MARKET MODELS OF FORWARD CDS SPREADS

Transcription

1 MARKET MODELS OF FORWARD CDS SPREADS Libo Li and Marek Rukowki School of Mahemaic and Saiic Univeriy of Sydney NSW 2006, Auralia 20 March 200 Abrac The paper re-examine and generalize he conrucion of everal varian of marke model for forward CDS pread, a fir preened by Brigo [0]. We compue explicily he join dynamic for ome familie of forward CDS pread under a common probabiliy meaure. We fir examine hi problem for ingle-period CDS pread under cerain implifying aumpion. Subequenly, we derive, wihou any rericion, he join dynamic under a common probabiliy meaure for he family of one- and wo-period forward CDS pread, a well a for he family of one-period and co-erminal forward CDS pread. For he ake of generaliy, we work hroughou wihin a general emimaringale framework. The reearch of M. Rukowki wa uppored under Auralian Reearch Council Dicovery Projec funding cheme (DP088460).

2 2 Marke Model of Forward CDS Spread Inroducion The marke model for forward LIBOR wa fir examined in paper by Brace e al. [6] and Muiela and Rukowki [23]. Their approach wa ubequenly exended by Jamhidian in [6, 7] o he marke model for co-erminal forward wap rae. Since hen, everal paper on alernaive marke model for LIBOR and oher familie of forward wap rae were publihed. Since modeling of (non-defaulable) forward wap rae i no preened here, he inereed reader i referred, for inance, Galluccio e al. [4], Pieerz and Regenmorel [25], Rukowki [26], or he monograph by Brace [5] or Muiela and Rukowki [24] and he reference herein. To he be of our knowledge, here i relaively carce financial lieraure in regard eiher o he exience or o mehod of conrucion of marke model for forward CDS pread. Thi apparen gap i a bi urpriing, epecially when confroned wih he marke praciioner approach o credi defaul wapion, which hinge on a uiable varian of he Black formula. The andard argumen which underpin he validiy of hi formula i he poulae of lognormaliy of credi defaul wap (CDS) pread, a dicued, for inance, in Brigo and Morini [], Jamhidian [8], Morini and Brigo [22], or Rukowki and Armrong [28]. In he commonly ued ineniy-baed approach o defaul rik, hi crucial propery of forward CDS pread fail o hold, however (ee, e.g., Bielecki e al. [2]), and hu a need for a novel modeling approach arie in a naural way. Recenly, aemp have been made o earch for explici conrucion of marke model for forward CDS pread in paper by Brigo [8, 9, 0] and Schlögl [29] (relaed iue were alo udied in Loz and Schlögl [20] and Schönbucher [30]). The preen work i inpired by hee paper, where under cerain implifying aumpion, he join dynamic of a family of CDS pread were derived explicily under a common probabiliy meaure and a conrucion of he model wa provided. Our main goal will be o derive he join dynamic for cerain familie of forward CDS pread in a general emimaringale eup. Our aim i o derive he join dynamic of a family of CDS pread under a common probabiliy. Firly, we will derive he join dynamic of a family of ingle period CDS pread under he poulae ha he inere rae i deerminiic. In he econd par, we will derive he join dynamic of a family of ingle period CDS pread under he aumpion ha he inere rae and defaul indicaor proce are independen. Laly, wihou any implifying aumpion, we will derive boh he join dynamic of a family of one- and wo-period CDS pread and a family of one-period and co-erminal CDS pread. We would alo like o menion ha, alhough no preened here, he join dynamic of a family of one-period and co-iniial CDS pread can alo be derived uing he echnique developed in Subecion 5.3 and 5.4. For each marke model, we will alo preen boh he boom-up approach and he op-down approach o he modeling of CDS pread. In he boom-up approach, one uually ar wih a credi rik model and, relying on he aumpion ha he Predicable Repreenaion Propery (PRP) hold, one how he exience of a family of volailiy procee for a family of forward CDS pread and derive heir join dynamic under a common probabiliy meaure. On he oher hand, in he op-down approach, for any given in advance family of volailiie, we focu on he direc derivaion of he join dynamic for a given family of forward CDS pread under a common probabiliy meaure. I i fair o poin ou, however, ha in hi paper we do no provide a fully developed credi rik model obained hrough he op-down approach, ince he conrucion of he defaul ime conien wih he derived dynamic of forward CDS pread i no udied. 2 Forward Credi Defaul Swap Le (Ω, G, F, Q) be a filered probabiliy pace, where F = (F ) [0,T ] i he reference filraion, which i aumed o aify he uual condiion. We work hroughou wihin he framework of he reducedform (i.e., ineniy-baed) mehodology. Le u fir ake he perpecive of he boom-up approach, ha i, an approach in which we pecify explicily he defaul ime uing ome alien probabiliic feaure, uch a he knowledge of i urvival proce or, equivalenly, he hazard proce. We hu aume ha we are given he defaul ime τ defined on hi pace in uch a way ha he F-urvival

3 L. Li and M. Rukowki 3 proce G = Q(τ > F ) i poiive. I i well known ha hi goal can be achieved in everal alernaive way, for inance, uing he o-called canonical conrucion of he random ime for a given in advance F-adaped ineniy proce λ. We denoe by G = (G ) [0,T ] he full filraion, ha i, he filraion generaed by F and he defaul indicaor proce H = {τ }. Formally, we e G = σ(h, F ) for every R +, where H = (H ) [0,T ] i he filraion generaed by H. I i well known ha for any 0 < u T and any Q-inegrable, F u -meaurable random variable X he following equaliy i valid (ee, for inance, Chaper 5 in Bielecki and Rukowki [] or Chaper 3 in Bielecki e al. [4]) E Q ( {τ>u} X G ) = {τ>} G E Q (G u X F ). () Finally, we aume ha an underlying defaul-free erm rucure model i given and we denoe by β(, u) = B Bu he defaul-free dicoun facor over he ime period [, u] for 0 u T, where in urn B = (B, [0, T ]) repreen he aving accoun. By aumpion, he probabiliy meaure Q will be inerpreed a he rik-neural meaure. The ame baic aumpion underpinning he boom-up approach will be mainained in Secion 5 where alernaive varian of marke model are preened. Le T = {T 0 < T < < T n } wih T 0 0 be a fixed enor rucure and le u wrie a i = T i T i. We oberve ha i i alway rue ha, for every i =,..., n, Q (τ > T i F ) Q (τ > T i F ). (2) When dealing wih he boom-up approach, we will make he ronger aumpion ha he following inequaliy hold, for every i =,..., n, Q (τ > T i F ) > Q (τ > T i F ). (3) We are in a poiion o formally inroduce he concep of he forward credi defaul wap. To hi end, we will decribe he cah flow of he wo leg of he ylized forward CDS aring a T i and mauring a T l, where T 0 T i < T l T n. We denoe by δ j [0, ) he conan recovery rae, which deermine he ize of he proecion paymen a ime T j, if defaul occur beween he dae T j and T j. Definiion 2.. The forward credi defaul wap iued a ime [0, T i ], wih he uni noional and he F -meaurable pread κ, i deermined by i dicouned payoff, which equal D i,l = P i,l κa i,l for every [, T i ], where in urn he dicouned payoff of he proecion leg equal P i,l = l ( δ j )β(, T j ) {Tj <τ T j } (4) and he dicouned payoff of he fee leg (alo known a he premium leg) per one uni of he pread equal l A i,l = a j β(, T j ) {τ>tj}. (5) Remark 2.. I hould be reed ha in he pecificaion of he wo leg we have, in paricular, deliberaely omied he o-called accrual paymen, ha i, a porion of he fee ha hould be paid if defaul occur beween wo enor dae, ay T j and T j. The inereed reader i referred o Brigo [8, 9, 0], Brigo and Mercurio [2], or Rukowki [27] for more deail. Specificaion (4) (5) mean ha we decided o adop here he poponed running CDS convenion propoed by Brigo [8, 9, 0]. Thi paricular choice of convenion i moivaed by he fac ha i appear o be he mo convenien for conrucing marke model of forward CDS pread. The value (or he fair price) of he forward CDS a ime i baed on he rik-neural formula under Q applied o he dicouned fuure payoff. Noe ha we only conider here he cae [, T i ], alhough an exenion o he general cae where [, T l ] i readily available a well.

