Affine credit risk models under incomplete information

Transcription

1 Affine credi risk models under incomplee informaion Rüdiger Frey 1, Cecilia Prosdocimi 2, Wolfgang J. Runggaldier 2 1 Deparmen of Mahemaics, Universiy of Leipzig Leipzig, Germany (ruediger.frey@mah.uni-leipzig.de). 2 Diparimeno di Maemaica Pura ed Aplicaa, Universiá di Padova, Via Triese 63, Padova, Ialy (cecilia.prosdocimi@gmail.com, runggal@mah.unipd.i). We consider he problem of compuing some basic quaniies such as defaulable bond prices and survival probabiliies in a credi risk model according o he inensiy based approach. We le he defaul inensiies depend on an exernal facor process ha we assume is no observable. We use sochasic filering o successively updae is disribuion on he basis of he observed defaul hisory. On one hand his allows us o capure aspecs of defaul conagion (informaion-induced conagion). On he oher hand i allows us o evaluae he above quaniies also in our incomplee informaion conex. We consider in paricular affine credi risk models and show ha in such models he nonlinear filer can be compued via a recursive procedure. This hen leads o an explici expression for he filer ha depends on a finie number of sufficien saisics of he observed inerarrival imes for he defauls provided one chooses an iniial disribuion for he facor process ha is of he Gamma ype. Key words: Credi risk, affine models, incomplee informaion, pricing of credi derivaives, nonlinear filering, finie dimensional filers. 1. Inroducion We consider he problem of evaluaing some basic quaniies such as defaulable bond prices and survival probabiliies in a credi risky environmen under incomplee informaion on he underlying model. We use he reduced-form or inensiy-based approach o credi risk wih defaul imes 1

2 2 modeled as he jump imes of a doubly sochasic Poisson process. In his model class defaul inensiies are driven by a common facor process X ; his is a convenien way for generaing dependence beween defaul evens of differen firms. Typically i is assumed ha X is no direcly observable, and his will also be he main seing here. In ha case he disribuion of X can be updaed on he basis of he observed defaul hisory and his leads o wha may be ermed as informaion-induced conagion (see e.g. 8]). In he nex secion 2 we discuss our underlying model and describe hree examples of basic quaniies in a credi risky environmen (prices of defaulable bonds wih and wihou recovery and survival probabiliies) ha we firs evaluae under he assumpion of full knowledge also of he facor process wih he main purpose of moivaing he filering problem ha arises in he evaluaion of hese same quaniies when only he defauls are observable, bu no he values of X. In secion 3 we discuss he filering problem and presen is general soluion. Alhough in explici form, his general soluion will in mos cases be difficul o compue and so he ineres arises o consider paricular classes of models, for which he soluion can acually be compued. One such class corresponds o he case when X is a finie-sae Markov chain and his is discussed in 4] in a more general conex. In his paper we shall concenrae on he so-called affine case and his is he subjec of secion 4, where we show ha he filer can be compued via a recursive algorihm. The acual compuaion of his recursive algorihm is discussed in he las secion 5, where we also show ha for a suiable choice of he iniial disribuion of he facor process he filer can be compued as an explici funcion of a finie number of sufficien saisics of he observed defaul inerarrival imes. In his paper we assume ha he informaion available o an agen comes only from observing he defaul hisory. More general informaion srucures can be envisaged such as in he case when agens can observe also prices of defaulable bonds. This generalizaion is considered in 4], bu no for he affine case. 2. The model and some basic examples Consider a marke wih m firms ha may defaul and denoe by τ j he defaul ime of firm j {1,, m. The defaul sae of he porfolio can be summarized by he defaul indicaor process (1) Y = (Y,1,, Y,m ) 0 wih Y,j = 1 {{τj Given a filered probabiliy space (Ω, F, G, P), all processes will be G adaped and τ j is an G sopping ime. Our inensiy-based defaul model implies ha here are no common defauls among he firms, so ha

