Static Models of Central Counterparty Risk

Transcription

1 Statc Models of Central Counterparty Rsk Samm Ghamam 1 Federal Reserve Board & UC Berkeley Center for Rsk Management Research Systemc Rsk and Fnancal Networks, IPAM March 25, The opnons expressed n ths presentaton are my own and do not necessarly reflect the vews of the Board of Governors or ts staff. Ghamam Statc Models of Central Counterparty Rsk 1 / 28

2 The 2009 G-20 Clearng Mandate Ghamam Statc Models of Central Counterparty Rsk 2 / 28 In response to the fnancal crss, the Group of Twenty (G-20) ntated a reform program n 2009 to reduce the systemc rsk from OTC dervatves. The G-20 s reform program ncludes the followng elements: All standardsed OTC dervatves should be cleared through central counterpartes (CCPs).

3 The 2009 G-20 Clearng Mandate Ghamam Statc Models of Central Counterparty Rsk 2 / 28 In response to the fnancal crss, the Group of Twenty (G-20) ntated a reform program n 2009 to reduce the systemc rsk from OTC dervatves. The G-20 s reform program ncludes the followng elements: All standardsed OTC dervatves should be cleared through central counterpartes (CCPs). Non-centrally-cleared dervatves should be subject to hgher captal requrements. In 2011, the G-20 agreed to add margn requrements on non-centrally cleared dervatves to the reform program.

4 The Clearng Mandate Ttle VII of the Dodd-Frank Act and the European Market Infrastructure Regulaton mplement the G-20 clearng mandate n the US and Europe. 2 Fnancal Stablty Board, OTC Dervatves Workng Group, 5th and 6th progress report on mplementaton. Ghamam Statc Models of Central Counterparty Rsk 3 / 28

5 The Clearng Mandate Ttle VII of the Dodd-Frank Act and the European Market Infrastructure Regulaton mplement the G-20 clearng mandate n the US and Europe. The CFTC and SEC fnalzed jont rules detalng the scope of Ttle VII n SEC s responsble for regulatng securty-based swaps or small baskets of them. The CFTC s responsble for all other swaps. 2 Fnancal Stablty Board, OTC Dervatves Workng Group, 5th and 6th progress report on mplementaton. Ghamam Statc Models of Central Counterparty Rsk 3 / 28

6 The Clearng Mandate Ttle VII of the Dodd-Frank Act and the European Market Infrastructure Regulaton mplement the G-20 clearng mandate n the US and Europe. The CFTC and SEC fnalzed jont rules detalng the scope of Ttle VII n SEC s responsble for regulatng securty-based swaps or small baskets of them. The CFTC s responsble for all other swaps. 42% of the global IRS market ($380 trllon at February 2013), 14% of the global CDS market ($22 trllon), about 12% of the commodty swaps ($2.6 trllon), 2% of the equty swaps ($6.3 trllon), and a de mnms share of FX swaps ($67.4 trllon) were beng cleared n February Fnancal Stablty Board, OTC Dervatves Workng Group, 5th and 6th progress report on mplementaton. Ghamam Statc Models of Central Counterparty Rsk 3 / 28

7 Dervatves CCPs Ghamam Statc Models of Central Counterparty Rsk 4 / 28 A dervatves CCP stands between a set of blateral counterpartes to dervatves transactons. To reman fnancally reslent, CCPs rely on ther default waterfall resources used n a pre-specfed order upon the default of clearng members.

8 Dervatves CCPs Ghamam Statc Models of Central Counterparty Rsk 4 / 28 A dervatves CCP stands between a set of blateral counterpartes to dervatves transactons. To reman fnancally reslent, CCPs rely on ther default waterfall resources used n a pre-specfed order upon the default of clearng members. The defaulter resources are varaton margn, ntal margn, and prefunded default fund contrbutons of the clearng member that has defaulted.

9 Dervatves CCPs Ghamam Statc Models of Central Counterparty Rsk 4 / 28 A dervatves CCP stands between a set of blateral counterpartes to dervatves transactons. To reman fnancally reslent, CCPs rely on ther default waterfall resources used n a pre-specfed order upon the default of clearng members. The defaulter resources are varaton margn, ntal margn, and prefunded default fund contrbutons of the clearng member that has defaulted. The potental loss exceedng the defaulter-pay resources are to be absorbed by the CCP s equty contrbutons and the survvor-pay prefunded default funds.

10 Dervatves CCPs Ghamam Statc Models of Central Counterparty Rsk 4 / 28 A dervatves CCP stands between a set of blateral counterpartes to dervatves transactons. To reman fnancally reslent, CCPs rely on ther default waterfall resources used n a pre-specfed order upon the default of clearng members. The defaulter resources are varaton margn, ntal margn, and prefunded default fund contrbutons of the clearng member that has defaulted. The potental loss exceedng the defaulter-pay resources are to be absorbed by the CCP s equty contrbutons and the survvor-pay prefunded default funds. The potental loss exceedng these layers of the default waterfall resources s to be absorbed by the survvors unfunded default funds.

