Some Approaches to Modeling Wrong-Way Risk in Counterparty Credit Risk Management and CVA

Transcription

1 Some Approaches o Modeling Wrong-Way Risk in Counerpary Credi Risk Managemen and CVA Alex Levin, Leon Shegalov alex.levin@rbccm.com, leon.shegalov@rbccm.com Quaniaive Finance Seminar Fields Insiue, Torono March 7, 212

2 Ouline * Pricing and measuremen of Counerpary Credi Risk (CCR) wih Wrong-Way Risk (WWR) / Righ-Way Risk (RWR) o Pricing of CCR: CVA wih WWR and Condiional EPE (CEPE) o CCR measure: Condiional Poenial Fuure Exposure (CPFE) wih WWR o Counerpary Credi Economic Capial (CCEC)-like measure wih WWR Unified mulifacor Gaussian and Jump-Diffusion defaul inensiy frameworks for CVA, CPFE and CCEC wih WWR and credi raing ransiions o New effecive calibraion procedure for a model problem wih Gaussian whie noise defaul inensiy based on Volerra inegral equaion o Mone Carlo based fiing of Gaussian and Jump-Diffusion defaul inensiies o arbirary survival probabiliy erm srucures o A simple approach for consisen join simulaion of defauls and credi raing ransiions in Gaussian hazard rae model A new Gamma-facor copula for improving defaul correlaions in Gaussian framework for porfolio Counerpary Credi Economic Capial and BCVA * The views expressed in he presenaion are of he auhors only and no necessarily of he Royal Bank of Canada Fields Quaniaive Finance Seminar 2

3 Pricing and Measuremen of Counerpary Credi Risk (CCR) wih Wrong-Way Risk (WWR) / Righ-Way Risk (RWR) The recen credi crisis has demonsraed he need o capure Wrong-Way Risk (WWR) in he Counerpary Credi Risk Managemen and pricing. One of he regulaory requiremens in Basel II and Basel III concerning he counerpary credi risk is he abiliy of financial insiuions o capure and manage WWR. General Wrong-Way Risk is defined in BIS (26) as he risk when he probabiliy of defaul of counerparies is posiively correlaed wih general marke risk facors ; or in BIS (21) as he risk where he exposure increases when he credi qualiy of he counerpary deerioraes. A so-called Righ-Way Risk (RWR) is opposie o he WWR. RWR represens he case when he exposure o he counerpary is negaively correlaed wih he counerpary s defaul probabiliy. Specific Wrong-Way Risk is defined in BIS (26) as he risk when he exposure o a paricular counerpary is posiively correlaed wih he probabiliy of defaul of he counerpary due o he naure of he ransacions wih he counerpary. An example of Specific WWR is a pu opion on he counerpary s own sock. Naurally, financial insiuions should be rewarded (in erms of CVA, CCR measures and counerpary credi capial) for doing Righ-Way Risk business, and penalized for doing Wrong-Way Risk business. Fields Quaniaive Finance Seminar 3

4 Pricing of he counerpary credi risk, i.e., calculaion of he Credi Value Adjusmen (CVA) and Bilaeral CVA (BCVA) should be performed in he risk neural measure. The counerpary credi risk measures (for example, Poenial Fuure Exposure (PFE)) are usually calculaed by Risk Managemen in he hisorical measure based on he parameers esimaed from he hisorical daa. As usually, change of measure from risk neural o hisorical can be performed wihin presened here reduced form framework by he corresponding change of drifs in sochasic processes for marke variables and hazard raes. Hisorical parameer esimaion for CCR requires very long ime series for he risk facors, canno accoun for possible fuure economic regime changes, and usually has insufficien accuracy, especially in he long-erm drif predicion. On he oher hand, regulaors allow for calculaion of he CCR exposures in boh risk-neural measure (i.e., based on he marke implied daa) and hisorical measure (i.e., based on he hisorical daa including he daa for sress periods) (see BIS (21), paragraph 98). For simpliciy of exposiion and possibiliy o compare CVA numbers wih he CCR measures, we consider all sochasic processes for boh CVA and CCR under he riskneural measure. Fields Quaniaive Finance Seminar 4

5 Pricing of CCR: CVA wih WWR and Condiional EPE (CEPE) We refer o he invesor and counerpary by index and 1. Le T be he mauriy of he porfolio. We denoe by τ and τ 1 he defaul imes of he invesor and counerpary, and by D(, s) he discoun facor a ime for mauriy s. Sochasic dynamics of all processes is considered in he risk-neural measure assuming sandard no-arbirage condiions. As we are ineresed in he defaul and marke facor simulaion model, for simpliciy, we will consider only he case of uncollaeralized counerparies. The price of credi risk wih WWR/RWR is defined by he following quaniies: Credi Valuaion Adjusmen (CVA) (1) CVA + ( ) = LGD Ε { D, τ ) NPV ( ) } 1 1 < τ1 T Debi Value Adjusmen (DVA) ( 1 τ 1 + (2) DVA( ) = LGD Ε { D, τ )( NPV) ( ) } Bilaeral CVA (3) BCVA 1 ( < τ T τ + ( ) = LGD1 Ε { 1 (, 1) ( 1) } < τ1 T1τ τ τ 1< τ D NPV + LGD Ε { 1 1 D(, τ )( NPV) ( τ )} < τ T The expecaions in he above expressions are aken over he join disribuion of he correlaed marke and credi facors. This allows for modeling WWR/RWR. τ < τ 1 Fields Quaniaive Finance Seminar 5

