Counterparty Risk and the Impact of Collateralization in CDS Contracts

Transcription

1 Conerpary Risk and he Impac of Collaeralizaion in CDS Conracs Tomasz R. Bielecki Deparmen of Applied Mahemaics, Illinois Insie of Technology, Chicago, IL, USA Igor Cialenco Deparmen of Applied Mahemaics, Illinois Insie of Technology, Chicago, IL, USA Ismail Iyignler Deparmen of Applied Mahemaics, Illinois Insie of Technology, Chicago, IL, USA April 13, 2011 Absrac We analyze he conerpary risk embedded in CDS conracs, in presence of a bilaeral margin agreemen. Firs, we invesigae he pricing of collaeralized conerpary risk and we derive he bilaeral Credi Valaion Adjsmen CVA, nilaeral Credi Valaion Adjsmen UCVA and Deb Valaion Adjsmen DVA. We propose a model for he collaeral by incorporaing all relaed facors sch as he hresholds, haircs and margin period of risk. We derive he dynamics of he bilaeral CVA in a general form wih relaed jmp maringales. We also inrodce he Spread Vale Adjsmen SVA indicaing he conerpary risk adjsed spread. Conerpary risky and he conerpary risk-free spread dynamics are derived and he dynamics of he SVA is fond as a conseqence. We finally employ a Markovian copla model for defal inensiies and illsrae or findings wih nmerical resls. TRB and IC acknowledge sppor from he NSF gran DMS

2 Conens 1 Inrodcion 2 2 Pricing Conerpary Risk: CVA, UCVA and DVA Dividend Processes and Marking-o-Marke Bilaeral Credi Valaion Adjsmen Unilaeral CVA and Deb Vale Adjsmen CVA via Credi Exposres Dynamics of CVA Dynamics of CVA when he immersion propery holds Fair Spread Vale Adjsmen SVA Dynamics Mlivariae Markovian Defal Model Resls Conclsion 23 1 Inrodcion No very long afer he collapse of presigios insiions like Long-Term Capial Managemen, Enron and Global Crossing, he financial indsry has again winessed dramaic downfalls of financial insiions sch as Lehman Brohers, Bear Searns and Wachovia. These recen collapses have sressed o he imporance of measring, managing and miigaing conerpary risk appropriaely. Conerpary risk is defined as he risk ha a pary in an over-he-coner OTC conrac will defal and will no be able o honor is conracal obligaions. Since he exchange-raded derivaive conracs are sbjec o clearing by he exchange, conerpary risk arises from OTC derivaives only. The main challenge in he conerpary risk assessmen and hedging is ha he exposres of OTC derivaives are sochasic and involve dependencies and sysemic risk facors sch as wrong way risks; he addiional level of complexiy is inrodced by risk miigaion echniqes sch as collaeralizaion and neing. Therefore, one needs o model poenial fre exposres and o price he conerpary risk appropriaely according o margin agreemens ha nderlie he collaeralizaion procedres. In his paper, we analyze he conerpary risk in a Credi Defal Swap CDS conrac in presence of a bilaeral margin agreemen. There are hree risky names associaed wih he conrac: he reference eniy, proecion seller he conerpary and he proecion byer he invesor. Conrary o he common approach which sars wih defining he Poenial Fre Exposre PFE and derives he Credi Valaion Adjsmen CVA as he price of he conerpary risk, we find he CVA as he difference beween he marke vales of a conerpary risk-free and a conerpary risky CDS conracs and dedc he relevan credi exposres accordingly. We consider he problem of bilaeral conerpary risk assessmen; ha is, we consider he siaion where he wo conerparies of he CDS conrac, i.e. he invesor and he conerpary, are sbjec o defal risk in a conerpary risky CDS conrac. 2

3 We focs on he collaeralized conracs, where, as a vial risk miigaion ool, a bilaeral margin agreemen is in force, and i reqires he conerpary and he invesor o pos collaeral in case heir exposre exceeds specific hreshold vales. We propose a model for he collaeral by incorporaing all relaed facors, sch as hresholds, margin period of risk and minimm ransfer amon. Then, we derive he dynamics of he bilaeral CVA which is essenial for dynamic hedge of he conerpary risk. We also compe he decomposiion of he fair spread for he CDS, and we analyze so called Spread Vale Adjsmen SVA. Essenially, SVA represens he adjsmen o be made o he fair spread o incorporae he conerpary risk ino he CDS conrac. Using he bilaeral CVA formla, we derive relevan formlas for assessmen of credi exposres, sch as PFE, Expeced Posiive Exposre EPE and Expeced Negaive Exposre ENE. In or model, he dependence beween defals and he wrong way risk is represened in a Markovian copla framework ha accons for simlaneos defals among he hree names represened in a CDS conrac. In his way, or model akes broader sysemic risk facors ino accon and qanifies he wrong way risk and he doble defals in a angible manner. This paper is organized as follows. In secion 2, we firs define he dividend processes regarding he conerpary risky and he conerpary risk-free CDS conrac in case of a bilaeral margin agreemen. We also define he CVA, UCVA and he DVA erms as well as he credi exposres sch as PFE, EPE and ENE. We hen prove he dynamics of he CVA in secion 3. Moreover, we find he fair spread adjsmen erm and is dynamics. In secion 4, we simlae he collaeralized exposres and he CVA sing or Markovian copla model of defal dependence. 2 Pricing Conerpary Risk: CVA, UCVA and DVA We consider a sandard CDS conrac, and we label by 1 he reference name, by 2 he conerpary he conerpary, and by 3 he invesor. Each of he hree names may defal before he mariy of he CDS conrac, and we denoe by τ 1, τ 2 and τ 3 heir respecive defal imes. These imes are modeled as nonnegaive random variables given on a nderlying probabiliy space Ω, G, Q. We le T and κ o denoe he mariy and he spread of or CDS conrac, respecively. We assme he recovery a defal covenan; ha is, we assme ha recoveries are paid a imes of defal. We inrodce righ-coninos processes H i by seing H i =I {τi } and we denoe by H i he associaed filraions so ha H i = σ H i : for i = 1, 2, 3. We assme ha we are given a marke filraion F, and we define he enlarged filraion G = F H 1 H 2 H 3, ha is G = σ F H 1 H 2 H 3 for any R+. For each R + oal informaion available a ime is capred by he σ-field G. In pariclar, processes H i are G-adaped and he random imes τ i are G-sopping imes for i = 1, 2, 3. Nex, we define he firs defal ime as he minimm of τ 1, τ 2 and τ 3 : τ = τ 1 τ 2 τ 3 ; he corresponding indicaor process is H = I {τ }. In addiion, we define he firs defal ime of he wo conerparies: τ = τ 2 τ 3, and he corresponding indicaor process H = I {τ }. We denoe by B he savings accon process, ha is B = e 0 r sds, 3

