Pricing CDOs with Correlated Variance Gamma Distributions

Transcription

1 Pricing CDOs wih Correlaed Variance Gamma Disribuions Thomas Moosbrucker Firs Version: Ocober 005 This Version: January 006 Deparmen of Banking, Universiy of Cologne, Alberus-Magnus-Plaz, 5093 Köln, Germany. Phone: +49 0) , Fax: +49 0) ,

2 Pricing CDOs wih Correlaed Variance Gamma Disribuions Absrac In his aricle, we propose a mehod for synheic CDO pricing wih Variance Gamma processes and disribuions. Firs, we exend a srucural model proposed by Luciano and Schouens [005] by allowing a more general dependence srucure. We show ha our exension leads o a correlaion smile as observed in liquid index ranches. Since his mehod is no adequae for pracical purposes, we exrac he dependence srucure ino a facor approach based on Variance Gamma disribuions. This approach allows for an analyical soluion for he porfolio loss disribuion. The model fis o prices of liquid CDS index ranches. I can be used o price bespoke CDOs in a consisen way. 1

3 1 Inroducion In he 1980s, Collaeralized Deb Obligaions CDOs) were inroduced for balance shee risk managemen. The emergence of credi derivaives in he 1990s offered he possibiliy of synheic risk ransfer of a porfolio of bonds or loans, oo. Since 003, credi risk of sandardised porfolios is raded in a liquid marke in he CDS indices iboxx and Trac-X. These indices merged ino itraxx in 004. Sandardised ranches ha are linked o hese indices sared o be acively quoed. Thus, he enire disribuion of porfolio loss as seen by marke paricipans) became an observable variable. This developmen poses new challanges o credi risk models. One should expec he mos common credi risk models o mach he marke implied loss disribuion. However, his is no rue. Using he common one facor Gaussian copula approach, differen correlaion parameers are needed o price differen ranches. Thus, he dependence srucure of defauls is no Gaussian. As an alernaive, we propose Variance Gamma VG) processes and disribuions for pricing liquid CDS index ranches. The following secion briefly describes he Gaussian copula approach and he problems relaed o his mehod. We give a survey over he possible soluions o hese problems in he lieraure and moivae he approach of his aricle. Secion 3 exends a srucural model proposed by Luciano and Schouens [005]. The abiliies of his model in explaining he dependence srucure implied by liquid ranches of DJ itraxx are examined. In secion 4, we propose a facor copula approach ha replicaes he dependence srucure of he srucural model and ha is analyically racable. Secion 5 concludes. Appendix A gives he resuls concerning VG processes and disribuions. We pospone proofs o Appendix B. Valuaion of CDOs A Collaeralized Deb Obligaion is a securiisaion of a porfolio of bonds or loans. The underlying porfolio is ransfered o a Special Purpose Vehicle ha issues securiies on he porfolio in several ranches. Each ranche is defined by an aachmen poin L a and a deachmen poin L d. For a percenual loss of L porfolio of he underlying porfolio, he

4 ranche suffers a percenual loss of L ranche = max{minl porfolio, L d ) L a, 0}. The lowes ranche has L a = 0 and is called he equiy ranche. Since i already suffers from he firs loss in he porfolio, i is he riskies ranche and has o pay he highes spread o invesors. For aachmen poins beween 3 and 7 percen, ranches are called mezzanine, while he highes ranches are called senior or super-senior. In synheic CDOs, porfolio credi risk is ransfered via Credi Defaul Swaps. Wih hese insrumens, no all ranches need o be sold o invesors. On he basis of sandardised ranches wo paries agree o ac as proecion buyer and proecion seller for his paricular single ranche. Sandard porfolios exis in he indices DJ CDX NA for eniies in Norhern America and DJ itraxx for European eniies. 1 The main indices consis of 15 equally weighed eniies. The aachmen and deachmen poins are 0%, 3%, 7%, 10%, 15% and 30% for CDX NA and 0%, 3%, 6%, 9%, 1% and % for DJ itraxx. There is also he possibiliy o rade he whole index. This corresponds o a ranche wih L a = 0% and L d = 100%. Spreads are quoed in basispoins per year for all ranches. The only excepion is he equiy ranche, where spread is quoed as a percenage upfron paymen plus 500 bp running premium. Table 1 shows marke quoes of DJ itraxx on June 4, 005. Tranche 0 3% 3 6% 6 9% 9 1% 1 % Index Spread 30.0% 98bp 34bp 0bp 14bp 40.0bp Table 1: Marke quoes of DJ itraxx 5 year on June 4, 005. Spread of he equiy ranche is quoed as a percenage upfron plus 500bp running premium. The oher ranches are quoed as bp per year. Source: Nomura Fixed Income Research..1 One Facor Gaussian Copula The Gaussian copula model has become he sandard marke model for valuing synheic CDOs. In is basic form, for every eniy i in he porfolio a sandard normal random 1 See Amao and Gynelberg [005] for deailed descripions of hese indices. 3