11 CCPs, Systemc Rsk, and the Fnancal System Ghamam Statc Models of Central Counterparty Rsk 5 / 28 The state of research on CCPs s not conclusve: Blateral vs Multlateral Nettng: Would CCPs ncrease net exposure? Cont and Kokholm [2012]: The mpact of central clearng on total expected exposure s hghly senstve to assumptons on heterogenety across asset classes n terms of rskyness and correlaton of exposures across asset classes. (Compare wth the results of Duffe and Zhu [2011].)

12 CCPs, Systemc Rsk, and the Fnancal System Ghamam Statc Models of Central Counterparty Rsk 5 / 28 The state of research on CCPs s not conclusve: Blateral vs Multlateral Nettng: Would CCPs ncrease net exposure? Cont and Kokholm [2012]: The mpact of central clearng on total expected exposure s hghly senstve to assumptons on heterogenety across asset classes n terms of rskyness and correlaton of exposures across asset classes. (Compare wth the results of Duffe and Zhu [2011].) Are CCPs good for the fnancal system? Prrong [2013]: Greater nettng and collateral redstrbutes rsk n the system. CCPs transform counterparty rsk nto lqudty rsk, whch can be more systemcally destablzng. Makng one part of the fnancal system nvulnerable does not make the system as a whole safer.

13 CCP Rsk Management and the CCP Rsk Captal CCPs collectvely take a mathematcal-model-based approach on margn requrements. The remanng layers of the default waterfall are specfed broadly. Post fnancal crss, ths fragmented CCP rsk measurement approach has been manly shaped by nternatonal standard settng bodes (SSBs) responsble for fnancal market nfrastructures (FMIs). 3 3 The commttee on payments and market nfrastructures (CPMI) and the techncal commttee of the nternatonal organzaton of securtes commssons (IOSCO) Ghamam Statc Models of Central Counterparty Rsk 6 / 28

14 CCP Rsk Management and the CCP Rsk Captal CCPs collectvely take a mathematcal-model-based approach on margn requrements. The remanng layers of the default waterfall are specfed broadly. Post fnancal crss, ths fragmented CCP rsk measurement approach has been manly shaped by nternatonal standard settng bodes (SSBs) responsble for fnancal market nfrastructures (FMIs). 3 Post clearng mandate, SSBs have devsed formulac methods based on whch clearng members are to hold captal aganst ther exposures to CCPs. The CCP rsk captal depends on all layers of the default waterfall resources; partcularly prefunded and unfunded default funds. 3 The commttee on payments and market nfrastructures (CPMI) and the techncal commttee of the nternatonal organzaton of securtes commssons (IOSCO) Ghamam Statc Models of Central Counterparty Rsk 6 / 28

15 CCP Rsk Management and the CCP Rsk Captal In the absence of a coherent CCP rsk measurement framework: CCPs may engage n non-unfable rsk management practces. There wll be nconsstences n the CCP rsk captal calculatons; t would be mpossble to develop a rsk senstve method. The central clearng of OTC dervatves may be dsncentvzed. 4 4 Regulatory reform of OTC dervatves: an assessment of ncentves to clear centrally, BIS[2014]. Ghamam Statc Models of Central Counterparty Rsk 7 / 28

16 CCP Rsk Management and the CCP Rsk Captal In the absence of a coherent CCP rsk measurement framework: CCPs may engage n non-unfable rsk management practces. There wll be nconsstences n the CCP rsk captal calculatons; t would be mpossble to develop a rsk senstve method. The central clearng of OTC dervatves may be dsncentvzed. 4 We ntroduce a model-based framework for the default waterfall of typcal dervatves CCPs: It can be vewed as a common ground for CCPs, ther drect clearng members, bank regulators, and CCP regulators. It can be used for a rsk senstve defnton of the CCP rsk captal based on whch non-model-based methods can be evaluated. 4 Regulatory reform of OTC dervatves: an assessment of ncentves to clear centrally, BIS[2014]. Ghamam Statc Models of Central Counterparty Rsk 7 / 28

17 How Should Banks be Captalzed Aganst Ther Exposures to CCPs? Ghamam Statc Models of Central Counterparty Rsk 8 / 28 Losses to a drect clearng member due to the CCP s default: The replacement cost of the dervatves portfolo t has cleared wth the CCP; losng the IM posted f not held n a bankruptcy remote manner; losng the prefunded default fund (?)