3 3 we may inroduce he ordered defaul imes 0 = T 0 < T 1 < < T m. One may hen also consider wha can be called he defaul-ideniy process ξ n ha denoes he ideniy of he firm defauling a T n. The defaul observaion hisory H G can hen be given he following wo equivalen represenaions (2) H = σ{y s ; s = σ{(t n, ξ n ) ; T n The facor process X R may be any Markov process (a specific such process will be considered below, see (6)). We assume ha defaul imes are condiionally independen, doubly sochasic random imes (see 7], Secion 9.6); he defaul inensiy of firm j a ime is given by λ j (X ) for some funcion λ j : R (0, ). Formally, his means ha defaul imes are independen given F X = σ(x : 0) wih condiional survival probabiliies given by P(τ j > F X ) = exp ( λ j (X s )ds ) The affine case We shall say ha we are in he affine case if X saisfies a diffusion equaion (3) dx = µ(x )d + σ(x )dw wih (4) µ(x ) = α X + β σ 2 (X ) = γ X + δ Furhermore, assuming for sake of generaliy ha also he shor rae is driven by he facor process, i.e. r = r(x ), (5) λ j (X ) = λ j X, λ j > 0 r(x ) = r X, r > 0 In paricular, for he process X we shall consider a Cox-Ingersoll-Ross (CIR)-ype model, i.e. (6) dx = (a bx ) d + σ X dw wih a, b, σ > 0 and a σ2 2 so ha X > 0 a.s. which, by (5), will hen also imply ha λ j (X ) > 0, r(x ) > 0 a.s. For his affine case in wha follows

4 4 we shall be able o derive explici expressions boh in he case of full as well as of parial informaion. To his effec we recall here he following proposiion, which in is general form can e.g. be found in 6], secion by making he following idenificaions : = T, λ = β, µ = λ, ψ(t ) = B(, T), aφ(t ) = A(, T). The paricular case for β = 0 can also be found in 7], secion The derivaion is based on he Kolmogorov equaion for funcionals of Markov processes. Proposiion 1. Le X saisfy (6) and define (7) F(, x) := E,x {e βx exp T λx s ds for a generic β 0 and λ > 0. In he presen affine case his funcion F(, x) admis he represenaion (8) F(, x) = exp A(, T) B(, T)x] where, for given T, he funcions A(, T), B(, T) saisfy he following firs order ordinary differenial equaions in 0, T] (9) B (, T) = b B(, T) σ2 B 2 (, T) λ, B(T, T) = β A (, T) = B(, T), A(T, T) = 0 and hey have as soluions (10) B(, T) = βγ + b + eγ(t ) (γ b)] + 2 λ(e γ(t ) 1) βσ 2 (e γ(t ) 1) + γ b + e γ(t ) (γ + b) A(, T) = 2a σ log 2γe (T )(γ+b) 2 2 βσ 2 (e γ(t ) 1) + γ b + e γ(t ) (γ + b) wih γ := b 2 + 2σ 2 λ. 2.2 Examples Of he following hree examples he firs wo concern pricing under full informaion and so he underlying probabiliy measure P has o be seen as a pricing (maringale) measure. The hird one concerns survival probabiliies and here P represens hen he hisorical/real world probabiliy measure. The basic quaniies in hese hree examples may be considered as building blocks for more imporan credi risky producs. ]

5 Example 1. Defaulable zero-coupon bond on firm j wih mauriy T and zero recovery Using sandard resuls for pricing defaulable claims in models wih doubly-sochasic defaul imes (see e.g. 7], secion 9.4.3) he price a ime T of a zero recovery bond on firm j can be expressed as (11) where p j (, T) = E (e ) T r(x s )ds (1 Y T,j ) G ( = (1 Y,j )E X e ) T R j (X s ) ds := Π j 1 (X, Y ) (12) R j (X ) := r(x ) + λ j (X ) I is hus a funcion Π j 1 (X, Y ), paramerized by, T ha for simpliciy we drop from he noaion, of he curren values of he facor and he defaul indicaor processes. From he previous secion 2.1 i is easily seen ha in he affine case he funcion Π j 1 (X, Y ) akes he following exponenially affine form (13) Π j 1 (X, Y ) = (1 Y,j ) exp α j (, T) β j (, T) x ] where, for X saisfying he CIR model (6), he coefficiens in (13) are given by he formulae in (10) wih β j (, T) given by he expression for B(, T) here and α j (, T) by ha for A(, T). Furhermore, for he presen case he coefficiens in he righ hand side of (10) have o be chosen as follows (we may consider, T as fixed): a, b, σ come from (6), β = 0, λ = λ j + r; γ = b2 + 2σ 2 λ Example 2. Recovery paymen Denoe by Z j τ j 1 {τj T he recovery paymen a he ime τ j of defaul of he j h firm, where Z j X is an F adaped process. I is well-known ha he value in of he recovery paymen is given by (14) ( (1 Y,j )E e ) τ j r(x s )ds Z j τ j 1 {τj T G = (1 Y,j )E X ( T Zsλ j j (X s )e s R(X u ) du ds ) := Π j 2 (X, Y ) ; see again 7] for a proof. From hese building blocks he price of many credi derivaives is obained in a sraighforward manner. For insance, he price of a zero-coupon