5 variable X i is defined by X i = ρm + 1 ρz i. 1) M and Z i are sandard normally disribued and ρ 1. The facor M represens a sysemaic and Z i an idiosyncraic risk facor of eniy i. For i j, he correlaion of X i and X j is given by ρ. Eniy i defauls, if X i is smaller han some defaul hreshold C i. The defaul hreshold is deermined so ha he risk neural defaul probabiliy Q i τ) of eniy i for every ime τ is given by Q i τ) = Φ 1 C i ), where Φ is he cumulaive disribuion funcion of a sandard normal random variable. Then he disribuion of he number of defauls can be obained. 3 If one assumes consan recovery raes, his disribuion implies a disribuion of porfolio loss. By assuming a zero mark-o-marke value for each ranche, spreads on he ranches can be calculaed. The main advanage of he model is he independence of defauls when condiioned o he common risk facor. This allows a simple implemenaion and fas compuaions. However, when we apply he model o liquid ranches of he credi indices DJ CDX or itraxx, i fails o fi marke prices of he ranches. Differen correlaion parameers are needed o fi he prices of differen ranches. For equiy and senior ranches his implied correlaion is higher han for mezzanine ranches. This phenomenon is known as he correlaion smile of implied correlaion. Table shows he implied correlaions corresponding o he quoes of able 1. 4 Tranche 0 3% 3 6% 6 9% 9 1% 1 % implied correlaion Table : Implied correlaions of DJ itraxx 5 year on June 4, 005. Marke prices of liquid index ranches are used o calibrae models for he valuaion of bespoke CDOs. If a model prices all liquid ranches correcly using he same parameer se, hen a bespoke CDO ranche wih non-sandard aachmen and deachmen poins can The risk neural defaul probabliy can be deermined from sinlge name credi defaul swaps. 3 See e.g. Gibson [004] for deails. 4 These implied correlaions slighly depend on cerain assumpions abou he porfolio srucure. We have assumed an infiniely large porfolio of idenical eniies. 4

6 be priced in a consisen manner. If his is no he case as in he Gaussian framework one needs o develop furher echniques. Wihin he Gaussian framework, one of hese echniques consiss of calculaing base correlaions. These are implied correlaions of hypoheical equiy ranches i.e. ranches wih aachmen poin L a = 0) wih varying deachmen poins. The advanage of base correlaions over implied correlaions is ha hey are monoonically increasing along wih he deachmen poin. One can herefore inerpolae beween base correlaions in order o price non-sandard ranches. However, his approach is raher an ad hoc mehod han a consisen way of CDO pricing. I does no resolve he fundamenal inconsisency in using he Gaussian copula approach. The search for models ha can price all ranches wih using one single parameer se has herefore been an acive field of research in recen ime. In he nex subsecion, we give a shor survey over he mehods proposed so far.. Furher Mehods An naural idea is o ry oher copulas han he Gaussian. The choices proposed so far include he suden, double, Clayon and Marshall-Olkin copula. Burschell, Gregory and Lauren [005] provide an overview over he resuls of calibraing hese copula approaches o marke spreads of DJ itraxx. They find ha he double copula fis he observed spreads bes. This copula has a small numerical disadvanage: if he risk facors M and Z i in 1) and -disribued, hen he disribuion of X i depends on ρ and has o be calculaed numerically. Recenly, Kalemanova, Schmid and Werner [005] proposed a facor copula approach based on Normal Inverse Gaussian disribuions. They show ha calibraion o liquid index ranches is as good as by he double copula. I was his idea ha inspired us o he use of he VG copula in secion 4. Empirical sudies of de Servigny and Renaul [004] and Das e al. [004] show ha defaul correlaions increase in imes of a recession. In he las monhs, several auors have proposed models ha incorporae his fac. Andersen and Sindenius [005] exend he Gaussian copula model as hey correlae ρ and M in equaion 1) negaively. 5