18 How Should Banks be Captalzed Aganst Ther Exposures to CCPs? Ghamam Statc Models of Central Counterparty Rsk 8 / 28 Losses to a drect clearng member due to the CCP s default: The replacement cost of the dervatves portfolo t has cleared wth the CCP; losng the IM posted f not held n a bankruptcy remote manner; losng the prefunded default fund (?) Losses to a drect clearng member due to the default of other clearng members: losng all or part of the prefunded default fund contrbutons (?) potental losses due to the CCP s unfunded default fund captal calls.

19 How Should Banks be Captalzed Aganst Ther Exposures to CCPs? Ghamam Statc Models of Central Counterparty Rsk 8 / 28 Losses to a drect clearng member due to the CCP s default: The replacement cost of the dervatves portfolo t has cleared wth the CCP; losng the IM posted f not held n a bankruptcy remote manner; losng the prefunded default fund (?) Losses to a drect clearng member due to the default of other clearng members: losng all or part of the prefunded default fund contrbutons (?) potental losses due to the CCP s unfunded default fund captal calls. It s mpossble to defne the second class of losses n a conceptually sound way n the absence of a unfyng default waterfall model for CCPs.

20 Statc Models of Dervatves CCPs Ghamam Statc Models of Central Counterparty Rsk 9 / 28 Consder a dervatves CCP wth n drect clearng members; CM defaults wth probablty p based on a fxed tme T > 0, = 1,..., n. Default ndcators Y s are dependent through t-copula threshold models: Y = 1{X > x } and X = a Z + 1 a 2 ξ, where Z and ξ s are ndependent standard normals, λ wth K χ 2 ν beng ndependent of Z and ξ s, and x s are chosen such that E[Y ] = p. λ K v

21 Statc Models of Dervatves CCPs Ghamam Statc Models of Central Counterparty Rsk 9 / 28 Consder a dervatves CCP wth n drect clearng members; CM defaults wth probablty p based on a fxed tme T > 0, = 1,..., n. Default ndcators Y s are dependent through t-copula threshold models: Y = 1{X > x } and X = a Z + 1 a 2 ξ, where Z and ξ s are ndependent standard normals, λ wth K χ 2 ν beng ndependent of Z and ξ s, and x s are chosen such that E[Y ] = p. λ K v Portfolo credt loss dstrbutons are skewed wth a relatvely heavy upper tal. These emprcal propertes can not be captured by normal copula models; they assgn very small probabltes to smultaneous defaults, (McNel et al. [2005]).

22 Credt (loan)-equvalent exposures Let C denote the CCP s collateralzed credt exposure to CM at ts default n the presence of the CM s VM and IM. We use dynamc counterparty credt rsk measures, e.g., EPE, to defne C s, T C (1 δ ) E[e (t)]dt, 0 where δ s the CM s recovery rate and, Ghamam Statc Models of Central Counterparty Rsk 10 / 28

23 Credt (loan)-equvalent exposures Ghamam Statc Models of Central Counterparty Rsk 10 / 28 Let C denote the CCP s collateralzed credt exposure to CM at ts default n the presence of the CM s VM and IM. We use dynamc counterparty credt rsk measures, e.g., EPE, to defne C s, T C (1 δ ) E[e (t)]dt, 0 where δ s the CM s recovery rate and, e (t) denotes the CCP s collateralzed exposure to CM at tme t, e (t) max{v + (t + ) V M (t) IM (t), 0}, where V + (t) max{v (t), 0}, V M (t) V + (t ˆ ), and IM (t) s the VaR or ES assocated wth V + (t + ) V + (t ˆ ).

24 CCP Losses n the One-Perod Model Ghamam Statc Models of Central Counterparty Rsk 11 / 28 Let U = (C DF ) + denote the CCP s exposure to CM n the presence of V M, IM, and DF.

25 CCP Losses n the One-Perod Model Ghamam Statc Models of Central Counterparty Rsk 11 / 28 Let U = (C DF ) + denote the CCP s exposure to CM n the presence of V M, IM, and DF. The CCP s counterparty credt loss n the presence of the defaulter-pay resources s, L (1) = n =1 U Y.

26 CCP Losses n the One-Perod Model Ghamam Statc Models of Central Counterparty Rsk 11 / 28 Let U = (C DF ) + denote the CCP s exposure to CM n the presence of V M, IM, and DF. The CCP s counterparty credt loss n the presence of the defaulter-pay resources s, L (1) = n =1 U Y. Survvng members prefunded default funds reduce L (1) to ( n L (2) = =1 U Y E DF s ) +, where, DF s DF n =1 DF Y.

