a double aucton to acheve effcency. They consder the case n whch a contnuum of agents could have dfferent valuatons for the defaulted bond. Therefore, n ther model, allocatve effcency becomes relevant. Chernove et al. (2013) document the same prce bas as do Gupta and Sundaram (2013). Takng nto account multple fnancal frctons n the market, they solve for equlbra of the two-stage aucton, assumng that agents have no prvate nformaton about the value of the defaulted bond. Assumng no prvate nformaton about the value of the defaulted bond, they use a CDS aucton to dscover a prce of the defaulted bond. Motvated by the observaton that the number of partcpants n CDS auctons rarely exceeds 15, 6 I consder a framework n whch there s small number of partcpants wth prvate sgnals about the common value of the defaulted bond. Thus, strategc behavor plays a more crucal role n my analyss. More mportantly, my paper s the only one (to the best of my knowledge) to take a mechansm desgn approach to analyze the CDS market. I complement the exstng lterature by characterzng the settlement mechansms that satsfy the key robustness propertes descrbed earler. I also contrbute to the broader lterature n game theory by developng a new noton of the core of a game of ncomplete nformaton. The noton of unravel-proofness under ncomplete nformaton can be nterpreted as a stablty condton. The set of mechansms that satsfy the property can be nterpreted as the core of the underlyng game of ncomplete nformaton. The notons of core and stablty have been generalzed to games of ncomplete nformaton n Wlson (1978), Dutta and Vohra (2005), Myerson (2007), Serrano and Vohra (2007), Yenmez (2013), Lu et al. (2014), and Pomatto (2015). The notons dffer across the above studes n the way that agents communcate ther prvate sgnals. Wlson (1978) consders two extreme cases: () all agents n a block share ther prvate nformaton completely (fne core) and () agents share no prvate nformaton. Dutta and Vohra (2005) and Myerson (2007) consder the blocks for whch the decson to jon the block comes from a Bayesan Nash Equlbrum. Lu et al. (2014) study the mplcatons of common knowledge of stablty of a two-sded match when one sde of the market has ncomplete nformaton about the other sde. 6 See 7

8 In my noton, a block exsts f the ext game has an equlbrum n whch a subset of agents wth a postve measure subset of types partcpates n the blockng mechansm. The ext game that I descrbe resembles the votng game n Holmström and Myerson (1983). In ther setup, all agents partcpate n the votng game, and they examne whether a mechansm can be Pareto mproved through reallocaton by an unanmously elected alternatve mechansm. My noton of unravel-proofness dffers n two mportant ways from notons of the core ntroduced n the lterature. Frst, n the pror notons of the core, the decson to jon the blockng mechansm comes after the realzaton of the grand mechansm s allocaton. Therefore, the blockng desgner or members of the blockng coalton take the allocaton of the grand mechansm as exogenous. Because agents have quaslnear preferences and heterogeneous belefs about a common value n my framework, the no-trade theorem mples that no subset of agents should agree to a reallocaton once the contracts are settled (see Mlgrom and Stokey (1982)). Therefore, f one apples the noton of stablty n the lterature to the envronment consdered n ths paper, all settlements would be stable and durable, as defned n Holmström and Myerson (1983). In my noton of the block, agents smultaneously choose whether they want to partcpate n the blockng mechansm. The second man dfference between my noton of the core and the notons n pror lterature s that, n my setup, the blockng mechansm and the settlement mechansm can coexst. Ths s because an agent may choose to partally partcpate n the settlement mechansm. None of the models of stablty n the lterature accommodates ths possblty. My paper s also related to a body of lterature that studes the ncentves of agents to partcpate n some centralzed clearnghouses. Ashlag and Roth (2014) study the ncentves of hosptals to partally enroll ther patent-donor pars n the kdney exchange program. Ekmekc and Yenmez (2015) study the ncentves of schools to partcpate n the centralzed school choce clearnghouse. Sönmez and Ünver (2015) propose an ncentve scheme n the kdney exchange program. There are two key dfferences between my framework and the frameworks analyzed by the above studes. Frst, unlke kdney exchange, monetary transfer s possble n the CDS settlement clearnghouse, and, second, partcpaton n the CDS settlement mechansm s the decson of (at least) two agents. 8

