Stochastic Intensity Models of Wrong Way Risk: Wrong Way CVA Need Not Exceed Independent CVA

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1 V OLUME 21 N UMBER 3 S PRNG Sochasc nensy Models of Wrong Way Rsk: Wrong Way CVA Need No Exceed ndependen CVA SAMM GHAMAM AND LSA R. GOLDBERG

2 Sochasc nensy Models of Wrong Way Rsk: Wrong Way CVA Need No Exceed ndependen CVA SAMM GHAMAM AND LSA R. GOLDBERG SAMM GHAMAM s an economs a he Board of Governors of he Federal Reserve Sysem n Washngon, DC, and a senor researcher a he Cener for Rsk Managemen Research a he Unversy of Calforna n Berkeley, CA. samm.ghamam@frb.gov LSA R. GOLDBERG s drecor of research a he Cener for Rsk Managemen Research and Adjunc Professor of Sascs a he Unversy of Calforna n Berkeley, CA. lrg@sa.berkeley.e Wrong way rsk can be ncorporaed n Cred Value Adjusmen (CVA) calculaons n a reced form model. Hull and Whe [212] nroced a CVA model ha capures wrong way rsk by expressng he sochasc nensy of a counerpary s defaul me n erms of he fnancal nsuon s cred exposure o he counerpary. We consder a class of reced form CVA models ha ncludes he formulaon of Hull and Whe and show ha wrong way CVA need no exceed ndependen CVA. hs resul s based on some general properes of he model calbraon scheme and a formula ha we derve for nensy models of dependen CVA (wrong or rgh way). We suppor our resul wh a sylzed analycal example as well as more realsc numercal examples based on he Hull and Whe model. We conclude wh a dscusson of he mplcaons of our fndngs for Basel CVA capal charges, whch are predcaed on he assumpon ha wrong way rsk ncreases CVA. Snce he cred crss, he emphass on counerpary cred rsk by boh global and U.S. regulaors has ncreased dramacally. Cred Value Adjusmen (CVA) s one of he mos mporan counerpary cred rsk measures; accordng o Basel, 1 banks are requred o hold regulaory capal based on CVA charges agans each of her counerpares (see, for nsance, Bohme e al. [211]). Consder a porfolo of dervave conracs ha a fnancal nsuon, such as a dealer, holds wh a counerpary. CVA s he dfference beween he porfolo value before and afer adjusmen for he rsk ha he counerpary mgh defaul; s he marke prce of counerpary cred rsk. 2 CVA s expressed n erms of he dealer s counerpary cred exposure, V, whch s he maxmum of zero and he fuure value of he porfolo. also depends on he maury,, of he longes ransacon n he porfolo and he defaul me,, of he counerpary. CVA can be expressed as a rsk-neural expeced dscouned loss: CVA = [(1 R) 1 ] (1) { } where D s he sochasc dscoun facor a me, 1{ } s an ndcaor funcon, and R s he fnancal nsuon s recovery rae. 3 Hereafer, for noaonal smplcy, we suppress he dependence of he CVA o he recovery rae, R. A wdely adoped assumpon s ha cred exposure, V, and he counerpary s defaul me,, are ndependen. hs leads o ndependen CVA, denoed CVA, and s expressed n erms of he densy, f, of : CVA = [ = ] f( ) d = E[ [ V ] f() d (2) SOCHASC NENSY MODELS OF WRONG WAY RSK SPRNG 214

3 where he las equaly follows from he ndependence of and V. n pracce, a counerpary s defaul me dsrbuon s approxmaed from counerpary cred spreads observed n he marke. Mone Carlo smulaon s hen used o esmae ndependen CVA by esmang E[D V ], based on a dscree me grd. he effcacy of ndependen CVA s lmed, snce here are mporan praccal cases where cred exposure, V, and he counerpary s defaul me,, are correlaed (see Gregory [21], Chaper 8). When cred exposure s negavely correlaed wh a counerpary s cred qualy, he exposure and s assocaed rsk measures are sad o be wrong way. Wrong way CVA, denoed CVA W, refers o CVA n he presence of wrong way rsk. When he correlaon s posve, he exposure and s assocaed rsk measures are sad o be rgh way. o smplfy he exposon, we concenrae on wrong way CVA. However, here are analogous resuls for rgh way CVA. A basc example of wrong way rsk occurs when a dervaves dealer akes a long poson n a pu opon on a sock of a company whose forunes are hghly posvely correlaed wh hose of s counerpary. Praconers wdely hold he vew ha wrong way rsk decreases a counerpary s cred qualy, and ha hs n urn ncreases CVA. hs s also evden from Basel CVA capal charges, where CVA W a CVA wh α > 1. s dffcul o buld any praccal nuon on he mpac of wrong way rsk on CVA n he absence of a mahemacal model capurng he correlaon beween cred exposure, V, and he counerpary s defaul me,, wh a well-defned calbraon scheme usng hsorcal