6 A defaul risk measure closely relaed o CVA is Expeced Posiive Exposure (EPE). In he case of independen marke and credi facors, he EPE a ime for he enor is defined as: + (4) () = { NPV ( ) } = M M EPE E NPV ( X ( )) g1( X ( )) X M M where he expecaion is aken over he disribuion of he marke facors X only. For independen marke and credi facors, CVA can be expressed via EPE as: (5) CVA = LGD T 1 + D(, s) EPE () s f1( s) ds where counerpary s defaul probabiliy densiy f 1 ( ) is calculaed from he survival probabiliy S ( 1 ) as f ) ( ) 1 = S1. Survival probabiliy S ( 1 ) is usually boosrapped from he CDS spread erm srucure by a sandard procedure (see JP Morgan (21)). ( ' Sandard EPE (4) and he corresponding CVA (5) do no require a join simulaion of he marke facors and hazard raes, bu hey do no capure WWR/RWR. An exension of he sandard EPE (4) ha accouns for he WWR is a so-called Condiional EPE (CEPE). CEPE was considered in Redon (26) in regards o modeling of WWR for Sovereign Risk (see also earlier works of Levy (1999) and Finger (2) ). Meron s framework was uilized in hese papers for modeling CEPE. dx M Fields Quaniaive Finance Seminar 6

7 In he mos general case, when he marke facors (including he discoun facor hrough he ineres rae facors) and credi facors are dependen, he CEPE a ime for he enor can be defined as he expeced exposure condiional on he counerpary s defaul: (6) CEPE = D ( {( 1 1 ) τ1 = } M + M { D( X ( )) NPV ( X ( )) } 1 τ = g1( 1 + () = D (, ) E D(, τ ) NPV ( τ ), ) 1 X M X h M h Here, g1( X ( ), X ( )) is a join PDF of he marke facors and D (, ) is a given iniial discoun facor erm srucure. The CVA wih WWR/RWR (1) can be expressed via CEPE as: (7) CVA = LGD T 1 D (, s) CEPE () s f1( s) ds X M ( ), X h ( )) dx M M X and credi facors We propose he use of he Condiional EPE and he corresponding Effecive Condiional EPE (ha naurally include WWR/RWR) insead of sandard EPE and Effecive EPE in calculaion of he Basel III Counerpary Credi Risk capial. This will reward RWR business and penalize WWR business. dx h h X, Fields Quaniaive Finance Seminar 7

8 CCR measure: Condiional Poenial Fuure Exposure (CPFE) wih WWR The mos popular risk measure in Financial Indusry for esimaing he Counerpary Credi Risk and monioring credi limis is Poenial Fuure Exposure (PFE). The Poenial Fuure Exposure profile PFE( ) is he maximum amoun of exposure NPV + () expeced o occur on he fuure dae wih a given degree of saisical confidence α (usually, α = 95% ). In oher words, PFE() is a α -percenile of he exposure disribuion: ( ) + (8) PFE( ) : = q α, NPV ( ) The Maximum (Peak) Poenial Fuure Exposure is he maximum of he ( ) he life of he porfolio. PFE over Exposure PFE Effecive EPE EPE Max PFE Time Fields Quaniaive Finance Seminar 8

9 The sandard PFE (8) requires simulaion only of he marke facors, bu i does no capure WWR/RWR. Similarly o exension of EPE o Condiional EPE, we propose o exend a sandard PFE (8) o a new CCR measure ha accouns for he WWR/RWR, which we call a Condiional PFE (CPFE). A CPFE() profile a ime for he enor is a α -percenile of he exposure disribuion condiional on he counerpary s defaul: (, τ ) + (9) ( ) = q α NPV ( τ ) CPFE = Calculaion of CPFE requires join simulaion of he correlaed marke and credi facors. I is he usual pracice of Credi Risk Deparmens no o include discouning in he PFE profile (as in (8) and (9)). However, o be more consisen wih he definiion of CVA (1) and CEPE (6), we modify he formula (9) for CPFE as : (, ( ) τ ) 1 + (1) ( ) = q α D (, ) D(, τ NPV ( τ ) CPFE = 1) where D (, ) is he iniial discoun facor erm srucure Auhors hank Terry Demopoulos of RBC QRA for his idea Fields Quaniaive Finance Seminar 9

10 Counerpary Credi Economic Capial (CCEC)-like measure wih WWR A drawback of he Sandard PFE and inroduced Condiional PFE is absence of he counerpary s credi qualiy in heir definiions. Curren pracice of Credi Risk Deparmens is o facor in he counerpary s credi raing in addiion o he PFE profile. This approach does no properly relae he full erm srucure of he counerpary s defaul probabiliy wih he PFE profile. To he conrary, CVA accouns for he full defaul probabiliy erm srucure. For his reason, we propose o complemen he CPFE for each counerpary wih a new CCR measure - Counerpary Credi Economic Capial (CCEC). The proposed CCEC is a quanile of a full counerpary s credi loss disribuion (while CVA is he expeced value of he discouned counerpary s credi loss disribuion). CCEC is similar o a Porfolio Credi Economic Capial, bu i is calculaed separaely for each counerpary: + (11) CCEC q α 1 NPV () : = [, T ], = τ1 ( ) CCEC (11) includes WWR/RWR for correlaed marke and credi facors and a full defaul probabiliy erm srucure of he counerpary. Fields Quaniaive Finance Seminar 1