27 CCP Losses n the One-Perod Model Ghamam Statc Models of Central Counterparty Rsk 11 / 28 Let U = (C DF ) + denote the CCP s exposure to CM n the presence of V M, IM, and DF. The CCP s counterparty credt loss n the presence of the defaulter-pay resources s, L (1) = n =1 U Y. Survvng members prefunded default funds reduce L (1) to ( n L (2) = =1 U Y E DF s ) +, where, DF s DF n =1 DF Y. L (2) or part of t s allocated to the survvng members n the form of unfunded default funds.

28 The PFMI and Prefunded Default Funds Ghamam Statc Models of Central Counterparty Rsk 12 / 28 CCPs often specfy ther DF based on the so called Cover 1/Cover 2 prncple of PFMI: 5 CCPs should mantan fnancal resources to cover the default of one or two partcpants that would potentally cause the largest aggregate credt exposures for the CCP n extreme but plausble market condtons. 5 Prncples for Fnancal Market Infrastructures [2012].

29 The PFMI and Prefunded Default Funds Ghamam Statc Models of Central Counterparty Rsk 12 / 28 CCPs often specfy ther DF based on the so called Cover 1/Cover 2 prncple of PFMI: 5 CCPs should mantan fnancal resources to cover the default of one or two partcpants that would potentally cause the largest aggregate credt exposures for the CCP n extreme but plausble market condtons. Compare the PFMI drven-df for CCPs wth large n and homogeneous portfolos and CCPs wth smaller n and heterogenous portfolos. 5 Prncples for Fnancal Market Infrastructures [2012].

30 The PFMI and Prefunded Default Funds Ghamam Statc Models of Central Counterparty Rsk 12 / 28 CCPs often specfy ther DF based on the so called Cover 1/Cover 2 prncple of PFMI: 5 CCPs should mantan fnancal resources to cover the default of one or two partcpants that would potentally cause the largest aggregate credt exposures for the CCP n extreme but plausble market condtons. Compare the PFMI drven-df for CCPs wth large n and homogeneous portfolos and CCPs wth smaller n and heterogenous portfolos. Credt qualty of the clearng members and ther correlaton do not play any role n specfyng DF. 5 Prncples for Fnancal Market Infrastructures [2012].

31 The PFMI and Prefunded Default Funds Ghamam Statc Models of Central Counterparty Rsk 12 / 28 CCPs often specfy ther DF based on the so called Cover 1/Cover 2 prncple of PFMI: 5 CCPs should mantan fnancal resources to cover the default of one or two partcpants that would potentally cause the largest aggregate credt exposures for the CCP n extreme but plausble market condtons. Compare the PFMI drven-df for CCPs wth large n and homogeneous portfolos and CCPs wth smaller n and heterogenous portfolos. Credt qualty of the clearng members and ther correlaton do not play any role n specfyng DF. The allocaton of DF to clearng members remans a subjectve matter. 5 Prncples for Fnancal Market Infrastructures [2012].

32 The Prefunded Default Funds n the One-Perod Model Ghamam Statc Models of Central Counterparty Rsk 13 / 28 Vew CCPs as fnancal nsttutons exposed to ther portfolo counterparty credt rsk; the portfolo consttuents are the members margned portfolos. The statc model s a one-perod approxmaton of the portfolo counterparty credt rsk. 6 See Tasche [1999], Tasche [2008], Denault [2001], and Kalkbrener [2005].

33 The Prefunded Default Funds n the One-Perod Model Ghamam Statc Models of Central Counterparty Rsk 13 / 28 Vew CCPs as fnancal nsttutons exposed to ther portfolo counterparty credt rsk; the portfolo consttuents are the members margned portfolos. The statc model s a one-perod approxmaton of the portfolo counterparty credt rsk. Defne the CCPs total DF based on a partcular rsk measure assocated wth L = n =1 C Y, DF ES α (L) = E[L L V ar α (L)]. 6 See Tasche [1999], Tasche [2008], Denault [2001], and Kalkbrener [2005].

34 The Prefunded Default Funds n the One-Perod Model Ghamam Statc Models of Central Counterparty Rsk 13 / 28 Vew CCPs as fnancal nsttutons exposed to ther portfolo counterparty credt rsk; the portfolo consttuents are the members margned portfolos. The statc model s a one-perod approxmaton of the portfolo counterparty credt rsk. Defne the CCPs total DF based on a partcular rsk measure assocated wth L = n =1 C Y, DF ES α (L) = E[L L V ar α (L)]. Allocate DF to clearng members based on the so called Euler captal allocaton prncple, 6 DF = C E[Y L V ar α (L)]. 6 See Tasche [1999], Tasche [2008], Denault [2001], and Kalkbrener [2005].