6 6 bond wih recovery is simply given by he sum of he price wihou recovery and he value of he recovery paymen, and also spreads of credi defaul swaps are easily compued. In he affine case also Π j 2 (X, Y ) can be given a more explici form ha is parly of he exponenially affine ype. We do no discuss his in deail here referring he reader o 7], secion or direcly o he original paper 3] Example 3. Survival probabiliies As already menioned, in his example he underlying probabiliy P is he hisorical/real world probabiliy measure. We wan in fac o compue he probabiliy, given our informaion, ha firm j does no defaul prior o a given ime T. A similar argumen as in he derivaion of (11) immediaely gives ] (15) P ( τ j > T G ) = (1 Y,j ) E X {exp T λ j (X s )ds := Π j 3 (X, Y ) Noice ha he expression of Π j 3 (X, Y ) is compleely analogous o ha of Π j 1 (X, Y ) in (11) so ha in he affine case i can be given an expression of he exponenially affine form like Π j 1 (X, Y ) in (13). 3. Incomplee informaion (he filering problem) In he examples of he previous secion we have seen ha, under full informaion also of he facor process X, he values of he basic quaniies of ineres can be expresses as an explici funcion Π(X, Y ) of he curren values of he facor and he defaul indicaor processes. If agens do no have access o he full informaion represened by he filraion G, bu only o ha corresponding o H, i.e. he informaion represened by he defaul hisory hen, based on ieraed condiional expecaions, i appears naural o consider as corresponding values for he basic quaniies he following (16) Π (Y ) := E { Π(X, Y ) H where he expecaion is under he measure P ha is a pricing (maringale) measure in he case of he firs wo examples and he hisorical measure in he hird. For he hird example here is no problem wih he definiion (16), bu in he case of he pricing examples one has o make sure ha (16) leads o arbirage-free prices. To his effec we can sae he following simple lemma Lemma 2. Assume he shor rae is H adaped, in paricular deerminisic. Then, aking as numeraire he money-marke accoun B := exp r 0 sds ], formula (16) leads o arbirage-free prices in he sense ha B 1 Π (Y ) is a (P, H ) maringale.

7 7 Proof : Le s ; hen E { Π (Y ) B H s = E { E { Π(X,Y ) B H Hs = E { Π(X,Y ) B H s = E { E { Π(X,Y ) B G s Hs = E { Π(X s,y s ) B s H s = Π s (Y s ) B s where we have used he fac ha P is a pricing (maringale) measure for he numeraire B in he sense ha B 1 Π(X, Y ) is a P maringale in he full filraion G. 3.1 The filering problem We have seen ha he problem of compuing values of risk-sensiive producs under incomplee informaion abou he facor process amouns o ha of compuing condiional expecaions as in (16). Since Y H, in wha follows we shall for simpliciy drop he dependence on Y so ha he righ hand side of (16) becomes of he form E { f (X ) H where f ( ) is a generic (bounded) funcion of he facor process. Denoing by π (dx) he condiional disribuion of X given H, (16) leads hen o he problem of compuing (17) π ( f ) := E { f (X ) H = f (x)π (dx) which is a nonlinear filering problem of a diffusion process, given poin process observaions. 3.2 General soluion of he filering problem Le us firs inroduce he global defaul inensiy m (18) λ(x, Y ) := (1 Y,j ) λ j (X ) j=1 which is he sum of he defaul inensiies of he sill surviving firms. Noe ha, for T n < T n+1, λ(x, Y ) is he inensiy of T n+1. In deriving our filering resuls we disinguish he cases beween defaul imes and a a defaul ime Filer beween defauls Le T n be he generic n h defaul ime and le T n, T n+1 ). I follows from he general filering equaions (Kushner-Sraonovich equaions) for poin process observaions by he so-called innovaions mehod (see 5], 2]) ha for π ( f ) as defined in (17) one has (19) π ( f ) = π Tn ( f ) + T n πs (L f ) π s ( λ f ) + π s ( λ) π s ( f ) ] ds