11 Unified mulifacor Gaussian wih jumps defaul inensiy framework for CVA, CPFE and CCEC wih WWR and credi raing ransiions In he reduced form (inensiy) framework, ime of he counerpary s defaul τ is modeled by he firs jump of he Cox process (see Lando (1998), Duffie and Singleon (1999)). Equivalenly, he defaul occurs when he inegraed defaul inensiy his for he firs ime he independen exponenial random boundary. Defaul inensiy λ () is usually chosen as posiive sochasic process (e.g., CIR process or CIR + Exponenial jumps, see Brigo and Pallavicini (28)). For posiive affine jump- diffusion defaul inensiies, he survival probabiliy of he counerpary can be found in a closed form (see Duffie and Singleon (1999), Duffie, Pan and Singleon (2)). Gaussian inensiy approach was considered in Schönbucher (23) and oher publicaions. However, negaive defaul inensiies lead o non-monoonic inegraed inensiies and cause he lack of affiniy and analyical racabiliy. Pracically, his approach was abandoned by researchers. In his presenaion, we consider Mone Carlo simulaion approach for Gaussian mean-revering Ornsein-Uhlenbeck (OU) inensiy wih Poisson jumps of arbirary sign framework wih Wrong-Way Risk, develop an effecive numerical calibraion procedure for fiing he drif of λ () ino he observed CDS spreads, and exend he model by consisen wih he hazard rae dynamics simple (CrediMerics-ype) simulaion of he credi raing ransiions for modeling credi riggers in he case of collaerized counerparies (see Yi (211) for credi riggers). Fields Quaniaive Finance Seminar 11

12 Gaussian hazard rae model for one name Similar o he Hull-Whie shor rae model (see Brigo and Mercurio (26)), we consider an addiive form of he OU process for possibly negaive defaul inensiy λ () : (12) h λ( ) = ϕ( ) + X ( ), [, T ] where ϕ () is a deerminisic funcion subjec o fiing ino he iniial erm srucure of he survival probabiliy boosrapped from he CDS spreads a ime zero, and X h () is a homogeneous OU process (13) h h h dx ( ) = κ X ( ) + σ dw, X ( ) = Here, W () is a sandard Wiener process in he risk neural measure ha can be correlaed wih marke variables in he case of Wrong/Righ-Way Risk (e.g., wih Hull- Whie ineres raes and log-normal FX raes in Amin and Jarrow (1991), equiy indices, commodiy prices; ec.) and defaul inensiies of oher names. We define he Gaussian inegraed sochasic inensiy Λ () and is maximum M () as: (14) Λ( ) = λ( s) ds = ϕ( s) ds + X h ( s) ds = Φ( ) + I h ( ), M ( ) = max s [, ] { Λ( s) } Fields Quaniaive Finance Seminar 12

13 If he defaul inensiy was a posiive sochasic process, hen he counerpary s survival (defaul) probabiliy S () ( P () ) would be expressed by a well-known Lando s formula: (15) S( ) = 1 P( ) = Ε Λ { ( e ) } In he case of non-posiive inensiy (e.g., Gaussian), he survival probabiliy is expressed M ( ) = max Λ( s) Λ ( (Jeanblanc e. al. (29), p. 42): via { } raher han ) (16) S { Λ( s) } max s [, ] ) = Ε e = Ε ( where f M (, y) is a densiy of M () a ime. M ( ) y { e } = e f (, y) dy + M.12 Defaul Time τ = (inf : Λ()=ε) for OU Inensiy λ() , years θ λ() Λ() M()=max(Λ()) ε Fig. 1. Pahs of OU and Inegraed OU inensiy hiing he exponenial barrier Fields Quaniaive Finance Seminar 13

14 In our mehodology for modeling WWR, we will sricly use Lando s approach, i.e., direcly simulae he defaul imes as hiing imes of he exponenial barrier y by Λ (). We will use he following connecion beween he defaul and hiing ime densiies. Lemma 1. Le Ty be he hiing ime of he boundary y > by Λ () and ft y (, y) be is densiy. Under some regulariy condiions, he defaul ime densiy ' (17) f d ( ) = P ( ) = S ( ) is given by he Laplace ransform evaluaed a 1 of he hiing ime densiy ft y (, y) wih respec o he barrier level y : (18) f d ( ) = + e y f T y ' (, y) dy Unforunaely, a closed analyical formula for he hiing ime densiy for he non- Markovian Inegraed OU process is unknown. A soluion of such problem is even unknown in he case of an arbirary drif ϕ () for much simpler (Markovian) drifed Wiener process! Therefore, here is a need for developing an effecive numerical mehod for calculaion of he drif ϕ () in defaul inensiy from he iniial erm srucure of CDS spreads. Fields Quaniaive Finance Seminar 14

15 A model problem wih whie noise defaul inensiy As illusraion of he problem, le us consider a model Cox process wih a drifed whie noise defaul inensiy λ ( ) = dw + ϕ( ), i.e. (19) Λ( ) = λ( s) ds= W +Φ( ), Φ() = As we see laer, he behavior for small ime of he Inegraed OU inensiy (14) and he corresponding funcion ϕ () is very differen from he behavior of Wiener inegraed inensiy (19) and is drif ϕ (). However, for large, boh inegraed inensiies have variances proporional o, and he shapes of boh drifs ϕ () are similar. If Φ ( ) = ν is linear, he hiing ime densiy is given by he Bachelier-Lévy formula: 2 y ( y ν) (2) ft y (, y) = exp, y > 3 2π 2 From Lemma 1, he corresponding defaul ime densiy () is: (21) f d ( ) ) 2 1 ν 1 2ν f d (, y) = exp + ( 1)exp ν 2 N ν π This densiy can also be direcly calculaed from he well-known CDF of he maximum of a Wiener process wih linear drif (see, for example, Jeanblanc e. al. (29)). Fields Quaniaive Finance Seminar 15