4 where he F-progressively measrable process r models he shor-erm ineres rae. We also poslae ha Q represens a maringale measre associaed wih he choice of he savings accon B as a discon facor or nmeraire. 2.1 Dividend Processes and Marking-o-Marke In his paper, all cash flows and he prices are considered from he perspecive of he invesor. We sar by inrodcing he conerpary-risk-free dividend process D, which describes all cash flows associaed wih a conerpary-risk-free CDS conrac; 1 ha is, D does no accon for he conerpary risk. Definiion 2.1 The cmlaive dividend process D of a conerpary risk-free CDS conrac maring a ime T is given as, D = for every [0, T ], where δ 1 : [0, T ] R is an F-predicable processes. ]0,] δdh 1 1 κ 1 H 1 d, 1 ]0,] Process δ 1 represens he loss given defal LGD; ha is δ 1 = 1 R 1, where R 1 is he fracion of he nominal ha is recovered in case of he defal of he reference name. We assme ni nominal, for simpliciy. The ex-dividend price processes of he conerpary risk-free CDS conrac, say S, describes he crren marke vale, or he mark-o-marke vale his conrac, Definiion 2.2 The ex-dividend price process S of a conerpary risk-free CDS conrac maring a ime T is given by, S = B E Q B 1 dd G, [0, T ]. 2 ],T ] Remark 2.3 Accordingly, we define he cmlaive dividend price process, say Ŝ, of a conerpary riskfree CDS conrac as Ŝ = S + B ]0,] dd, [0, T ]. Now, we are in posiion o define he dividend process D C of a conerpary risky CDS conrac, ha is he CDS conrac ha accons for he conerpary risk associaed wih he wo conerparies of he conrac. Definiion 2.4 The dividend process D C of a T -mariy conerpary-risky CDS conrac is given as D C = ]0,] + + C dh + δ 1 1 H dh 1 + δ 2 1 H dh 2 ]0,] ]0,] δ 3 1 H dh 3 + δ 4 1 H d [ H 2, H 3] ]0,] δ 5 1 H d [ H, H 1] κ 1 H d, [0, T ], ]0,] ]0,] ]0,] 3 1 We shall someimes refer o sch conrac as o he clean conrac. 4

5 where δ i : [0, T ] R is F-predicable processes for i = 1,..., 5 and C : [0, T ] R is a F-predicable process represening he collaeral amon kep in he margin accon. Margin accon is a conracal ool ha spplemens he CDS conrac so o redce poenial losses ha may be incrred by one of he conerparies in case of he defal of he oher conerpary, while he CDS conrac is sill alive. For he deailed descripion of he mechanics of he collaeral formaion in he margin accon we refer o secion 2.2 see also [BC11]. In case of any credi even, associaed wih he collaeralized CDS conrac, he firs cashflow ha akes place is ransfer of he collaeral amon; for example, in case when he nderlying eniy defals a ime = τ = τ 1, before any of he conerparies defals he collaeral in he margin accon is acqired by one of he conerparies depending on he sign of C τ. Ths, consisenly wih he convenions of he so called close-o cashflows cf. [BC11] we define δ i s as follows: We se δ 1 = δ 1 C. This is becase afer he collaeral ransfer he conerpary pays he remaining recovery amon δ 1 C. A ime = τ = τ 2, when he conerpary defals, hen, afer he collaeral ransfer akes place, if he ncollaeralized mark-o-marke MM of he CDS conrac is negaive, ha is if S +I {=τ 1 }δ 1 C < 0, 2 he invesor closes o he posiion by paying he defaling conerpary he ncollaeralized MM. If he ncollaeralized MM is posiive, he invesor closes o he posiion and receives a fracion R 2 of he ncollaeralized MM from he conerpary. Therefore, in his case, he close-o paymen is defined as, δ 2 = R 2 S + I {=τ 1}δ 1 C + S + I {=τ 1}δ 1 C. In case of invesor defal, ha is a ime = τ = τ 3, if he ncollaeralized MM is posiive, ha is if S + I {=τ 1}δ 1 C > 0, he conerpary closes o he posiion by paying he ncollaeralized MM. If he ncollaeralized MM is negaive, he conerpary receives a fracion R 3 of he ncollaeralized MM. Hence, he close-o paymen is defined as, δ 3 = S + I {=τ 1 }δ 1 C + R3 S + I {=τ 1 }δ 1 C. If he invesor and he conerpary defal simlaneosly a ime = τ = τ 2 = τ 3, if he ncollaeralized MM negaive, he conerpary receives a fracion R 3 of he ncollaeralized MM; however, if he ncollaeralized MM is posiive, he invesor receives a fracion R 2 of he ncollaeralized MM. Therefore, we se, δ 4 = S + I {=τ 1 }δ 1 C. If = τ = τ = τ 1, ha is when he invesor or he conerpary defal simlaneosly wih he reference eniy, invesor receives a fracion R 2 of he remaining recovery amon, δ 1 C +, when he conerpary defals. Likewise, if he invesor defals, he conerpary receives a porion R 3 of he remaining recovery amon, δ 1 C. The close-o paymen in join defals inclding he nderlying eniy has he form, δ 5 = δ 1 C. 2 The erm I {=τ 1 }δ 1 represens he exposre in case when he conerpary and he nderlying eniy defal simlaneosly. 5