35 The Unfunded Default Funds The CCPs loss n the absence of unfunded default funds becomes, L (2) = ( n ) + U Y E DF s = =1 ( n ) + C Y E DF, =1 where U = (C DF ) + and DF s DF n =1 DF Y. Ghamam Statc Models of Central Counterparty Rsk 14 / 28

36 The Unfunded Default Funds The CCPs loss n the absence of unfunded default funds becomes, L (2) = ( n ) + U Y E DF s = =1 ( n ) + C Y E DF, =1 where U = (C DF ) + and DF s DF n =1 DF Y. The DF -based or C-based loss allocaton rules: consder the CCP s unfunded default fund captal call on CM, DF Ȳ n j=1 DF jȳj L (2), and C Ȳ n j=1 C jȳj L (2), where Ȳj = 1 Y j, j = 1,..., n, the former has more desrable rsk management justfcatons. Ghamam Statc Models of Central Counterparty Rsk 14 / 28

37 The Unfunded Default Funds The CCPs loss n the absence of unfunded default funds becomes, L (2) = ( n ) + U Y E DF s = =1 ( n ) + C Y E DF, =1 where U = (C DF ) + and DF s DF n =1 DF Y. The DF -based or C-based loss allocaton rules: consder the CCP s unfunded default fund captal call on CM, DF Ȳ n j=1 DF jȳj L (2), and C Ȳ n j=1 C jȳj L (2), where Ȳj = 1 Y j, j = 1,..., n, the former has more desrable rsk management justfcatons. Ghamam Statc Models of Central Counterparty Rsk 14 / 28

38 The Unfunded Default Funds Consder the DF -based allocaton of, ( n L (2) = =1 C Y E DF ) +. Ghamam Statc Models of Central Counterparty Rsk 15 / 28

39 The Unfunded Default Funds Consder the DF -based allocaton of, ( n L (2) = =1 C Y E DF ) +. The uncapped case: DF uc L uc = DF Ȳ L (2) 0, DF s where DF s = n j=1 DF jȳj. Ghamam Statc Models of Central Counterparty Rsk 15 / 28

40 The Unfunded Default Funds Consder the DF -based allocaton of, ( n L (2) = =1 C Y E DF ) +. The uncapped case: DF uc L uc = DF Ȳ L (2) 0, DF s where DF s = n j=1 DF jȳj. The capped case: where β > 0. DF L = mn{l uc, βdf Ȳ }, Ghamam Statc Models of Central Counterparty Rsk 15 / 28

41 The Unfunded Default Funds: The Condtonal Case Defne the CM s unantcpated loss assumng ts tme-t survval, DF uc,s L uc,s = DF DF s, C j Y j E DF j +, where DF s, DF j DF jy j. Ghamam Statc Models of Central Counterparty Rsk 16 / 28

42 The Unfunded Default Funds: The Condtonal Case Defne the CM s unantcpated loss assumng ts tme-t survval, DF uc,s L uc,s = DF DF s, C j Y j E DF j +, where DF s, DF j DF jy j. The capped case from the CM s perspectve becomes, where β > 0. DF s L s = mn{l uc,s, βdf }, Ghamam Statc Models of Central Counterparty Rsk 16 / 28

43 Ghamam Statc Models of Central Counterparty Rsk 17 / 28 Total Losses to CM The uncapped case: L t,uc,s = L uc,s = DF DF s, C j Y j E DF j +.

44 Ghamam Statc Models of Central Counterparty Rsk 17 / 28 Total Losses to CM The uncapped case: L t,uc,s = L uc,s = DF DF s, C j Y j E DF j The capped case: let Ỹ denote the CCP s default ndcator from CM s perspectve assumng t s survval at tme T. The CM s total potental loss becomes, L t,s = L s + ŨỸ, where L s = mn{luc,s, βdf }, and Ũ denotes the CM s loan-equvalent exposure to the CCP at ts default. +.

45 Ghamam Statc Models of Central Counterparty Rsk 17 / 28 Total Losses to CM The uncapped case: L t,uc,s = L uc,s = DF DF s, C j Y j E DF j The capped case: let Ỹ denote the CCP s default ndcator from CM s perspectve assumng t s survval at tme T. The CM s total potental loss becomes, L t,s = L s + ŨỸ, where L s = mn{luc,s, βdf }, and Ũ denotes the CM s loan-equvalent exposure to the CCP at ts default. And, E[Ỹ] = P ccp, = P j C j Y j > E + DF + DF s + j +. DF j.

46 Total Losses to CM : Countng DF n Ghamam Statc Models of Central Counterparty Rsk 18 / 28 The uncapped case: L t,uc,s { } DF = mn DF, L (1) + L uc,s, DF s, where L (1) = ( j (C j DF j )Y j E) + represents the CCP s loss n the presence of defaulter-pay resources and condtonal on CM s tme-t survval.