16 In general case, when he survival probabiliy erm srucure S() is given from he marke (i.e., he defaul ime densiy f d () is known), he corresponding non-linear funcion Φ () can be found by he following effecive numerical mehod. Proposiion 1. The funcion Φ () solves he following Volerra-ype equaion: f d ( ) = 1 2π e 2 Φ ( ) 2 + ( Φ ' () 1) e Φ( ) 2 Φ() Ν (22) + Φ () Φ( s) s Φ ' () f d 2π ( s) ( s) e ( ( ) ( )) 2 Φ Φ s 2( s) ds, Φ() = Proof: A Volerra inegral equaion for he hiing ime densiy of a non-linear boundary a () by a Gaussian Markovian sochasic process was derived in Durbin (1985). Specifically, given he value y > of he exponenial random variable ε, he hiing ime densiy ft y (, y) of he boundary a( ) = y Φ( ) by he drifless Wiener process W () is he soluion of he following Volerra inegral equaion: Fields Quaniaive Finance Seminar 16

17 (23) f + T y (, y) = Φ () Φ( s) s 1 2π 3 e Φ ( Φ( ) y) 2 2 ' () f T ' ( y Φ() + Φ () ) y 2π ( s, y) ( s) e ( Φ( ) Φ( s) ) 2 2( s) ds From Lemma 1, by aking he Laplace ransform wih respec o y of he boh sides of his equaion (23), one obains he required equaion (22). The Volerra inegral equaion (22) gives a very effecive numerical algorihm for finding he drif Φ() from a given defaul ime densiy ( ) by he discreizaion mehod wih f d sequenial soluion of he corresponding non-linear equaion for ) previously found values Φ ), Φ( ),..., Φ( ). ( 1 i 1 Φ ( i given he Tesing resuls for his numerical mehod and direc Mone Carlo simulaion of he defaul imes are presened in Fig. 2 for he following esing defaul ime densiy ~ 1.5ν (24) ( ) = 2 α α fd e + (1 ν ) e β.5 2π π ( ν =.682, β =.671, α =.112) and linear-drif defaul ime densiy (2). Fields Quaniaive Finance Seminar 17

18 3 Calibraion of Φ() for Whie-Noise Inensiy.2 Simulaed vs. Inpu Defaul Densiies for Whie-Noise Inensiy Calibraed Φ() Linear Φ() MC Densiy for Linear Φ() Theoreical Densiy for Linear Φ() MC Densiy for Calibraed Φ() Inpu Defaul Densiy Fig. 2. Example of calibraion of Φ () for whie noise inensiy model fromvolerra inegral equaion and is verificaion by Mone Carlo simulaion of hiing imes. Fields Quaniaive Finance Seminar 18

19 Mone Carlo based calibraion for OU defaul inensiy There is no known inegral equaion for funcion Φ() for Inegraed OU process (14) (hings are complicaed by he fac ha Inegraed OU process is no a Markovian process). We develop a Mone-Carlo based fiing procedure using he idea of sequenial calculaion of he values of Φ () similar o soluion of he Volerra inegral equaion. We assume he survival probabiliy S() is calculaed from he observed CDS spreads using a sandard boosrap mehod (see JP Morgan (21)). We also assume he mean reversion speed κ and volailiy σ for he OU hazard rae process (14) are known. We simulae a significan number N of pahs for he inegraed OU sae variable I h () wih fine ime seps. Assuming piece-wise linear Φ (), we sequenially calculae Φ ( i ) from non-linear equaions using he previously found values Φ( ), Φ( 1),..., Φ( i 1) direcly from he sampling mean in he Lando s formula for he survival probabiliy: N 1 h (25) S( i) = exp max Φ( ) + X ( s) ds = N,..., i 1 The advanage of Gaussian framework is he highes performance for join simulaion of he Wiener, OU and Inegraed OU processes in closed form for significan number of pahs required for achieving he sufficien accuracy (see Glasserman (23)). Fields Quaniaive Finance Seminar 19

20 We presen in Fig. 3 wo examples for Mone Carlo based fiing of Φ () ino he piecewise consan iniial hazard rae erm srucures h ( ) : firs for fla hazard raes, and second for acual hazard raes of he BB and A raed companies. The calculaed ϕ () is compared wih he approximae Duffie-Singleon drif of he form 2 D S σ 1 exp( 2κ ) 1 exp( κ ) ϕ ( ) = 2 + h ( ) 2 + (26) κ 2κ κ ha corresponds o he affine survival probabiliy formula if one replaces he maximum of he Inegraed OU inensiy by he Gaussian Inegraed OU inensiy iself. 6% 5% 4% 3% 2% 1% % -1% -2% -3% Calibraion of Funcion φ() for Gaussian Hazard Rae Model y. Hazard Rae, h=5% Gauss. φ(), h=5%, σ=1% Gauss. φ(), h=5%, σ=5% D.-S. φ(), h=1%, σ=1% D.-S. φ(), h=1%, σ=5% D.-S. φ(), h=5%, σ=1% D.-S. φ(), h=5%, σ=5% Hazard Rae, h=1% Gauss. φ(), h=1%, σ=1% Gauss. φ(), h=1%, σ=5% 1% 9% 8% 7% 6% 5% 4% 3% 2% 1% % -1% -2% -3% -4% -5% Calibraion of Funcion φ() for BB and A Raed Counerparies y. BB Hazard Rae BB, D.-S. φ BB, Gauss. φ A Hazard Rae A, D.-S. φ A, Gauss. φ Fig. 3.Calibraion of Φ () using Mone Carlo mehod Fields Quaniaive Finance Seminar 2