6 We are now ready o define he price processes associaed wih a conerpary risky CDS conrac. Definiion 2.5 The ex-dividend price process S C of a conerpary risky CDS conrac maring a ime T is given as, S C = B E Q ],T ] dd C The cmlaive price process ŜC of a conerpary risky CDS conrac is given by, Ŝ C = S C + B ]0,] G, [0, T ]. 4 dd C, [0, T ]. The conerpary poss collaeral when he invesor makes a margin call, which happens when invesor s exposre exceeds he conerpary s hreshold pls he MTA. Likewise, he invesor delivers collaeral when he conerpary makes a margin call, which happens when proecion seller s exposre o he byer exceeds he byer s hreshold pls he MTA cf. [ISD05], pages Since we are doing or analysis from he poin of view of he byer, we se he conerpary s hreshold Γ cpy o be a non-negaive consan, and he invesor s hreshold Γ inv o be a non-posiive consan. In accordance wih he above discssion we define collaeral process as follows, Definiion 2.6 The collaeral process is given as, C = I {S>Γ cpy+mt A} S Γ cpy + I {S<Γ by MT A} S Γ inv, on he se { < τ}, and, C = I {Sτ >Γ cpy +MT A} S τ Γ cpy + I {Sτ <Γ by MT A} S τ Γ inv, on he se {τ < τ+ }. Remark 2.7 Noe ha he collaeral consrcion described above is cash based. The ne cash vale of he collaeral porfolio is deermined sing haircs. The hairc or, valaion percenage describes he amon ha will be charged from a pariclar collaeral asse. Effecive vale of he collaeral asse is deermined by sbracing he mark-o-marke vale of he asse mliplied by an appropriae hairc cf. [ISD05], page 67. Therefore, he haircs applied o collaeral asses shold reflec he marke risk on hose asses. The hairc is defined as a percenage, where 0% hairc implies complee mark-o-marke vale of he asse o be sed as collaeral wiho any disconing. Governmen secriies having high credi raing sch as Treasry bonds and Treasry bills are sally sbjeced o 1% o 10% hairc, while for more risky, volaile or illiqid secriies, sch as a sock opion, he hairc migh be as high as 30%. The only asse ha is no sbjeced o any hairc as collaeral is cash where sally boh paries mally agree he se of an overnigh index rae cf. [ISD10], page 27. The erm valaion percenage is also sed in Credi Sppor Annex CSA docmens. The valaion percenage defines he amon ha he marke vale of he asse is mliplied by o yield he effecive collaeral vale of he asse. Hence, he following relaion holds beween he hairc and he valaion percenage, V P = 1 h 6

7 where V P is he valaion percenage and h is he oal hairc a ime ha he collaeral asses are disconed by. We will no go ino he deails of he formaion of he hairc since i is eiher pre-deermined in he CSA docmens or relaed o marke risk measres sch as VaR of he collaeral asses relaed o marke risk measres sch as VaR of he collaeral asses. cf. [ISD05], page 68. The main prpose of he hairc is o miigae amorizaion or depreciaion in he collaeral asse vale a he ime of a defal and in he margin period of risk. Moreover, he hairc shold be pdaed as freqenly as i can be o reflec he changes in he volailiy or liqidiy of he collaeral asses cf. [ISD05], page 63. Therefore, he oal vale of he collaeral porfolio a ime is eqal o 1 + h C, where h is he appropriae hairc applied o he collaeral porfolio. 2.2 Bilaeral Credi Valaion Adjsmen In his secion, we shall compe he CVA on a CDS conrac, sbjec o a bilaeral margin agreemen. Definiion 2.8 The bilaeral Credi Valaion Adjsmen process on a CDS conrac maring a ime T is defined as for every [0, T ]. CVA = S S C, 5 We now presen an alernaive represenaion for he bilaeral CVA, which is convenien for compaional prposes. Proposiion 2.9 The bilaeral CVA process on a CDS conrac maring a ime T saisfies for every [0, T ]. CVA = B E Q I {<τ=τ 2 T } τ 1 R 2 S τ + I {τ=τ 1 }δ 1 τ C τ + G Proof. We begin by observing ha for i = 1, 2, 3. Conseqenly, B E Q I {<τ=τ 3 T } τ 1 R 3 S τ + I {τ=τ 1 }δ 1 τ C τ G, 6 ],T ] ],T ] dd C = τ δ i 1 H dh i = Bτ 1 δ τ i I {<τ=τ i T }, + τ + Bτ 1 κ δ τ 1 I {<τ=τ 1 T } + Bτ 1 δ τ 2 I {<τ=τ 2 T } δ τ 3 I {<τ=τ 3 T } + Bτ 1 δ τ 4 I {<τ=τ 2 =τ 3 T } δ 5 τ I {<τ=τ =τ 1 T } + τ C τ I {<τ T } ],T ] I {τ>} d. 7 7

8 Using he definiions of he close-o cash-flows δ i τ, i = 1,..., 5, we ge from 7 ],T ] dd C = Bτ 1 δ 1 τ C τ I{<τ=τ 1 T } κ + τ ],T ] B 1 I {τ>} d + Bτ 1 C τ I {<τ T } 8 R 2 Sτ +I {τ=τ 1 }δ 1 τ C τ + S τ + I {τ=τ 1 }δτ 1 C τ I {<τ=τ 2 T } Sτ + Bτ 1 + I {τ=τ 1}δτ 1 + C τ R 3 Sτ + I {τ=τ 1}δτ 1 C τ I {<τ=τ 3 T } Bτ 1 Sτ + I {τ=τ 1}δτ 1 C τ I{<τ=τ 2=τ 3 T } Bτ 1 δ 1 τ C τ I{<τ=τ =τ 1 T }. Since I {<τ T } eqaliy = I {<τ=τ 1 T } + I {<τ=τ 2 T } + I {<τ=τ 3 T } I {<τ=τ 2=τ 3 T } I {<τ =τ 1 T }, sing he R i S τ C τ + S τ C τ + C τ = S τ + 1 R i S τ C τ and observing ha I {τ=τ 1 }S τ = 0, we can rearrange he erms in 8 as follows, ],T ] Now, combining 9 wih 1 we see ha dd C = Bτ 1 δτ 1 I {<τ=τ 1 T } κ B 1 I {τ>} d 9 ],T ] + τ S τ I{<τ=τ 2 T } + I {<τ=τ 3 T } I {<τ=τ 2 =τ 3 T } I{τ τ 1 } τ 1 R 2 S τ + I {τ=τ 1 }δ 1 τ C τ + I{<τ=τ 2 T } + τ 1 R 3 S τ + I {τ=τ 1 }δ 1 τ C τ I{<τ=τ 3 T }. I{<τ=τ S C = B E Q 1 T } + I {τ>t } B 1 dd G ],T ] From here, observing ha + B E Q I{<τ=τ 2 T } + I {<τ=τ 3 T } I {<τ=τ 2=τ 3 T } I{τ τ 1 } EQ B 1 dd G τ G ]τ,t ] B E Q I {<τ=τ 2 T } τ 1 R 2 S τ + I {τ=τ 1 }δ 1 τ C τ + G + B E Q I {<τ=τ 3 T } τ 1 R 3 S τ + I {τ=τ 1 }δ 1 τ C τ G. I {τ } + I {τ>t } + I {<τ=τ 1 T } + I {<τ=τ 2 T } + I {<τ=τ 3 T } I {<τ=τ 2=τ 3 T } I{τ τ 1 } = 1, 8

