Structural jump-diffusion model for pricing collateralized debt obligations tranches

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1 Appl. Mah. J. Chnese Unv. 010, 54): Srucural jump-dffuson model for prcng collaeralzed deb oblgaons ranches YANG Ru-cheng Absrac. Ths paper consders he prcng problem of collaeralzed deb oblgaons ranches under a srucural jump-dffuson model, where he asse value of each reference eny s generaed by a geomerc Brownan moon and jump wh an asymmerc double exponenal dsrbuon. Condoned on he common facor of ndvdual eny, hs paper ges he condonal dsrbuon, and furher obans he loss dsrbuon of he whole reference porfolo. Based on he sem-analyc approach, he far spreads of collaeralzed deb oblgaons ranches,.e., he prces of collaeralzed deb oblgaons ranches, are derved. 1 Inroducon Collaeralzed deb oblgaons rggered a worldwde fnancal crss n 008. The reason was manly due o he rraonal prces of he collaeralzed deb oblgaons ha resuled from he porfolo of reference enes jumped down heavly. Ths paper nroduces he srucural jump-dffuson model o descrbe he jump feaure of he ndvdual asse value and furher dscuss he prcng problem of he collaeralzed deb oblgaons ranches. More specfcally, a collaeralzed deb oblgaon, or CDO, consss of a porfolo of reference enes e.g. bonds, loans, resdenal and commercal morgages) whose cred rsk s sold o nvesors who, n reurn for an agreed paymen usually a perodc fee), wll bear he losses of he porfolo derved from he defaul of he reference enes. Through a securzaon echnque, CDOs repackage a porfolo cred rsk no ranches wh varyng senory. Durng he lfe of he ransacon he resulng losses affec frs he so called equy pece and hen, afer he equy ranche has been exhaused, he mezzanne ranches. Furher losses, due o cred evens on a large number of reference enes, are suppored by senor and super senor ranches. The cred rsk of he porfolo underlyng he CDO s sold n hese ranches. Generally, a ranche Receved: MR Subjec Classfcaon: 91B70, 91B8. Keywords: Srucural jump-dffuson model, Brownan moon, asymmerc double exponenal dsrbuon, collaeralzed deb oblgaons, loss dsrbuon. Dgal Objec IdenferDOI): /s y. Suppored by he Naonal Naural Scence Foundaon of Chna ), he Naural Scence Foundaon of Shandong Provnce 009ZRB019AV), Mahemacal Subjec Consrucon Funds and he Key Laboraory of Fnancal Informaon Engneerng of Ludong Unversy 008).

2 YANG Ru-cheng. Srucural jump-dffuson model for prcng collaeralzed deb oblgaons ranches 41 s defned by a lower and an upper aachmen pons. The buyers of he ranche wh lower aachmen pon K L and upper aachmen pon K U wll bear all losses n he porfolo value n excess of K L, and up o K U, percen of he nal value of he porfolo. CDO ranchng allows he holders of each ranche o lm her loss exposure o K U K L percen of he nal porfolo value. The cenral problem for prcng CDO ranches s he calculaon of loss dsrbuons of he reference porfolo over dfferen me horzons, and he key ssue s how o model he ndvdual asse value, especally n srucural models. The earler works for modellng he ndvdual asse value can be found n [1] where he evoluon of asse value was descrbed by a dffuson process drven by a geomerc Brownan moon. Bu he emprcal applcaon of a dffuson approach had yelded very dsapponng resuls for fng he marke daa, so some beer models were proposed. Examples of hese models can be found n [-9]. Albrecher e al. [] addressed a Lévy model for CDO prcng. Zhou [3] used a jump-dffuson approach wh a Posson process o analyze cred spreads, and Wllemann [4] used he jump-dffuson model o analyze a CDO prcng. [5] o [8] consdered he CDO prcng under he copula model. Wang e al. revewed he CDO heory n recen years. They derved some relavely good resuls for machng he marke daa of asse value. However, n hese models, here sll exss a lepokurc feaure ha he reurn dsrbuon of asses may have a hgher peak and wo asymmerc heaver als. For analyzng an opon prcng, Kou [10] presened a jump-dffuson asse model wh asymmerc double exponenal dsrbuon and derved some deal resuls. Movaed by hs, we nroduce he dea of Kou no CDO prcng. Usng he asse process n [10] and [11], we analyze n deal he loss dsrbuon of he reference porfolo and derve he far spreads of CDO ranches. The paper s organzed as follows. Secon summarzes a sem-analyc approach for prcng CDO ranche. Secon 3 presens he srucural jump-dffuson model wh asymmerc double exponenal dsrbuon, and derves he loss dsrbuon of he whole reference porfolo. Sem-analyc approach ForprcngaCDOranche[K L,K U ], ha s, fndng he far spread of he ranche, we mus analyze he values of wo legs: defaul leg DL) and premum leg PL). The value of DL presens he value of ranche losses rggered by cred evens durng he CDO lfeme, and he value of PL s he premum paymens weghed by he ousandng asse orgnal ranche amoun mnus accumulaed losses). Under he rsk-neural measure, he expeced value of boh legs should be equal,.e., E [ [ DL [KL,K U ]] E PL[KL,KU ]], 1) from whch we can derve he far spread of ranche [K L,K U ]. Now le us descrbe n deal he mehod o calculae he values of DL and PL for CDO ranche [K L,K U ]. For convenence, we assume ha he reference porfolo consss of N enes. Le 1. R he recovery rae of he -h reference eny, R [0, 1];