21 Mone Carlo simulaion of EPE and CPFE profiles wih WWR For calculaion of CVA, CPFE and CCEC wih WWR, we use brue force join Mone Carlo simulaion of he correlaed Gaussian inegraed defaul inensiies and relevan o he counerpary s porfolio marke variables (e.g., Hull-Whie ineres raes) wih explici calculaion of he defaul imes as hiing imes of he exponenial random boundary by Inegraed OU inensiies and revaluaion of he porfolio MM a he defaul ime. Example 1.1. Sock price disribuions condiional on defaul and classical example of Specific WWR ransacion pu opion on he counerpary s own sock PDF of Sock Price Disribuion a 5 years wih WWR, RWR and Zero Correlaion Uncondiional ρ= ρ=-1 ρ=+1 $ $5 $1 $15 $2 $25 $3 $45 $4 $35 $3 $25 $2 $15 $1 $5 $ PFE, CPFE, EPE and CEPE Profiles for 5-year ATM Pu Opion PFE - ρ= EPE - ρ= CPFE - ρ=-.9 CEPE - ρ= y. 5 Fields Quaniaive Finance Seminar 21

22 ρ= ρ=-.9 WWR Raio CPFE 95% $38.26 $ CCEC 95% $18.34 $ CCEC 99% $31.78 $ Unilaeral CVA $2.72 $ Bilaeral CVA (BCVA) $2.53 $ Table 1. CPFE, CCEC, CVA and BCVA for he 5-year pu opion (LGD=6%) Example 1.2. CPFE/CEPE for RWR ransacion Call Calendar Spread on counerpary s sock - 5-yr. long call srike $13 and 56-mo. shor call srike $126 PFE Profiles for Call Calendar Spread EPE Profiles for Call Calendar Spread $1 $16. $9 $8 $7 $6 $14. $12. $1. $5 $4 $3 $2 $1 $8. $6. $4. $2. $ y. $ y. 95% PFE - ρ= 95% PFE - ρ=-.9 EPE - ρ= EPE - ρ=-.9 For his RWR ransacion, he proposed Economic Capial-like measure CCEC makes more sense han PFE, because he peak exposure is achieved during a very shor period, which corresponds o a very low probabiliy of defaul. Fields Quaniaive Finance Seminar 22

23 Example 2. CPFE/EPE profiles wih he corresponding PDF/CDF of he condiional credi loss disribuion and CVA/CCEC values wih WWR/RWR for he 1-year payer ineres rae swap $12 PFE & EPE Profiles for 1-year Payer IR Swap.7 PDF of Loss Disribuions for 1-year Payer IR Swap 1% CDF of Loss Disribuions for 1-year Payer IR Swap $1.6 95% $8.5 9% $6.4 85%.3 $4.2 8% $2.1 75% $ y. PFE ρ=. - Undiscouned EPE ρ=. PFE ρ=. - Frwd measure EPE ρ=+.9 PFE ρ=+.9 EPE ρ= -.9 PFE ρ= -.9. $. $5. $1. ρ=. ρ=+.9 ρ= -.9 CVA=$.61 (ρ=.) CVA=$1.1 (ρ=+.9) CVA=$.16 (ρ= -.9) 7% $ $5 $1 ρ=. ρ=+.9 ρ= -.9 CCEC 95%=$5.2, ρ=. CCEC 95%=$7.2, ρ=+.9 CCEC 95%=$1.42, ρ= -.9 Noe. Undiscouned PFE is based on disribuion of -1 + PFE is based on disribuion of D(,) [ D() MM ()] MM + (), Forward measure, where D(, ) is he iniial discoun facor erm srucure and D() is a simulaed sochasic discoun facor. Forward measure PFE is more consisen wih CVA formula. Fields Quaniaive Finance Seminar 23

24 Example 3. CPFE/EPE profiles wih he corresponding CDF of he condiional credi loss disribuion and CCEC values wih WWR/RWR for he 1-year USD/EUR cross currency basis swap $3 PFE & EPE Profiles for 1-year Cross Currency Basis Swap 1% CDF of Loss Disribuions for 1-year XCcy Basis Swap $25 95% $2 $15 $1 $5 $ y. PFE ρ=. EPE ρ=. EPE ρ=+.9 PFE ρ=+.9 EPE ρ= -.9 PFE ρ= -.9 9% 85% 8% 75% 7% $ $5 $1 $15 ρ=. ρ=+.9 ρ= -.9 CCEC 95%= $8.96, ρ=. CCEC 95%=$12.61, ρ=+.9 CCEC 95%= $2.49, ρ= -.9 UCVA=$.74 (ρ=), WWR CVA=$1.18 (ρ=.9), RWR CVA=$.32 (ρ=-.9) (for LGD=1%) Fields Quaniaive Finance Seminar 24