9 we ge S C = B E Q B 1 dd G ],T ] B E Q I {<τ=τ 2 T }Bτ 1 1 R 2 S τ C τ + G + B E Q I {<τ=τ 3 T } τ 1 R 3 S τ C τ G, 10 which is This proves he resl. S C = S B E Q I {<τ=τ 2 T } τ 1 R 2 S τ C τ + G + B E Q I {<τ=τ 3 T } τ 1 R 3 S τ C τ G. Remark 2.10 The above resls shows ha he vale of he bilaeral CVA is he same as he sm of he vale of a long posiion in a zero-srike call opion on he ncollaeralized amon and he vale of a shor posiion in a zero-srike p opion on he ncollaeralized amon Unilaeral CVA and Deb Vale Adjsmen The bilaeral nare of he conerpary risk is a conseqence of possible defal of he conerpary and he possible defal of he invesor. The vales of poenial losses associaed wih hese wo componens are called nilaeral CVA UCVA and deb vale adjsmen DVA, respecively, and defined below. Definiion 2.11 The Unilaeral Credi Vale Adjsmen is defined as, UCVA = B E Q I {<τ=τ 2 T } τ 1 R 2 S τ + I {τ=τ 1 }δ 1 τ C τ + G, [0, T ], and symmerically he Deb Vale Adjsmen is defined as DVA = B E Q I {<τ=τ 3 T } τ 1 R 3 S τ + I {τ=τ 1 }δ 1 τ C τ G, [0, T ]. Remark 2.12 DVA accons for he risk of invesor s own defal, and i represens he vale of any poenial osanding liabiliies of he invesors ha will no be honored a he ime of he invesor s defal: In fac, a ime of his/her defal, he invesor only pays o he conerpary he recovery amon, ha is R 3 Sτ + I {τ=τ 1 }δ 1 τ C τ. Therefore, he invesor gains he remaining amon, which is eqal o 1 R 3 S τ + I {τ=τ 1}δ 1 τ C τ, on his/her osanding liabiliies by defaling. Risk managemen of his componen is of grea imporance for financial insiions. When considering he nilaeral conerpary risk DVA is se o zero. In view of Proposiion 2.9 and of he above definiion we have ha CVA = UCVA DVA, [0, T ]. 9

10 Noe ha he bilaeral CVA amon may be negaive for he invesor de o own s defal effec. This also indicaes ha he price S C conerpary risk-free conrac. Remark 2.13 Upfron CDS Conversion of conerpary risky CDS conrac may be greaer han he price S of Afer he CDS Big Bang cf. [Mar09] a process has been originaed o replace sandard CDS conracs wih so called pfron CDS conracs. An pfron CDS conrac is composed of an pfron paymen, which is an amon o be exchanged pon he incepion of he conrac, and of a fixed spread. The fixed spread, say κ, will be 100bps for invesmen grade CDS conracs, and 500bps for high yield CDS conracs. The recovery rae is also sandardized o wo possible vales: 20% or 40%, depending on he credi worhiness of he reference name. The corresponding cmlaive dividend process of a conerpary-risk-free CDS conrac is described in he following definiion. Definiion 2.14 The cmlaive dividend process D of a conerpary-risk-free pfron CDS conrac, maring a ime T, is given as D = ]0,] δdh 1 1 UP κ 1 H 1 d, [0, T ], ]0,] where UP is he pfron paymen, and κ is he fixed spread. Reacall ha he spread κ 0 of a sandard CDS conrac is se sch ha he proecion leg P L 0 and fixed leg κ 0 DV 01 0 are eqal a iniiaion making he price of he conrac o be zero. Similarly, in he case of an pfron CDS conrac, wih κ being fixed, he pfron paymen UP is chosen sch ha he conrac has zero vale a iniiaion. I is easy o conver he convenional spread κ 0 ino an pfron paymen PU and vise versa. Indeed, direcly from he Definiion 2.14, and definiions of P L 0 and DV 01 0, we have P L 0 UP κdv 01 0 = P L 0 κ 0 DV 01 0 = 0, which implies he following represenaions UP = κ 0 κ DV 01 0, κ 0 = UP DV κ. In view of he conversion formlae presened above he discssion of CVA, DVA and UCVA done for sandard CDS conracs can be adoped o he case of he pfron CDS conracs in a sraighforward manner CVA via Credi Exposres Credi exposre is defined as he poenial loss ha may be sffered by eiher one of he conerparies de o he oher pary s defal. Here, we discss some measres commonly sed o qanify credi exposre, sch as Poenial Fre Exposre PFE, Expeced Posiive Exposre EPE and Expeced Negaive Exposre ENE, and heir relaion o CVA. Poenial Fre Exposre is he basic measre of credi exposre: 10