3 4 Appl. Mah. J. Chnese Unv. Vol. 5, No. 4. M he noonal of he -h reference eny, M > 0; 3. a he lower defaul barrer of he he -h reference eny, a > 0; 4. τ he defaul me for he -h reference eny; 5. T he maury me; 6. r he rsk-free dscoun rae, r>0. The cumulave loss a me s gven by N L) M 1 R )1 {τ }, 1 where he ndcaor funcon 1 { } s gven by { 1, x { }, 1 { } 0, oherswse, and he cumulave ranche loss of [K L,K U ]s L [Kl,K U ]) mn{l),k U } mn{l),k L }. For he defaul leg, we assume ha 0 0 < 1 < < M 1 < M T s he spread paymen daes. Then, under he rsk neural probably measure, he expeced value of he defaul leg for he ranche [K L,K U ]s E [ ] M DL [KL,K U ] e ru d EL [KL,K U ] u ) 0 [ M E e rm L [KL,K U ] m ) L [KL,K U ] m 1 ) )]. ) m1 On he oher hand, he expeced value of he premum leg of ranche [K L,K U ] s he presen value of all expeced spread paymens, and s gven by E [ ] [ M PL [KL,K U ] E S CDO Δ m e rm mn { } ] max[k U L m ), 0], K U K L, 3) m1 where Δ m m m 1. Subsung ) and 3) no 1) yelds [ M E e rm L[KL,KU ] m ) L [KL,K U ] m 1 ) )] m1 S CDO [ M ]. 4) E Δ m e rm mn{max[k U L [KL,K U ] m ), 0],K U K L } m1 Eq.4) shows ha he key ssue for prcng he CDO ranche [K L,K U ] s o compue he loss dsrbuon L [KL,K U ]) of he reference porfolo. The nex secon wll presen a srucural jump-dffuson form o model he ndvdual asse value and analyze he loss dsrbuon of he underlyng porfolo.