daa o esmae he model parameers. n hs arcle, workng whn he wdely used recedform modelng framework, we show ha wrong way rsk does no necessarly ncrease CVA,.e., CVA could exceed CVA W. Our sarng pon s he model nroced by Hull and Whe [212], summarzed n he followng secon. n ha model, he logarhm of he counerpary s defaul me nensy s an affne funcon of he dealer s exposure o he counerpary. n Sochasc nensy Models of CVA, we consder a class of nensy models of CVA ha ncludes he formulaon of Hull and Whe [212]. We show ha he calbraon scheme of nensy models mples ha he model-mpled cred qualy s supposed o mach he marke-mpled cred qualy. hs holds regardless of how he exogenous relaonshp beween V and s specfed. Le λ denoe he counerpary s defaul me sochasc nensy. As shown n he subsecons under Sochasc nensy Models of CVA, hs mporan mplcaon of he calbraon scheme gves us a useful expresson for CVA : CVA = λ [ ] e d Dervng he followng formula for CVA n he presence of wrong way rsk, CVA W = λ e enables us o drecly compare CVA W and CVA and conclude ha wrong way CVA need no exceed ndependen CVA. ha s, usng reced form modelng, we derve a formula for CVA W and a calbraon-mpled formula for CVA, so ha CVA W and CVA become comparable. We shall emphasze ha n he absence of such a framework,.e., a dependen CVA model wh a well-defned calbraon scheme, no praccal comparson can be made beween wrong way CVA and ndependen CVA. n Numercal Examples we provde numercal examples, based on he Hull and Whe model, showng ha CVA can exceed CVA W. 4 We dscuss he regulaory mplcaons of our resul n Regulaory reamen of Wrong Way Rsk. 5 HE HULL AND WHE SOCHASC NENSY MODEL OF CVA Hull and Whe [212] ncorporae wrong way rsk n a CVA model by formulang a counerpary s defaul nensy n erms of a dealer s cred exposure o he counerpary. hey assume ha he sochasc nensy of a counerpary s defaul me,, denoed by λ, s gven by: λ = e bv +aa d (3) where b s a consan and α s a deermnsc funcon of me. he parameer b governs he ype and level of dependen rsk, and s calbraed by subjecve judgmen n Hull and Whe [212]. A posve value for b ndcaes wrong way rsk, and a negave value ndcaes rgh way rsk. 6 Le s denoe he counerpary s maury- cred spread, and le R denoe he recovery SPRNG 214 HE JOURNAL OF DERVAVES

4 rae. 7 Gven b, he pecewse consans a are sequenally chosen o sasfy: e s 1 = E e (4) as closely as possble. Hull and Whe [212] use he lef sde of Formula (4) as an approxmaon of he counerpary s survval probably up o me >,.e., P( > ). he appendx of Hull and Whe [212] deals how a s sequenally specfed usng Formula (4); he expecaon on he rgh sde s esmaed by Mone Carlo smulaon afer me dscrezaon of he negral of he nensy process, λ. SOCHASC NENSY MODELS OF CVA Movaed by he Hull and Whe model, we consder nensy models of CVA n whch a counerpary s defaul nensy, λ, s drven by a sngle rsk facor, V. he real-valued process {V } s defned on a flered probably space (Ω, F, {G }, P), where {G } denoes he flraon generaed by V. o ncorporae wrong way rsk, he nensy, λ, s defned as an ncreasng funcon of exposure, V. n hs seng, he defaul me,, adms a sochasc nensy, λ. A consequence of hs s an expresson for survval probables (under echncal condons summarzed n Appendx A): P( ( > ) = Ee and condonal survval probables: s P( ( > > ) = Es e (5) (6) where < s < and E s denoes expecaon condonal on all avalable nformaon a me s. 8 Also, Formula (5) mples ha he densy of he defaul me s gven by: = λ f ) e ( λ (7) Remark 1. he resuls of hs arcle hold when λ s drven by more han one rsk facor. hs becomes evden from Remark 2 n Appendx A and a common mplcaon of calbraon schemes n reced form models as dscussed n he followng secons, Calbraon of Sochasc nensy Models and Wrong Way CVA Need No Exceed ndependen CVA. Defnng λ as a funcon of a sngle rsk facor, V, merely faclaes he communcaon of our resuls; smplfes he noaon and resembles he Hull and Whe model. Calbraon of Sochasc nensy Models Many of he reced form models n he cred leraure benef from he compuaonal convenence of affne nensy modelng, by assumng ha λ s an affne funcon of a gven Markov process X, such ha he condonal expecaon n Formula (6) can be wren as: λ( P( ( > > ) = E e Xu ) s = e s α (, ) +β (, ). X s (8) where coeffcens α and β depend only on s and, < s < (see Duffe and Sngleon [23] and Duffe e al. [2]). he Markov process X can be muldmensonal. However, here, for smplcy, we hnk of X as a