25 Consisen Simulaion of Defauls and Credi Raing Transiions in Gaussian Hazard Rae Model The considered pah-wise Mone Carlo simulaion framework allows for implemenaion of full collaeral logic wih hresholds for collaeralized counerparies (see Brigo e. al. (211), Gregory (21), Pykhin and Zhu (27) ). However, he majoriy of agreemens wih collaerized counerparies include credi riggers ha depend on credi raing ransiions. A defaul inensiy framework deermines only defaul imes. Therefore, here is a need for consisen exension of he hazard rae framework by credi raing ransiion modeling. One of such approaches was considered in Lando (1998). However, ha model includes a full marix of sochasic inensiies for defauls and credi raing ransiions, and i is oo complicaed for calibraion and pracical use. On he oher hand, praciioners widely use a simple CrediMerics approach based on he Markov ransiion marix model of Jarrow, Lando and Turnbull (1997). A presened Gaussian hazard rae framework allows for a simple reasonable exension ha consisenly combines he OU defaul inensiy model wih he CrediMerics credi raing ransiion approach. Assume, he annual credi raing ransiion/defaul marix A is given. A proposed join OU defaul inensiy/credi raing ransiion model is as follows: Because he defaul imes in our model are fully deermined by he defaul inensiy, we recalculae a reduced credi raing ransiion marix A ~ condiional on no-defaul from he iniial full ransiion/defaul marix A Fields Quaniaive Finance Seminar 25

26 We pre-calculae a sequence of he corresponding roos of he condiional on i no-defaul credi raing ransiion marix A ~ for each ime sep Δ i using, for example, Markov generaor approach (see Israel (2)) and conver he ransiion probabiliies ino he normal quaniles In he original CrediMerics mehod, a sandard normal random variable (inerpreed as asse reurn ) is generaed for each ime sep and compared wih he normal quaniles. A defaul or credi raing ransiion occurs when his normal random variable falls ino he corresponding bucke. In our approach, we joinly simulae he Gaussian defaul inensiy and negaively correlaed wih i asse Wiener process: dx ( ) = κ X ( ) + σ dw (27) a h 2 ξ dw = ρ dw + 1 ρ dw h h ξ ( W and W are independen Wiener processes; he correlaion ρ should be close o -1, because he credi spread reurns are srongly negaively correlaed wih he asse reurns) If defaul did no occur for a given ime sep Δ i (i.e., he inegraed defaul inensiy I h () did no hi he exponenial barrier), hen he sandard normal asse reurn variable a i i h Δ W Δ is compared wih he quaniles of he condiional on no-defaul credi raing ransiion marix h i A ~ and he corresponding credi raing is assigned Fields Quaniaive Finance Seminar 26

27 Mulifacor Gaussian and Jump-Diffusion frameworks Gaussian framework allows for easy and efficien implemenaion of mulifacor models wih housands of correlaed counerparies and marke facors. In pracice, all correlaions beween counerpary credi spreads are no available. A sandard indusry pracice is o rely on he CAPM-like regression approach: each counerpary credi spread is regressed on a se of credi indices and marke facors conribuing o he Wrong-Way Risk for his counerpary. Gaussian framework is ideal for regressions. Proposed Gaussian defaul inensiy framework is easily exendable o jump-diffusion model wih no resricion on he sign of jumps (o he conrary, he affine jump diffusion framework requires posiive jumps and posiive diffusion processes, e.g., Brigo, Pallavicini and Papaheodorou (211) use square-roo jump-diffusion process wih posiive exponenial jumps). In addiion o Wiener processes, independen compound Poisson (or Lévy jump) processes are inroduced and defaul inensiy h X i of he counerpary i is modeled as: h h (28) d ( ) = κ i X ( ) + σ i ai jdwj + b i i, i X, k j k a i, j, bi, k where coefficiens define he correlaion beween counerparies. The Mone Carlo based fiing procedure for Φ () says he same, because inegraed inensiy is a coninuous funcion of and he roos for equaion (25) can be found. J i dj k Fields Quaniaive Finance Seminar 27

28 Comparison of Gaussian and Kou jump models As an example of jump defaul inensiy model wih no resricion on he sign of jumps, we consider mean reversion hazard rae and logarihm of he FX rae driven by linear combinaion of independen compound Poisson processes wih double-exponenial jump size disribuions, i.e. so-called Kou (22) model. The coefficiens wih respec o common compound Poisson processes define he FX/defaul inensiy correlaions and resul in WWR/RWR. The compensaor for he FX rae in Kou model is known, and he drif Φ( ) in he defaul inensiy (25) is calculaed from he CDS spreads by he Mone- Carlo based fiing procedure described earlier in he presenaion (see Fig 4). I is well known ha jumps have shor-erm impac compared o diffusions (e.g., Lando (24))..2 Funcion φ() for Gaussian and Jump Defaul Inensiy Models y Gaussian model Kou model, downward jumps Kou model, symmeric jumps Kou model, upward jumps Fig. 4. Calibraion of Φ () for jump model using Mone Carlo mehod Fields Quaniaive Finance Seminar 28

29 Figure 5 illusraes more significan Wrong/Righ-Way Risk for jump model compared o Gaussian OU defaul inensiy model (for shor one-year horizon, where impac of jumps is significan)..6 One-year PDF for FX Rae Gauss., ρ= Gauss., ρ=1 Gauss., ρ=-1 Jump, ρ= Jump, ρ=1 Jump, ρ=-1 Fig. 5. Condiional FX Rae disribuions for Gaussian and Kou defaul inensiies Fields Quaniaive Finance Seminar 29