47 Total Losses to CM : Countng DF n Ghamam Statc Models of Central Counterparty Rsk 18 / 28 The uncapped case: L t,uc,s { } DF = mn DF, L (1) + L uc,s, DF s, where L (1) = ( j (C j DF j )Y j E) + represents the CCP s loss n the presence of defaulter-pay resources and condtonal on CM s tme-t survval. The capped case: L t,s { } DF = mn DF, L (1) + L s + DF ŨỸ, s, where L s = mn{luc,s, βdf }.

48 The CCP Rsk Captal Ghamam Statc Models of Central Counterparty Rsk 19 / 28 The CCP rsk captal based on expected losses: [ { }] E[L t,s DF ] = E mn DF, L (1) + E[L s ] + DF ŨP ccp,, s, where L s DF = mn DF s, C j Y j E DF j +, βdf.

49 The CCP Rsk Captal Ghamam Statc Models of Central Counterparty Rsk 19 / 28 The CCP rsk captal based on expected losses: [ { }] E[L t,s DF ] = E mn DF, L (1) + E[L s ] + DF ŨP ccp,, s, where L s DF = mn DF s, C j Y j E DF j +, βdf. The CCP rsk captal based on unexpected losses: V ar α (L t,s ) E[L t,s ].

50 The CCP Rsk Captal Ghamam Statc Models of Central Counterparty Rsk 19 / 28 The CCP rsk captal based on expected losses: [ { }] E[L t,s DF ] = E mn DF, L (1) + E[L s ] + DF ŨP ccp,, s, where L s DF = mn DF s, C j Y j E DF j +, βdf. The CCP rsk captal based on unexpected losses: V ar α (L t,s ) E[L t,s ]. Basel II s credt rsk captal s based on unexpected loss. Basel III s counterparty credt rsk captal does not accept a well-defned expected or unexpected loss characterzaton.

51 The Regulatory CCP Rsk Captal The BCBS has often developed both non-model-based and model-based methods for rsk captal requrements. For nstance: The Basel II s non-model-based Standardzed approach and the model-based Foundaton Internal Ratngs Based (IRB)/Advanced IRB approaches for credt rsk captal. 7 7 See Gordy [2003] for model-theoretc aspects of Basel II s credt rsk framework. Ghamam Statc Models of Central Counterparty Rsk 20 / 28

52 The Regulatory CCP Rsk Captal The BCBS has often developed both non-model-based and model-based methods for rsk captal requrements. For nstance: The Basel II s non-model-based Standardzed approach and the model-based Foundaton Internal Ratngs Based (IRB)/Advanced IRB approaches for credt rsk captal. 7 The regulatory CCP rsk captal s not model-based. Snce the CCP rsk captal depends on the default waterfall n a non-straghtforward way, non-model based methods wll not be conceptually sound and logcally consstent. 7 See Gordy [2003] for model-theoretc aspects of Basel II s credt rsk framework. Ghamam Statc Models of Central Counterparty Rsk 20 / 28

53 The Regulatory CCP Rsk Captal The BCBS has often developed both non-model-based and model-based methods for rsk captal requrements. For nstance: The Basel II s non-model-based Standardzed approach and the model-based Foundaton Internal Ratngs Based (IRB)/Advanced IRB approaches for credt rsk captal. 7 The regulatory CCP rsk captal s not model-based. Snce the CCP rsk captal depends on the default waterfall n a non-straghtforward way, non-model based methods wll not be conceptually sound and logcally consstent. The nterm framework for the CCP rsk captal was frst publshed by BCBS n A consultatve document modfed the nterm framework n BCBS-CPMI-IOSCO publshed ther fnalzed CCP rsk captal requrements n Aprl See Gordy [2003] for model-theoretc aspects of Basel II s credt rsk framework. Ghamam Statc Models of Central Counterparty Rsk 20 / 28

54 The Regulatory CCP Rsk Captal Ghamam Statc Models of Central Counterparty Rsk 21 / 28 The CM s CCP rsk captal: Default fund captal charges + Trade exposure captal charges

55 The Regulatory CCP Rsk Captal Ghamam Statc Models of Central Counterparty Rsk 21 / 28 The CM s CCP rsk captal: Default fund captal charges + Trade exposure captal charges The one-perod-model s average-based CCP rsk captal: [ { }] E[L t,s DF ] = E mn DF, L (1) + E[L s ] + DF ŨP ccp,, s, where L s DF = mn DF s, C j Y j E DF j +, βdf.

56 The Regulatory CCP Rsk Captal Ghamam Statc Models of Central Counterparty Rsk 21 / 28 The CM s CCP rsk captal: Default fund captal charges + Trade exposure captal charges The one-perod-model s average-based CCP rsk captal: [ { }] E[L t,s DF ] = E mn DF, L (1) + E[L s ] + DF ŨP ccp,, s, where L s DF = mn DF s, C j Y j E DF j +, βdf. What are the regulatory Default fund captal charges supposed to represent?