one-dmensonal process, e.g., a square-roo dffuson. Suppose ha he condonal survval probables on he lef sde of Formula (8) are markempled. For nsance, hey may be approxmaed from corporae bond spreads. Gven he convenen form of he condonal expecaon n Formula (8), and gven ha X has usually well-known dsrbuonal properes, sascal esmaes of he parameers of X and λ are ofen based on (approxmae) maxmum lkelhood esmaon mehods or he Kalman fler. (See Duffe e al. [2], Duffe and Sngleon [23] Appendx B, and Lando [24]. For examples of papers usng an approxmae maxmum lkelhood esmaon mehod and Kalman fler, see Duffe e al. [23] and Duffee [1999], respecvely.) n CVA sochasc nensy modelng, he unknown parameers of λ are also o be esmaed va Formula (5) or Formula (6), assumng ha survval probables or condonal survval probables are marke-mpled. Hull and Whe [212] use Formula (5) and approxmae survval probables based on CDS spreads. Corporae bond spreads can be used o approxmae condonal survval probables (see Appendx B). ha s, Formula (6) can also be used for he calbraon of an nensy model of CVA. SOCHASC NENSY MODELS OF WRONG WAY RSK SPRNG 214

5 n CVA nensy models consdered n hs arcle, λ s a funcon of he cred exposure process V, whch s he maxmum of zero and he value of a dervaves porfolo conssng of possbly housands of dervaves conracs. So he sochasc process governng he dynamcs of V canno be assumed as gven a pror, and affne nensy modelng canno be appled here. ha s, when he dsrbuonal properes of V are no gven a pror, he parameers of λ canno be specfed by benefng from convenen expressons smlar o he one on he rgh sde of Formula (8) and by usng well-known sascal parameer esmaon mehods. n hs sense, he erm calbraon, as opposed o sascal esmaon, s more suable for CVA nensy modelng. We shall emphasze ha regardless of he sophscaon and he mechancs of sascal esmaon or calbraon schemes, he parameers of λ are o be esmaed or approxmaed such ha he model-mpled survval probables: λ u Ee mach he marke-mpled survval probables, or, smlarly, he model-mpled condonal survval probables: λ u s s E e mach he marke-mpled condonal survval probables, where < s <. ha s: he sascal esmaon or calbraon scheme of sochasc defaul nensy models s o ensure ha model-generaed (condonal) survval probables mach marke-mpled (condonal) survval probables. Hereafer, for smplcy we focus on Formula (5) and survval probables. he above observaon has mporan mplcaons for CVA calculaons n he presence of wrong way-rgh way rsk. n wha follows, we furher elaborae on hs by revsng he Hull and Whe calbraon scheme. Consder he Hull and Whe model agan, where bv a λ = e +a. Le < 1 < < n denoe a dscree me grd, and se P( > ) p, = 1, 2,, n. Suppose ha n marke-mpled survval probables p 1,, p n, approxmaed based on maury- CDS spreads wh s 1 1 nsn 1 1 e,..., e R, are gven. Suppose ha b s gven, and he model s unknown parameers are a 1,, a n, on he abovemenoned me grd; a a. he Hull and Whe calbraon scheme sequenally esmaes a s by esmang: 1 Ee wh Mone Carlo smulaon and makng hese Mone Carlo esmaes equal o p, for = 1,, n. For nsance, gven b and p 1, he calbraon scheme uses: p 1 = E e 1 bv + a e 1 a s frs sep o specfy a 1. hs s done by replacng he expecaon above wh s Mone Carlo esmae, based on samplng from V, and hen numercally solvng for a 1. ha s, he calbraon scheme approxmaes a s sequenally, by makng he survval probables generaed by he Hull and Whe model equal o marke-mpled survval probables, p 1,, p n. Model-mpled Counerpary Cred Qualy Suppose ha a counerpary s survval probables P( > ), for >, are consdered o be a measure of s cred qualy. Wrong way exposures are defned by Canabarro and Duffe [23] as cred exposures ha are negavely correlaed wh he cred qualy of he counerpary. n wha follows, we show ha sochasc nensy models of CVA capure hs basc defnon. However, o reerae he resul of he prevous secon: he calbraon scheme equaes a counerpary s model-mpled cred qualy o he counerpary s markempled cred qualy. n oher words, wrong way rsk does no affec a counerpary s cred qualy. n he presence of wrong way rsk, he sochasc nensy, λ, of a counerpary s defaul me,, s defned as an ncreasng funcon of he cred exposure V. Condonal on a gven sample pah of he cred exposure process n [, ], we can wre: P( ( > ) = e λ ( V ) G u (9) SPRNG 214 HE JOURNAL OF DERVAVES