9 Agent 1 Agent 1 Agent Agent 3 Agent 2 Agent 2 Agent 3 Case 1 Case 2 Case 3 Fgure 1: An arrow from agent to j means that has sold contracts to j. 3 Leadng Example I llustrate how the CDS contracts work and the man theoretcal contrbuton of the paper wth the followng example. The reader may omt readng ths example and start from Secton 4. Leadng Example: There are three agents, 1, 2, and 3. There s a bond wth a face value of 100. Assume that the ssuer of the bond has defaulted and that the value of the defaulted bond s v(s), where s = (s 1, s 2, s 3 ) s the agents sgnal profle. Agent 1 s a protecton seller, and Agents 2 and 3 are protecton buyers. Agents 2 and 3 may each have 10 CDS contracts wth the protecton seller. These homogeneous CDS contracts are on the bond. There are three possble cases (see Fgure 1): 1. Agents 2 and 3 have 10 CDS contracts wth Agent Agent 2 has 10 CDS contracts wth Agent 1, and Agent 3 has no CDS contracts. 3. Agent 3 has 10 CDS contracts wth Agent 1, and Agent 2 has no CDS contracts. Denote the number of CDS contracts that agent has n case j by n j. Assume n j > 0 f agent s a protecton buyer n case j, n j < 0 f he s a protecton seller, and n j = 0 f he does not have any CDS contracts. For example, n 1 1 = 20 and n1 2 = n1 3 = 10. These contracts are settled by ether physcal settlement or cash settlement. In the case of physcal settlement, the protecton buyer hands the defaulted bond to the 9

10 protecton seller and, n return, receves 100. Therefore, the protecton buyer s payoff from the physcal settlement of one contract s 100 v(s), and the protecton seller s payoff from the physcal settlement s (100 v(s)). In the case of cash settlement, the protecton seller pays the loss to the protecton buyer(s) n the form of monetary transfer. Therefore, f p s a prce for the defaulted bond, the protecton seller pays 100 p to the protecton buyer to settle one CDS contract. If q j s the number of agent s contracts that are settled through cash settlement, and p j s the cash settlement prce, agent s payoff s as follows: u ((n j q j j )(100 v(s)) + q (100 p j )). where u : R R s agent s utlty functon. Agents sgnals about the value of the defaulted bond s ether 0 or 1, s {0, 1} for {1, 2, 3}. Sgnals are ndependently dstrbuted, and s = 1 wth probablty 1. The value of the defaulted bond condtonal 2 on sgnals s as follows: v(s) = 21(2s 1 + s 2 + s 3 ). Agents 1 and 2 each possess nne defaulted bonds. Therefore, some of the contracts must be settled through cash settlement. I descrbe a drect settlement mechansm. A descrpton of a mechansm s a prce and a quantty functon for each agent n each network. The quantty s the number of CDS contracts that are settled by cash settlement, and the prce s the cash settlement prce. Let q j and p j denote the quantty of cash settlement and cash settlement prce, respectvely, for agent n network j. Consder the followng settlement mechansm: 10

11 q 1 1 (s) = 6 + 4s 1 s 2 s 3, p 1 1 (s) = 28 28s 1 + 8s 2 + 8s s 1 s s 1 s s 2s s 1s 2 s 3, q 1 2 (s) = 3 2s 1 + s 2, p 1 2 (s) = 28 28s s s 1s s 3, q 1 3 (s) = 3 2s 1 + s 3, p 1 3 (s) = 28 28s s s 1s s 2, q 2 1 (s) = 4, q2 2 (s) = 4, q2 3 (s) = 0, p2 1 (s) = p2 2 (s) = 42, q 3 1 (s) = 3.5, q3 3 (s) = 3.5, q3 2 (s) = 42, p3 1 (s) = p3 3 (s) = 42. These prces and quanttes guarantee ex-post ncentve compatblty. Moreover, the followng holds: s {0, 1} 3 and j {1, 2, 3} : s {0, 1} 3 and j {1, 2, 3} : 3 =1 3 j=1 q j (s) = 0, (1) q j (s)(100 p j (s)) = 0. (2) Equaton (1) s a market-clearng condton. Note that q j (100 p j ) s the cash transfer that agent receves n network j; therefore, equaton (2) s the budget-balanced condton. In addton to these propertes, f U j (s) s agent s payoff from the settlement mechansm n case j, the followng holds: E s [q j (s)(100 p j (s))] = E s[n j (100 v(s))]. Ths condton s called unbased prcng. Ths means that, from an ex-ante pont of vew, that all contracts are settled by physcal settlement or by cash settlement, wth the prce equal to the value of the defaulted bond. I study agents ncentves to partcpate n the settlement mechansm when an arbtrary group of agents can form coaltons and settle some of ther contracts wth an arbtrary blockng mechansm. As an llustraton, consder the settlement mechansm that I descrbed above. I consder a block by Agents 1 and 3 (see Fgure 2). In ths 11