4 YANG Ru-cheng. Srucural jump-dffuson model for prcng collaeralzed deb oblgaons ranches 43 3 Loss dsrbuon of reference porfolo of CDO 3.1 Srucural jump-dffuson model Referrng o he model n [10] or [11], we assume ha he -h asse value Z ) ofhe reference porfolo s gven by dz ) Z ) μ d + σ dw )+d Ñ) ) V j 1), 1,...,N, where W ) s a sandard Brownan moon, Ñ) s a Posson process wh nensy λ, and {V j, j 1,...,N)} s a sequence of ndependen dencally dsrbued..d.) nonnegave random varables such ha Υ lnv j ) has an asymmerc double exponenal dsrbuon wh he densy f Υ y) pη 1 e η1y 1 {y 0} +1 p)η e ηy 1 {y<0}, η 1 > 0, η > 0, where p 0and1 p 0 represen he probables of upward and downward jumps respecvely. In oher words, { ξ +, wh probably p, lnv j )Υ ξ, wh probably 1 p, where ξ + and ξ are exponenal random varables wh means 1/η 1 and 1/η, respecvely. In he model, all sources of randomness, Ñ), W ), and Υ, are assumed o be ndependen. Then E[V j ]E[e Υ ]p η 1 η p) η η +1, η 1 > 1, η > 0. The requremen η 1 > 1 s needed o ensure ha ev j ) < + and ez )) <. Le a be he lower barrer of he -h eny s asse. Then defaul s defned o occur when Z ) a, ha s, he defaul me s τ nf{ 0:Z ) a }, 1,...,N. Applyng he generalzed Iô formula see [11]) o Z ) yelds { ) } Z ) Z 0)exp μ σ Ñ) + σ W ) V j,.e., ) ln Z ) lnz 0) + μ σ Ñ) + σ W )+ ln V j. The rsk-neural dynamcs for he asse value Z ) aregvenby Z ) ) ) ln r σ Ñ) + σ W )+ ln V j. Z 0) Le X ) Z) a.then ln X ) ) ln X 0) ) ) + r σ Ñ) + σ W )+ ln V j

5 44 Appl. Mah. J. Chnese Unv. Vol. 5, No. 4 wh X 0) Z0) a, and he defaul me τ of he -h eny can be expressed as τ nf{ 0:X ) 1} nf{ 0:ln X ) ) 0}. 5) Now we decompose he drvng sandard Brownan moon W ) ofhe-h reference asse as W ) ρ W C )+ 1 ρ W ), where ρ [0, 1], and W ) andw C ) are ndependen sandard Brownan moons. Here ρ ρ l s he correlaon beween he asse value processes of wo enes k and l, dw C ) she macroeconomc facor n whch W C s he common rsk facor) of he asse value whch s he same for all enes, and dw ) s he connuous change of dosyncrac facor n whch W s he dosyncrac unceran facor) of he -h eny asse value. So he dynamcs of lnx )) s ln X ) ) ln X 0) ) + r σ Ñ) ) + σ ρ W C )+σ 1 ρ W )+ ln V j. 6) 3. Condonal defaul probably of ndvdual asse Condonng he common facor W C, we wll derve he condonal probably dsrbuon of defaul me for an ndvdual asse n hs secon. Followng he deas of Kou [11], we gve he compuaon mehod for condonal defaul probably dsrbuon gven W C as follows. Le Mx) 1 σ 1 ρ )x + r σ + 1 ) 1 p)η ρ x + λ pη ) 1 1. x + η x η 1 For any wce connuously dfferenable funcon ux), he operaor G s defned by Gux) 1 σ 1 ρ )u x)+ r σ + 1 ) ρ u [ ] x)+λ ux + y) ux) fυ y)dy. Lemma 3.1. For any s 0, + ), here exs wo posve roos θ 1,s and θ,s of Mx) s wh θ,s < η <θ 1,s < 0. Le { B 1 e θ1,sx +1 B 1 )e θ,sx, x > 0, vx) 1, x 0 wh B 1 θ,sη + θ 1,s ) η θ,s θ 1,s ). Then vx) s connuous on 0, ) and for x>0, Gvx) svx) 0. Proof. The frs saemen s obvous because Mx) <swhen x 0andMx) >swhen x η,andmx) >swhen x and Mx) <swhen x η. NowweshowhaGvx) svx) 0forx>0. In fac, Gvx) svx) λ vx + y)f Υ y)dy + B 1 e θ1,sx[ 1 σ 1 ρ )θ1,s + r σ + 1 ) ] ρ θ 1,s s +1 B 1 )e θ,sx[ 1 σ 1 ρ )θ,s + r σ + 1 ρ ) ] θ,s s, 7)