7 Hull, Predescu and Whie [005] have developed a srucural model where firm value processes are correlaed Brownian moions. The degree of correlaion may depend on he sysemaic par of he process and herefore on he sae of he economy. When hey assume a negaive dependence of correlaion and he sysemaic par of he firm value processes, he auhors show ha spreads of CDX NA and itraxx are fied significanly beer han by a consan correlaion. Anoher aemp o generae a dependence srucure as observed in he marke consiss of he inroducion of a sochasic business ime. This idea has been used for he valuaion of equiy derivaives for a long ime. In a firm value approach, sochasic business ime leads o varying volailiies of he firm value process. If business ime goes fas, firm values vary more and are herefore more likely o hi he defaul barrier. Thus, a fas business ime corresponds o a bad economic environmen. Giesecke and Tomecek [005] model defaul imes in a porfolio as imes of jumps of a Poisson process. The ime scale of his process is varied depending on incoming informaion like economic environmen and defauls. In his way, conagion effecs can be considered. Joshi and Sacey [005] use Gamma processes o calibrae an inensiy model o marke prices of liquid CDO ranches. When business ime is he sum of wo Gamma processes, hey show ha heir model can fi o he correlaion smile of DJ itraxx. Cariboni and Schouens [004] model firm value processes and use Brownian moions subordinaed by Gamma processes. The resuling Variance Gamma processes VG processes) are calibraed o single name credi curves. Luciano and Schouens [005] exend he approach of Cariboni and Schouens [004] for he valuaion of defaul baskes. All firm value processes follow he same Gamma process and hus he same business ime. For every eniy in he porfolio, hey model is firm value process as an exponenial of a Variance Gamma process. The parameers of hese VG processes are deermined by he credi curves of he corresponding single name CDS. Since all firm value processes follow he same business ime and since he Brownian moions are independen, he complee dependence srucure is deermined by hese parameers. The model we propose in his paper exends he approach of Luciano and Schouens 6

8 [005] by allowing a more general dependence srucure. We have chosen Variance Gamma processes and heir disribuions for our model, since hey have a number of good mahemaical properies and since hey have proven o explain a number of economic findings. Mahemaically, he disribuions have nice properies such as lepokursis and fai ails. Their densiies are known in closed form and he class of VG disribuions is closed under scaling and convoluion if parameers are chosen suiably. Economically, Cariboni and Schouens show ha heir model fis o a variey of single name credi curves. The idea of a sochasic business ime leads o an increase of defaul correlaions in recessions, which has proven o creae correlaion smiles. Finally, VG processes have shown o explain he volailiy smile in equiy opions see Madan e al. [1998]). Thus, Variance Gamma processes and disribuions are a naural candidae for explaining he correlaion smile. 3 The Srucural Variance Gamma Model We briefly describe he srucural model of Luciano and Schouens [005]. This serves as a base case for hree exensions in he following subsecions. In he las subsecion, we examine he abiliy of he model and is exensions o explain he observed correlaion smile in liquid index ranches. We provide he properies of VG processes needed for his aricle in Appendix A. 3.1 Base Case: Idenical Gamma Processes, independen Brownian Moions Le N be he number of eniies in he porfolio and for every i {1,..., N} le X i) = θ i G + σ i W i) G ) be a VG process wih parameers θ i, ν, σ i ). This means ha G ) 0 is a Gamma process wih parameers ν 1, ν) 5 and for every i he process W i) ) 0 is a sandard Brownian moion. For i j, hese Brownian moions are independen. The firm value process S i) ) 0 of eniy i is given by 5 The reason why he parameer ν does no depend on i will be explained in he nex subsecion. 7

9 ) S i) = S i) 0 exp r + X i) + ω i. 3) In equaion 3), r denoes he risk free ineres rae a and ω i = 1 1 ν log 1 ) σ i ν θ i ν a parameer o ensure he maringale propery of he discouned firm value Appendix B for deails). Eniy i defauls a ime τ > 0 if τ = min T {Si) < L i) }. S i) expr ) see In his base case, all firm value processes 3) follow he same Gamma subordinaor G ) 0. Luciano and Schouens argue ha all firms are subjec o he same economic environmen and hus informaion arrival should affec business ime of all eniies. The Brownian moions are independen, however. This means ha all correlaion involved is caused by he common business ime. Figure 1 shows wo pahs of VG processes wih idenical Gamma processes and independen Brownian moions. Jumps occur a idenical imes, bu heir direcions are condiionally independen Figure 1: VG processes wih idenical Gamma processes We exend his correlaion srucure in he following subsecions. Firs, we allow business ime o be correlaed and no idenical for all eniies. We herefore allow changes in business ime o be caused by sysemaic or idiosyncraic informaion. Second, we 8