12 Agent 1 Agent 1 Agent Agent 3 Agent 2 Agent 2 Agent 3 Case 1 Case 2 Blockng Mechansm Fgure 2: Agent 1 and Agent 3 sde settle ten contracts. blockng mechansm, seven contracts are settled by physcal settlement, and three contracts are settled by cash settlement. The cash settlement prces for Agents 1 and 3 are 30 and 38.5, respectvely. The followng nequaltes hold (see the appendx for the calculatons.): E s2 [U 1 1 (0, s 2, 0)] E s2 [U e 1 (0, s 2, 0)], E s2 [U 1 1 (1, s 2, 0)] E s3 [U e 1 (1, s 2, 0)], E s2 [U 1 3 (0, s 2, 0)] E s2 [U e 3 (0, s 2, 0)], E s2 [U 1 3 (0, s 2, 1)] E s2 [U e 3 (0, s 2, 1)]. (3) Therefore, there exsts a Bayesan Nash Equlbrum n whch Agents 1 and 3 choose the ext opton when ther sgnals are 0. In ths blockng, when Agents 1 and 3 vst the blockng mechansm,.e., when (s 1, s 3 ) = (0, 0), the blockng desgner s payoff s 3( ), whch s postve. Gven ths model of agents partcpaton, n ths paper, I answer the followng two questons. Frst, whch settlement mechansm ensures that all agents wll partcpate wth all of ther contracts and s unbased and budget balanced? Second, f I allow agents to settle a number of ther contracts wth some blockng mechansms and take nto account agents payoff from blockng mechansms, whch settlement mechansm s unbased and budget balanced? As I wll show, the answer to both questons s a mechansm whereby the desgner sets constant prce and quantty. 12

13 4 Model Wthout loss of generalty, I assume the face value of the defaulted bond s 100. Each CDS contract has a protecton buyer and a protecton seller. In the case of a default, the protecton buyer should be compensated for the loss on the reference asset (bond) by the protecton seller. These CDS contracts are homogeneous, and each corresponds to one bond. I assume that the default has happened, and I consder the contract settlement problem. Let K be the set of all agents. These agents may have CDS contracts on the bond between each other. A contract matrx specfes the number of contracts of a par of agents. In a contract matrx N = [n, j ], agents, j K have net n, j contracts. Assume n, j > 0 f j s a protecton seller and s a protecton buyer, n, j = 0 f they do not have any CDS contracts, and n, j < 0 f s the protecton seller. 7 Throughout ths paper, I use the words network and contract matrx nterchangeably. Set n to be the net number of contracts that agent j has, n = j K n, j. Each agent has a number of defaulted bonds; assume agent has b 0 defaulted bonds. Each agent has a prvate sgnal for the value of the defaulted bond. Agent s sgnal s drawn from S = [0, 1]. Gven s [0, 1] k, a profle of agents sgnals, the expected value of the defaulted bond s v(s). I assume v(s) s non-decreasng and contnuous n agents sgnals. If A K s a subset of agents, set S A = S. Gven B A K and s S A, let π B (s) S B be the projecton of s on ts B elements. For economy of exposton, I use the notaton s and s, j for π K\{} (s) and π K\{, j} (s) respectvely. CDS contracts are settled by ether physcal settlement or cash settlement. In the case of physcal settlement, the protecton buyer hands n the defaulted bond to the protecton seller and, n return, receves 100. Therefore, the protecton buyer s payoff from the physcal settlement of one contract s 100 v(s), and the protecton seller s payoff from the physcal settlement s (100 v(s)). In the case of cash settlement, the protecton seller pays the loss to the protecton buyers. Therefore, f p s the prce of the defaulted bond, and q s the number of agent s contracts that are settled through 7 Note that n, j + n j, = 0. 13