8 8 where L is he generaor ha corresponds o he diffusion process X. Furhermore, noicing ha for T n, T n+1 ) one has Y = Y Tn, whenever T n, T n+1 ) we shall consider λ as a funcion of x alone, i.e. (20) λ(x) = λ(x, Y Tn ) wih λ(x, y) as in (18). Proposiion 3. Assume he condiions for he uniqueness of he soluion of (19) as described e.g. in Appendix 2 of 2] hold. The soluion o (19) is hen for T n, T n+1 ) and an inegrable (bounded) f given by (21) π ( f ) = ϱ ( f ) ϱ (1) where ϱ ( f ), he unnormalized condiional expecaion of f, can be obained as (22) ϱ ( f ) = ψ (T n, x)( f ) π Tn (dx) wih (23) ψ (T n, x)( f ) := E Tn,x and λ(x ) according o (20). { f (, X ) e λ(x s )ds Tn Remark 4. This proposiion shows ha, beween defauls, he filer evolves deerminisically and is evoluion is deermined by he Markovian semigroup ψ (T n, x)( f ) in (23). Proof : Alhough he proof can be obained from ha of Proposiion 3.4 in 2], we presen here a direc derivaion. By Io s formula we have f (, X ) e λ(x s Tn )ds f (T n, X Tn ) = e s λ(x u )du Tn T (L f )(s, Xs ) λ(x s ) f (s, X s ) ] ds n + e s λ(x u Tn )du σ f X T s n x (s, X s)dw s Assuming he condiions for applying Fubini s heorem are saisfied, le us ake on he lef and righ hand sides he expecaion condiional on X Tn = x

9 9 hus obaining { f (, X )e λ(x s )ds Tn f (T n, x) E Tn,x = E T Tn,x {e s λ(x u )du Tn (L f )(s, Xs ) λ(x s ) f (s, X s ) ] ds n Using he definiion of ψ in (23) his las relaion can be rewrien as ψ (T n, x)( f ) ψ Tn (T n, x)( f ) = T n ψs (T n, x) (L f ) ψ s (T n, x) ( λ f ) ] ds Inegraing wih respec o π Tn (dx), aking ino accoun (22) and applying once more Fubini, one arrives a ϱ ( f ) ϱ Tn ( f ) = T n ϱs (L f ) ϱ s ( λ f ) ] ds From here one obains hen immediaely ( ) ϱ ( f ) dπ ( f ) = d = π (L f ) d π ( λ f ) d + π ( f ) π ( λ) d ϱ (1) from which he resul follows by he assumed uniqueness of he soluion of (19) Filer a a defaul Consider now a generic defaul ime T n. Again from he general filering equaions of he innovaions approach (5],2]) one has (24) π Tn ( f ) = π T n (λ ξ n f ) π T n (λ ξn ), where we implicily use ha π T n (λ ξn ) > 0 a.s. The expression π T n ( f ) denoes here he lef hand limi of π Tn ( f ) in T n, which exiss by (19). Concluding his secion 3.2 one sees ha he crucial poin o obain a soluion of he filering equaions is he possibiliy of explicily compuing he semigroup ψ in (23) and in he nex Secion 4 we shall address his issue in he case of an affine model. 4. Filering in affine models As menioned a he end of he previous secion, in his secion we shall derive an explici represenaion of he semigroup ψ in (23) for affine models and his will hen lead o an explici soluion of he filering problem.