6 Hereafer, when condonng on a gven sample pah of he exposure process n [, ], we suppress he dependence of λ G on G, and we refer o he survval probables on he lef sde of Formula (9) as pahdependen survval probables. Consder wo gven ( k sample pahs, { ) ; }, k = 1, 2, for whch: u (2) λ( ) u hs mples ha he counerpary s cred qualy s lower along he second sample pah,.e., he counerpary s pah-dependen survval probably s lower along he second sample pah: e (2) (1) λ( Vu ) ) λ( u < e n oher words, wrong way rsk affecs a counerpary s cred qualy on a pah-wse bass,.e., lowers he cred qualy along some pahs. However, he calbraon sraegy ha uses Formula (5) equaes he average of pah-dependen survval probables wh he markempled survval probables: P( ( > ) = E[ [ ( > G )] = Marke mpled me Survval Probably An analogous argumen shows ha rgh way rsk does no affec he cred qualy of he counerpary. Wrong Way CVA Need No Exceed ndependen CVA n Lemma 1 of Appendx A, we derve he followng formula for dependen CVA (rgh or wrong way), whch assumes ha he sochasc nensy of he counerpary s defaul me,, s a funcon of he dealer s cred exposure, V: CVA W = λ e d (1) Focusng on wrong way CVA, we now show ha CVA W need no exceed CVA n sochasc nensy models of CVA. Our resul s based on comparng he wrong way CVA formula wh a calbraon-mpled ndependen CVA formula nroced below. he calbraon-mpled expresson for ndependen CVA holds for all nensy models whose calbraon scheme uses Formula (5) or Formula (6). We furher suppor our resul by consrucng a sylzed example a he end of hs secon and n Numercal Examples. Recall ha o calculae CVA, CVA = [ D V ] f( d ) he probably densy funcon (pdf), f, of a counerpary s defaul me s marke-mpled and approxmaed from CDS or bond spreads. he calbraon scheme of sochasc nensy models equaes he marke-mpled (condonal) survval probables o he model-mpled (condonal) survval probables as suggesed by Formulas (5) and (6). hs mples ha he marke-mpled pdf of counerpary s defaul me, f(), s supposed o mach he model-mpled pdf: E λ e for all [, ] as also suggesed by Formula (7). hs gves he useful calbraon-mpled expresson for CVA : CVA = [ λ ] e d (11) whch enables us o compare CVA W and CVA drecly, regardless of he mechancs and sophscaon of he model calbraon sraegy. Hereafer, for smplcy, assume ha he sochasc dscoun facor D s consan or ndependen of λ and V. A comparson of he calbraon-mpled CVA (rgh-hand sde of Formula (11)) and CVA W (rgh-hand sde of Formula (1)) suggess ha wrong way CVA need no exceed ndependen CVA. Noe ha snce he sochasc defaul nensy process s defned as an ncreasng funcon of he cred exposure process, λ, and V are posvely correlaed. ha s, E[ [ λ ] E [ ] E [λ λ ] However, hs has no mplcaon for he par of erms: λ [ λ ED u V λ e and E[ ] E e (12) SOCHASC NENSY MODELS OF WRONG WAY RSK SPRNG 214

7 or for he me negrals of hose erms. We end hs secon by consrucng a sylzed example for whch we analycally prove ha CVA CVA W n some pars of he parameer space. n Numercal Examples, we gve more realsc numercal examples for whch CVA CVA W n he Hull and Whe model. Example 1. Le X denoe a [, 1] unform random varable. Defne he exposure V n he nerval [, ] based on X as follows: X < 1 / V = nx < 1 2 where n s a posve consan. Le λ be he sochasc nensy of a counerpary s defaul me,, and suppose: λ = + u bk a (13) for = 1, 2 and K 1 = X, K 2 = nx. n Formula (13), b s a posve consan and he parameers a 1 and a 2 are calbraed o marke cred spreads. Noe ha snce he me negral of he sochasc nensy s an ncreasng funcon of he exposure, he defnon of wrong way rsk s capured n hs sylzed example. Le p 1 and p 2 denoe he marke-mpled survval probables of he counerpary by me 1 and 2, respecvely. he calbraon scheme ha uses Formula (5) specfes he unknown parameers a 1 and a 2 based on: bk a p = P( ( > ) = E[ ] (14) for = 1, 2 and K 1 = X and K 2 = nx. We show ha for large n: CVA he proof s n Appendx C. Dscusson of Our Resuls C VA (15) Our sudy challenges he premse ha wrong way rsk ncreases CVA and shows ha ndependen CVA can exceed wrong way CVA. Reced-form modelng enables he modeler o exogenously correlae cred exposures and he defaul me of a counerpary, by makng W he defaul me s nensy an ncreasng funcon of cred exposures. he calbraon scheme of any nensy model equaes he model-mpled counerpary s cred qualy wh he marke-mpled counerpary s cred qualy derved from, for nsance, CDS prces. n he followng secon, Model-mpled Counerpary Cred Qualy, hs saemen has been rephrased as n nensy