14 cash settlement, the agent s payoff at sgnal profle s S K s as follows: 8 u (b v(s) + (n q )(100 v(s)) + q (100 p )). where u : R R s agent s utlty functon. The utlty functon, u (.), s strctly ncreasng and s normalzed such that u (0) = 0. Note that one can rewrte the payoff of agent as follows: u (b v(s) + n (100 v(s)) + q (v(s) p )). where the second term nsde u (.) s hs payoff f all of the contracts are physcally settled or f the prce s equal to the value of the defaulted bond.. The thrd term can be thought of as the bas. In general, a settlement mechansm s a reallocaton of the defaulted bonds and monetary transfer. Note that combnatons of physcal settlement and cash settlement can generate any allocaton of the defaulted bonds and monetary transfer. Let K(N) be the set of agents who have some CDS contracts n network N, formally, K(N) = { K n, j 0 for some j K}. In ths envronment, a drect settlement mechansm takes the network and the profle of reported sgnals as nputs and returns a cash settlement quantty and a cash settlement prce for each agent. A drect settlement mechansm conssts of functons q N : S K R and p N : S K R for all agents K and network N. The cash settlement quantty s q N, and pn s the cash settlement prce for agent n network N. Let p N = (p N (.)) K and q N = (q N (.)) K be the profle of prce and quantty functons when the network s N and P = (p N ) and Q = (q N ) be the prce and quantty profles. Note that I allow agents to have dfferent cash settlement prces; n other words, I am not restrctng the case to p N = p N j for all, j K. Therefore, any reallocaton of money and the defaulted bonds can be generated by cash settlement and physcal settlement. The defaulted bonds that are used for physcal settlement must be cleared, formally, for 8 Agent gves/receves n q of hs defaulted bonds and receves/pays the face value of the bond; n addton, he receves/pays hs loss for the rest of the contracts. 14

15 all networks N and s S K : (n q N (s)) = 0. K Ths s equvalent to K q N (s) = 0. Ths mechansm s ex-post ncentve compatble f, for all networks N, K, and s = (s, s ) & s = (s, s ) S K, the followng holds: u (b v(s) + (n q N (s))(100 v(s)) + qn (s)(100 pn (s))) u (b v(s) + (n q N (s ))(100 v(s)) + q N (s )(100 p N (s ))). (4) Ths means that agent wth prvate nformaton s should not fnd t proftable to msreport hs sgnal as s, when all other agents sgnal profles are s. Because the utlty functon s ncreasng, nequalty (4) s equvalent to: (n q N (s))(100 v(s)) + qn (s)(100 pn (s)) (n q N (s ))(100 v(s)) + q N (s )(100 p N (s )). I use the notaton (P, Q, U) for a settlement mechansm wth prce, quantty, and payoff functons p N (.), qn (.), and U N (.) for all networks N and agents K. Agent s payoff n network N, when all agents are reportng ther sgnals truthfully, s as follows: 9 U N (s) = u (b v(s) + n (100 v(s)) + q N (s)(v(s) pn (s))). For economy of exposton, I defne Λ N (s) = n (100 v(s))+q N (s)(v(s) pn (s)) to be the rsk-neutral payoff of agent from settlng the CDS contracts when the network s N; therefore, U N (s) = u (b v(s) + Λ N (s)).10 Note that the cash settlement part of the rsk-neutral payoff, q N (s)(v(s) pn (s)), s a monetary transfer to the agent. The mechansm s ex-post budget balanced f, for all networks, N and s S K, the equalty K q N (s)(v(s) pn (s)) = 0 holds for all sgnal profles s. Because n s sum to zero, ths s equvalent to K Λ N (s) = 0 for all s. It s ex-post weakly budget balanced f, for all networks, N and s S K, the nequalty K Λ N (s) 0 holds. It s ex-ante budget balanced f E ρ [ K Λ N (s)] = 0 for all networks N and sgnal profles s. I defne 9 In general U N (s) = u (X (s) + n (100 v(s)) + q N (s)(v(s) pn (s))) where X (s) s agent s payoff from hs all other assets. Ths generalzaton would not change the subsequent results n the paper. 10 I am not restrctng attenton to rsk-neutral agents. 15

16 ex-ante weakly budget-balanced mechansms smlarly. I restrct attenton to ex-post ncentve compatble and ex-ante weakly budget-balanced settlement mechansms. A mechansm has no short sell f q (s) n b for all K and s S K. Agents pror about the sgnals s denoted by µ; the probablty of observng sgnal profle s s µ(s). The desgner does not know µ, but she holds a pror ρ about µ. Let κ be the support of the desgner s belef. Let ρ be the desgner s pror about the agents pror, that s, ρ s a probablty dstrbuton over agents prors, µ. I use E ρ to refer to the expected value symbol, gven the desgner s nformaton. I mantan the followng assumpton throughout the paper. Assumpton 4.1. Full Rank Belef: If functon x : S R satsfes E µ [x(s)] = 0 for almost all µ n support of the desgner s belef, then x(s) = 0 for all s S. Ths assumpton means that the desgner s suffcently unaware of the agents pror. Ths assumpton s volated f, for example, the desgner knew the expected value of the agents sgnals. Ths assumpton would be satsfed, however, f the desgner beleves that the agents pror s a small perturbaton of some dstrbuton. An agent s short sellng f he ends up wth a net negatve number of defaulted bonds after the settlement, that s, b < n q. The problem s trval f the market facltates short sellng; n ths event, the agent should smply set q N (s) = 0 and settle all contracts wth physcal settlement. However, short sellng n ths market s generally dffcult or mpossble. Therefore, any settlement should satsfy the no short sell constrant, whch s q n b for all K. 5 Desred Propertes 5.1 Partcpaton As of 2009, all CDS contracts are pegged to the result of the centralzed CDS settlement mechansm. ISDA has argued that such polcy ensures certanty, consstency, enhanced transparency, and lqudty. Even though CDS contracts are hardwred to the outcome of the settlement mechansm, f both partes of a CDS contract agree, they are allowed 16