11 Definiion 2.15 Poenial Fre Exposre of a CDS conrac wih a bilaeral margin agreemen is defined as follows, P F E = I {τ=τ 2 } 1 R 2 S τ + I {τ=τ 1 }δτ 1 + C τ I {τ=τ 3 } 1 R 3 S τ + I {τ=τ 1 }δτ 1 C τ. Remark 2.16 Observe ha he CVA is relaed o PFE as follows, CV A = B E Q I{<τ T } τ P F E G, [0, T ]. Expeced Posiive Exposre is defined as he expeced amon he invesor will lose if he conerpary defal happens a ime, and Expeced Negaive Exposre is defined as he expeced amon he invesor will lose if his own defal happens a ime. Noe ha here is no disconing involved and he losses are condiional on defal a ime. EPE and ENE are necessary qaniies o price and hedge conerpary risk. Definiion 2.17 The Expeced Posiive Exposre of a CDS conrac wih a bilaeral margin agreemen is defined as, EP E = E Q 1 R 2 S τ + I {τ=τ 1}δτ 1 + C τ τ = τ2 =, and he Expeced Negaive Exposre is defined as, for every [0, T ]. Remark 2.18 I can be shown cf. ENE = E Q 1 R 3 S τ + I {τ=τ 1 }δτ 1 C τ τ = τ3 = process can be represened in erms of EPE and ENE as follows for every [0, T ]. CV A = B T [ABCJ11] ha in case of a deerminisic discon facor he CVA B T Bs 1 EP E s G 1 Q τ = τ 2 ds Bs 1 ENE s G 1 Q τ = τ 3 ds 2.3 Dynamics of CVA In his secion we derive he dynamics for CVA. This is imporan for deriving formlae for dynamic hedging of conerpary risk, he isse ha will be discssed in a differen paper. We begin wih defining some axiliary sopping imes, ha will come handy laer on: 11

12 τ {1} τ 1 if τ 1 τ 2, τ 1 τ 3 :=, τ {2} τ 2 if τ 2 τ 1, τ 2 τ 3 :=, oherwise oherwise τ {3} τ 3 if τ 3 τ 1, τ 3 τ 2 :=, τ {4} τ 2 if τ 2 = τ 3, τ 2 τ 1 :=, oherwise oherwise τ {5} τ 1 if τ 1 = τ 2, τ 1 τ 3 :=, τ {6} τ 1 if τ 1 = τ 3, τ 1 τ 2 :=, oherwise oherwise τ {7} τ 1 if τ 1 = τ 2 = τ 3 :=. oherwise Accordingly, we define he defal indicaor processes: H {1} := I {τ1,τ 1 τ 2,τ 1 τ 3 } = I {τ {1},}, H{2} := I {τ2,τ 2 τ 1,τ 2 τ 3 } = I {τ {2},}, H {3} := I {τ3,τ 3 τ 1,τ 3 τ 2 } = I {τ {3},}, H{4} := I {τ2 =τ 3,τ 1 τ 2 } = I {τ {4},}, H {5} := I {τ1 =τ 2,τ 1 τ 3 } = I {τ {5},}, H{6} := I {τ1 =τ 3,τ 1 τ 2 } = I {τ {6},}, H {7} := I {τ1 =τ 2 =τ 3 } = I {τ {7} }. Remark 2.19 Noe ha one can represen processes H {i}, i = 1,..., 7, as follows H {7} H {5} = [[ H 1, H 2], H 3], H{6} = [ H 1, H 3] H{7}, = [ H 1, H 2] H{7}, H {4} = [ H 2, H 3] H{7}, H {3} = H 3 H {4} H {6} H {7}, H {2} = H 2 H {4} H {5} H {7}, H {1} = H 1 H {5} H {6} H {7}. In pariclar, hese processes are G-adaped processes. Le G = Q τ > F be he srvival probabiliy process of τ wih respec o filraion F. I is a F spermaringale and i admis niqe Doob-Meyer decomposiion G = µ ν where µ is he maringale par and ν is a predicable increasing process. We assme ha G is a coninos process and v is absolely coninos wih respec o he Lebesge measre, so ha dν = v d for some F-progressively measrable, non-negaive process v. We denoe by l he F-progressively measrable process defined as l = G 1 v. Finally, we assme ha all F maringales are coninos. We assme ha hazard process of each sopping ime τ {i} admis an F, G-inensiy process q i for every i = 1,..., 7, so ha he process M {i}, given by he formla, M {i} = H {i} 0 1 H {i} q i d 12

13 is a G-maringale for every [0, T ] and i = 1,..., 7. We now have he following echnical resl, Lemma 2.20 The processes τ M i := M {i} τ = H {i} τ 0 ld, i 0, i = 1, 2,..., 7, and where are G-maringales τ M := H τ l d, 0, 0 l i = I {τ } q i and l = l i 0, i = 1, 2,..., 7, Proof. Fix i = 1,..., 7. Process M i follows a G-maringale, since i is G-maringale M {i} sopped a he G sopping ime τ. Moreover, we have ha M = 7 M i, so ha process M is also a G-maringale. We shall now proceed wih deriving some sefl represenaions for he processes S C and S. Lemma 2.21 The ex-dividend price process S C of a conerpary risky CDS conrac, given in 4, can be represened as follows, where S C = B E Q τ I {<τ=τ {i} T} δi τ κ ],T ] δ 1 = δ 1, δ 2 = S 1 R 2 S C + δ 3 = S + 1 R 3 S C, Proof. Le s rewrie 9 in he following form, S C = B E Q δ 4 = S 1 R 2 S C R 3 S C I {τ>} d G δ 5 = δ 1 1 R 2 δ 1 C +, δ 6 = δ R 3 δ 1 C δ 7 = δ 1 1 R 2 δ 1 C R3 δ 1 C. B τ 1 δτ 1,5,6,7 I {<τ=τ {i} T} + B 1 τ S τ τ 1 R 2 S τ + I {τ=τ 1}δ 1 τ C τ + + Bτ 1 1 R 3 S τ + I {τ=τ 1 }δτ 1 C τ I {<τ=τ {i} T} κ i=3,4,6,7 ],T ] i=2,4,5,7 i=2,3,4 I {τ>} d G, I {<τ=τ {i} T} I {<τ=τ {i} T} 11 13