4 4 Marke Model of Forward CDS Spread Definiion 2.2. The value of he forward credi defaul wap for he proecion buyer equal, for every [, T i ], S i,l (κ) = E Q (D i,l G ) = E Q (P i,l G ) κ E Q (A i,l G ). (6) In he econd equaliy in (6) we ued he definiion of he proce D i,l and he poulaed propery ha he pread κ i F -meaurable, and hu alo G -meaurable for every [, T i ]. Le u oberve ha A i,l = {τ>ti }A i,l and P i,l = {τ>ti }P i,l o ha alo D i,l = {τ>ti }D i,l. Uing formula (), i i hu raighforward o how ha he value a ime [, T i ] of he forward CDS aifie S i,l (κ) = {τ>} G E Q (D i,l F ) = Si,l {τ>} (κ), (7) where he pre-defaul price aifie i,l S (κ) = P i,l κãi,l, where we denoe P i,l = G E Q (P i,l F ), à i,l = G E Q (A i,l F ). (8) More explicily, he pre-defaul value a ime [0, T i ] of he fee leg per one uni of pread, ha i, of he defaulable annuiy i given by à i,l = l a j G E Q (β(, T j ) {τ>tj } F ). (9) Similarly, he pre-defaul value a ime [0, T i ] of he proecion leg equal P i,l = l ( δ j )G E Q (β(, T j ) {Tj<τ T j} F ). (0) Since he forward CDS i erminaed a defaul, he concep of he fair (or par) forward CDS pread i only meaningful prior o defaul. I i alo poible, and in fac more convenien, o inroduce he noion of he pre-defaul fair forward CDS pread, which can be formally defined for any dae [0, T 0 ]. To hi end, we ue he following definiion, in which he iuance dae of a forward CDS i irrelevan, and hu i define he F-adaped proce κ i,l = (κ i,l, [0, T i ]). Definiion 2.3. For any dae [0, T i ], he pre-defaul fair forward CDS pread a ime i he F -meaurable random variable κ i,l i,l uch ha S (κ i,l ) = 0. For breviy, he pre-defaul fair forward CDS pread will imply be called forward CDS pread i,l in wha follow. Recall ha S (κ i,l i,l ) = P κ i,l à i,l, where he procee Ãi,l and P i,l are given by (9) and (0), repecively. The forward pread for he CDS aring a T i and mauring a T l i hu given a he raio of pre-defaul value of he fee and proecion leg, ha i, κ i,l = i,l P à i,l, [0, T i ]. () Wihin he boom-up approach, he main goal i o analyze he dynamic of he forward CDS pread κ i,l for a given in advance pecificaion of he defaul ime. In paricular, we are inereed in he exience of a maringale meaure for hi proce and explici compuaion of he volailiy proce for κ i,l. The laer ak appear o be raher difficul and hu i can only be performed for ome relaively imple ochaic model of defaul ineniy (ee Bielecki e al. [3] who examine he hedging of a credi defaul wapion in he CIR defaul ineniy model i examined). Moreover, an explici conrol of he volailiy proce for forward CDS pread i no feaible. Therefore, he valuaion of opion on a forward CDS in he defaul ineniy model hrough cloed-form oluion i a very difficul ak wihin he claic reduced-form approach, in which one uually pecifie he model by elecing he defaul ineniy proce.

5 L. Li and M. Rukowki 5 By conra, in he op-down approach we poulae ha he volailiie for a given family of forward CDS pread are predeermined (and hu hey can be choen, for inance, a deerminiic funcion) and we aim o derive he join dynamic of hee procee under a common probabiliy meaure ha, ypically, i choen o be a maringale meaure for one of hee procee. Since hi work focue on he concep of a marke model, which i obained in hi way, i i clear ha we will be mainly inereed in he op-down approach. However, a deailed analyi of he boom-up approach i alo ueful, ince i provide imporan guidance abou he alien feaure of forward CDS pread. Before we proceed o he main opic of hi work, we will preen and examine cerain abrac emimaringale eup ha will prove ueful in Secion 5. I i noable ha abrac reul eablihed wihin ome paricular eup may be laer ued o obain dynamic wihin differen applied framework. For inance, Seup (A) inroduced in Subecion 4. cover imulaneouly he cae of (defaul-free) forward LIBOR under ochaic inere rae and one-period forward CDS pread under deerminiic inere rae. 3 Preliminarie We need fir o recall ome definiion and reul from ochaic calculu, which will be ued in wha follow. In hi ecion, he mulidimenional Iô inegral hould be inerpreed a he vecor ochaic inegral (ee, for inance, Shiryaev and Cherny [3]). We fir quoe a verion of Giranov heorem (ee, e.g., Brémaud and Yor [7] or Theorem in Jeanblanc e al. [9]). Le (Ω, G, F, P) be a given filered probabiliy pace. Noe ha i i no poulaed in Secion 4 ha he σ-field F 0 i rivial. For arbirary real-valued emimaringale X and Y, we denoe by [X, Y ] heir quadraic covariaion; i i known o be a finie variaion proce. Propoiion 3.. Le P and P be equivalen probabiliy meaure on (Ω, F T ) wih he Radon- Nikodým deniy proce Z = d P, [0, T ]. (2) F If M i a (P, F)-local maringale hen he proce i a ( P, F)-local maringale. M = M d[z, M] (0,] Z Le M loc (P, F) (M(P, F), repecively) and for he cla of all (P, F)-local maringale ((P, F)- maringale, repecively). Aume ha Z i a poiive (P, F)-maringale uch ha E P (Z 0 ) =. Alo, le an equivalen probabiliy meaure P be given by (2). Then he linear map Ψ Z : M loc (P, F) M loc ( P, F), which i defined by he formula Ψ Z (M) = M d[z, M], (0,] Z M M loc (P, F), i called he Giranov ranform aociaed wih he Radon-Nikodým deniy proce Z. By he ymmery of he problem, he proce Z := /Z i he Radon-Nikodým deniy proce of P wih repec o P. The correponding Giranov ranform Ψ Z : M loc ( P, F) M loc (P, F) aociaed wih Z i hu given by he formula Ψ Z ( M) = M Z d[z, M], M M loc ( P, F). (0,]