17 to sde settle ther contracts by a concerted settlement procedure. Hence, I develop a model to understand how agents partcpate n the settlement mechansm. 11 Lack of partcpaton harms the transparency settlement procedure snce t drves lqudty away from the aucton. Moreover, agents may manpulate the result of aucton through lack of partcpaton n the clearnghouse. Ths s mportant because lqudty and transparency were among the man reasons offered for a central mechansm to settle CDSs n the frst place. An agent who does not have any CDS contracts s not oblgated to partcpate n the settlement mechansm; however, he can choose to partcpate f there s a postve payoff. Ths motvates the followng defnton: A mechansm s ex-post ndvdually ratonal for agents wthout contracts f an agent wthout a contract leaves the aucton wth a non-negatve payoff. In other words, for all networks N, all sgnal profles s S K, all agents K that satsfy n, j = 0 j K, the nequalty Λ N (s) 0 holds. It s nterm ndvdually ratonal f all agents wthout CDS contract leave the clearnghouse wth a non-negatve expected utlty; n other words, the nequalty E s [U N (s)] E s [u N (b v(s))] holds for all agents K that satsfy n, j = 0 j K. I formally model the partcpaton decson of agents who have CDS contracts. In standard mechansm desgn, agents can choose whether to partcpate n the mechansm. They partcpate when they have a non-negatve payoff from partcpatng n the mechansm. Partcpaton n ths envronment s dfferent for an mportant reason. Agents wth CDS contracts are requred to partcpate by default; however, f both partes of a CDS contract agree, they can settle some of ther contracts through another mechansm. In ths envronment, agents outsde optons are no longer exogenous; rather, they depend on ther sgnals as well as other agents sgnals. In other words, f an agent agrees to settle a CDS contract though another mechansm, t reveals nformaton about hs own prvate sgnal. I do not assume the number of contracts that a par of agents has s prvate nformaton; rather, agents are legally allowed to not brng a number of ther contracts to the settlement mechansm f all partes of these contracts agree See 12 Snce contracts are hardwred to a centralzed clearnghouse, the desgner of the mechansm knows that network of contracts. 17

18 All of the results of the paper are proven when I only consder sde settlements by a par of agents. To provde a more general model of partcpaton, however, I consder sde settlements by multple pars of agents. Due to blateral decsons that groups of agents may make pror to partcpatng n the mechansm about the number of contracts, the desgner may face contract matrces that are dfferent from the orgnal network of contracts. When the contract matrx s N, f a group of agents choose to settle some of ther contracts outsde of the settlement mechansm, the desgner faces a new contract matrx, namely M. In ths case, M s a reducton of N. Formally, M = [m, j ] s a reducton of N = [n, j ] f, for all, j K, the nequalty m, j n, j holds. I use the notaton M N, f M s a reducton of N. Let A be the set of all agents who choose to settle some of ther contracts outsde of the settlement mechansm. Note that A = K(M N), where M N s a contract matrx n whch agents and j have m, j n. j contracts. A blockng mechansm can be vewed as a settlement mechansm when the set of agents s A and the network of contracts s N M. The man dfferences are that t does not have to be budget balanced and does not have to clear the number of defaulted bonds used. Nevertheless, the blockng mechansm must provde the desgner wth an expected postve surplus. A blockng mechansm has an mportant role: It settles all contracts that were not brought to the settlement mechansm. I use the notaton (P, Q, U ) for the blockng mechansm. Let U e be the payoff of an agent from jonng the blockng mechansm,.e.: U e (s) = u [b v (s) + Λ (s) + Λ M (s)] (5) where Λ (s) = (n m q (s))v(s) + q (s)(100 p (s)). I present two models for the blockng: () complete nformaton case and () ncomplete nformaton case Complete Informaton Case Agents n A, for a subset of ther types, block the settlement mechansm and reduce the network from N to M f there exsts a blockng mechansm (P, Q, U ) and a prescrbed non-zero measure subset of types S S for agents n A, such that the followng holds: 18