30 Figure 6 also confirms larger Wrong-Way Risk for jump model compared o Gaussian OU defaul inensiy model for shorer horizons. $45 $4 $35 $3 $25 $2 $15 $1 $5 $ PFE & EPE Profiles for 1-year Cross-Currency Basis Swap y. 1 Gaussian 95% PFE Profile Jump 95% PFE Profile Gaussian EPE Profile Jump EPE Profile $7 $6 $5 $4 $3 $2 $1 $ PFE & EPE Profiles for 1-year Cross- Currency Basis Swap y. 1 Gaussian 95% PFE Profile Jump 95% PFE Profile Gaussian EPE Profile Jump EPE Profile Fig. 6. WWR for Gaussian and Kou defaul inensiies Fields Quaniaive Finance Seminar 3

31 The use of new Gamma-Facor Copula (GFC) for improving defaul correlaions in he Porfolio Credi Economic Capial I is well documened in credi risk lieraure ha radiional Gaussian correlaions beween hazard rae processes are no able o generae sufficien defaul correlaions for Porfolio credi risk modeling. The applicaion of copulas o relae marginal defaul probabiliies of differen names is considered as alernaive approach o diffusion correlaions. In general, here are hree alernaive approaches for modeling dependen defauls: o Correlaed defaul inensiy processes wih independen exponenial hresholds o Dependen exponenial hresholds wih independen inensiy processes o Correlaed defaul inensiy processes wih dependen exponenial hreshold Our preference is he hird approach. Everyday correlaions beween credi spreads of he counerparies are observed in he marke. Gaussian copula and Suden- copula wih ν > degrees of freedom are he mos popular choices for a copula. However, he corresponding correlaed mulivariae Gaussian and disribuions are defined on a whole space, while a desired mulivariae exponenial disribuion is defined on he posiive n dimensional ocan, i.e., hese nonlinear copula ransformaions are no naural for he problem a hand. Gaussian copula has zero ail-dependence (see McNeil(25), p. 211) and poorly models exreme lowprobabiliy join defauls observed in he marke during credi. Though a -copula does have ail-dependence, i requires a very low number of degrees of freedom and high correlaion coefficiens o provide sufficien ail-dependence Fields Quaniaive Finance Seminar 31

32 Bu, because a mulivariae disribuion is a normal-mixure disribuion by he 2 χ common ν -disribued variance, he componens of a -disribued vecor are dependen even for zero correlaion coefficiens. This means a -copula is unable o model highly correlaed exponenial hresholds wih ail-dependence for a group of required names and simulaneously independen exponenial hresholds for anoher group of required names. We propose a new consrucion naural for correlaed exponenial random variables hrough he decomposiion of he exponenial random variables ino he sums of some independen Gamma random variables ( Gamma-facors").We call he copula ha is implicily defined by his consrucion a Gamma-facor" copula. Le ξ, ξ, 1 2 K, ξm denoe m independen gamma random variables ξ i ~ Γ( α,1), α [,1], wih he same scaling parameer 1 and probabiliy densiy funcions 1 α i 1 y (29) fα ( y) = y e, y > i Γ( αi ) We define n ( n m) dependen uni exponenial random variables Υ, K 1, Υn as follows: r r r T r T r T Υ = Aγ, Υ = [ Υ1, K, Υn ], γ = [ γ1, K, γ n], α = [ α1, K, αn] r A = [ a [ ] T i, j ]( n m), Aα = 1, K,1 ( n 1) The enries of he n m load marix A are eiher zero or one. Wihin hese m Gamma facors, here are m n common facors (i.e., he corresponding columns of he marix A Fields Quaniaive Finance Seminar 32

33 have more han one enries equal o 1) and n idiosyncraic facors wih all zero enries excep one equal o 1 in he corresponding columns of he marix A. The example of marix A is shown in Table 2. The common Gamma random variables can be associaed wih cerain economic facors such as secors, indusries, regions, credi raings, ec. This idea is similar o Moody's Analyics Global Correlaion Facor and CrediMerics approaches for modeling asse correlaions using some global and idiosyncraic facors. Marke Ind.A Ind.B Ind.C Cnry 1 Cnry 2 Idio1 Idio2 Idio3 Idio4 Name Name Name Name Table 2. Example of Gamma-Facor Copula load marix A Le us consider wo names for calculaion of he defaul correlaion. Le ˆ ξ, ξ1, ξ2 be hree independen gamma random variables common gamma-facor ˆ ξ ~ Γ(α,1), and wo idiosyncraic gamma-facors ξ1 ~ Γ (1 α,1), ξ2 ~ Γ (1 α,1), all wih he same scaling parameer β = 1, where he dependency parameer α [,1]. The correlaed exponenial random hresholds Υ 1 and Υ 2 for wo given names are defined as (3) Υ1 = ˆ ξ + ξ ˆ 1, Υ2 = ξ + ξ2 When α =, and Υ are independen; when α = 1 hey become perfecly correlaed. Υ1 2 Fields Quaniaive Finance Seminar 33

34 Bivariae exponenial disribuion Bivariae copula Fields Quaniaive Finance Seminar 34

35 The impac of he Gaussian correlaion and parameer α of Gamma-Facor Copula on he defaul correlaion erm srucures for wo counerparies Auhors hank Chuang Yi, formerly of RBC Risk Mehodology, for performing his Mone Carlo invesigaion Fields Quaniaive Finance Seminar 35