6 6 Marke Model of Forward CDS Spread Propoiion 3.2. Le P be a probabiliy meaure equivalen o P on (Ω, F T ) wih he Radon-Nikodým deniy proce Z. Then, for any ( P, F)-local maringale Ñ, here exi a (P, F)-local maringale N uch ha Ñ = N d[z, Z N]. (3) (0,] Le u e L = ÑZ. Then he proce N i given by he formula L N = Ñ0 + dl (0,] Z (0,] Z 2 dz. (4) From Propoiion 3.2, i follow ha he proce N given by (4) belong o he e (Ψ Z ) (Ñ). In fac, we have ha ha N = (Ψ Z ) (Ñ), a he following reul how. Lemma 3.. Le Ñ be any ( P, F)-local maringale and le he proce N be given by formula (4) wih L = ÑZ. Then he proce N i alo given by he following expreion N = Ñ Z d[z, Ñ]. (5) (0,] Corollary 3.. The linear map Ψ Z : M loc (P, F) M loc ( P, F) i bijecive and he invere map (Ψ Z ) : M loc ( P, F) M loc (P, F) aifie (Ψ Z ) = Ψ Z. The following definiion i andard (ee, for inance, Jacod and Yor [5]). Definiion 3.. We ay ha an R k -valued (P, F)-maringale M = (M,..., M k ) ha he predicable repreenaion propery (PRP) wih repec o F under P, if an arbirary (P, F)-local maringale N can be expreed a follow N = N 0 + ξ dm, [0, T ], (0,] for ome R k -valued, F-predicable, M-inegrable proce ξ = (ξ,..., ξ k ). We hen ay ha M = (M,..., M k ) i he panning (P, F)-maringale. The following well-known reul examine he predicable repreenaion propery under an equivalen change of a probabiliy meaure. For he ake of compleene, we provide he proof of hi reul. Propoiion 3.3. Aume ha he PRP hold under P wih repec o F wih he panning (P, F)- maringale M = (M,..., M k ). Le P be a probabiliy meaure equivalen o P on (Ω, F T ) wih he Radon-Nikodým deniy proce Z. Then he PRP hold under P wih repec o F wih he panning ( P, F)-maringale M = ( M,..., M k ), where for every l =,..., k, M l = M l (0,] Z d[z, M l ]. Hence for any ( P, F)-local maringale Ñ here exi an Rk -valued, F-predicable, M-inegrable proce ξ uch ha under P Ñ = Ñ0 + ξ d M. (0,] Proof. By Propoiion 3.2, here exi N M loc (P, F) wih N 0 = 0 uch ha Ñ Ñ0 = Ψ Z ( N) = N d[z, Z N]. (0,]

7 L. Li and M. Rukowki 7 Since he PRP i aumed o hold under P wih repec o F wih he panning maringale M = (M,..., M k ), here exi ome R k -valued, F-predicable, M-inegrable proce ξ = (ξ,..., ξ k ) uch ha Ñ Ñ0 = = (0,] (0,] ξ dm (0,] ξ dψ Z (M). ξ Z d[z, M] Therefore, he claim hold by eing ξ = ξ and M = Ψ Z (M). Noe ha he proce ξ i M-inegrable under P, ince i i Ψ Z (M) inegrable under P. For he definiion and properie of he ochaic exponenial (alo known a he Doléan-Dade exponenial) and ochaic logarihm, we refer o Secion in Jeanblanc e al. [9]. Definiion 3.2. The ochaic exponenial E(X) of a real-valued emimaringale X i he unique oluion Y o he following ochaic differenial equaion Y = + Y dx. (6) (0,] Le X c be he coninuou maringale par of X, ha i, he unique coninuou local maringale X c uch ha X0 c = X 0 and X X c i a purely diconinuou emimaringale. Then he unique oluion o equaion (6) i given by he following expreion Y = E(X) = e X X 0 2 [Xc,X c ] ( + X )e X. 0< The map X E(X) can be invered if almo all pah of he proce E(X) and i lef-coninuou verion E(X) do no hi zero (ee, e.g., Choulli e al. [3]). Recall ha a ube of Ω R + i called evanecen if i projecion on Ω ha null probabiliy. Le S be he pace of all real-valued emimaringale. Conider he following ubpace: S 0 = { X S { X = } i evanecen } and S = { Y S {Y Y = 0} i evanecen }. Definiion 3.3. The ochaic logarihm of Y S i defined by he formula L(Y ) = dy. (7) Y (0, ] I i clear ha for any non-zero conan emimaringale Y = c 0, he ochaic logarihm vanihe, ha i, L(Y ) = L(c) = 0. We alo have he following reul, which how, in paricular, ha he map E : S 0 {X 0 = 0} S {Y 0 = } i a bijecion and i invere i given by he map L : S {Y 0 = } S 0 {X 0 = 0}. Propoiion 3.4. (i) For every X S 0, he equaliy (L E)(X) = X X 0 hold. (ii) For every Y S, he equaliy Y = Y 0 (E L)(Y ) hold. The following reul ummarize he crucial properie of he ochaic logarihm. Propoiion 3.5. (i) For any wo emimaringale Y and Y 2 belonging o S, he produc Y Y 2 belong o S and L(Y Y 2 ) = L(Y ) + L(Y 2 ) + [L(Y ), L(Y 2 )]

8 8 Marke Model of Forward CDS Spread o ha (ii) For every Y S, he proce /Y belong o S and Therefore, L(Y Y 2 ) c = L(Y ) c + L(Y 2 ) c. (8) L(/Y ) = L(Y ) + [Y, /Y ]. L(/Y ) c = L(Y ) c. (9) (iii) For every Y S he coninuou maringale par L(Y ) c of he ochaic logarihm L(Y ) aifie ( L(Y ) c = (0, ] ) c dy = dy c. (20) Y (0, ] Y Conequenly, for any coninuou local maringale M we have ha [L(Y ) c, M] = d[y c, M]. (2) Y (0,] 4 Abrac Semimaringale Seup The goal of hi ecion i o examine a few alernaive abrac emimaringale eup and o eablih ome generic reul ha will be ubequenly ued in Secion 5 o derive he join dynamic of CDS pread under variou e of aumpion. In fac, he reul of hi ecion can alo be ued o deal wih modeling of non-defaulable forward wap rae, bu we do no deal wih hi iue in hi work, ince i wa reaed elewhere (ee, for inance, Galluccio e al. [4], Jamhidian [6] or Rukowki [26]). Le (Ω, G, F, P) be a given filered probabiliy pace. We conider an F-adaped ochaic procee wih he ime parameer [0, T ] for ome fixed T > 0. We poulae ha he PRP hold wih repec o F under P wih he panning (P, F)-maringale M = (M,..., M k ). I i well known (ee Propoiion 3.3) ha he PRP i preerved under an equivalen change of meaure. Specifically, le P be any probabiliy meaure equivalen o P on (Ω, F T ), wih he Radon-Nikodým deniy proce Z = d P, [0, T ]. F Then an arbirary ( P, F)-maringale Ñ can be expreed under P a Ñ = Ñ0 + ξ dψ(m), [0, T ], (0,] for ome proce ξ, where he R k -valued ( P, F)-maringale Ψ(M) := (Ψ(M ),..., Ψ(M k )) aifie, for l =,..., k, Ψ(M l ) = M l d[z, M l ], [0, T ]. (0,] Z The following auxiliary reul will prove o be ueful in he equel. Of coure, he probabiliy meaure P i inroduced in par (i) in Lemma 4. i imply defined by eing P i (A) = E P (Z i T A) for every A F T. Recall ha M c = (M,..., M k ) c = (M,c,..., M k,c ). Lemma 4.. (i) Le (Ω, G, F, P) be a filered probabiliy pace and le for any i =,..., n he proce Z i be a poiive (P, F)-maringale wih E P (Z i 0) =. Then, for each i =,..., n, here exi