19 1. For all A and s A S, the followng nequalty holds: E s A [U N (s A, s A )] E s A [U e (s A, s A )] (6) where U e s agent s payoff from jonng the blockng mechansm; see equaton (5). Agents n A jon the coalton when ther types are n the prescrbed subset of types. Inequalty (6) means that, f all sgnals of agents n A are n the prescrbed sets, then the expected payoff of all agents A from the settlement mechansm wth network N s not larger than an agent s total payoff from the blockng mechansm and the payoff from the settlement mechansm wth network M. Ths gves agent an ncentve to jon the coalton when all blockng agents sgnals are n the prescrbed sets. 2. For all s A S K, such that π A\{} (s A ) j A\{} S j and π {}(s) S \S, the followng nequalty holds: E s A [U N (s A, s A )] E s A [U e (s A, s A )]. (7) Inequalty (7) means that f agent s sgnal s not n S, and the sgnal of all other agents n A are n the prescrbed sets, then agent s expected payoff from the settlement mechansm wth network N s not smaller than the agent s total payoff from the blockng mechansm and hs payoff from the settlement mechansm wth network M. 3. The blockng desgner has an expected postve payoff. Formally, the followng nequalty must hold: E[ Λ (s) π A (s) S A ] > 0. (8) Inequaltes (6) and (7) mean that agents n A, for a subset of ther prvate sgnals, may form a coalton and settle some of ther contracts wth the blockng mechansm. Agents n the coalton, A, choose between (P N, Q N, U N ) and (P M, Q M, U M ) plus the blockng mechansm. Consder a game n whch agents n A choose whether to ext the mechansm. The block s formed only when all agents choose to ext. If agents choose 19

20 the strategy of extng only f the type s n the prescrbed set, then these nequaltes guarantee that agent A, upon learnng the types of all agents n A \ {}, would not regret hs partcpaton decson. The mechansm s ex-post unraveled f blockng exsts. To understand nequalty (8), thnk of the blockng desgner as an agent. Note that, n general, the blockng mechansm does not have to balance the budget or clear the number of defaulted bonds that are used for physcal settlement. Because the blockng mechansm desgner may buy defaulted bonds from agents n A. Inequalty (8) means that the blockng desgner s expected payoff, condtonal on the event that the block s formed, must be postve. The frst term that appears n the summaton s the blockng desgner s payoff from defaulted bonds, and the second term s hs payoff from the monetary transfer. The strct nequalty (8) guarantees that no mechansm s blocked by the null mechansm. 13 A settlement mechansm (P, Q, U) s ex-post unravel-proof f, for any par of contract matrces M and N and subset of agents A K, where agents n A reduced N to M, agents n A cannot form a block for a subset of ther types. A settlement mechansm s weakly ex-post unravel-proof f there s no par of agents who could form a block. One can strengthen the noton of unravel-proof to consder blocks by more than two agents. The results of the paper, however, are correct n ether case. The followng proposton provdes suffcent condtons for unravel proofness. Proposton 5.1. If agents are rsk neutral and the followng nequalty holds, then there s no ex-post unravelng n whch agents n A reduce the network from N to M. E s A [Λ N (s A, s A )] E s A [Λ M (s A, s A )] for almost all s A S A (9) Proof. See the Appendx for a proof. Note that nequalty ((9)) means that the sum of agents rsk-neutral expected payoffs that are n A s greater when the network s N compared to that of network M. Snce sum of agents payoff n the sde settlement s negatve, the nequalty mples 13 A mechansm where all the quanttes are equal to zero. 20

21 that total payoff of agents n A does not mprove f they choose to sde settle. Hence, sde settlement s not benefcal for all agents n A. The contrbuton of ths paper s that I consder the possblty that the outsde opton of an agent depends on the set of other agents who exercse ther outsde opton. Ths s because the agents who choose to ext may decde to band together and settle some of ther contracts among themselves through a dfferent mechansm. Such a possblty was frst consdered n the lterature on cooperatve games and culmnated n the noton of the core. The noton of unravelng presented n ths paper s related to the block n matchng theory and the block n cooperatve game theory. The dfference between blockng n matchng theory and the noton of unravelng s that, n my setup, the network s predetermned and only prce and quantty of cash settlement are chosen through a mechansm. Unravel-proofness s a property of a mechansm that s defned over networks, but stablty s defned over a possble match. Unlke smlar concepts n corporate game theory, the noton of unravel-proofness can be naturally extended to envronments wth ncomplete nformaton. A generalzaton of the unravel-proofness noton to envronments wth ncomplete nformaton s presented n the followng secton Incomplete Informaton Case I extend the blockng mechansm defnton to envronments n whch agents who partcpate n a block do not know each other s sgnals but share a pror. When agents make decsons about whether to jon the blockng mechansm, they update ther belef upon observng other agents decsons. An agent s choce to partcpate n a blockng mechansm reveals nformaton about hs prvate sgnal. Other agents take ths nto account when makng ther decsons. The noton of nterm blockng s defned as follows: For all A, subsets S S are called the prescrbed sets. Let event E be defned as follows: E = {s S K π A\{} (s) S j, π {} (s) S \ S }. j A\{} 21