57 The BCBS-CPMI-IOSCO s Fnalzed Rule Ghamam Statc Models of Central Counterparty Rsk 22 / 28 Default Fund-Exposure Captal Charges The CCP s hypothetcal captal requrements: ( n ) K ccp = (captal rato) RW (C DF ) +, =1 the rsk weght s 20%, and the mnmum captal rato has been hstorcally set equal to 8%.

58 The BCBS-CPMI-IOSCO s Fnalzed Rule Ghamam Statc Models of Central Counterparty Rsk 22 / 28 Default Fund-Exposure Captal Charges The CCP s hypothetcal captal requrements: ( n ) K ccp = (captal rato) RW (C DF ) +, =1 the rsk weght s 20%, and the mnmum captal rato has been hstorcally set equal to 8%. The CM s CCP rsk captal: { } DF K cm = max DF K ccp, 8% 2% DF

59 The BCBS-CPMI-IOSCO s Fnalzed Rule Ghamam Statc Models of Central Counterparty Rsk 22 / 28 Default Fund-Exposure Captal Charges The CCP s hypothetcal captal requrements: ( n ) K ccp = (captal rato) RW (C DF ) +, =1 the rsk weght s 20%, and the mnmum captal rato has been hstorcally set equal to 8%. The CM s CCP rsk captal: { } DF K cm = max DF K ccp, 8% 2% DF K cm can not be ratonalzed.

60 Margns Procyclcalty and the One-Perod Model Ghamam Statc Models of Central Counterparty Rsk 23 / 28 Durng tmes of fnancal stress, the market volatlty ncreases. Ths n turn ncreases the margn requrements. Hgh margn requrements affect the fundng and market lqudty; lqudty dry-ups further ncrease the market volatlty. 8 8 Brunnermeer and Pedersen [2009].

61 Margns Procyclcalty and the One-Perod Model Ghamam Statc Models of Central Counterparty Rsk 23 / 28 Durng tmes of fnancal stress, the market volatlty ncreases. Ths n turn ncreases the margn requrements. Hgh margn requrements affect the fundng and market lqudty; lqudty dry-ups further ncrease the market volatlty. 8 Consder the CCPs expected losses n the presence of all layers of the default waterfall, E[L (3) ] = E[( n C Y E DF α DF ) + ]. =1 8 Brunnermeer and Pedersen [2009].

62 Margns Procyclcalty and the One-Perod Model Ghamam Statc Models of Central Counterparty Rsk 23 / 28 Durng tmes of fnancal stress, the market volatlty ncreases. Ths n turn ncreases the margn requrements. Hgh margn requrements affect the fundng and market lqudty; lqudty dry-ups further ncrease the market volatlty. 8 Consder the CCPs expected losses n the presence of all layers of the default waterfall, E[L (3) ] = E[( n C Y E DF α DF ) + ]. =1 The mx of VM, IM, E, and DF can be chosen such that margns procyclcalty be reduced whle the CCPs mantan the same level of fnancal reslency. 8 Brunnermeer and Pedersen [2009].

63 Concluson Ghamam Statc Models of Central Counterparty Rsk 24 / 28 CCPs are supposed to brng transparency to the OTC dervatves market. Internatonal regulatory rsk management standards nfluence and shape the dynamcs of fnancal markets.

64 Concluson Ghamam Statc Models of Central Counterparty Rsk 24 / 28 CCPs are supposed to brng transparency to the OTC dervatves market. Internatonal regulatory rsk management standards nfluence and shape the dynamcs of fnancal markets. The current central clearng envronment s fragmented: The mathematcal-model-based treatment of margn requrements; The PFMI s prncple-based treatment of the guarantee fund; The formulac CCP rsk captal rules.

65 Concluson Ghamam Statc Models of Central Counterparty Rsk 24 / 28 CCPs are supposed to brng transparency to the OTC dervatves market. Internatonal regulatory rsk management standards nfluence and shape the dynamcs of fnancal markets. The current central clearng envronment s fragmented: The mathematcal-model-based treatment of margn requrements; The PFMI s prncple-based treatment of the guarantee fund; The formulac CCP rsk captal rules. The proposed CCP rsk measurement framework, whch models the CCPs default waterfall coherently, can mprove the status quo. It can be used for a rsk senstve and conceptually sound defnton of the CCP rsk captal.