10 allow he Brownian moions o be dependen. This means ha he direcions of jumps are correlaed. Finally, we inegrae hese wo ideas ino a hird exension. In secion 4 we show ha he las exension may be solved analyically under some simplifying assumpions. 3. Exension A: Correlaed Gamma Processes, independen Brownian Moions In his exension, changes in business ime may be caused by informaion concerning he enire economy or by informaion abou he individual firm. If we decompose G ) 0 ino a sysemaic and an idiosyncraic par, we can incorporae his idea ino he model. For every i, we seperae G i) ) 0 ino a sum of wo Gamma processes dg i) = df + du i). In his decomposiion, F ) 0 and U i) ) 0 are independen Gamma processes wih parameers a F, b F ) = aν 1, ν) and a U i), b U i)) = 1 a)ν 1, ν) wih 0 a 1. I follows ha G i) ) 0 is a Gamma process wih parameers 1 a)ν 1 + aν 1, ν) = ν 1, ν). 6 For a 1, his exension coincides wih he base case. The parameer a conrols for he pairwise correlaion CorrX i) 1, X j) 1 ). In Appendix B, we show ha for independen Brownian moions W i) and W j) and for i j we have For a homogeneous porfolio i.e. CorrX i) 1, X j) θ i θ j ν 1 ) = a. 4) θ i ν + σi θj ν + σ j idenical credi curves) all eniies have idenical parameers θ i = θ, σ i = σ and all correlaions in he inerval [0; The upper bound is obained for a 1, which is he base case. reached for a 0, i.e. for independen Gamma processes. θ ν ] can be reached. θ ν+σ The lower bound is One should bear in mind, however, ha his correlaion has o be reaed wih care. For example, if θ i = θ j = 0, he processes X i) and X j) are uncorrelaed, bu no necessarily 6 For wo independen Gamma variables X Γa X, λ) and Y Γa Y, λ) heir sum is Gamma disribued wih X + Y Γa X + a Y, λ). This is also he reason for which we chose he parameer ν o be idenical for all i. 9

11 independen. For a beer undersanding of he dependence srucure, copulas have o be regarded. Figure shows sample pahs of correlaed VG processes for a = 0.5. There are imes where only one process has a large jump. These jumps are caused by he idiosyncraic facor U i). A oher imes, boh realisaions jump. Those jumps are caused by he sysemaic facor F Figure : VG processes wih correlaed Gamma processes 3.3 Exension B: Idenical Gamma Processes, dependen Brownian Moions The exension proposed in he las subsecion correlaes imes of high aciviy of he VG processes. The direcions of hese jumps are condiionally independen since he Brownian moions W i) and W j) are independen. If jumps are caused by informaion concerning he sae of he economy, his informaion should be more or less) good or bad for all companies. The model reflecs his fac if he Brownian moions are correlaed. The correlaion of he Brownian moions is modelled via dw i) = bdm + 1 bdz i). In his exension, we choose he same Gamma subordinaor for each eniy. Then, for he correlaion of he VG processes we find for i j see Appendix B) 10

12 CorrX i) 1, X j) 1 ) = θ i θ j ν + σ i σ j b. 5) θ i ν + σi θj ν + σ j For a homogeneous porfolio wih idenical parameers θ i = θ, σ i = σ all correlaions θ in he inerval [ ν, 1] can be reached, if posiive values for b are considered. The lower θ ν+σ bound corresponds o b = 0. If b = 1, he Variance Gamma processes are idenical and herefore have a correlaion of 1. Figure 3 shows wo pahs for b = 0.5. Large jumps occur a idenical imes and heir direcions are correlaed Figure 3: VG processes wih idenical Gamma subordinaors and correlaed Brownian moions 3.4 Exension C: Sum of wo independen VG Processes While in he above exensions we decompose eiher he Gamma or he Brownian par of he VG process, we now decompose he whole VG process ino wo VG processes. We resric ourselves o he case where all VG processes X i) ) 0 have he same parameers θ, ν, σ). We choose 0 < c < 1 and decompose X i) ) 0 ino X i) = cm + 1 c Z i) 6) 11

13 wih θ M, ν M, σ M ) = θ Z i), ν Z i), σ Z i)) = cθ, ν ) c, σ, 7) ) 1 ν c θ, 1 c ), σ. 8) From he resuls of Appendix A i is clear ha he sum X i) ) 0 is again a VG process and is parameers are given by θ X i), ν X i), σ X i)) = θ, ν, σ). We show in Appendix B ha for i j. CorrX i) 1, X j) 1 ) = c 9) Noe ha his exension is no he juncion of exensions A and B for a = b = c. Neverheless, he correlaion of X i) correlaion. 3 ) 0 and X j) ) 0 sems from boh Gamma and Brownian Figure 4: Correlaed VG processes Figure 4 shows wo pahs of correlaed VG processes for c = 0.5. Jumps occur for sysemaic and idiosyncraic reasons. The direcions of sysemaic jumps are correlaed. Since all firm value processes possess idenical parameers, he eniies in he porfolio are assumed o have idenical credi curves. This simplificaion and he fac ha all processes are of VG ype allow for an analyical soluion ha is given in he nex secion. 1