10 10 Recall ha for an affine model we have posulaed an affine dynamics for X ha we ake here as given by he CIR model (6). Furhermore, as in (5), we shall assume λ j (X ) = λ j X, which by (18) and (20) hen implies ha also λ(x, Y ) = λ(y ) X ; for he shor rae we assume ha i is consan, i.e. r(x ) r. Finally, we shall assume f (, x) o be exponenially affine, analogously o F(, x) in (8), namely of he form (25) f (, x) = exp α(, T) β(, T)x ] and is specific form depends on he specific problem a hand (see he examples in secion 2.2). Noice nex ha wih f (, x) of he form (25) he semigroup ψ becomes (26) ψ (T n, x)( f ) = E Tn,x = e α(,t) E Tn,x {e α(,t) β(,t)x e λ Tn X sds {e β(,t)x e λ Tn X s ds where we have simply wrien λ for λ(y ) since Y = Y Tn. The crucial quaniy becomes herefore he second facor in he righmos expression in (26). We have now he following proposiion, he proof of which follows immediaely from Proposiion 1 Proposiion 5. Under he assumpions of his secion one has (27) E Tn,x {e β(,t)x e λ Tn X s ds = exp A(T n, ; β) B(T n, ; β)x ] β=β(,t) where A(T n, ; β) and B(T n, ; β) are given by (28) A(T n, ; β) = 2a σ log 2γe ( Tn)(γ+b) 2 2 βσ 2 (e γ( Tn) 1) + γ b + e γ( Tn) (γ + b) B(T n, ; β) = βγ + b + eγ( T n) (γ b)] + 2 λ(e γ( T n) 1) βσ 2 (e γ( T n) 1) + γ b + e γ( T n) (γ + b) wih γ := b 2 + 2σ 2 λ 4.1 Filer beween defauls Combining Proposiions 3 and 5 and noicing ha he consan 1 can be expressed as he funcion f (, x) in (25) for α(, T) = β(, T) = 0, one immediaely obains

11 11 Proposiion 6. The filer π ( f ) is, for T n, T n+1 ), given by e B(T n,;β(,t))x π Tn (dx) (29) π ( f ) = η(t n ;, T) e B(T n,;0)x π Tn (dx) where (30) η(t n ;, T) = e α(,t) exp A(T n, ; β(, T)) A(T n, ; 0) ] Remark 7. I follows from (28) ha B(T n, ; β) is, for T n, nonnegaive whenever β β() where 2 λ(e (31) β() γ( Tn) 1) = min γ + b + e γ( Tn) (γ b), γ b + ] eγ( Tn) (γ + b) σ 2 (e γ( Tn) 1) Thus β() is sricly posiive for > T n and i is equal o zero for = T n. From he examples in secion 2.2 i follows ha he value of β(, T) o be used for he parameer β in he numeraor of he righ hand side in (29) (as well as in (34) below) is sricly posiive, while in he denominaor of he same formulae i is equal o zero. Consequenly he corresponding value of B(T n, ; β) is always nonnegaive. In (34) below we shall also need he derivaive of B(T n, ; β) wih respec o β, evaluaed a β = β(t n+1, T) and a β = 0. Since T n+1 > T n and hus β(t n+1 ) < 0, no only β = β(t n+1, T), bu also β = 0 are inerior poins of he domain of posiiviy for B(T n, ; β) and in his domain i is easily seen from (28) ha B(T n, ; β) is differeniable wih respec o β. Recall now also ha X as given by (6) for a σ2 2 has is a-priori suppor in he posiive half line and his implies ha also all condiional disribuions π (dx) have heir suppor in he posiive half line. I follows ha he wo inegrals in he righ hand side of (29) are well defined and finie. From he above Remark 7 we have ha, for all values of ineres for β, B(T n, ; β) is sricly posiive. Given ha X > 0 anyway, his leads us o define he momen generaing funcion of he condiional disribuion π (dx) of X by ( ) (32) χ (φ) := π e φ X φ > 0 From Proposiion 6 one hen obains immediaely Corollary 8. For T n, T n+1 ) and f (, x) as in (25) we have ha he filer value π ( f ) is given by (33) π ( f ) = η(t n ;, T) χ T n (B(T n, ; β(, T)) χ Tn (B(T n, ; 0)) where η( ) is as in (30), χ Tn ( ) as in (32) and B(T n, ; β) is given by (28).