9 L. Li and M. Rukowki 9 a probabiliy meaure P i equivalen o P on (Ω, F T ) wih he Radon-Nikodým deniy proce given by i = Z i F, [0, T ]. (22) (ii) Le u aume, in addiion, ha he PRP hold wih repec o F under P wih he panning (P, F)-maringale M = (M,..., M k ). Then, for any i =,..., n, he panning (P i, F)-maringale Ψ i (M) = (Ψ i (M ),..., Ψ i (M k )) i given by he following expreion, for every l =,..., k, Ψ i (M l ) = M l d[z i, M l ] (23) or, equivalenly, (0,] Z i Ψ i (M) = M [ L(Z i ) c, M c] The equaliy Ψ i (M) c = M c hold for every i =,..., n. Z i 0< Z M i. (24) Proof. The fir par of he lemma i raher clear. The econd par in Lemma 4. follow from Propoiion 3. wih P = P i excep for equaliy (24). To derive (24) from (23), we ar by recalling ha [Z i, M l ] = [Z i,c, M l,c ] + Z M i. l 0< By combining hi equaliy wih (23), we obain Ψ i (M l ) = M l d[z i,c, M l,c ] (0,] Z i Formula (2) yield, for every l =,..., k, [ L(Z i ) c, M l,c] = (0,] Z i Z i 0< d[z i,c, M l,c ], Z M i. l and hu (24) follow. The equaliy Ψ i (M) c = M c i an immediae conequence of (24). In wha follow, we will wrie [M c ] and o denoe he marix of quadraic variaion procee for he R k -valued proce M c, ha i, [ M,c, M,c] [... M,c, M k,c] [M c ] =.... [.. (25) M k,c, M,c] [... M k,c, M k,c] 4. Seup (A) Throughou hi ubecion, we will work under he following anding aumpion, referred o a Seup (A) in he equel. Le (Ω, G, F, P) be a given filered probabiliy pace. Aumpion 4.. We poulae ha: (i) he proce X = (X,..., X n ) i F-adaped, (ii) he procee + a j X j, j =,..., n belong o S, where a,..., a n are non-zero conan, (iii) for every i = 0,..., n, he proce Z X,i, which i given by he formula (by he uual convenion, Z X,n = ) Z X,i = n ( + a j X j ), (26) i a poiive (P, F)-maringale, (iv) he PRP hold under (P, F) wih he panning (P, F)-maringale M = (M,..., M k ).

10 0 Marke Model of Forward CDS Spread The following lemma reveal an inereing feaure of Seup (A). Lemma 4.2. For every i =,..., n, he proce Z X,i X i i a (P, F)-maringale. Proof. We fir oberve ha Z X,n = + a n X n where a n i a non-zero conan. Hence X n i a (P, F)-maringale and, ince Z X,n =, hi alo mean ha Z X,n X n i a (P, F)-maringale. Noe alo ha (26) yield, for i =,..., n 2, Z X,i = ( + a i+ X i+ )Z X,i+ = Z X,i+ + a i+ X i+ Z X,i+, where, by aumpion, he procee Z X,i and Z X,i+ are (P, F)-maringale. Since a i+ i a non-zero conan, we conclude ha he proce Z X,i+ X i+ i a (P, F)-maringale a well. Le he probabiliy meaure P i, i =,..., n be defined a in Lemma 4., wih he Radon- Nikodým deniy procee Z i, i =,..., n given by Z i = c i Z X,i = c i n ( + a j X j ), (27) where he poiive conan c,..., c n are choen in uch a way ha E P (Z i 0) = for i =,..., n. Noe ha, in paricular, we have ha P n = P ince Z n =. Lemma 4.3. For every i =,..., n, he proce X i i a (P i, F)-maringale and i admi he following repreenaion under P i X i = X0 i + ξ i dψ i (M) = X0 i + (0,] (0,] l= ξ i,l dψ i (M l ), (28) where ξ i = (ξ i,,..., ξ i,k ) i an R k -valued, F-predicable proce and he (P i, F)-maringale Ψ i (M) i given by (24), o ha he equaliy Ψ i (M) c = M c hold for every i =,..., n. Proof. Le u fix i =,..., n. I follow eaily from Lemma 4.2 ha he proce X i i a (P i, F)- maringale. Furhermore, in view of par (ii) in Lemma 4., he PRP hold wih repec o F under P i wih he panning (P i, F)-maringale Ψ i (M). Thi immediaely implie he aed propery. Remark 4.. I i worh reing ha par (iv) in Aumpion 4.2 i only ued o deduce he exience of procee ξ,..., ξ n uch ha repreenaion (28) hold for procee X,..., X n. In Secion 5, when dealing wih he conrucion of marke model, we will no aume ha he PRP hold, and we will poulae inead ha for a given family of forward CDS pread he counerpar of repreenaion (28) i valid for ome volailiy procee, which can be choen arbirarily. A imilar remark applie o oher abrac eup conidered in he foregoing ubecion. The following reul underpin, on he one hand, he marke model of forward LIBOR (ee, for inance, Jamhidian [7]) and, on he oher hand, he marke model of one-period forward CDS pread under he aumpion ha inere rae are deerminiic (ee Secion 5. below). In he proof of Propoiion 4., par (iv) in Aumpion 4.2 i only ued indirecly hrough formula (28). Propoiion 4.. For every i =,..., n, he emimaringale decompoiion of he (P i, F)-maringale Ψ i (M) under he probabiliy meaure P n = P i given by Ψ i (M) = M (0,] a j ξ j d [M c ] + a j X j Z i 0< Z M i. (29)

11 L. Li and M. Rukowki More explicily, for every i =,..., n and l =,..., k, Ψ i (M l ) = M l a j (0,] + a j X j Z M i. l Z i 0< m= ξ j,m d[m l,c, M m,c ] (30) For every i =,..., n, he dynamic of he proce X i wih repec o he (P, F)-maringale M are dx i = l= ξ i,l Z i Z i dm l l= ξ i,l M l. a j + a j X j l,m= ξ i,l ξ j,m d[m l,c, M m,c ] (3) Proof. We will apply par (ii) of Lemma 4.. We ar by noing ha (7), (8), (20) and (26) yield ( L(Z i ) c = L( + a j X j ) c a j dx j ) c a j dx j,c = (0, ] + a j X j = (0,] + a j X j, where we alo ued par (ii) in Aumpion 4.. In view of (24) and (28), we have ha a j dx j,c a j ξ j (0,] + a j X j = (0,] + a j X j dψ j (M) c a j ξ j = (0,] + a j X j dm c, where he econd equaliy follow from he fac ha he equaliy Ψ j (M) c = M c hold for every j =,..., n (ee Lemma 4.). Conequenly, [ [ L(Z i ) c, M c] n = ] a j ξ j (0, ] + a j X j dm c, M c a j ξ j d[m c ] = (0,] + a j X j. Combining he la formula wih formula (24) in Lemma 4., we obain he deired equaliy (29). Formula (30) i an immediae conequence of (29). Finally, in order o derive he dynamic of X i under P, i uffice o combine (28) wih (30). 4.2 Seup (B) In hi ubecion, we work under he following exenion of Aumpion 4., which i herafer ermed Seup (B). We aume ha we are given wo independen filraion, denoed a F and F 2, and we e F = F F 2, meaning ha for every we have ha F = σ(f, F 2 ). We conider he R n -valued procee X and Y given on he filered probabiliy pace (Ω, G, F, P). Aumpion 4.2. We poulae ha: (i) he procee X = (X,..., X n ) and Y = (Y,..., Y n ) are adaped o he filraion F and F 2, repecively, (ii) he procee + a j X j, j =,..., n and + b j Y j, j =,..., n belong o S, where a,..., a n and b,..., b n are non-zero conan, (iii) for every i = 0,..., n, he procee Z X,i and Z Y,i, which are given by (by convenion, Z X,n = Z Y,n = ) Z X,i = Z Y,i = n n ( + a j X j ), ( + b j Y j ),