14 3.5 Applicaion o Index Tranches We now apply he model o daa on ranches of DJ itraxx 5 year from June 4, 005 o December 9, 005. For each of hese daes, we also have index spreads of DJ itraxx for he mauriies of 3, 5, 7 and 10 years. Our daa se consiss of 5 weekly quoes. The index spread curves are used o deermine he parameers θ i, ν, σ i ). Since our daa se does no include credi curves for he single eniies, we assume ha he porfolio is homogeneous wih respec o defaul probabiliies. Thus all processes have idenical parameers. For each dae, he calibraion is accomplished in wo seps: Firs, we deermine he parameers θ, ν, σ) o fi he index spread curve given by quoes of he indices wih differen mauriies. In he base case, he enire correlaion srucure is deermined and we calculae ranche spreads. In our exensions, we conduc a second sep: we deermine he correlaion paramers a, b and c such ha he spread of he equiy ranche is mached. In all he compuaions of his subsecion, we have se he iniial firm value o S 0 = 100, he defaul barrier o L = 50 and recovery raes o R = 40%. We simulaed he firm value process on a discree ime grid wih d = The risk-free zero curve is he EUR zero curve. We calculae par spreads in he usual way. We assume ha all paymens fee and coningen legs) occur on quarerly paymen daes T i, i = 1,..., n. The expeced defaul loss ELT i ) on he ranche up o paymen dae T i is given by ELT i ) = 1 0 max minx1 R), L d ) L a, 0) f LossTi )x)dx, where R is he consan) recovery rae, L a and L d denoe he ranche s aachmen and deachmen poins and f LossTi ) is he probabiliy densiy funcion of losses unil T i. The ranche s mark-o-marke from he invesors view) for a spread per annum of s 7 We have esed he model o robusness wih respec o he assumpions on L, R and d. Qualiaively, he resuls were he same, alough compuaion speed obviously depends on d. 13

15 is hen given by 8 MTMs) = Fee Coningen n = s i e rt i L d L a ) ELT i )) i=1 n e rt i ELT i ) ELT i 1 )), i=1 where i is he accrual facor for paymen dae i. The par spread is obained by MTMs par ) = 0. Table 3 shows he resuls of he calibraion o he index spread curve. Calibraion was done o minimise he absolue pricing error APE). In his calibraion, he APE denoes AP E = mauriies spread Marke Mauriy spread Model Mauriy We have calibraed he full model o spread curves as well as some cases, where we resriced eiher o θ = 0 or ν = 1. If θ = 0, we ge unskewed disribuions. If ν = 1, he Gamma disribuions are exponenial disribuions. We find ha all cases lead o a good fi o observed spreads. This resul confirms he findings of Cariboni and Schouens [004]. Mauriy 3 years 5 years 7 years 10 years average APE Marke resr. o θ = ) 0.77).87) 8.93) resr. o ν = ) 0.70) 0.8) 3.88) unresr ) 0.53) 0.8) 0.88) Table 3: Calibraion o spread curves of DJ itraxx from June 4, 005 o December 9, 005. Numbers in paranheses denoe average pricing errors. Marke quoes are mid quoes obained by Inernaional Index Company. 8 See for example Gibson [004] for deails.. 14

16 Given he parameers from his calibraion, we can calculae model implied ranche spreads in he base case and he exensions. In he base case, here is no addiional correlaion parameer, so ranche spreads are already deermined by he values of θ, ν, σ). We find ha in all cases, he spread of he equiy ranche is significanly underesimaed. This shows ha implied correlaion of his approach is oo high. Thus, we can use exension A and C o calibrae he correlaion parameers a and c o mach he equiy ranche. We have deermined a and c such ha he spread of he equiy ranche is mached. For comparison of he over all fi of he models, we have included he average APE ino he able, where he APE is given by AP E = ranches spread Marke Tranche spread Model Tranche. 10) Since exension B furher increases correlaion compared o he base case, we canno calibrae his exension o mach he equiy ranche. We show he calibraion resuls of he base case and exensions A and C o ranche spreads in able 4. For comparison, we have also included he values for he double copula wih 4 and 5 degrees of freedom. Burschell e al [005] find he double 4) copula o mach ranches bes when compared o several oher copulas. We find ha in he base case, he model spread of he equiy ranche is much lower han observed in he marke. The negaive spread of 1.8 for he case resriced o ν = 1 indicaes ha he model implied spread is even smaller han he 500bp running premium. In he exensions, he upfron premium of he equiy ranche can be mached. In exension A, pricing errors are smaller han for he Gaussian Copula, which indicaes a model implied correlaion smile. APEs are sill quie large compared o he double 4) copula for θ = 0 and in he unresriced case. The bes fi is achieved in he case resriced o ν = 1. Exension C leads o a fi ha is comparable o he double 4) copula in all hree cases. 15