36 Comparison of impac on Bilaeral CVA of he Gamma-Facor Copula and Gaussian defaul inensiy correlaion We compare he impac of he Gaussian correlaion ρ λ, λ1 beween he invesor s (BAC) and counerpary s (F) defaul inensiies and he dependency parameer α of he GFC on he Bilaeral CVA of a 1-year Ineres Rae Swap wih noional $1. Marke daa is as of Oc. 2, 211, he Hull-Whie risk-neural parameers for he USD ineres rae were calibraed by a sandard procedure, mean-reversion parameers and volailiies of he hazard raes were esimaed from he hisorical daa, recovery raes are 4%. BCVA α\ρ $1.15 $1.13 $ $1.1 $1.8 $ $1.1 $.97 $.9 Relaive Impac w.r.. α=, ρ= α\ρ % 99% 97%.5 95% 94% 92% 1. 88% 85% 78% (CVA = $1.77, DVA = $.48) Fields Quaniaive Finance Seminar 36

37 References 1. Amin, K., and Jarrow, R. (1991) Pricing foreign currency opions under sochasic ineres raes. Journal of Inernaional Money and Finance, 1, BIS (26) Basel II: Inernaional Convergence of Capial Measuremen and Capial Sandards: A Revised Framework. Basel Commiee on Banking Supervision. 3. BIS (21) Basel III: A global regulaory framework for more resilien banks and banking sysems. Basel Commiee on Banking Supervision. 4. Brigo, D., and Capponi, A. (21) Bilaeral counerpary risk wih applicaion o CDSs. Risk Magazine, March, Brigo, D., e. al. (211) Collaeral margining in arbirage-free counerpary valuaion adjusmen including re-hypohecaion and neing. Working paper. 6. Brigo, D., e. al. (28) Counerpary risk valuaion for energy-commodiies swaps: impac of volailiies and correlaions. Fich Soluions, Brigo, D., and Mercurio, F. (26) Ineres Rae Models - Theory and Pracice: Wih Smile, Inflaion and Credi. Springer. 8. Brigo, D., and Pallavicini, A. (28) Counerpary risk and CDSs under correlaion. Risk Magazine, February, Brigo, D., e. al. (211) Bilaeral counerpary risk valuaion for ineres-rae producs: impac of volailiies and correlaions. Inernaional Journal of Theoreical and Applied Finance, 14(6), Duffie, D., Pan, J., and Singleon, K. (2) Transform analysis and asse pricing for affine jump-diffusions. Economerica, 68, Fields Quaniaive Finance Seminar 37

38 11. Duffie, D. and Singleon, K. (1999). Modeling erm srucures of defaulable bonds. The Review of Financial Sudies, 12(4), Durbin, J. (1985) The firs-passage densiy of a coninuous Gaussian process o a general boundary. Journal of Applied Probabiliy, 22, Finger, C. (2) Toward a beer esimaion of wrong-way credi exposure. RiskMerics Journal, May, Glasserman, P. (23). Mone Carlo mehods in financial engineering. Springer. 15. Gregory, J. (21) Counerpary Credi Risk: The new challenge for global financial markes. Wiley. 16. Gupon, G., e. al. (1997) CrediMerics Technical Documen. JPMorgan. 17. Hull, J. and Whie, A. (212) CVA and Wrong-Way Risk, Universiy of Torono working paper. 18. Israel, R., e. al. (2) Finding generaors for Markov chains via empirical ransiion marices, wih applicaions o credi raings. Mahemaical Finance, 11, Jackson, K., Kreinin, A. and Zhang, W. (29) Randomizaion in he firs hiing ime problem. Saisics and Probabiliy Leers, 79, Jarrow, R., Lando, D., and Turnbull, S. (1997) A Markov model for he erm srucure of credi risk spreads. Review of Financial Sudies, 1(2), Jeanblanc, M., Yor, M., and Chesney, M. (29) Mahemaical Mehods for Financial Markes, Springer. 22. JP Morgan (21) Par Credi Defaul Swap spread approximaion from defaul probabiliies. JPMorgan. Fields Quaniaive Finance Seminar 38

39 23. Kou, S. (22) A jump diffusion model for opion pricing, Managemen Science 48(8), Lando, D. (1998) On Cox processes and credi risky securiies. Review of Derivaives Research, 2, Lando, D. (24) Credi Risk Modeling: Theory and Applicaions. Princeon Universiy Press. 26. Levy, A., (1999) Wrong-Way Exposure. Are firms underesimaing heir credi risk? Risk Magazine, July. 27. McNeil, A., Frey, R., and Embrechs, P. (25) Quaniaive Risk Managemen: Conceps, Techniques, and Tools. Princeon Universiy Press. 28. Peskir, G. and Shiryaev, A. (26) Opimal sopping and free-boundary problems Springer 29. Pykhin, M., and Zhu, S. (27) A Guide o modeling Counerpary Credi Risk, GARP Risk Review, July, Redon, C. (26) Counerpary risk - Wrong-way risk modeling. Risk Magazine, April, Schönbucher, P. (23) Credi Derivaives Pricing Models: Model, Pricing and Implemenaion. Wiley. 32. Yi, C. (211) Dangerous knowledge: Credi Value Adjusmen wih credi riggers. Inernaional Journal of Theoreical and Applied Finance, 14(6), Fields Quaniaive Finance Seminar 39