12 2 Marke Model of Forward CDS Spread are poiive (P, F)-maringale, (iv) he PRP hold under (P, F) wih he panning (P, F)-maringale M = (M,..., M k ). The following auxiliary lemma i raher andard and hu i proof i omied. Lemma 4.4. Aume ha F and F 2 are independen filraion and le F = F F 2. (i) Le X and Y be real-valued ochaic procee adaped o he filraion F and F 2, repecively. Then we have ha, for every 0, E P (X Y F ) = E P (X F ) E P (Y F ) = E P (X F ) E P (Y F 2 ). (ii) Le M and N be real-valued maringale wih repec o he filraion F and F 2, repecively. Then he produc MN i a (P, F)-maringale. Remark 4.2. I i enough o aume in condiion (iii) in Aumpion 4.2 ha he proce Z X,i (Z Y,i, repecively) i a (P, F )-maringale (a (P, F 2 )-maringale, repecively). Indeed, if he proce Z X,i i a (P, F )-maringale hen i alo follow a (P, F)-maringale, by he aumed independence of filraion F and F 2. Of coure, he ymmeric argumen can be applied o he proce Z Y,i. We are in a poiion o prove he following generalizaion of Lemma 4.2. Lemma 4.5. Under Aumpion 4.2 he following properie hold, for every i =,..., n, (i) he procee Z Y,i Z X,i X i and Z Y,i Z X,i Y i are (P, F)-maringale, (ii) le he proce Z X,Y,i be defined by he formula Z X,Y,i := c i Z X,i Z Y,i = c i n where c i i a poiive conan; hen Z X,Y,i i a poiive (P, F)-maringale. ( + a j X j )( + b j Y j ), (32) Proof. To prove par (i), we fir argue, a in he proof of Lemma 4.2, ha he procee Z X,i X i and Z Y,i Y i are (P, F)-maringale. Since Z X,i X i (Z Y,i Y i, repecively) i F -adaped (F 2 -adaped, repecively) we conclude ha Z X,i X i (Z Y,i Y i, repecively) i alo a (P, F )-maringale ((P, F 2 )- maringale, repecively). We alo oberve ha he proce Z Y,i i a (P, F 2 )-maringale and hu, in view of par (ii) in Lemma 4.4, we conclude ha he produc Z Y,i Z X,i X i i a (P, F)-maringale. By he ame oken, he produc Z X,i Z Y,i Y i i a (P, F)-maringale. The proof of he econd aemen i baed on imilar argumen and hu i i omied. In he equel, i i aumed ha he conan c i are choen in uch a way ha E P (Z X,Y,i 0 ) = for every i =,..., n. Le he probabiliy meaure P i, i =,..., n be defined a in par (i) in Lemma 4. wih he Radon-Nikodým deniy procee Z i = Z X,Y,i. In paricular, we have ha P n = P ince, by he uual convenion, Z X,Y,n =. The following reul i a ligh exenion of par (ii) in Lemma 4.. Lemma 4.6. For every i =,..., n, he procee X i and Y i are (P i, F)-maringale and hey have he following inegral repreenaion under P i X i = X0 i + ξ i dψ i (M), (33) and Y i (0,] = Y0 i + ζ i dψ i (M), (34) (0,] where ξ i = (ξ i,,..., ξ i,k ) and ζ i = (ζ i,,..., ζ i,k ) are R k -valued, F-predicable procee and he R k -valued (P i, F)-maringale Ψ i (M) i given by. Ψ i (M) = M [ L(Z X,Y,i ) c, M c] The equaliy Ψ i (M) c = M c hold for every i =,..., n. 0< Z X,Y,i Z X,Y,i M. (35)

13 L. Li and M. Rukowki 3 Proof. In view of Lemma 4.5, he procee X i and Y i are manifely (P i, F)-maringale. Moreover, by virue of par (ii) in Lemma 4., he PRP hold wih repec o F under P i wih he panning (P i, F)-maringale Ψ i (M). Thi in urn implie ha equaliie (33) and (34) hold for ome uiably inegrable procee ξ i and ζ i. Equaliy (35) follow immediaely from (24). The la aemen i a conequence of (35). Propoiion 4.2. Suppoe ha Aumpion 4.2 are aified. Then for every i =,..., n he emimaringale decompoiion of he (P i, F)-maringale Ψ i (M) under he probabiliy meaure P n = P i given by he following expreion Ψ i a j ξ j d[m c ] b j ζ j d[m c ] (M) = M (0,] + a j X j (0,] + b j Y j (36) Z X,Y,i M. 0< Z X,Y,i The dynamic of X i and Y i wih repec o he (P, F)-maringale M are given by and dx i = dy i = l= l= ξ i,l ζ i,l dm l b j + b j Y j dm l b j + b j Y j a j + a j X j l,m= l,m= ξ i,l ξ j,m d[m l,c, M m,c ] ξ i,l ζ j,m d[m l,c, M m,c ] a j + a j X j l,m= l,m= ζ i,l Z X,Y,i ξ j,m d[m l,c, M m,c ] ζ i,l ζ j,m d[m l,c, M m,c ] Z X,Y,i Z X,Y,i Z X,Y,i l= l= ξ i,l M l ζ i,l M l. Proof. The proof i analogou o he proof of Propoiion 4.. We ar by noing ha L(Z X,Y,i ) c = L( + a j X j ) c + L( + b j Y j ) c = (0,] a j dx j,c + a j X j + (0,] b j dy j,c + b j Y j. Recall ha Ψ j (M) c = M c for every j =,..., n. In view of (33) and (34), by imilar compuaion a in he proof of Propoiion 4., we hu obain [ [ L(Z X,Y,i ) c, M c] n = ] a j ξ j (0, ] + a j X j dm c, M c [ ] b j ζ j + (0, ] + b j Y j dm c, M c and hi in urn yield [ L(Z X,Y,i ) c, M c] = n (0,] a j ξ j d[m c ] + a j X j + (0,] b j ζ j d[m c ] + b j Y j. (37) By combining (35) wih (37), we obain (36). To eablih he dynamic of procee X i and Y i under P, i uffice ubiue (36) ino (33) and (34), repecively.

14 4 Marke Model of Forward CDS Spread 4.3 Seup (C) Throughou hi ubecion, we will work under he following e of aumpion, which are aified by he procee X and Y defined on a filered probabiliy pace (Ω, G, F, P). Aumpion 4.3. We poulae ha: (i) he procee X = (X,..., X n ) and Y = (Y 2,..., Y n ) are F-adaped, (ii) for every i =,..., n, he proce Z X,i, which i given by he formula (by he uual convenion, Z X,n = ) Z X,i = c i n Y j X j X j Y j, (38) i a poiive (P, F)-maringale and c,..., c n are conan uch ha E P (Z X,i 0 ) =, (iii) for every i = 2,..., n, he proce Z Y,i, which i given by he formula Z Y,i = c i ( Z X,i + Z X,i ) X = i X i ci X i Y i Z X,i, (39) i a poiive (P, F)-maringale and c 2,..., c n are conan uch ha E P (Z Y,i 0 ) =, (iv) for every i =,..., n, he proce X i i a (P i, F)-maringale, where he Radon-Nikodým deniy of P i wih repec o P equal Z X,i, (v) for every i = 2,..., n, he proce Y i i a ( P i, F)-maringale, where he Radon-Nikodým deniy of P i wih repec o P equal Z Y,i, (vi) he PRP hold under (P, F) wih he panning (P, F)-maringale M = (M,..., M k ). When dealing wih Seup (A) and (B), we explicily aed he aumpion ha cerain procee are in he pace S, o ha heir ochaic logarihm are well defined. In he cae of Seup (C) and (D), we make inead a general aumpion ha whenever he noion of he ochaic logarihm i employed, he underlying ochaic proce i aumed o belong o he pace S. Noe alo ha he probabiliy meaure P i and P i are defined a in par (i) in Lemma 4., wih he Radon-Nikodým deniy procee Z X,i and Z Y,i, repecively. In paricular, we have ha P n = P ince Z X,n =. Lemma 4.7. For every i =,..., n, he proce X i admi he following repreenaion under P i X i = X0 i + ξ i dψ i (M), (40) (0,] where ξ i = (ξ i,,..., ξ i,k ) i an R k -valued, F-predicable proce and he R k -valued (P i, F)-maringale Ψ i (M) i given by Ψ i (M) = M [ L(Z X,i ) c, M c] 0< Z X,i Z X,i M. (4) For every i = 2,..., n, he proce Y i ha he following repreenaion under P i Y i = Y0 i + ζ i d Ψ i (M), (42) (0,] where ζ i = (ζ i,,..., ζ i,k ) i an R k -valued, F-predicable proce and he R k -valued ( P i, F)-maringale Ψ i (M) i given by Ψ i (M) = M [ L(Z Y,i ) c, M c] 0< Z Y,i Z Y,i M. (43) The equaliie Ψ i (M) c = M c, i =,..., n and Ψ i (M) c = M c, i = 2,..., n are valid.