12 Filer a a defaul ime Based on he general formula (24) for he filer a a generic defaul ime T n, we can now show he following (since for he case beween defauls we considered he inerval T n, T n+1 ), here we ake T n+1 o denoe he generic defaul ime) Proposiion 9. Assuming he condiions are fulfilled o differeniae under he inegral sign, a he generic defaul ime T n+1 we have (34) π Tn+1 ( f ) = e α(t β e A(T n,t n+1 ;β) χ Tn (B(T n, T n+1 ; β)) ] β=β(tn+1,t) n+1,t) β e A(T n,t n+1 ;β) χ Tn (B(T n, T n+1 ; β)) ] β=0 where A(T n, T n+1 ; β) and B(T n, T n+1 ; β) are given by (28), β(t n+1, T) corresponds o he exponenially affine represenaion of f (, x) in (25) and χ Tn ( ) is as defined in (32). Proof : Saring from (24) for he defaul ime T n+1 and noicing ha π T n+1 ( f ) is he limi, for T n+1, of he filer π ( f ) when T n, T n+1 ), using Proposiion 3 wih f (, x) as in (25) and wih he semigroup ψ ( ) as in (26) combined wih Proposiion 5, we obain he following sequence of equaliies, where we use differeniaion under he inegral sign in wo insances π Tn+1 ( f ) = π T n+1 {λ ξ n+1 X Tn+1 f (T n+1,x Tn+1 ) π T n+1 {λ ξn+1 X Tn+1 = e α(t n+1,t) E Tn,x { = e α(t n+1,t) = e α(t n+1,t) X Tn+1 e β(t n+1,t) X T λ T n+1 n+1 e Tn ETn,x {X Tn+1 e λ T n+1 Xudu Tn π Tn (dx) { β E Tn,x e β X T λ T n+1 n+1 e Tn { β E Tn,x e β X T λ T n+1 n+1 e Tn Xu du Xu du π Tn (dx) π Tn (dx) β= β(t n+1,t) Xu du π Tn (dx) β=0 β {expa(t n,t n+1 ;β) B(T n,t n+1 ;β) x] β=β(tn+1,t) π Tn (dx) β{expa(t n,t n+1 ;β) B(T n,t n+1 ;β) x] β=0 π Tn (dx) = e α(t n+1,t) βe A(Tn,T n+1 ;β) χ Tn (B(T n,t n+1 ;β))] β=β(tn+1,t) βe A(Tn,T n+1 ;β) χ Tn (B(T n,t n+1 ;β))] β=0 Noice ha, by he consideraions in Remark 7 all he above quaniies are well defined.

13 13 Remark 10. Noice ha in he filer updae a T n+1 only he informaion abou he iming T n+1 of he (n + 1) s defaul is being used and no also ha of he defauling firm ξ n+1 ; in fac, he facor λ ξn+1, which conains his informaion, drops ou due o he normalizaion. This happens however only because of our affine form of he dependence of he defaul inensiy on he facor process and he fac ha he laer is aken o be scalar. In any case, he informaion abou ξ n+1 is no los as i becomes par of Y Tn+1. From Corollary 8 and Proposiion 9 i now follows ha he crucial quaniy for compuing he filer in he affine case is he momen generaing funcion χ (φ) in (32), which we need o compue only for he values of corresponding o he defaul imes and for any φ > 0. Noice also ha χ (φ) is nohing bu he filer π ( f ) for f (, x) when his laer funcion has he exponenially affine form in (25) wih α(, T) = 0 and β(, T) = φ. Combining hese remarks wih he resuls of Proposiion 9 one obains immediaely Corollary 11. Under he assumpions of Proposiion 9 one has β e A(T n,t n+1 ;β) χ Tn (B(T n, T n+1 ; β)) ] β=φ (35) χ Tn+1 (φ) = e A(T n,t n+1 ;β) χ Tn (B(T n, T n+1 ; β)) ] β=0 β On he basis of he previous resuls we can now wrie down an algorihm o compue recursively he filer in he affine case, which we do in he nex subsecion. 4.3 Filer algorihm i) Sar a T 0 = 0 wih a given χ 0 (φ). ii) A he generic T n+1 compue (see Corollary 11) β e A(T n,t n+1 ;β) χ Tn (B(T n, T n+1 ; β)) ] β=φ χ Tn+1 (φ) = e A(T n,t n+1 ;β) χ Tn (B(T n, T n+1 ; β)) ] β=0 β iii) For T n, T n+1 ) and wih f (, x) = expα(, T) β(, T)x] he filer is hen given by (see Corollary 8) π ( f ) = η(t n ;, T) χ T n (B(T n, ; β(, T)) χ Tn (B(T n, ; 0)) where η(t n ;, T) is given in (30) and B(T n, ; β) in (28). iv) For = T n+1 he filer is (see Proposiion 9 and Corollary 11) π Tn+1 ( f ) = e α(t n+1,t) χ Tn+1 (β(t n+1, T))