22 Defne event E as E = {s S K π A (s) S j}. j A Note that E s the event that the prvate sgnals of all agents n A, except for agent, are n the prescrbed sets. Event E s the event that the prvate sgnals of all agents n A are n the prescrbed sets. The nequaltes n the blockng mechansm defnton, nequaltes (6), (7), and (8), change to the followng nequaltes: E s [U N (s) E] E s [U e (s) E], (10) E s [U N (s) E ] E s [u e (s) E ]. (11) E[ Λ (s)) E] > 0. (12) To nterpret nequaltes (10) and (11), magne a game whose players are agents n A. These agents, after observng ther prvate sgnals, choose whether to partcpate n the blockng mechansm. If all of these agents decde to partcpate n the blockng mechansm, ther payoff s that of the blockng mechansm plus that of the settlement mechansm when the network s M. If some decde not to partcpate n the blockng mechansm, ther payoff s only that of the settlement mechansm when the network s N. The mechansm s unraveled f ths game has a Bayesan Nash Equlbrum n whch agents n A, for a subset of ther types, choose the blockng mechansm. Wth the new defnton of a block, unravel-proofness s naturally redefned. A mechansm s nterm unravel-proofness f there s no par of agents who could form an nterm block. One can strengthen the noton of unravel-proofness to consder blocks by more than two agents. The results of the paper, however, are correct n ether case. Proposton 5.2. A settlement mechansm (P, Q, U) that settles all contracts wth cash settlement wth a constant prce,.e., p N = p and q N = n, s nterm unravel-proof. Moreover, f agents are rsk neutral and the followng nequalty holds, then there s no 22

23 nterm unravelng n whch agents n A reduce the network from N to M: E s A [Λ N (s A, s A )] E s A [Λ M (s A, s A )] for almost all s A S A (13) Proof. The proof s an adaptaton of the proof of proposton 5.1 and, hence, t s omtted. 5.2 Unbasedness As I stated n Secton 2, several authors have crtczed the current settlement mechansm for underprcng the underlyng bond. The current mechansm sets a based prce, whch results n a dfference between physcal settlement and cash settlement. Because t s not known at the tme of contractng a CDS whether the CDS contract wll be settled by cash settlement or physcal settlement, the based prcng results n uncertanty and, hence, effcency loss. I look for mechansms that overcome ths ssue. Due to the nformaton rent, based on agents prvate nformaton about the value of the defaulted bond, as dscussed n Secton 1, ex-post correct prcng s not possble unless all contracts are physcally settled. I defne unbasedness n two ways, as seen below Weakly Unbased A weakly-unbased mechansm s one that does not msprce the defaulted bond n expectaton. Formally, mechansm (P, Q, U) s weakly unbased f, for all networks N and agents K: E[Λ N (s)] = [n (100 v(s))]. (14) Equaton (14) means that, from an ex-ante perspectve, the agents rsk-neutral payoff from the settlement mechansm s the same as ther payoff from physcal settlement of all contracts or cash settlement wth the prce equal to the value of the defaulted bond. Note that, because both prce and quantty may depend on the sgnal profle, ths condton s not equvalent to E[p N (s)] = E[v]. Observaton 5.1. If E[Λ N (s)] = n (100 E[v]), then E ρ [ K Λ N (s)] = ( K n )(100 23