15 L. Li and M. Rukowki 5 Proof. Le u fix i =,..., n. By par (iv) and (vi) in Aumpion 4.3, he proce X i i a (P i, F)- maringale. In view of par (ii) in Lemma 4., he PRP hold wih repec o F under P i wih he panning (P i, F)-maringale Ψ i (M). Thi yield he inegral repreenaion of X i under P i. The proof of he econd aemen i imilar, wih par (v) and (vi) in Aumpion 4.3 ued. The following reul will be employed in he join modeling of forward LIBOR and forward CDS pread. Propoiion 4.3. For every i =,..., n, he emimaringale decompoiion of he (P i, F)-maringale Ψ i (M) under he probabiliy meaure P n = P i given by [ Ψ i (ζ j ξ (M) = M ) j d[m c ] ] (ξ j ζ j (0,] Y j X j ) d[m c ] (0,] X j Y j 0< Z X,i Z X,i M. For every i = 2,..., n, he emimaringale decompoiion of he ( P i, F)-maringale Ψ i (M) under he probabiliy meaure P n = P i given by Ψ i (ξ i ξ (M) = M ) i d[m c ] (ξ i ζ (0,] X i + ) i d[m c ] X i (0,] X i Y i [ (ζ j ξ ) j d[m c ] ] (ξ j ζ j (0,] Y j X j ) d[m c ] (0,] X j Y j 0< Z Y,i Z Y,i M. Proof. Uing (8), (20) and (38), we obain L(Z X,i ) c = = [ L(Y j X j ) c L(X j Y j ) c ] [ (0,] dy j,c dx j,c Y j X j (0,] dx j,c X j dy j,c Y j ]. Conequenly, [ L(Z X,i ) c, M c] = n (0,] (0,] d[y j,c, M c ] d[x j,c, M c ] Y j X j d[x j,c, M c ] d[y j,c, M c ] X j Y j. Uing equaion (40) and (42) and he equaliie Ψ j (M) c = Ψ j (M) c = M c, which were eablihed in Lemma 4.7, we conclude ha [ n [ L(Z X,i ) c, M c] = (ζ j ξ) j d[m c ] ] (ξ j ζ j (0,] Y j X j ) d[m c ] (0,] X j Y j. By combining he formula above wih (4), we obain he deired expreion for he proce Ψ i (M). To eablih he formula for Ψ i (M), we fir noe ha L(Z Y,i ) c = L(X i X i ) c L(X i Y i ) c + L(Z X,i ) c,

16 6 Marke Model of Forward CDS Spread which in urn implie ha [ L(Z Y,i ) c, M c] = [ L(X i X i ) c L(X i Y i ) c, M c] + [ L(Z X,i ) c, M c]. Since he decompoiion of he proce [L(Z X,i ) c, M c ] wa already found above, i uffice o focu on he proce L(X i X i ) c L(X i Y i ) c. In view of (20), we have ha L(X i X i ) c L(X i Y i ) c = (0,] dx i,c X i dx i,c X i (0,] dx i,c By making again ue of inegral repreenaion (40) and (42), we hu obain [ L(X i X i ) c L(X i Y i ) c, M c] = (ξ i ξ) i d[m c ] (0,] X i X i X i (ξ i (0,] dy i,c Y i X i. ζ) i d[m c ]. Y i Afer combining he decompoiion of [ L(Z X,i ) c, M c] and [ L(X i X i ) c L(X i Y i ) c, M c], we conclude ha Ψ i (ξ i ξ (M) = M ) i d[m c ] (ξ i ζ (0,] X i + ) i d[m c ] X i (0,] X i Y i [ (ζ j ξ ) j d[m c ] ] (ξ j ζ j (0,] Y j X j ) d[m c ] (0,] X j Y j which i he deired formula. 0< Z Y,i Z Y,i M, 4.4 Seup (D) The eup conidered in hi ubecion i imilar o Seup (C) and hu i will be preened raher uccincly. We work here under he following anding aumpion. Aumpion 4.4. We poulae ha: (i) he procee X = (X 0,..., X n2 ) and Y = (Y 0,..., Y n ) are F-adaped, (ii) for every i = 0,..., n, he proce Z Y,i, which i given by he formula (by he uual convenion, Z Y,0 = ) Z Y,i = c i i j=0 Y j Y j+ X j X j, (44) i a poiive (P, F)-maringale and c,..., c n are conan uch ha E P (Z Y,i 0 ) =, (iii) for every i = 0,..., n 2, he proce Z X,i, which i given by he formula Z X,i = c i Y i+ Y i Y i+ X i ZY,i, (45) i a poiive (P, F)-maringale and c 0,..., c n2 are conan uch ha E P (Z X,i 0 ) =, (iv) for every i = 0,..., n 2, he proce X i i a (P i, F)-maringale, where he Radon-Nikodým deniy of P i wih repec o P equal Z X,i, (v) for every i = 0,..., n, he proce Y i i a ( P i, F)-maringale, where he Radon-Nikodým deniy of P i wih repec o P equal Z Y,i, (vi) he PRP hold under (P, F) wih he panning (P, F)-maringale M = (M,..., M k ). The probabiliy meaure P i and P i are defined a in par (i) in Lemma 4., wih he Radon- Nikodým deniy procee Z X,i and Z Y,i, repecively. Hence P 0 = P ince Z Y,0 =.

17 L. Li and M. Rukowki 7 Lemma 4.8. (i) For every i = 0,..., n 2, he proce X i admi he following repreenaion under P i X i = X0 i + ξ i dψ i (M), (46) (0,] where ξ i = (ξ i,,..., ξ i,k ) i an R k -valued, F-predicable proce and he R k -valued (P i, F)-maringale Ψ i (M) equal Ψ i (M) = M [ L(Z X,i ) c, M c] 0< Z X,i Z X,i M. (ii) For every i = 0,..., n, he proce Y i ha he following repreenaion under P i Y i = Y0 i + ζ i d Ψ i (M), (47) (0,] where ζ i = (ζ i,,..., ζ i,k ) i an R k -valued, F-predicable proce and he R k -valued ( P i, F)-maringale Ψ i (M) equal Ψ i (M) = M [ L(Z Y,i ) c, M c] 0< Z Y,i Z Y,i M. The equaliie Ψ i (M) c = M c, i = 0,..., n 2 and Ψ i (M) c = M c, i = 0,..., n are valid. Proof. Le u fix i = 0,..., n 2. By par (iv) and (vi) in Aumpion 4.4, he proce X i i a (P i, F)-maringale. In view of par (ii) in Lemma 4., he PRP hold wih repec o F under P i wih he panning (P i, F)-maringale Ψ i (M). Thi prove he fir par of he lemma. The proof of he econd aemen i imilar, wih par (v) and (vi) in Aumpion 4.4 ued. To derive he formulae of Propoiion 4.4, i uffice o make a uiable change of he index and o permue he procee X i and Y j in he proof of Propoiion 4.3. Propoiion 4.4. For every i = 0,..., n 2, he emimaringale decompoiion of he (P i, F)- maringale Ψ i (M) under he probabiliy meaure P 0 = P i given by Ψ i (M) = M (0,] [ i j=0 0< (0,] Z X,i (ζ i+ Y i+ ζ) i d[m c ] Y i (ζ j ξ j ) d[m c ] Y j X j Z X,i M. (ζ i+ + (0,] (ζ j+ (0,] ξ) i d[m c ] X i ] ξ) j d[m c ] Y i+ Y j+ X j For every i = 0,..., n, he emimaringale decompoiion of he ( P i, F)-maringale Ψ i (M) under he probabiliy meaure P 0 = P i given by [ i Ψ i (M) = M 0< j=0 Z Y,i (0,] (ζ j ξ j ) d[m c ] Y j X j Z Y,i M. (ζ j+ (0,] ξ j ) d[m c ] Y j+ X j ]