14 which, afer rearranging erms, leads o S C = B E Q + τ + τ + τ + τ + Bτ 1 κ This proves he resl. Bτ 1 δ 1 τ I {<τ=τ {1} T} + B 1 τ S 1 R 2 S τ C τ + I {<τ=τ {2} T} S τ + 1 R 3 S τ C τ I {<τ=τ {3} T} S τ 1 R 2 S τ C τ R 3 S τ C τ I {<τ=τ {4} T} δτ 1 1 R 2 δτ 1 + C τ ],T ] δ 1 τ + 1 R 3 δ 1 τ C τ I {<τ=τ {5} T} I {<τ=τ {6} T} δ 1 τ 1 R 2 δ 1 τ C τ R3 δ 1 τ C τ I {τ>} d G. I {<τ=τ {7} T} In case when R 2 = R 3 = 1 process S is he same as process S C. Ths, we obain from he above Corollary 2.22 The ex-dividend price process S of a conerpary risk-free CDS conrac, can be represened as follows, S = B E Q τ I {<τ=τ {i} T}ˆδ τ i κ ],T ] where ˆδ 1 = ˆδ 5 = ˆδ 6 = ˆδ 7 = δ 1, and ˆδ 2 = ˆδ 3 = ˆδ 4 = S. Ths, S = B E Q κ ],T ] Bτ 1 I {<τ=τ 1 T }δτ 1 + Bτ 1 I {τ>} d G. I {τ>} d G, 12 4 I {<τ=τ {i} T} S τ 13 i=2 3 The following resl is borrowed from [BJR08] see Lemma 3.1 herein Lemma 2.23 The following eqaliy holds Q a.s. B E Q for every [0, T ]. I {<τ=τ {i} T} B 1 τ δ i τ G = I {<τ} B G E Q T lδ i i G d F, 14 The pre-defal ex-dividend price processes, say S C and S C, are defined as he niqe F-adaped processes cf. [BJR08] sch ha S C = I {<τ} SC, S = I {<τ} S. 3 We noe ha formla 13 provides a represenaion of S, which is convenien for or prposes. The radiional represenaion of S, ypically sed in he conex of conerpary risk free CDS conracs is S = B E Q Bτ 1 1 I {<τ1 T }δτ 1 1 κ B 1 I {τ1 >}d ],T ] G. 14

15 In view of he above we hs obain he following resl Lemma 2.24 We have ha, for every [0, T ], and S C = B G E Q S = B G E Q Proof. From Lemma 2.21 we have ha S C = B E Q τ Now, in view of 14 we see ha B E Q τ I {<τ=τ {i} T} δi T B 1 G lδ i i κ d F, 15 T B 1 G lˆδ i i κ d F. 16 I {<τ=τ {i} T} δi G G κb E Q ],T ] = I {<τ} B G E Q T Le s now fix 0, and define Y s := κ ],s] B 1 d for s. Ths, we ge κb E Q ],T ] I {τ>} d G = B E Q I{τ>T } Y T G + B E Q I{<τ T } Y τ G. I {τ>} d G. lδ i i G d F. I is known from [BJR08], ha and B B E Q I{<τ T } Y τ G = I{<τ} G E Q T Y dg F B E Q I{τ>T } Y T G = I{<τ} G E Q GT Y T F. Finally, since Y is of finie variaion, 15 follows by applying he inegraion by pars formla T T T GY T Y s dg s = G s dy s = κ Eqaliy 16 is obained as a special case of 15, by seing R 2 = R 3 = 1. B G s B 1 d. We are ready now o derive dynamics of he pre-defal price processes, ha we shall se in order o derive he dynamics of he CVA process. Lemma 2.25 i The pre-defal ex-dividend price of a conerpary risky CDS conrac follows he dynam- 15

16 ics given as d S C = r + l S C lδ i i κ d + G 1 B dm C S C dµ + G 2 SC d µ B d µ, m C, [0, T ], where m C T = E Q B 1 G lδ i i κ d F 0 ii The pre-defal ex-dividend price of a conerpary risk-free CDS conrac follows he dynamics given as where d S = r + l S lˆδ i i κ d + G 1 B dm S dµ + G 2 S d µ B d µ, m, [0, T ], T m = E Q B 1 G lˆδ i i κ d F. 0 Proof. The argmen below follows he one in he proof of Proposiion 1.2 in [BJR08]. where In view of 15 we may wrie S C as U = m C S C = B G 1 U, Since G = µ v, hen applying Iô s formla one obains Conseqenly, d G 1 U = G 1 dm C 0 B 1 G lδ i i κ d. lδ i i κ d + U G 3 d µ G 2 dµ dv G 2 d µ, m C, 16

17 d S C = B G 1 dm C lδ i i κ d + B U G 3 d µ G 2 dµ l Gd B G 2 d µ, m C + r B G 1 U d = r + l S C lδ i i κ d + G 1 B dm C S dµ + G 2 which verifies he resl saed in i. S d µ B d µ, m C, Saring from 16, and sing compaions analogos o he ones done in i, one can derive he resl saed in ii. The dynamics of he CVA process are easily derived wih help of he above lemma, Proposiion 2.26 The bilaeral CVA process saisfies he following SDE, dcva = r CVA d CVA dm 1 H lξ i i d + 1 H B G 1 dn G 1 CVA dµ + G 2 CVA d µ 1 H G 2 B d µ, m d µ, m C, where wih T n = E Q B 1 G lξ i i d F, [0, T ], 0 ξ 1 = 0, ξ 2 = 1 R 2 S C +, ξ 3 = 1 R 3 S C, ξ 4 = 1 R 2 S C + 1 R 3 S C, ξ 5 = 1 R 2 δ 1 C +, ξ 6 = 1 R 3 δ 1 C, ξ 7 = 1 R 2 δ 1 C + 1 R3 δ 1 C. Proof. Applying he inegraion by pars formla we ge ha dcva = 1 H d S d S C S S C dh. 17