66 Appendx: The Legacy Rato-Tranches Methods The CCP s hypothetcal captal requrements: n K ccp = (captal rato) RW ( C ), the rsk weght s to represent the credt qualty of the clearng members, and the mnmum captal rato has been hstorcally set equal to 8%. =1 Ghamam Statc Models of Central Counterparty Rsk 25 / 28

67 Appendx: The Legacy Rato-Tranches Methods Ghamam Statc Models of Central Counterparty Rsk 25 / 28 The CCP s hypothetcal captal requrements: n K ccp = (captal rato) RW ( C ), the rsk weght s to represent the credt qualty of the clearng members, and the mnmum captal rato has been hstorcally set equal to 8%. The Rato approach: =1 K cm = DF DF K ccp.

68 Appendx: The Legacy Rato-Tranches Methods Ghamam Statc Models of Central Counterparty Rsk 25 / 28 The CCP s hypothetcal captal requrements: n K ccp = (captal rato) RW ( C ), the rsk weght s to represent the credt qualty of the clearng members, and the mnmum captal rato has been hstorcally set equal to 8%. The Rato approach: K cm = DF DF K ccp. The Tranches approach: K ccp f DF < K ccp K cm = DF DF K ccp +.16Kccp(DF Kccp) DF =1 f K ccp < DF

69 Appendx: The CCP Rsk Captal Ghamam Statc Models of Central Counterparty Rsk 26 / 28 Consder the Rato Approach: K cm =.08 RW DF n DF ( C ) =1

70 Appendx: The CCP Rsk Captal Ghamam Statc Models of Central Counterparty Rsk 26 / 28 Consder the Rato Approach: Recall: L uc,s K cm =.08 RW DF n DF ( C ) =1 ( = DF ) + DF s, j C jy j DF

71 Appendx: The CCP Rsk Captal Consder the Rato Approach: Recall: L uc,s K cm =.08 RW DF n DF ( C ) =1 ( = DF ) + DF s, j C jy j DF The normal-copula-based approxmaton of K cm n the exchangeable case for large n: ( ) E[L uc,s DF n n ] θ 1 C + θ 2 ( C 2 ) 1 2, DF (1 p) + pdf where θ 1 and θ 2 depend on p, the correlaton between normals underlyng the copula model, and the confdence level assocated wth DF. For our numercal examples the order of θ 1 vares from 10 2 to 10 4, and the order of θ 2 vares from 10 3 to Ghamam Statc Models of Central Counterparty Rsk 26 / 28 =1 =1

72 Appendx: Monte Carlo CCP Rsk Measurement Estmate the CCP s credt-equvalent exposures to the clearng members: T C (1 δ ) E[e (t)]dt, 0 where e (t) max{v + (t + ) V M (t) IM (t), 0}, V M (t) V + (t ˆ ), and IM (t) s the VaR or ES assocated wth V + (t + ) V + (t ˆ ); = 1,..., n. 9 9 See Ghamam and Zhang [2014] for effcent Monte Carlo CCR n the absence of IM. Ghamam Statc Models of Central Counterparty Rsk 27 / 28

73 Appendx: Monte Carlo CCP Rsk Measurement Estmate the CCP s credt-equvalent exposures to the clearng members: T C (1 δ ) E[e (t)]dt, 0 where e (t) max{v + (t + ) V M (t) IM (t), 0}, V M (t) V + (t ˆ ), and IM (t) s the VaR or ES assocated wth V + (t + ) V + (t ˆ ); = 1,..., n. 9 Ths wll be computatonally ntensve; partcularly for hgh confdence level-im s. It would be challengng to develop effcent Monte Carlo schemes for estmaton of C s as V s the value of the CM s portfolo consstng of possbly thousands of dervatves transactons. 9 See Ghamam and Zhang [2014] for effcent Monte Carlo CCR n the absence of IM. Ghamam Statc Models of Central Counterparty Rsk 27 / 28

74 Appendx: Monte Carlo CCP Rsk Measurement Ghamam Statc Models of Central Counterparty Rsk 28 / 28 Use Monte Carlo to estmate q V ar α (L), L = n =1 C Y.

75 Appendx: Monte Carlo CCP Rsk Measurement Ghamam Statc Models of Central Counterparty Rsk 28 / 28 Use Monte Carlo to estmate q V ar α (L), L = n =1 C Y. Use the Monte Carlo estmate of q n place of the true value of q to estmate DF = E[L L > q] and DF = C E[Y L > q], = 1,..., n.

76 Appendx: Monte Carlo CCP Rsk Measurement Ghamam Statc Models of Central Counterparty Rsk 28 / 28 Use Monte Carlo to estmate q V ar α (L), L = n =1 C Y. Use the Monte Carlo estmate of q n place of the true value of q to estmate DF = E[L L > q] and DF = C E[Y L > q], = 1,..., n. For effcent estmaton of DF and DF s, recall, Y = 1{X > x }, X = a Z + 1 a 2 ξ, E[L L > q] = λ E[L1{L > q}] P (L > q) and use the mportance samplng algorthm of Bassamboo et al. [2008].