models, wrong way rsk does no affec a counerpary s cred qualy, o furher emphasze hs mporan mplcaon of he calbraon scheme. Usng hs, we derve a calbraon-mpled expresson for he ndependen CVA formula o make drecly comparable wh dependen CVA, whose formula s derved n Appendx A. (See he rgh sdes of Equaons (11) and (1), respecvely.) hen follows ha here s no reason ha one should exceed he oher. s no he purpose of our arcle o numercally expermen wh a fxed model n order o aach fnancal nerpreaons o dfferen pars of he parameer space o formulae a rule prescrbng where CVA W could exceed CVA. A dfferen nensy model of CVA,.e., a dfferen funconal relaon beween λ and V, could lead o dfferen numercal resuls, whch would n urn lead o dfferen ses of fnancal rules and nerpreaons. s he purpose of hs arcle o show ha CVA can exceed CVA W for a broad class of reced form models. Example 1 n he prevous secon s a sylzed seng n whch we show ha CVA exceeds CVA W n some par of he parameer space. On he bass of our sudy, one could argue ha he dependence of a counerpary s cred qualy on cred exposures s already refleced, for nsance, n CDS prces, whch are ndcaors of he cred qualy. n fac, when CDS prces are beleved o reflec all he nformaon on a counerpary s cred qualy, one could queson he need for dependen CVA, whch s hen o be compared wh ndependen CVA. Afer all, n a dependen CVA nensy model, afer exogenously fxng a relaon beween a counerpary s defaul nensy and cred exposures, one should f he model o he marke-mpled cred qualy, whch s also presen n he ndependen CVA formula. NUMERCAL EXAMPLES hs secon s a summary of our numercal examples based on he Hull and Whe [212] model. hey demonsrae ha ndependen CVA can exceed wrong SPRNG 214 HE JOURNAL OF DERVAVES

8 way CVA. here are many praccal nsances where Mone Carlo esmaes of CVA and CVA W are close, bu he former exceeds he laer. We consder conrac level exposures for forward ype conracs and pu opons. n wha follows, we assume ha he rsk-free rae, r, s consan. ha s, he dscoun facor s D = e r and ndependen and wrong way CVA are: CVA [V V ] f( d ) and CVA W = DEV λ e where V denoes he me value of he dervave conrac and s he maury of he conrac. Also, λ s he sochasc nensy proposed by Hull and Whe,.e., λ = exp(bv + a ). Assumng ha b s gven, he pecewse consan deermnsc funcon a s approxmaed based on he counerpary s -maury cred spreads, s, and Formula (4) (see he deals n Hull and Whe [212], Appendx). he expeced exposures, E[V ], are wh respec o he physcal measure n our numercal examples. here s no consensus n counerpary cred rsk around choces of measure for CVA calculaons (see Gregory [29] and Chapers 7 and 9 of Gregory [21] for dscussons on he use of rsk-neural and physcal measure n CVA calculaons). 9 Mone Carlo CVA Esmaon Mone Carlo esmaors of CVA and CVA W, denoed θˆ and θˆw, are defned as follows. Consder he me grd, < 1 < < n, n θ ˆ = = 1 = 1 n d ξ Δ where Δ 1, and, V = m 1 m = V, j 1 j wh V j beng he j h Mone Carlo realzaon of V V. Smlarly, ξ s he m-smulaon-run average of λ Δ k=1 k k V λe, wh Δ k = k k1 beng defned based on a fner me grd, < 1<... < l l, > n. Le {S } denoe a geomerc Brownan moon, X S = S e, where {X } s a Brownan moon wh drf μ and volaly σ. We sample from he rsk facor S based on he physcal measure. hen, gven he Mone Carlo realzaon of S, he valuaon s based on he rsk-neural measure. 1 hs mples V = e r( ) E [S S ] = S for a forward ype conrac. For he pu opons, we smply se V = e r( ) E[(K S ) + S ]. he cred curve s assumed o be fla a s. So, n he ndependen case, he defaul me,, s an exponenal random varable wh mean 1/s. hs leads o he followng closed-form formula for ndependen CVA n ss he forward conrac case, CVA = α ( p( α ) 1) wh 2 σ α=μ + r s. 2 Numercal Resuls CVA esmaes n he followng numercal examples are based on m = 1 5 smulaon runs. 11 We assume a recovery rae of R =, a consan rsk-free rae of r =.1, and an annualzed volaly of 25%. he cred qualy of he counerpary s nvesmen grade wh a fla spread curve a 1 bass pons. he famly of forward conracs presened n Exhb 1 and he famly of n-he-money pu opons analyzed n Exhb 2 are boh examples where ndependen CVA and wrong way CVA are close, bu CVA exceeds CVA W a each maury. he coeffcen b =.2 n boh Exhbs 1 and 2 ndcaes a relavely low dependence of sochasc nensy on exposure. Exhb 3 presens anoher 2% n-he-money pu opon example where CVA W exceeds E XHB 1 Forward Conrac CVA numbers and esmaes are of order 1 3, m = 1 5, b =.2, μ =, σ =.25, S = 2, spread =.1, Δ = 5 ΔΔ=,. 1 for = 1,.8,.6,.4, and Δ=. 1 for =.1,.2. E XHB 2 Pu Opon CVA esmaes are of order 1 3, m = 1 5, b =.2, μ =, σ =.25, S = 1, K = 12, spread =.1, Δ = 5 ΔΔ=,. 1 for = 1,.8,.6,.4, and Δ =. 1 for =.1,.2. SOCHASC NENSY MODELS OF WRONG WAY RSK SPRNG 214