17 0 3% 3 6% 6 9% 9 1% 1 % average APE Marke 7.1% Gauss 7.1% ) 18.6) 4.4) 6.8) double 4) 7.1% ) 9.6) 7.1).4) double 5) 7.1% ) 11.7) 6.0) 0.6) VG Base case, θ = % ) 195.1) 61.8) 9.4) 6.7) VG Base case, ν = 1-1.8% ) 56.6) 88.3) 83.5) 65.3) VG Base case, unres. 6.3% ) 19.0) 17.3) 76.5) 4.3) VG Ex. A, θ = 0 7.1% ) 15.3) 5.6) 7.9) VG Ex. A, ν = 1 7.1% ) 8.4) 3.0).6) VG Ex. A, unres. 7.1% ) 7.) 10.4).0) VG Ex. C, θ = 0 7.1% ) 6.) 3.8) 1.4) VG Ex. C, ν = 1 7.1% ) 7.5) 3.3) 1.7) VG Ex. C, unres. 7.1% ) 10.9) 7.).1) Table 4: Average Marke spreads and average model implied spreads for DJ itraxx 5yr beween June 4, 005 and December 9, 005. Numbers in paranheses denoe average pricing errors. 16

18 4 VG Copula Model We showed in he las secion ha he correlaion smile in liquid index ranches may be explained by he srucural variance gamma model. For pracical purposes however, here are some disadvanages of he srucural model. Firs, we have o use Mone Carlo simulaions o calibrae he model o marke quoes, which leads o slow implemenaions. Furhermore, we have used he parameers θ, ν, σ) solely for he calibraion o he index spread curve and no for he calibraion o ranche prices. We herefore implemen he idea of exension C above ino a facor copula approach. We assume a homogeneous porfolio and consan defaul inensiies for he eniies. This defaul inensiy is deermined by he 5 year index spread. Thus, we can use he parameers of he VG disribuion and a correlaion parameer o calibrae he approach o ranche quoes. Since VG disribuions possess a fourh parameer µ in general see Appendix A) and we resric our disribuions o have zero mean and uni variance, his gives us degrees of freedom besides correlaion. Anoher assumpion used in our examples is he porfolio o consis of an infinie number of eniies. This allows us o use he large homogeneous porfolio LHP) approximaion 9, which is a common procedure for oher copula approaches. As wih he Gaussian copula, his assumpion may be relaxed. In his case, a semi-analyical approach similar o he ones for he Gaussian copula has o be conduced. However, he LHP mehod allows us o calculae an analyical soluion for he porfolio loss disribuion. 4.1 VG Copula In analogy o he Gaussian Copula we define he one facor VG copula by X i = cm + 1 c Z i, 11) 9 For he Gaussian copula model, his approximaion was inroduced by Vasicek [1987]. 17

19 where M and Z i are independenly VG disribued random variables. The disribuion parameers are given by 10 M V G cθ, ν c, ) 1 νθ, cθ, Z i V G 1 c θ, ν 1 c, 1 νθ, 1 c θ Noe ha he disribuions of M and Z i correspond o he disribuions of M 1 and Z i) 1 in 7) and 8) in secion 3.4 wih 1 νθ and µ = θ o obain zero mean and uni variance. Appendix A, we find Using he resuls abou scaling and convoluing VG variables a he end of ). cm V G 1 c Z i V G The disribuion of X i is c θ, ν ) c, c 1 νθ, c θ, 1 c )θ, ν 1 c, 1 c 1 νθ, 1 c )θ X i = cm + 1 c Z i V G θ, ν, ) 1 νθ, θ. The hird and fourh parameers were chosen such ha all variables have zero mean and uni variance. We show in Appendix B ha he correlaion of X i and X j for i j is given by ). CorrX i, X j ) = c. 1) Using 11) and he LHP approximaion we can deduce he loss disribuion of he porfolio. If we denoe he common) defaul hreshold by C and condiion on he sysemaic facor M, hen eniy i defauls if Z i < C cm 1 c. The condiional defaul probabiliy is given by ) C cm P X i < C M = m) = F Zi, 1 c 10 See Appendix A for deails on VG disribuions and heir properies as far as we need hem for his aricle. 18