18 This ogeher wih Lemma 2.25 implies dcva = S S C dm + 1 H r S S C l i ˆδi δ i d + 1 H B G 1 dm dm C 1 H G 1 S S C dµ + 1 H G 2 S S C d µ 1 H G 2 B d µ, m d µ, m C, which proves he resl Dynamics of CVA when he immersion propery holds Here we adap he resls derived above o he case when he immersion propery holds beween filraions F and G, ha is he case when every F-maringale is a G-maringale nder Q. In his case, he coninos maringale µ in he Doob-Meyer decomposiion of G vanishes, so ha he srvival process G is a nonincreasing process represened as G = v. Freqenly, he immersion propery is referred o as Hypohesis H. For an excellen discssion of he immersion propery we refer o [JLC09]. Assmpion 2.27 Hypohesis H holds beween he filraions F and G nder Q. In view of he resls and he noaion from Proposiion 2.26 we obain Corollary 2.28 Assme ha Assmpion 2.27 is saisfied. Then, dcva = r CVA d CVA dm 1 H lξ i i d + 1 H B G 1 dn, [0, T ]. Remark 2.29 If we assme ha he filraion F is generaed by a Brownian moion, hen, in view of he Brownian maringale represenaion heorem, here exiss an F-predicable process ζ sch ha dn = ζ dw and dcva = r CVA d CVA dm 1 H lξ i i d + 1 H B G 1 ζ dw. 2.4 Fair Spread Vale Adjsmen Le fix [0, T ], and le s denoe by κ he marke spread of he conerpary risk-free CDS conrac a ime ; ha is, κ is his level of spread ha makes he pre-defal vales of he wo legs of a conerpary risk-free CDS conrac eqal o each oher a ime, S κ =

19 I is convenien o wrie he above eqaion in he form ha is common in pracice: P L κ DV 01 = 0, 18 where P L and DV 01 are processes represening pre-defal vales of he proecion leg and he risky anniy, respecively, so ha 4 P L = B G 1 E Q ],T ] G 1 δ 1,5,6,7 l i d F, 19 and where Therefore, we ge, DV 01 = B G 1 E Q ],T ] G 1 = Q τ 1 > F. G 1 d F, 20 E Q ],T ] B 1 G 1 δ 1,5,6,7 li d F κ = E Q ],T τ 1 ] B 1 G d F We denoe by κ C he spread which makes he vales of he wo pre-firs-defal legs of a conerpary risky CDS conrac eqal o each oher a every [0, T ] as S C κ C = P L C κ C DV 01 C = Similarly, we se he spread κ C 0 risky CDS conrac a any ime [0, T ]. Using Lemma 3.1, κ C every [0, T ], where and iniiaed a ime = 0 in order o compe he fair price of a conerpary P L C = B G E Q DV 01 C = B G E Q We may now inrodce he following definiion, κ C = P LC DV 01 C, T B 1 G lδ i i d F ],T ] admis he following represenaion for 23 G d F We noe ha formla 19 provides a represenaion of P L, which is convenien for or prposes. The radiional represenaion of P L, ypically sed in he conex of conerpary risk free CDS conracs is P L = B G 1 E Q B 1 G 1 δλ 1 1 d ],T ] F, where λ 1 is he F inensiy of τ 1. 19

20 Definiion 2.30 The Spread Vale Adjsmen process of a conerpary risky CDS conrac maring a ime T is defined as, SVA = κ κ C for every [0, T ]. Monioring SVA is of grea imporance since i provides a more pracical way o qanify he conerpary risk. Moreover, he spread difference is a very sefl indicaor for he rading decisions in pracice cf. [Gre09]. Proposiion 2.31 The SVA of a conerpary risky CDS conrac maring a ime T eqals, SVA = CVA B G 1 E Q ],T ] B 1 G d F for every [0, T ], where he pre-firs-defal bilaeral Credi Valaion Adjsmen process CVA is given as CVA = S S C, 25 for every [0, T ]. Proof. Le s rewrie P L C as P L C = P L C κ DV 01 C + κ DV 01 C by a simple modificaion. Now, sing 5 and 22, we conclde ha P L C = S C κ + κ DV 01 C = S κ CVA + κ DV 01 C. Since S κ = 0, hen κ C has he following form, κ C = CVA + κ DV 01 C DV 01 C, which is κ C = CVA DV 01 C + κ SVA Dynamics Applying Iô formla one obains he dynamics of he fair spread process and of he conerpary risk adjsed spread process as 20

21 1 dκ = DV 01 B 1 G 1 κ δ 1 l 1 1 DV 01 d η 1, η 2 + d + κ DV 01 d η dη 1 κ dη 2, [0, T ], DV 01 where and where wih and DV 01 := E Q ],T ] G 1 d F, η 1 := E Q B 1 G 1 δl 1 d 1 ]0,T ] F, η 2 = E Q B 1 G 1 d ]0,T ] F = DV 01 + B 1 G 1 d, ]0,] dκ C 1 = DV 01 C G κ C δ l i i d + κc DV 01 C d ζ 2 1 DV 01 C d ζ 1, ζ dζ 1 DV 01 C κ C dζ 2, ζ 2 = E Q ]0,T ] DV 01 = E Q ],T ] G 1 d F, ζ 1 = E Q B 1 G lδ i i d ]0,T ] F, G d F = DV 01 C + ]0,] B 1 G d. 27 Combining he above resls, we find he dynamics of he SVA process: d SVA = dκ dκ C, [0, T ]. Dynamics of he SVA of grea imporance for observing he behavior of he difference beween he fair spread and he conerpary risk adjsed spread. Conerpary risk dynamics can be assessed in a more iniive manner by comping he SVA dynamics. 21

22 3 Mlivariae Markovian Defal Model In his secion, we propose an nderlying sochasic model following he lines of [BCJZ11]. Towards his end we define a Markovian model of mlivariae defal imes wih facor processes X = X 1, X 2, X 3 which will have he following key feares, The pair X, H is Markov in is naral filraion, Each pair X i, H i is a Markov process, A every insan, eiher each conerpary defals individally or simlaneosly wih oher conerparies. Noe ha he second propery grans qick valaion of he CDS and independen calibraion of each model marginal X i, H i, whereas he hird propery will allow s o accon for dependence beween defals. We presen here some nmerical resls as an applicaion of above heory. The inensiies of defal are assmed o be of he affine form l i, X i = ai + X i, where a i is a consan and X i is a homogenos CIR process generaed by, dx i = ζ i µi X i d σi XdW i i, for i = 1, 2, 3. Each collecion of he parameers ζ i, µ i, σ i may ake vales corresponding o a low, a medim or a high regime which are given as follows. Credi Risk Level ζ µ σ X 0 Low Medim High follows Moreover, following he mehodology in [BCJZ11], we specify he marginal defal inensiy processes as q 1 = l 1 + l 5 + l 6 + l 7, q 2 = l 2 + l 4 + l 5 + l 7, q 3 = l 3 + l 4 + l 6 + l 7 where he relaed srvival probabiliies are fond as Q τ i > = E Q e 0 qi d and Q τ > = E Q e 0 l d. A deailed discssion inclding implemenaion and he calibraion of he model can be fond in [ABCJ11] and [BCJZ11]. 3.1 Resls Or aim here is o assess by means of nmerical experimens he impac of collaeralizaion on he conerpary risk exposre. We presen nmerical resls for differen collaeralizaion regimes disingished by 22