6 YANG Ru-cheng. Srucural jump-dffuson model for prcng collaeralzed deb oblgaons ranches 45 and Snce x vx + y)f Υ y)dy 0 1 p)η e ηy dy x 1 p)η B1 e θ1,sx+y) +1 B 1 )e θ,sx+y)) e ηy dy pη 1 B1 e θ1,sx+y) +1 B 1 )e θ,sx+y)) e η1y dy 1 B 1η 1 B ) 1)η 1 1 p)e ηx p)η + B 1 pη ) 1 e θ1,sx θ 1,s + η θ,s + η θ 1,s + η θ 1,s η 1 1 p)η +1 B 1 ) pη ) 1 e θ,sx. 8) θ,s + η θ,s η 1 B 1η θ 1,s+η + 1 B1)η θ,s+η 1 0, subsung 8) no 7) yelds Gvx) svx) B 1 e θ1,sx[ 1 σ 1 ρ )θ 1,s + r σ + 1 ) 1 p)η ρ θ 1,s s + λ +1 B 1 )e θ,sx[ 1 σ 1 ρ )θ,s + r σ + 1 ρ pη 1 θ 1,s + η ) θ,s s + λ )] 1 θ 1,s η 1 1 p)η pη 1 1 θ,s + η θ,s η 1 B 1 e θ1,sx Mθ 1,s ) s ) +1 B 1 )e θ,sx Mθ,s ) s ) 0. Le { τ nf 0: z + r σ Ñ) ) + σ 1 ρ W } )+ ln V j 0 9) wh z ln X 0) ) + σ ρ w. From Lemma 3.1, we have Theorem 3.1. For any s>0, we have E[e s τ ]B 1 e θ1,sz +1 B 1 )e θ,sz, z > 0, 10) and he Laplace ransform of P { τ } s gven by e s P { τ }d 1 E[e s τ ], 1,...,N. s 0 Furhermore, wh F s) 1E[e s τ s ],hen P { τ } L 1 F s)) 11) where L 1 s he Laplace nverson. Proof. From Lemma 3.1, we have Gvz) svz) 0 for any z>0. Applyng he generalzed Iô formula o he process e s vlnx ))) n whch lnx )) s gven by 6) and akng expecaon yeld E [ e s τ) v lnx τ )) )] vz)+e τ Snce lnx u)) > 0foru 0, τ ), we have E [ e s τ) vlnx τ )) ] vz). Leng yelds E [ e s τ] vz) B 1 e θ1,sz +1 B 1 )e θ,sz. 0 e su[ Gv lnx u)) ) sv lnx u)) )] du. )]

7 46 Appl. Mah. J. Chnese Unv. Vol. 5, No. 4 Applyng he Laplace ransform o P { τ } yelds e s P { τ }d 1 e s dp { τ } 1 E[e s τ ]. 0 s 0 s The equaly 11) nvolvng he Laplace nverson s obvous see [1]). Remarks 3.1. Theorem 3.1 shows how o ge he defaul probably P { τ } easly. We only have o fnd E[e s τ ] by 10) and hen apply 11). Snce W C s ndependen of W and Υ, from 5), 6) and 9) we have Theorem 3.. Le P W C be he condonal defaul dsrbuon gven on W C for he -h reference eny. Then P W C P {τ W C ) w} P { τ }, 1,...,N. 3.3 Loss dsrbuon of he whole porfolo In hs secon we are gong o consder he algorhm for compung he loss dsrbuon of he whole reference porfolo. Condonng he common facor W C, he ndvdual defaul me τ for 1,...,N are ndependen. Then he condonal characersc funcon of L) gvenw C can be expressed by E [ e IuL) W C w ] [ E e N IuM 1 R )1 {τ } 1 ] W C w N E [ e IuM1 R)1 {τ } W C w ], where I s he magnary un wh I 1. Thus E [ e IuL) W C w ] N E [ e IuM1 R)1 { τ } W C w ]. 1 1 Noe ha he ndvdual condonal characersc funcon for he -h reference eny s E [ e IuM1 R)1 { τ } W C w ] e IuM1 R) p W C + 1 p W C ) 1+ e IuM1 R) 1 ) p W C. Hence E [ e IuL) W C w ] N ] [1+e IuM1 R) 1)p W C. 1 By akng ou he common facor W C we ge he uncondonal characersc funcon E [ e IuL)] N ] [1+e IuM1 R) 1)p W C f W C w)dw, 1 where f W C w) s he densy of he sandard Brownan moon W C ) wh f W C w) 1 e w. π Le Hu) E[e IuL) ]. Then by he nverson of characersc funcon see [13]), he prob-