20 where F Zi denoes he cumulaive disribuion funcion of Z i, which is idenical for all i. We find ha he loss disribuion funcion of he porfolio is ) 1 c F 1 Z F porfolio loss x) = F i x) C M, 13) c if he recovery raes are zero. The proof from Vasicek [1987] applies here, oo. We provide i in Appendix B. 4. Applicaion o CDS Index Tranches In his subsecion, we calibrae he VG Copula model o he marke quoes we have used in secion 3.5. The defaul hreshold C is deermined o mach he index spread. We can hus use he parameers θ, ν and c o calibrae he model o ranche spreads. The calibraion is done by a minimisaion of he absolue princing error AP E = ranches spread Marke Tranche spread Model Tranche while keeping he equiy spread fixed o mach he marke spread. As in secion 3.5, we assume a consan recovery rae of 40% for all eniies. The risk-free zero curve is he EUR zero curve. Firs, we fix θ = 0 which leads o unskewed disribuions. In a second sage, we calibrae over all parameers. The resuls are given in able 5. Boh he resriced θ = 0) and he unresriced case lead o a beer fi han he double 4) copula. The fi is slighly improved in he unresriced case. In he unresriced case, he average APEs of all ranches excep he [3 6%]-ranche are below 4bp, which is abou he bid ask spread on a ypical day. Comparing he absolue pricing errors of able 4 srucural model) o hose of able 5 facor copula), we find ha we could reduce his pricing errors by his analyical model. This is possible since we assumed a consan defaul inensiy and ignored he CDS indices wih mauriies differen han 5 years. One should no compare parameer values of ρ, θ and ν, since we have resriced ourselves o uni variances for all disribuions in his secion. Furher differences occur since defauls are no riggered by a consan defaul barrier L as in he las secion.. 19

21 0-3% 3-6% 6-9% 9-1% 1-% average APE ρ Average Marke Quoes 7.1% Gauss 7.1% ) 18.6) 4.4) 6.8) double 4) 7.1% ) 9.6) 7.1).4) double 5) 7.1% ) 11.7) 6.0) 0.6) VGν) 7.1% ) 11.7) 9.7) 4.5) VGν,θ) 7.1% ) 3.6) 1.6).3) Table 5: Calibraion o DJ itraxx 5yr Series 5 for weekly prices beween June 4, 005 and December 9, 005. VGν) denoes he VG Copula model resriced o θ = 0 and VGν,θ) he unresriced model. Average pricing errors for he ranches are given in paranheses. The values of ρ denoe correlaion averages. 5 Conclusion In his aricle, we proposed a valuaion mehod for CDS index ranches by means of Variance Gamma processes and disribuions. We showed ha exensions of a srucural model developed by Luciano and Schouens [005] can generae a correlaion smile as observed in he marke. Since compuaions wihin his model are ime-consuming, we exraced he resuling dependence srucure ino a VG copula. This model shares he advanages of he one facor Gaussian copula of condiional independence and he LHP approximaion. Ye, he VG Copula urns ou o be more flexible and leads o a dependence srucure ha fis o observed ranche spreads. We can herefore price bespoke CDO ranches in a consisen way. 0

22 Appendix A: Variance Gamma processes and disribuions A random variable X is said o be Variance Gamma disribued wih parameers θ, ν, σ, µ X V Gθ, ν, σ, µ)) if is densiy is given by f X x) = exp ) θx σ ν 1 ν πσγ 1 ) ν x µ) σ ν + θ ) 1 ν 1 4 ) ) 1 σ K 1 ν 1 x µ) σ ν + θ. K is he modified Bessel funcion of he hird kind, K ν x) = 1 y ν 1 exp 1 ) xy + y 1 ) dy. The parameer domain is resriced o µ, θ R and ν, σ > Since his disribuion is infiniely divisible, here exiss a Lévy Process X ) 0 such ha X 1 has he above disribuion. X ) 0 is called Variance Gamma process and can be represened by X = µ + θ G + σw G, where G ) 0 is a Gamma process parameers ν 1, ν) and W ) 0 is a sandard Brownian moion. In his aricle, we have se µ = 0 when we consider VG processes. If X V Gθ, ν, σ, µ), he Laplace ransform L X of X is given by L X z) = e µz We herefore obain he momens: E[X] = µ + θ, Var[X] = νθ + σ, 1 θνz 1 ) 1 νσ z ν, z R, 14) S[X] = θν 3σ + νθ σ + νθ ), 3/ K[X] = 31 + ν νσ 4 νθ + σ ) ). Besides infinie divisibiliy, he class of VG disribuions is closed under scaling and convoluion if parameers are chosen suiably: 11 This definiion is aken from Bibby and Sørensen [003] and we used he parameer ransformaion µ µ, α νσ + θ σ,β θ 4 σ,λ 1 ν. 1