18 8 Marke Model of Forward CDS Spread 5 Marke Model for Forward CDS Spread We are in he poiion o apply he abrac eup developed in Secion 4 o he modeling of finie familie of forward CDS pread in variou configuraion. A menioned in he inroducion, we will fir conider he imple cae of a deerminiic inere rae. Subequenly, we will proceed o he udy of more general, and hu more pracically relevan, marke model. The aim in all model i o derive he join dynamic of a given family of forward CDS pread under a ingle meaure by fir finding expreion for all raio of defaulable annuiie (annuiy deflaed wap numéraire) in erm of forward CDS pread from a predeermined family. In hi ecion, i will be aumed hroughou ha he σ-field F 0 i rivial. Alhough hi aumpion i no neceary for our furher developmen, i i commonly ued in financial modeling, ince i fi well he inerpreaion of he σ-field F 0 a repreening he (deerminiic) marke daa available a ime Model of One-Period CDS Spread Following he paper by Brigo [9, 0] and Schlögl [29], we will fir examine he iue of properie and conrucion of a marke model for a family of one-period forward CDS pread under he aumpion ha he inere rae i deerminiic. I i worh noing ha he iuaion conidered in hi ubecion i formally imilar o he modeling of a family of dicree-enor forward LIBOR (ee, for inance, Brace [5], Jamhidian [6, 7] and Muiela and Rukowki [23, 24]). However, we need alo o examine here he iue of exience of a defaul ime conien wih a poulaed dynamic of forward CDS pread. Needle o ay ha hi iue doe no arie in he problem of modeling defaul-free marke inere rae, uch a: forward LIBOR or, more generally, forward wap rae. 5.. Boom-Up Approach By he boom-up approach we mean a modeling approach in which one ar wih ome credi rik model (where he defaul ime τ and he aociaed urvival proce G are defined) a well a a erm rucure of inere rae. All oher quaniie of inere are ubequenly compued from hee fundamenal, in paricular, he join dynamic of a family of forward CDS pread. In hi ene, we are building up aring from he boom. We refer o Secion 2 for he decripion of a generic forward CDS and he definiion of he aociaed fair forward CDS pread. Recall ha we conider a enor rucure T = {T 0 < T < < T n } wih T 0 0 and we wrie a i = T i T i. We ar by noing ha, uing formulae (9) and (0), one can derive he following expreion for he pre-defaul forward CDS pread κ i := κ i,i correponding o he one-period forward CDS aring a T i and mauring a T i + â i κ i = E Q(β(, T i ) {τ>ti} F ) E Q (β(, T i ) {τ>ti} F ), [0, T i], (48) where we denoe â i = a i /( δ i ) (recall ha δ i i a non-random recovery rae). In addiion o he family of one-period forward CDS pread κ,..., κ n, we alo conider in Subecion 5. and 5.2 he family of forward LIBOR L,..., L n, which are defined by he formula + a i L i = B(, T i) B(, T i ), (49) where for every T > 0 we denoe by B(, T ) he price a ime of he defaul-free uni zero-coupon bond mauring a ime T. Throughou Subecion 5., we work under he anding aumpion ha he inere rae i deerminiic, o ha + â i κ i = Q(τ > T i F ) Q(τ > T i F ) [0, T i ], (50) I hen follow immediaely from (3) and (50) ha he proce κ i i poiive, for i =,..., n.

19 L. Li and M. Rukowki 9 We oberve ha he righ-hand ide in (50) can be repreened a follow, for every i =,..., n and [0, T i ], + â i κ i = a iβ(, T i ) à i2,i a i β(, T i ) à i,i = a i a i + a i L i à i2,i à i,i. (5) Formula (5) how ha + â i κ i i equal, up o a deerminiic funcion, o he raio of defaulable annuiie correponding o he forward CDS pread κ i and κ i. Since he inere rae are now aumed o be deerminiic, we find i eaier o work direcly wih (50), however. Le u menion ha he cae of random inere rae i examined in Subecion 5.2. Le u define he probabiliy meaure P equivalen o Q on (Ω, F Tn ) by eing, for every [0, T n ], η = = Q(τ > T n F ). dq F Q(τ > T n ) Lemma 5.. For every i = 0,..., n, he proce Z κ,i, which i given by he formula n Z κ,i ( = + âj κ j ), [0, Ti ], i a (P, F)-maringale where, by he uual convenion, we e Proof. I uffice oberve ha from (50) we obain Z κ,n =. Z κ,i = Q(τ > T i F ) Q(τ > T n F ). (52) I i now eaily een ha he proce Z κ,i η i a (P, F)-maringale, which in urn implie ha he aered propery i valid. In he nex ep, for any i =,..., n, we define he probabiliy meaure P i equivalen o P by eing i n = Z κ,i ( := c i Zκ,i + âj κ j ), (53) F where c i = Q(τ>T n) Q(τ>T) i he normalizing conan and he ymbol mean ha he equaliy hold up o a normalizing conan. Noe ha he equaliy Z κ,n = hold and hu P n = P. Lemma 5.2. For any fixed i =,..., n, he one-period forward CDS pread κ i i a poiive (P i, F)- maringale. Proof. I i enough o how ha + â i κ i i a (P i, F)-maringale. To hi end, i uffice o combine formulae (50) and (52). The relaionhip beween variou maringale meaure for he family of one-period forward CDS pread are ummarized in he following diagram where Q n dq P n n dq n n P n n2 n... = Q(τ > T n F ), F Q(τ > T n ) 2 3 P 2 i i F = ĉ i ( + â i κ i ) Q(τ > T i F ) Q(τ > T i F ), 2 P where ĉ i = Q(τ>Ti) Q(τ>T i ) i he normalizing conan.