23 differen hreshold vales. The nmerical experimens below have been done sing he hree facor 2F paramerizaion given in [BCJZ11], he recovery raes are fixed o 40%, he risk-free rae r is aken as 0 and he mariy is se o T = 5 years. Table 3.1 shows he vales of CVA 0 and SVA 0 for differen hreshold regimes. Threshold vales are chosen as a fracion of he noional cf. [Pyk09]. Compaions are done assming ha refer o Table 3 he nderlying eniy, he conerpary, and he invesor has high risk levels. Simlaed fair spread wiho conerpary risk is fond as 153bps. Case A represens he ncollaeralized regime where here is no collaeral exchanged his is done by seing he hresholds infiniy, whereas oher Case F corresponds o he fll collaeralizaion where he hresholds are se o 0. In each case, compaions are done by seing MT A o zero and assming here is no margin period. One can observe ha decreasing hreshold vale dramaically decreases he iniial CVA and herefore he SVA vales. Γ cpy Γ inv CVA 0 SVA 0 Case A Case B Case C Case D Case E Case F In Figre 1, we presen he EP E and ENE crves for each case A o F, and we also plo he mean collaeral vales. Compaions are carried o by rnning 10 4 Mone Carlo simlaions. I is apparen ha he behavior of he EP E and EN E vales decreases as a resl of increased collaeralizaion. Noe ha here are peaks in he collaeral vale in he very beginning and hrogh he mariy. This effec can be explained as follows: Observe from Table 1 ha he invesor has lower hreshold han he conerpary in each cases from A o F. As a resl, having he lower hreshold vale, invesor will be posing collaeral before he conerpary. Therefore, nil he conerpary s exposre reaches he hreshold, he collaeral vale remains negaive; meaning ha here will be margin calls for he invesor before he conerpary. Figre 2 plos he mean of sample CVA pahs. Saring from CVA 0 we compe he mean sample pahs in each case. The behavior of CVA as a credi hybrid opion, as indicaed in Remark 2.10, can be clearly observed in he graphs. CVA vales decrease over ime as a resl of ime decay since he expeced loss decreases close o he expiraion. The effec of collaeralizaion on he CVA vales is apparen in he graphs. Observe ha increased iniial hreshold vales are of grea imporance since one can significanly redce he fre CVA vales by appropriaely seing he collaeral hresholds. Moreover, one can also se dynamic hresholds by linking he hreshold vales o he conerparies defal inensiies or credi raings. In his way, conerparies will have more conrol on he fre vales of he CVA of he CDS conrac and dynamically manage he CVA since he collaeral hresholds will be reacing o he changes in he defal inensiies or credi raings. This approach will be frher invesigaed in a fre research. 4 Conclsion In his paper, we discssed he modeling of conerpary risk in he presence of bilaeral margin agreemens. We defined an appropriae collaeral process which akes varios margin agreemen parameers ino accon. The dynamics of he conerpary risk adjsmen, CVA, has been fond for he bilaeral case. This achieve- 23

24 EPE ENE Collaeral EPE ENE Collaeral EPE ENE Collaeral EPE ENE Collaeral EPE ENE Collaeral EPE ENE Collaeral Figre 1: EPE, ENE and he Collaeral crves for Case A, B, C, D, E, and F men helps s o beer ndersand and monior he behavior of he bilaeral CVA as well as he nilaeral CVA and he DVA. We observed he impac of collaeral agreemens on conerpary risk adjsmens as well as he credi exposres sch as he EPE and he ENE. The presence of simlaneos defals in or model represens he wrong way risk involved in he CDS conracs. We formlae he fair spread vale adjsmen, which is named as SVA, ha indicaes he addiional spread vale o incorporae he conerpary risk ino he fair spread vale. Moreover, we derive he dynamics of he fair spread and he conerpary risky spread and herefore he spread vale adjsmen, SVA. Finally, as in [ABCJ11] and [BCJZ11] we presen or nmerical resls sing a Markovian model of conerpary credi risk. References [ABCJ11] S. Assefa, T.R. Bielecki, S. Crepey, and M. Jeanblanc. CVA Compaion for Conerpary Risk Assessmen in Credi Porfolios. In Credi Risk Froniers: Sbprime Crisis, Pricing and Hedging, CVA, MBS, Raings, and Liqidiy. Bloomberg, [BC11] T. R. Bielecki and S. Crepey. Dynamic Hedging of Conerpary Exposre. In T. Zariphopolo, M. Rkowski, and Y. Kabanov, ediors, The Msiela Fesschrif. Spring,

25 Figre 2: Forward CVA crves for Case A, B, C, D, E, and F [BCJZ11] T.R. Bielecki, S. Crepey, M. Jeanblanc, and B. Zargari. Valaion and Hedging of CDS Conerpary Exposre in a Markov Copla Model. Forhcoming in Inernaional Jornal of Theoreical and Applied Finance, [BJR08] [Gre09] Tomasz R. Bielecki, Moniqe Jeanblanc, and Marek Rkowski. Pricing and rading credi defal swaps in a hazard process model. Annals of Applied Probabiliy, 186: , John Gregory. Conerpary Credi Risk: The New Challenge for Global Financial Markes. Wiley, [ISD05] ISDA. ISDA Collaeral Gidelines [ISD10] ISDA. ISDA Marke Review of OTC Derivaive Bilaeral Collaeralizaion Pracices [JLC09] [Mar09] [Pyk09] M. Jeanblanc and Y. Le Cam. Immersion propery and credi risk modelling. In Opimaliy and Risk - Modern Trends in Mahemaical Finance, pages Springer, Marki. The CDS Big Bang: Undersanding he Changes o he Global CDS Conrac and Norh American Convenions M. Pykhin. Modeling Credi Exposre for Collaeralized Conerparies. Jornal of Credi Risk, 54:3 27,