14 14 Sep ii) above is a recursive formula o compue he momen generaing funcion χ Tn ha hen leads o an explici filer soluion according o Seps iii) and iv). Alhough recursive in naure, his Sep ii) may become increasingly difficul o compue since he analyic expression for χ Tn (φ) migh become more and more involved wih every sep. A naural quesion hen arises o see wheher, wih a suiable choice of he iniial χ 0 (φ), he recursions remain a a level ha allows for feasible compuaions. This will be he subjec of he nex Secion 5, where we also show ha, alhough a priori he filer is no finie-dimensional in he radiional sense, for a suiable choice of he iniial disribuion i can be paramerized by a finie number of sufficien saisics. Remark 12. We conclude his secion by poining ou ha our filer resuls may have a wider scope han only wha we have described here by showing ha he filer can be compued by compuing he condiional momen generaing funcion and ha his can be done recursively according o (35). In fac, since his condiional momen generaing funcion is relaed o he Laplace ransform of he condiional (filer) disribuion, one can, a leas in heory, inver his condiional momen generaing funcion hereby recovering he filer disribuion iself. This would hen allow o compue condiional expecaions no only of exponenially affine funcions of he facor process bu also of any inegrable funcion hereof. 5. Finie dimensional compuaion of he filer In his secion we exhibi a choice for χ 0 (φ) ha allows Sep ii) in he filer algorihm of he previous subsecion 4.3 o remain compuable a every sep. For his purpose recall ha he recursions in Sep ii) correspond o he recursive formula (35), where he coefficiens A(T n, T n+1 ; β), B(T n, T n+1 ; β) are given in (28). Inroduce he shorhand noaions (36) R n+1 = γ + b + e γ(t n+1 T n ) (γ b) U n+1 = σ ( 2 e γ(t n+1 T n ) 1 ) S n+1 = 2 λ ( e γ(t n+1 T n ) 1 ) V n+1 = γ b + e γ(t n+1 T n ) (γ + b) W n+1 = 2 γ e (T n+1 Tn) (γ+b) 2 ha, since γ = b σ 2 λ, are all posiive quaniies. The coefficiens A(T n, T n+1 ; β) and B(T n, T n+1 ; β) from (28) can hen be given he following

15 15 represenaion (37) A(T n, T n+1 ; β) = 2 a log ( ) W n+1 σ 2 β U n+1 +V n+1 B(T n, T n+1 ; β) = β R n+1+s n+1 β U n+1 +V n+1 Choosing as disribuion for he iniial value X 0 of he facor process a specific Gamma-ype disribuion, we shall now prove he following heorem ha gives an explici compuable represenaion for χ (φ) a he various defaul imes = T n. Theorem 13. Le (38) χ 0 (φ) = ( φ ) 2a σ 2 = (1 + φ) 2a σ 2, φ > 0 which according o (32) corresponds o he momen generaing funcion of a Gamma disribuion for X 0 wih parameers ( 2a σ 2, 1 ). Then (39) χ Tn (φ) = c n (φ H n + K n ) 2a σ 2 n p n (φ) where H n and K n saisfy he recursions H n = R n H n 1 + U n K n 1, H 0 = 1 (40) K n = S n H n 1 + V n K n 1, K 0 = 1 he coefficien c n is given by (41) c n = ] K 2a σ 2 n 1 n p n (0) and p n (φ) is a polynomial of degree n 1 given by (42) 1 for n = 0 and n = 1 ( 2a n + 1 ) H σ 2 n (φu n + V n ) p n (φ) p n (φ) = +U n (φh n + K n ) p n (φ) + (φh n + K n )(φu n + V n ) φ p n(φ) wih ( ) φrn (43) p n (φ) = (φu n + V n ) n 2 + S n p n 1, n 2 φu n + V n for n 2