23 1. If X V Gθ, ν, σ, µ) and c > 0, hen cx V Gcθ, ν, cσ, cµ).. If X 1 V Gθ 1, ν 1, σ 1, µ 1 ) and X V Gθ, ν, σ, µ ) such ha σ 1ν 1 = σ ν and θ 1 σ 1 = θ σ hen X 1 + X V G θ 1 + θ, ν ) 1 + ν, σ 1 + σ, µ 1 + µ. ν 1 ν Appendix B Maringale propery of S i) in 3) in secion 3.1 [ E S i) ] [ ] = S i) 0 expr + ω i ) E expx i) ) = S i) 0 exp r + 1ν log 1 1 ) ) σ i ν θ i ν 1 exp ν log 1 1 ) )) σ i ν θ i ν = S i) 0 expr). The second equaion holds as he characerisic exponen of Appendix A) Ψ X i)u) = 1 1 ν log + 1 ) u σi ν iuθ i ν X i) ) 0 is given by see and since E [ ] expx i) ) = exp Ψ i) ). Proof of 4) in secion 3. If X i) = θ i G i) X j) = θ j G j) + σ i W i) G i), + σ j W j) G j)

24 as in secion 3. hen CovX i, X j ) = E k=i,j = θ i θ j Cov +θ j σ i E θ k G k) ) + σ i G i) G j) ), G j) + θ i σ j E G i) +σ i σ j E G i) G j) = θ i θ j Cov = θ i θ j Cov = θ i θ j VarF ) = θ i θ j aν. G i) F + U i) ), G j) ) EW j) ) }{{} =0 G k) W k) G i) ) EW i) )EW j) ) }{{} ), F + U j) =0 ) ) ) G j) ) EW i) ) }{{} =0 In he same way or using he resuls of Appendix A) i can be shown ha VarX i) ) = θ i ν + σ i ). We herefore see ha he correlaion for all is: CorrX i), X i) θ i θ j ν ) = a. θ i ν + σi θj ν + σ j Proof of 5) in secion 3.3 If X i) = θ i G + σ i W i) G, X j) = θ j G + σ j W j) G 3

25 as in secion 3.3 hen Cov X i) ), X j) = E k=i,j θ k G EG )) + σ k G W k) ) = θ i θ j VarG ) + θ i σ j E G G ) +θ j σ i E G G EW j) ) }{{} =0 = θ i θ j VarG ) +σ i σ j E bf + 1 bu i) EW i) ) }{{} =0 ) ) +σ i σ j EG )E W i) W j) ) bf + 1 bu j) ) )) = θ i θ j ν + σ i σ j bef ) = θ i θ j ν + σ i σ j b). Proof of 9) in secion 3.4 and 1) in secion 4.1. If X i) = cm + 1 c Z i), X j) = cm + 1 c Z j) as in secion 3.4 hen ) Cov X i) 1, X j) 1 = Cov cm c Z i) 1, cm 1 + ) 1 c Z j) 1 = c Var M 1 ) = c ν c cθ) + σ) ) = c νθ + σ ). Since we ge VarX i) 1 ) = νθ + σ ) 1, CovX i) 1, X j) 1 ) = c. The proof of 1) is idenical. Proof of 13) in secion 4.1 4

26 This proof is idenical o he one proposed by Vasicek [1987] for he Gaussian copula. Le N be he number of eniies in he porfolio. Condiional on M = m, he probabiliy of 0 k N defauls is hen N P k M = m) = k N = k ) PX i < C M = m)) k 1 PX i < C M = m)) N k ) )) k )) N k C mc C mc F Zi 1 F Zi. 1 c 1 c We suppress he index i of Z i in he following, since all Z i are idenically disribued. The uncondiional probabiliy is herefore P k = ) N )) k )) N k C mc C mc F Z 1 F Z f M m)dm. k 1 c 1 c If we subsiue ) C mc s = F Z, 1 c we find for he percenage porfolio loss no exceeding x: F N x) = = [Nx] P k k=0 [Nx] 1 k=0 0 s k 1 s) N k dw s) wih Since W s) = F M 1 c lim N [Nx] k=0 1 c F 1 Z s) C ) ). ) N s k 1 s) N k = 0 if x < s k = 1 if x > s, he cumulaive disribuion funcion in he limi N is F porfolio loss x) = W x) = F M 1 c 1 c F 1 Z x) C ) ). 5

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