9 E XHB 3 Pu Opon CVA esmaes are of order 1 3, m = 1 5, b = 1, μ =, σ =.25, S = 1, K = 12, spread =.1, Δ = 5 ΔΔ=,. 1 for = 1,.8,.6,.4, and Δ =. 1 for =.1,.2. CVA a each maury; noe ha he dfference s mos pronounced for = 1. he coeffcen b = 1 n Exhb 3 ndcaes a relavely hgher dependence of nensy on exposure. We also came across unrealsc cases of pu opons where CVA exceeds CVA W n a more pronounced way. For nsance, consder he case where he cred spread s fla, a 1 6 bass pons,.e., s = 1. hs gves CVA =.169 and CVA W =.57 for = 1. ha s, ndependen CVA s roughly hree mes larger han wrong way CVA. 12 Noe ha θˆ and θˆw are based esmaors of CVA and CVA W e o he me dscrezaon. deally, he mean square error of hese esmaors should be esmaed. hs s compuaonally exremely expensve n our seng. o ge a feel for he sascal effcency of our esmaors, we noe ha for he forward conrac example presened n Exhb 1, CVA s analycally calculaed, and Mone Carlo esmaes of CVA concde wh he exac values. Snce Mone Carlo esmaon of CVA s compuaonally nensve, a valuable lne of research s o develop effcen Mone Carlo esmaors of CVA. (See Ghamam and Zhang [213] for effcen Mone Carlo ndependen CVA esmaon.) REGULAORY REAMEN OF WRONG WAY RSK Basel s counerpary cred rsk (CCR) regulaory capal charges conss of counerpary defaul rsk (carred over from Basel ) and CVA capal charges for blaeral dervaves ransacons (see BCBS [211]). For cenrally cleared dervaves ransacons, he Basel Commee on Bankng Supervson (BCBS) has recenly devsed capal charges on banks for her cenral counerpary cred rsk (see BCBS [212]). n all hese CCR regulaory capal charges, he BCBS assumes ha wrong way rsk ncreases dfferen measures of CCR, CVA beng one of hem. hen approxmaes a wrong way CCR measure by ncreasng he ndependen CCR measure usng he so-called α mulpler, whch s ofen se o 1.4. ha s, n he case of CVA, wrong way CVA s ofen approxmaed by he ndependen CVA mes 1.4. should be noed ha capurng wrong way rsk s no he only purpose of he BCBS s α mulplers (see Pykhn and Zhu [26], Secon 4.2, on α mulplers and he references here). Smlar o he vew ofen held by praconers n he fnancal nsry, he BCBS s premse n CVA calculaons s ha wrong way rsk ncreases CVA. Our fndngs challenge hs premse. Our resuls would be useful when revewng he mehodology underlyng CCR capal charges ha ncorporae dependen rsk (wrong or rgh way). Hsorcally, BCBS has aken relavely smple and conservave approaches n areas where mahemacal modelng becomes challengng; he alpha-mulpler approach o wrong way CVA esmaon was o provde smple and conservave wrong way CVA esmaes. Fnancal nsuons ha prove o be suffcenly sophscaed n erms of her quanave capables are usually approved by regulaors o use her own nernally developed rsk-sensve models. Our resuls are also useful for regulaors when fnancal nsuons CVA models are beng evaluaed for replacemen by he BCBS s less rsk-sensve proposed mehods. CONCLUSON A mahemacal model s requred o ncorporae he dependency beween a counerpary s cred qualy and cred exposures so as o compare ndependen CVA and dependen CVA (wrong or rgh way). he calbraon scheme of he model plays a crcal role n quanfyng hs comparson. n hs arcle, we focus on sochasc nensy models of CVA ha nclude he formulaon of Hull and Whe [212]. We derve a formula for CVA and show ha he general properes of he calbraon scheme, regardless of s level of sophscaon, mply ha dependen CVA may or may no exceed ndependen CVA. Usng he Hull and Whe model, we generae numercal examples ha confrm our resul for wrong way and ndependen CVA. BCBS s regulaory CCR capal charges assume ha wrong SPRNG 214 HE JOURNAL OF DERVAVES