24 E[v]) = 0. Therefore, a mechansm s weakly ex-ante budget balanced f t s weakly unbased Unbased If the mechansm s not strategy-proof, some agents may settle some of ther contracts outsde of the settlement mechansm. Therefore, agents total payoff s not only the payoff from the settlement mechansm; t also should nclude the payoff from the blockng mechansms. A mechansm s unbased f the agents total payoff, ncludng the payoff from the settlement mechansm and the blockng mechansms, from an axante perspectve, s equal to the agents payoff from physcal settlement of all contracts or cash settlement wth the correct prce. To formally defne unbased mechansms, I frst defne the noton of partcpatonchoce. For some networks, a par of agents may fnd t proftable to settle some of ther contracts wth a blockng mechansm. Usng the partcpaton model ntroduced n Secton 5.1, I allow pars of agents to take these actons; a mechansm s unbased f t s weakly unbased, regardless of these actons. The results of the paper are vald f one consders only coaltons by par of agents n the defnton of unbasedness. To provde a rcher model, however, I consder groups of agents. Consder an ex-post ncentve-compatble settlement mechansm, namely (P, Q, U), whch may not be unravel-proof. Let Ω be the set of all possble networks. I allow for several coaltons to coexst; let P N be the set of blockng coaltons when the true network of contract s N. Hence, a partcpaton-choce s a collecton of sets (P N ) N Ω, whereby elements of P N capture the sub-networks that jon a coalton f the true network of the contract s N. For all networks N, each element of c P N has a network c N N, a subset of agents c A K(c N ), a set of type profles S c S for all c A, and a blockng mechansm for the c N network, ( ˆP c, ˆQ c, Û c ). Let c t = c A S c for all c A. Ths partcpaton-choce should satsfy three condtons: 1. Let A N (s) be the set of all coaltons that are formed when the sgnal profle s s S K. Formally, A N (s) = {c A N π ca (s) c t }. It must be that N(s) = c A N (s) c N N. Ths means that the network that s left 24

25 after coaltons are formed s a reducton of N. 2. Agent s payoff from ths partcpaton-choce when the network s N s u (Λ P N (s)), where: Λ P N (s) = c A N (s): c A ˆΛ c (s) + ΛN N(s) (s). Jonng the coaltons for the prescrbed types must be a Bayesan Nash equlbrum. Formally, for all networks N and c A N, let events E c and E c be defned as: E c = {s S K π Ac (s) E c = {s S K π Ac (s) j K(c N )\{} j K(c N ) For all c A, the followng nequaltes should hold: S c j }, S c j, π {}(s) S \ S c }. E s [u P N (s) E c ] E s [u P N\{c} (s) E c ], E s [u P N (s) Ec] E s [u P N\{c} (s) Ec]. 3. Gven a coalton c A N, when the sgnal profle s s S K, agent K(c N ) enters the coalton c f π ca (s) S c t. The blockng desgner s expected payoff from the blockng mechansm should be postve, condtonal on the event that all agents n c A jon the blockng mechansm. Ths s smlar to nequalty (12), whch ensures that the blockng mechansms are self-sustanng. Consder a partcpaton-choce for each agents pror µ. The mechansm s unbased f, for all networks N, all agents K, and all partcpaton-choces, the followng holds: E ρ [Λ P N (s)] = E ρ [n (100 v(s))]. (15) Ths condton ndcates that, from an ex-ante perspectve, agent s total payoff from the blockng mechansm and the settlement mechansm for all possble partcpatonchoces s equal to the agent s payoff from physcal settlement of all contracts. Note that, because I allow some contracts to be settled outsde of the settlement mechansm, 25

26 the noton of budget balancedness must be modfed. The mechansm s weakly budget balanced regardless of agents partcpaton-choce f, for all networks N and all partcpaton-choces, the followng holds: E ρ [ K(N) Λ N N(s) (s)] Robustness Wth Respect to Network The current mechansm n use takes only the net number of contracts, and not the detals of the network of contracts, as an nput. To elaborate, consder the followng example: Agent 1 has sold 500 CDSs to Agent 2, Agent 2 has sold 400 CDSs to Agent 3 and Agent 3 has sold 300 CDSs to Agent 1. The clearnghouse offsets 300 contracts n ths loop, hence ths network s treated as the case where agent 2 has 200 long CDS postons (CDS buyer) and agent 1 and 3 each have 100 short postons (CDS sellers). Ths motvates the property of robust wth respect to network. Here s the formal defnton: A mechansm s robust wth respect to the network f, for all pars of networks M = [m, j ] and N = [n, j ] that satsfy j K m, j = j K n, j for all K, the followng equaltes hold: K, q N (s) = qm (s) and p N (s) = pm j (s). Snce nettng cancel out some of the postons, ths property lowers the transactons costs and contemporary rsk. 6 Results I provded a model of partcpaton when groups of agents can jon the blockng mechansm. Nevertheless, all of the results of the paper are proven when only a par of agents engage n sde settlement. I look for mechansms that satsfy a combnaton of propertes that I have ntroduced n the prevous secton. Before presentng the characterzaton results, I ntroduce a class of mechansms called posted-prce mechansms. A posted prce mechansm sets prces and quanttes that do not depend on agents sgnals. More over there exsts q n for all n Z, satsfyng q n + q n = 0 for all n Z, 26