20 20 Marke Model of Forward CDS Spread Le u aume, in addiion, ha he PRP hold under P = P n wih he panning (P, F)-maringale M. Then hi propery i alo valid wih repec o he filraion F under any probabiliy meaure P i for i =,..., n wih he panning (P i, F)-maringale Ψ i (M). Therefore, for every i =,..., n, he poiive (P i, F)-maringale κ i admi he following inegral repreenaion under P i κ i = κ i 0 + κ i σ i dψ i (M), (54) (0,] where σ i i an R k -valued, F-predicable proce. For he ake of convenience, we will henceforh refer o σ i a he volailiy of he forward CDS pread κ i. To um up, we have ju hown he exience of he volailiy procee for one-period forward CDS pread. I will be now no difficul o derive he join dynamic of hee procee under a ingle probabiliy meaure, for inance, he maringale meaure P = P n. However, ince he join dynamic of he one-period CDS pread are he ame a in he op-down approach preened in he foregoing ubecion, we do no provide he explici formula here and we refer inead he reader o equaion (60) Top-Down Approach A wa explained above, in he andard boom-up approach o modeling of CDS pread, we ar by chooing a erm rucure model and we combine i wih he pecificaion of he defaul ime. Togeher wih he aumpion ha he PRP hold, we can ubequenly derive he join dynamic and how he exience of a family of volailiy procee. By conra, in he op-down approach o a marke model, we ake a inpu a family of volailiy procee and a family of driving maringale. We hen derive he join dynamic of a given family of forward CDS pread under a ingle probabiliy meaure and how ha hey are uniquely pecified by he choice of he volailiy procee and driving maringale. Thu, inead of doing compuaion from a given credi and erm rucure model, one can direcly pecify he join dynamic of a family of CDS pread hrough he choice of he volailiy procee and driving maringale. Thi juifie he name op-down approach. In principle, i i alo poible o conruc he correponding defaul ime, bu hi iue i no deal wih in hi paper. In hi ubecion, we will how how o apply he op-down approach in order o conruc he model of one-period forward CDS pread under he aumpion ha he inere rae i deerminiic. To achieve hi goal, we will work under he following anding aumpion, which are moivaed by he analyi of he boom-up approach. Aumpion 5.. We are given a filered probabiliy pace (Ω, G, F, P) and we poulae ha: (i) he iniial value of procee κ,..., κ n are given, (ii) for every i =,..., n, he proce Z κ,i, which i given by he formula (by he uual convenion, Z κ,n = ) Z κ,i = c i n ( + â j κ j ), (55) i a poiive (P, F)-maringale, where c i i a conan uch ha Z κ,i 0 =, (iii) for every i =,..., n, he proce κ i i a (P i, F)-maringale, where he Radon-Nikodým deniy of P i wih repec o P equal Z κ,i, o ha, in paricular, P n = P, (iv) for every i =,..., n, he proce κ i ha he following repreenaion under (P i, F) κ i = κ i 0 + κ i σ i dñ, i (56) (0,] where Ñ i i an R k i -valued (P i, F)-maringale and σ i i an R k i -valued, F-predicable, Ñ i -inegrable proce.

21 L. Li and M. Rukowki 2 In view of par (iii) in Aumpion 5., he following definiion i raher obviou. Definiion 5.. The probabiliy meaure P i i called he maringale meaure for he one-period forward CDS pread κ i. We hould make ome imporan commen here. According o he op-down approach, he procee κ,..., κ n are in fac no given in advance, bu hould be conruced. Alo, he defaul ime i no defined, o he forward CDS pread canno be compued uing formulae (9), (0) and (). Therefore, he volailiy procee σ,..., σ n appearing in formula (56), a well a he driving maringale Ñ,..., Ñ n, can be choen arbirarily by he modeler. Pu anoher way, he volailiy procee and driving maringale hould now be een a model inpu. A an oupu in he op-down conrucion of a marke model, we obain he join dynamic of he forward CDS pread κ,..., κ n under a common probabiliy meaure. The form of he join dynamic obained below hould be een a neceary condiion for he arbirage-free propery of he model. Unforunaely, ufficien condiion for he arbirage-free propery are more difficul o handle. Remark 5.. I i worh menioning ha aumpion (56) can be replaced by he following one κ i = κ i 0 + σ i dñ, i (0,] which, in fac, i more general han (56). All foregoing reul can be eaily adjued o hi alernaive poulae. Similar commen apply o oher varian of marke model of forward CDS pread (and forward LIBOR), which are examined in he foregoing ubecion, o hey will no be repeaed in wha follow. Lemma 5.3. Le k = k + + k n. There exi an R k -valued (P, F)-maringale M uch ha he proce κ i admi he following repreenaion under he maringale meaure P i κ i = κ i 0 + κ i σ i dψ i (M), (57) (0,] where σ i i an R k -valued, F-predicable proce and he proce Ψ i (M) i given by (24). Proof. Uing he bijeciviy of he Giranov ranform, for every i =,..., n one can eablih he exience of an R k i -valued (P, F)-maringale N i uch ha Ñ i = Ψ i (N i ) = Ψ Z κ,i(n i ). Hence (56) yield κ i = κ i 0 + κ i σ i dψ i (N i ). (0,] To complee he proof, i uffice o define he R k -valued (P, F)-maringale M by eing M = (N,..., N n ) and o formally exend each proce R k i -valued, F-predicable proce σ i o he correponding R k -valued, F-predicable proce σ i by eing o zero all irrelevan componen of σ i. I i worh noing ha Ñ n = N n ince Z κ,n =. Conequenly, for i = n equaion (57) become κ n = κ n 0 + κ n σ n dm. (0,] Of coure, hi i conien wih he inerpreaion of P n a he maringale meaure for κ n. By conra, for i =,..., n i i expeced ha he dynamic (57) of κ i under P n will alo have a non-zero drif erm, which i due o appear, ince he proce Ñ i i no a (P n, F)-maringale, in general. The main reul of hi ubecion, Propoiion 5., furnihe an explici expreion for he drif erm in he dynamic (57) of κ i under P n.

22 22 Marke Model of Forward CDS Spread Remark 5.2. In pracice, for inance, when he procee Ñ i are Brownian moion under he repecive maringale meaure P i, he acual dimenion of he underlying driving (P, F)-maringale can ypically be choen o be le han k + + k n. For more deail on explici conrucion of marke model for forward CDS pread driven by correlaed Brownian moion, we refer o paper by Brigo [0] and Rukowki [27], where he common volailiie-correlaion approach i preened. In view of Lemma 5.3, we may reformulae Aumpion 5. in he impler, and more pracically appealing, manner. Aumpion 5.2. We are given a filered probabiliy pace (Ω, G, F, P) and we poulae ha: (i) he iniial value of procee κ,..., κ n are given, (ii) for every i =,..., n, he proce Z κ,i, which i given by he formula (by he uual convenion, Z κ,n = ) Z κ,i = c i n ( + â j κ j ), (58) i a poiive (P, F)-maringale, where c i i a conan uch ha Z κ,i 0 =, (iii) for every i =,..., n, he proce κ i ha he following repreenaion under (P, F) κ i = κ i 0 + κ i σ i dψ i (M), (59) (0,] where M i an R k -valued (P, F)-maringale, σ i i an R k -valued, F-predicable proce and he proce Ψ i (M) i given by formula (24) wih Z i = Z κ,i. By comparing Aumpion 5.2 wih Aumpion 4. and Lemma 4.3, we ee ha he join dynamic of κ,..., κ n under P = P n can now be compued by applying reul obained for Seup (A). Of coure, in order o proceed, we need o make an implici aumpion ha all relevan procee belong o he pace S inroduced in Secion 3. The following lemma i a raher raighforward conequence of Propoiion 4.. Recall ha for he R k -valued (P, F)-maringale M we have ha and he marix [M c ] i given by formula (25). M c = (M,..., M k ) c = (M,c,..., M k,c ) Lemma 5.4. Under Aumpion 5.2, for every i =,..., n he emimaringale decompoiion of he (P i, F)-maringale Ψ i (M) under he maringale meaure P n = P i given by Ψ i (M) = M (0,] â j κ j σ j d[m c ] + â j κ j 0< Z κ,i Z κ,i M, where σ j = (σ j,,..., σ j,k ). More explicily, for every i =,..., n and l =,..., k, Ψ i (M l ) = M l (0,] â j κ j + â j κ j m= σ j,m d[m l,c, M m,c ] Z κ,i Z κ,i M. l 0< The following reul, which follow eaily by combining formula (57) wih Lemma 5.4, furnihe an explici expreion for he join dynamic of he family of one-period forward CDS pread under deerminiic inere rae. An imporan concluion from Propoiion 5. i ha he join dynamic of one-period forward CDS pread are uniquely pecified, under Aumpion 5., by he choice of he driving (P i, F)-maringale Ñ,..., Ñ n and he volailiy procee σ,..., σ n or, more pracically, under Aumpion 5.2, by he choice of he driving (P, F)-maringale M under P and he volailiy procee σ,..., σ n.