16 16 Proof : The saemen is clearly rue for n = 0. We show i firs for n = 1 and hen inducively for all n 2. i) he case n = 1 : by (35),(36), (37) and he recursions in (40) we have (44) χ T1 (φ) = = ( ) 2a ( ) W 1 σ 2 βr1 +S β βu 1 +V χ 1 T0 1 βu 1 +V 1 β=φ ( ) 2a ( ) W 1 σ 2 βr1 +S β βu 1 +V χ 1 T0 1 βu 1 +V 1 β=0 ( ) 2a ( ) W 1 σ 2 2a βh 1 +K 1 σ 2 β βu 1 +V 1 βu 1 +V 1 β=φ ( ) 2a ( ) W 1 σ 2 2a βh 1 +K 1 σ 2 β βu 1 +V 1 βu 1 +V 1 β=0 = (φh 1+K 1 ) 2a σ 2 1 K 2a σ which indeed corresponds o (39) wih (41) and (42). ii) he general case n 2 : assume (39) holds for n 1. Then, always by (35), (36), (37) and (40) we obain (45) χ Tn (φ) = = { ( ) 2a ( Wn σ 2 βrn+sn ) β χ βun+vn Tn 1 βun+vn β { ( ) 2a Wn βun+vn σ 2 χ Tn 1 ( βrn+sn βun+vn { ( ) 2a ( 1 σ 2 βhn+kn β βun+vn βun+vn β { ( ) 2a 1 βun+vn σ 2 ( βhn+kn βun+vn β=φ ) β=0 ) 2a σ 2 n+1 ( βrn+sn ) p n 1 βun+vn β=φ ) 2a σ 2 n+1 ( βrn+sn ) p n 1 βun+vn Taking ino accoun ha, by (43), ( ) βrn + S n (46) p n 1 = (βu n + V n ) 2 n p n (β) βu n + V n he numeraor in he righmos expression of (45) becomes (47) { β (βun + V n )(βh n + K n ) 2a n+1 p σ 2 n (β) β=φ = (βh n + K n ) 2a σ 2 n{ U n (βh n + K n ) p n (β) β=0 + ( 2a n + 1 ) H σ 2 n (βu n + V n ) p n (β) + (βu n + V n )(βh n + K n ) β p n(β) β=φ = (φ H n + K n ) 2a σ 2 n p n (φ)

17 17 where we have used he definiion of p n (φ) in (42). Reurning o (45) and recalling ha he denominaor in he righmos expression of (45) is he same as he numeraor excep for puing β = 0, one finally obains (48) χ Tn (φ) = (φ H n + K n ) 2a σ 2 n p n (φ) K 2a σ 2 n n p n (0) = c n (φ H n + K n ) 2a σ 2 n p n (φ) Remark 14. I follows from Theorem 13 ha, for a choice of he iniial disribuion corresponding o χ 0 (φ) in (38), he sequence χ Tn (φ) and herefore (see Seps iii) and iv) in Secion 4.3) he enire filer is parameerized by a same finie number of sufficien saisics, namely he pairs (H n, K n ) and he polynomial funcions p n (φ), all of which can be compued recursively on he basis of he funcions R n, S n, U n, V n of he inerarrival imes of he defauls. References 1. Bielecki, T. R., M. Jeanblanc, & M. Rukowski (2004). Modeling and Valuaion of Credi Risk. In : Sochasic Mehods in Finance (M. Frielli and W. Runggaldier, eds), Lecure Noes in Mahemaics. Vol 1856, Springer Verlag. 2. Ceci,C., & A. Gerardi (2006). A Model for High Frequency Daa under Parial Informaion : a Filering Approach. Inernaional Journal of Theoreical and Applied Finance (o appear). 3. Duffie, D., & N. Gârleanu (2001). Risk and Valuaion of Collaeralized Deb Obligaions. Financial Analyss Journal. 57, Frey, R., & W. J. Runggaldier ( 2006). Credi Risk and Incomplee Informaion : a Nonlinear Filering Approach. Preprin. 5. Kliemann,W. H., G. Koch, & F. Marchei (1990). On he Unnormalized Soluion of he Filering Problem wih Couning Process Observaions. IEEE Transacions on Informaion Theory. 36, Lamberon,D., & B. Lapeyre (1995). An Inroducion o Sochasic Calculus Applied o Finance. Chapman and Hall. 7. McNeil A. J., R. Frey, & P. Embrechs (2005). Quaniaive Risk Managemen. Princeon Universiy Press. 8. Schönbucher, P. (2004) Informaion-driven Defaul Conagion. Preprin, Deparmen of Mahemaics, ETH Zürich.