10 way rsk ncreases CVA, and CVA W s approxmaed by α CVA, where α s ofen assumed o be 1.4. Our resuls would be useful when revewng he regulaory CVA capal charge ha ncorporae dependen rsk (wrong or rgh way). A PPENDX A DEFAUL MES WH SOCHASC NENSY AND HE PROOF OF HE DEPENDEN CVA FORMULA s well known ha a defaul me,, defned on a flered probably space (Ω, F, {F }, P), adms a sochasc nensy, λ, when he process, 1{ } λ s a marngale (where mn{, }). o make he marngale propery precse, he flraon s o be specfed. (For he general case, see Bremaud [1981], Chaper 2.) n wha follows, we do hs for our seng. A consequence of he exsence of an nensy s he deny: = P( ( > ) Ee u whch s used hroughou hs arcle and n he proof Lemma 1. Doubly Sochasc Random mes Le be a defaul me on a flered probably space (Ω, F, {F }, P). Le {H } denoe he flraon generaed by he defaul ndcaor process 1{ }. Suppose ha he dsrbuon of depends on addonal nformaon denoed by {G }. Se F G H where F s he smalles σ-algebra ha conans G and H. 13 he defaul me,, s called doubly sochasc when for all >, 14 P(( ) = P( ) and when condonal on G, λ u s srcly ncreasng. 15 n our seng, {G } s he flraon generaed by he exposure process V. he frs condon mples ha gven he pas values, u, of V, he fuure, s > does no conan any exra nformaon for predcng he probably ha occurs before. 16 he cred exposure process, V, could have jumps e o he expraon of rades pror o he maury of he longes nsrumen n he porfolo. n hs case, where V has pons of dsconnuy, may no be doubly sochasc. Bu can be shown ha sll adms a sochasc nensy λ (see Bremaud [1981], Defnon D7 and heorem D8). Lemma 1. Consder a real-valued process V defned on he probably space (Ω, F, P). Le {G }, denoe he flraon generaed by V,.e., G = σ{v s ; s }, he smalles σ-feld wh respec o whch V s s measurable for every s [, ], and le G G F. Le D denoe a real-valued process ha s adaped o {G }. Le denoe a counerpary s defaul me, whch adms he sochasc nensy λ ha s adaped o {G }. For, ( ) ( P( > = e and P( > ) = Ee hen he followng holds for any gven : [ = }] E[ 1{ λe Proof. Condonal on G we can wre, E[ [ } ] [ ] ) 1{ = DV, = f G( d (A-1) = λ λ DV e u d d where f G s he condonal densy of and s derved based on he lef sde of Formula (A-1). hen he Lemma follows by nong ha: and E[ [ ] = [ ] E λ d λ e Remark 2. We would lke o emphasze ha he dependen CVA formula of Lemma 1 also apples o mulfacor sengs. ha s, when λ s defned based on more han one rsk facor, he proof works by {G } denong he flraon generaed by all he rsk facors. d SOCHASC NENSY MODELS OF WRONG WAY RSK SPRNG 214

11 A PPENDX B APPROXMANG CONDONAL SURVVAL PROBABLES FROM ZERO-COUPON BOND SPREADS Here we use a sylzed seng o show how condonal survval probables can be approxmaed from zero-coupon bond spreads. Le δ(, ) denoe he rsk-neural prce of a maury- defaul-free zero-coupon bond a me >. s well known ha: δ (, ) = E e r u where r s he shor rae process and E denoes he rskneural expecaon condonal on nformaon avalable by me (see, for nsance, Bjork [29]). Le d(, ) denoe he rsk-neural prce of a maury- zerorecovery defaulable zero-coupon bond a me >. Reced-form deb prcng for a defaul me wh he rsk-neural defaul nensy process λ gves ( +λ d, ( ) = E e ) +λ ) as shown by Lando [1998]. Noe ha n a sylzed seng where λ and r are ndependen, condonal survval probables are easly obaned from he defaulable and defaul free bond prces: λ ( > [ u d (, ) P( > ) = E ] = δ (, ) More realsc corporae bond reced-form prcng models also allow he modeler o esmae condonal survval probables from marke daa (see Duffe and Sngleon [23] Chaper 6 and he references heren). A PPENDX C PROOF OF HE RESUL OF EXAMPLE 1 Assume zero shor rae, whch gves D 1. Frs consder CVA : CVA = [ V 1{ }] = E[ ] P( ) + E [ ] P( ) where A = ( -1, ], = 1, 2. Noe ha 1 2 bk a b P( ( ) = E[ [ ] E[ [ K a ] where K = = a =, K 1 = X, and K 2 = nx. Usng he rgh sde of he above n he CVA formula, we can wre CVA [X = ]( + E[ [ ] ( ( ) (C-1) Now, consder CVA W and recall he proof of Lemma 1: CVA = [ V 1{ }] = E[ [ 1{ } } ]] W V Consder he condonal expecaon on he rgh sde above; furher condonng on he defaul me gves [ 1{ ] = E[ } bx a 1 1 e bx a1 + nx e e and so, CVA = 1 e 1 + n e W bnxa 2 + e bx a b X X a bnxa 2 (C-2) Usng Formula (C-1), Formula (C-2), and smple algebrac manpulaons, we have CVA = e a 1 (1 n b CVAW ) X, e X a ( ) + ( Xe, bnx ) (C-3) Smple calculaons show bn bn bnx 1 1 e e ncov (, ) = + 2 2b b n 2b bn 2 hs erm converges o a consan as n. Noe ha, usng Chebyshev s algebrac nequaly, Coυ(X, e bx ) < when b >. So, CVA CVA W for large values of n. ENDNOES We are graeful o Rober Anderson, Sephen Fglewsk, ravs Nesmh, and an anonymous referee for her commens and helpful dscussons. 1 Basel s a global regulaory sandard on bank capal adequacy, sress esng and marke lqudy rsk agreed upon by he members of he Basel Commee on Bankng Supervson n , and scheled o be nroced from 213 unl 218. SPRNG 214 HE JOURNAL OF DERVAVES