8 YANG Ru-cheng. Srucural jump-dffuson model for prcng collaeralzed deb oblgaons ranches 47 ably dsrbuon F L) x) ofhelossl) s 1 U F L) x) P L) x) lm Hu) e Ixu U π U Iu du. Ths compuaon of he loss dsrbuon of he whole reference porfolo s no only suable for nhomogeneous porfolo, bu also for homogeneous one,.e., when ρ ρ [0, 1], a a>0, M M>0andR R [0, 1] for all 1,,...,N. However, here s a smpler compuaon for homogeneous porfolo. Theorem 3.3. Assume ha he reference porfolo s homogeneous. Le L k k N 1 R)M. Then he loss dsrbuon P {L) L k } of he he whole reference porfolo s gven by ) N W P {L) L k } P C ) k W 1 P C ) N kfw C w)dw, k lnx 0)) and he dsrbuon funcon F L) of he cumulave loss s k F L) L k )P {L) L k } P {L) L l }. Proof. Snce L k k N 1 R)M s equal o he probably ha exacly k ou of N enes defaul, and he defauls are ndependen when condoned on W C ), hen ) N P P {L) L k W C W ) w} C ) k W 1 P C ) N k k and P {L) L k } P {L) L k W C ) w}f W C w)dw. From he fac ha P W C P {L) L k } l0 1forw, lnx 0))], we have lnx0)) + P {L) L k W C ) w}f W C w)dw lnx 0)) P {L) L k W C ) w}f W C w)dw ) N W P C ) k W 1 P C ) N kfw C w)dw. k lnx 0)) The cumulave loss follows readly from he defnon of cumulave dsrbuon funcon. So far we have been compung he loss dsrbuon of he whole reference porfolo. Once we have found he loss dsrbuon of L), we can easly ge he far spreads S CDO of CDO ranche [K L,K U ]by4). References [1] H Albrecher, S A Ladoucee, W Schouens. A generc one-facor Lévy model for prcng synhec CDOs, Adv Mah Fnanc, 007, 3: [] F Black, J Cox. Valung corporae secures: some effecs of bond ndenure provson, J Fnanc, 196, 31:

9 48 Appl. Mah. J. Chnese Unv. Vol. 5, No. 4 [3] X Burschell, J Gregory, J P Lauren. A comparave analyss of CDO prcng models, Workng paper, ISFA Acuaral School, Unversy of Lyon & BNP-Parbas, 005. [4] R Durre. Probably: Theory and Examples, Duxbury Press, [5] W H Flemng, H M Soner. Conrolled Markov and Vscosy Soluons, Sprnger-Verlag, [6] J Hull, A Whe. Valuaon of a CDO and an n-h o defaul CDS whou Mone Carlo smulaon, J Dervaves, 004, : 8-3. [7] S G Kou. A jump-dffuson model for opon prcng, Manage Sc, 00, 48: [8] S G Kou, H Wang. Frs passage mes of a jump dffuson process, Adv Appl Prob, 003, 35: [9] J P Lauren, J Gregory. Baske defaul swaps, CDOs and facor copulas, J Rsk, 005, 7: [10] D Z Wang, S T Rachev, F J Fabozz. Prcng ranches of a CDO and a CDS ndex: recen advances and fuure research, Workng paper, Unversy of Calforna, 006. [11] S Wllemann. Fng he CDO correlaon skew: A racable jump model, J Cred Rsk, 00, 3: [1] G Xu. Exendng Gaussan copula wh jumps o mach correlaon smle, Wachova Secures, Workng paper, 006. [13] C Zhou. The erm srucure of cred spreads wh jump rsk, J Bank Fnanc, 001, 5: School of Mahemacs and Informaon, Ludong Unversy, Yana 6405, Chna Emal: yang-rucheng@163.com