12 2 hroughou hs arcle, we consder he unlaeral CVA. See Chaper 7 of Gregory [21] for dscussons on Blaeral CVA. 3 A dervaon of Formula (1) s n Chaper 7 of Gregory [21]. 4 n pracce, dealer porfolos are complex, and here are almos always collaeral and neng agreemens assocaed wh posons. However, n order o effecvely communcae our man resuls, we consder uncollaeralzed conrac-level exposure n our numercal examples. 5 We use he followng erms nerchangeably n he sequel: counerpary cred exposures and cred exposures; also, sochasc defaul nensy models, nensy models, and reced-form models. 6 Hull and Whe [212] dscuss, bu do no mplemen, an esmaon scheme based on hsorcal observaons of he exposure V and cred spread of he counerpary. 7 hs s he recovery rae assocaed wh he cred defaul swap conrac on he counerpary, and may or may no be equal o he recovery rae ha appears n he CVA formula. he recovery rae n he CVA formula refers o he fracon of loss ha s recovered by he fnancal nsuon (a dervaves dealer) f he counerpary defauls. 8 Clearly, Formula (5) follows from Formula (6) by akng s =. We have presened hem separaely o smplfy he exposon, as each expresson s used o calbrae an nensy model o dfferen ypes of hsorcal daa. We dscuss hs n Calbraon of Sochasc nensy Models and n Appendx B. 9 Noe ha n he above seng {D V } s a marngale under he rsk-neural measure. We have chosen he physcal measure merely o avod he rval case, V o = E[D V ] and so CVA = V o P( ), resulng from {D V } beng a marngale. 1 hs s he common and well-known pracce n rsk managemen: samplng from he rsk facors based on he physcal measure and hen rsk-neural valuaon (see, for nsance, Chaper 9 of Glasserman [24]). 11 We use MALAB o proce he resuls. 12 he remanng parameers for hs unrealsc example are σ =.3, b = 2, μ =, S = 1, K = 1.5. Also, Δ = 1. and Δ = By defnon, s an H -soppng me. Noe ha s also an F = G H -soppng me for any {G }. 14 represens he frs even me of a condonal or doubly sochasc Posson process. 15 See, for nsance, Chaper 9 of McNel e al. [25]. 16 Many of he sochasc nensy models n he cred leraure work under hs doubly sochasc framework (see, for nsance, Duffe and Sngleon [23]). REFERENCES BCBS. Basel : A Global Regulaory Framework for More Reslen Banks and Bankng Sysems. Bank for nernaonal Selemens, Basel, Swzerland, Capal Requremens for Bank Exposures o Cenral Counerpares. Bank for nernaonal Selemens, Basel, Swzerland, 212. Bjork,. Arbrage heory n Connuous me. UK: Oxford Unversy Press, 29. Bohme, M., D. Charella, P. Harle, M. Neukrchen,. Poppenseker, and A. Raufuss. Day of Reckonng: New Regulaon and s mpac on Capal Marke Busnesses. McKnsey workng paper on rsk, 211. Bremaud, P. Pon Processes and Queues. New York: Sprnger- Verlag, Canabarro, E., and D. Duffe. Measurng and Markng Counerpary Rsk. n Asse/Lably Managemen for Fnancal nsuons, eded by L. lman, London, 23. Duffee, G. Esmang he Prce of Defaul Rsk. Revew of Fnancal Sudes, 12 (1999), pp Duffe, D., J. Pan, and K. Sngleon. ransform Analyss and Asse Prcng for Affne Jump-Dffusons. Economerca, 68 (2), pp Duffe, D., L.H. Pedersen, and K.J. Sngleon. Modelng Soveregn Yeld Spreads: A Case Sudy of Russan Deb. he Journal of Fnance, Vol. 58, No. 1 (23), pp Duffe, D., and K.J. Sngleon. Cred Rsk. Prnceon, NJ: Prnceon Unversy Press, 23. Ghamam, S., and B. Zhang. Effcen Mone Carlo Counerpary Cred Rsk Prcng and Measuremen. Workng paper, Cener for Rsk Managemen Research, UC Berkeley, 213. Glasserman, P. Mone Carlo Mehods n Fnancal Engneerng. Sprnger, 24. Gregory, J. Beng wo-faced Over Counerpary Cred Rsk. Rsk, 22 (29), pp SOCHASC NENSY MODELS OF WRONG WAY RSK SPRNG 214

13 Gregory, J. Counerpary Cred Rsk. Wley Fnance, 21. Hull, J., and A. Whe. CVA and Wrong Way Rsk. Fnancal Analyss Journal, Vol. 68, No. 5 (212), pp Lando, D. On Cox Processes and Cred Rsky Secures. Revew of Dervaves Research, 2 (1998), pp Lando, D. Cred Rsk Modelng: heory and Applcaons. Prnceon Seres n Fnance, Prnceon, NJ: Prnceon Unversy Press, 24. McNel, A.J., R. Frey, and P. Embrechs. Quanave Rsk Managemen. Prnceon Seres n Fnance, Prnceon NJ: Prnceon Unversy Press, 25. Pykhn, M., and S. Zhu. Measurng Counerpary Cred Rsk for radng Procs Under Basel. London: Rsk Books, 26. o order reprns of hs arcle, please conac Dewey Palmer a dpalmer@journals.com or SPRNG 214 HE JOURNAL OF DERVAVES