STOCHASTIC METHODS IN CREDIT RISK MODELLING, VALUATION AND HEDGING

STOCHASTIC METHODS IN CREDIT RISK MODELLING, VALUATION AND HEDGING Tomasz R. Bielecki Deparmen of Mahemaics Norheasern Illinois Universiy, Chicago, USA T-Bielecki@neiu.edu (In collaboraion wih Marek Rukowski) 1. STRUCTURAL METHODOLOGIES: VALUE-OF-THE-FIRM APPROACH 2. REDUCED METHODOLOGIES: INTENSITY-BASED APPROACH 3. MODELLING OF DEPENDENT DEFAULTS AND MIGRATIONS 4. DEFAULTABLE TERM STRUCTURES 2003 CIME-EMS Summer School on Sochasic Mehods in Finance Bressanone, July 6-12, 2003

OUTLINE 1 Model s Inpus 1.1 Term Srucure of Credi Spreads 1.1.1 Credi Classes 1.1.2 Credi Spreads 1.1.3 Spo Maringale Measure P 1.1.4 Zero-Coupon Bonds 1.1.5 Condiional Dynamics of Bonds Prices 1.2 Recovery Schemes 2 Credi Migraion Process 3 Defaulable Term Srucures 3.1 Single Credi Raings Case 3.1.1 Credi Migraions 3.1.2 Maringale Dynamics of a Defaulable ZCB 3.1.3 Risk-Neural Represenaions 3.2 Muliple Credi Raings Case 3.2.1 Credi Migraions 3.2.2 Maringale Dynamics of a Defaulable ZCB 3.2.3 Risk-Neural Represenaions 3.3 Saisical Probabiliy 3.3.1 Marke Prices for Risks 3.3.2 Saisical Defaul Inensiies

4 Defaulable Coupon Bond 5 Credi Derivaives 5.1 Defaul Swap 5.2 Toal Reurn Swap 6 Defaulable Lévy Term Srucures

SELECTED REFERENCES R. Jarrow, D. Lando and S. Turnbull (1997): A Markov model for he erm srucure of credi risk spreads. Review of Financial Sudies 10, 481 523. M. Kijima and K. Komoribayashi (1998): A Markov chain model for valuing credi risk derivaives. Journal of Derivaives 6, Fall, 97 108. D. Lando (2000): Some elemens of raing-based credi risk modeling. In: Advanced Fixed-Income Valuaion Tools, J. Wiley, Chicheser, pp. 193 215. D. Lando (2000): On correlaed defauls in a raing-based model: common sae variables versus simulaneous defauls. Preprin, Universiy of Copenhagen. P. Schönbucher (2000): Credi risk modelling and credi derivaives. Docoral disseraion, Universiy of Bonn,. T.R. Bielecki and M. Rukowski (2000): Defaulable erm srucure: Condiionally Markov approach. Preprin. T.R. Bielecki and M. Rukowski (2000): Muliple raings model of defaulable erm srucure. Mahem. Finance 10. R. Douady and M. Jeanblanc (2002): A raing-based model for credi derivaives, Preprin. E. Eberlein and F. Özkian (2003): The defaulable Lévy erm srucure: raings and resrucuring, Mahem. Finance 13

1. MODEL S INPUTS Sandard inensiy-based approach (as, for insance, in Jarrow and Turnbull (1995) or Jarrow, Lando and Turnbull (1997)) relies on he following assumpions: exisence of he maringale measure Q is posulaed, he relaionship beween he saisical probabiliy P and he risk-neural probabiliy Q derived via calibraion, credi migraions process is modelled as a Markov chain, marke and credi risk are separaed (independen). The HJM-ype model of defaulable erm srucures wih muliple raings was proposed by Bielecki and Rukowski (2000) and Schönbucher (2000). This approach: formulaes sufficien consisency condiions ha ie ogeher credi spreads and recovery raes in order o consruc a riskneural probabiliy Q and he corresponding risk-neural inensiies of credi evens, shows how he saisical probabiliy P and he risk-neural probabiliy Q are conneced via he marke price of ineres rae risk and he marke price of credi risk, combines marke and credi risks.

1.1 Term Srucure of Credi Spreads We are given a filered probabiliy space (Ω, F, P) endowed wih a d-dimensional sandard Brownian moion W. Remark: We may assume ha he filraion F = F W. For any fixed mauriy 0 < T T he price of a zero-coupon Treasury bond equals B(, T ) = exp ( T f(, u) du ), where he defaul-free insananeous forward rae f(, T ) process is subjec o he sandard HJM posulae. (HJM) The dynamics of he insananeous forward rae f(, T ) are given, for T, as he unique srong soluion o f(, T ) = f(0, T ) + 0 α(u, T ) du + 0 σ(u, T ) dw u for some deerminisic funcion f(0, ) : [0, T ] R, and some F-adaped sochasic processes α : A Ω R, σ : A Ω R d, where A = {(u, ) 0 u T }.

1.1.1 Credi Classes Suppose here are K 2 credi raing classes, where he K h class corresponds o he defaul-free bond. For any fixed mauriy 0 < T T, he defaulable insananeous forward rae g i (, T ) corresponds o he raing class i = 1,..., K 1. We assume ha: (HJM i ) The dynamics of he insananeous defaulable forward raes g i (, T ) are given by, for T, g i (, T ) = g i (0, T ) + 0 α i(u, T ) du + 0 σ i(u, T ) dw u for some deerminisic funcions g i (0, ) : [0, T ] R, and some F-adaped sochasic processes α i : A Ω R, σ i : A Ω R d 1.1.2 Credi Spreads We assume ha g K 1 (, T ) > g K 2 (, T ) >... > g 1 (, T ) > f(, T ) for every T. Definiion 1 For every i = 1, 2,..., K 1, he i h forward credi spread equals s i (, T ) = g i (, T ) f(, T ).

1.1.3 Spo Maringale Measure P The following condiion excludes arbirage across defaul-free bonds for all mauriies T T and he savings accoun: (M) There exiss an F-adaped R d -valued process γ such ha E P { exp ( T 0 γ u dw u 1 2 T and, for any mauriy T T, we have 0 γ u 2 du )} = 1 α (, T ) = 1 2 σ (, T ) 2 σ (, T )γ where α (, T ) = T σ (, T ) = T α(, u) du σ(, u) du. Le γ be some process saisfying Condiion (M). Then he probabiliy measure P, given by he formula dp dp = exp ( T 0 γ u dw u 1 2 T 0 γ u 2 du ), P-a.s., is a spo maringale measure for he defaul-free erm srucure. [We ll see ha discoun bonds B(, T ) of all mauriies are maringales under he measure P (afer discouning wih he money marke process B ).]

1.1.4 Zero-Coupon Bonds The price of he T -mauriy defaul-free zero-coupon bond (ZCB) is given by he equaliy B(, T ) := exp ( T f(, u) du ). Formally, he Treasury bond corresponds o credi class K. Condiional value of T -mauriy defaulable ZCB belonging a ime o he credi class i = 1, 2,..., K 1, equals D i (, T ) := exp ( T g i (, u) du ). We consider discouned price processes Z(, T ) = B 1 B(, T ), Z i (, T ) = B 1 D i (, T ), where B is he usual discoun facor (savings accoun) B = exp ( 0 f(u, u) du). Le us define a Brownian moion W under P by seing W = W 0 γ u du, [0, T ].

1.1.5 Condiional Dynamics of Bonds Prices Lemma 1 Under he spo maringale measure P, for any fixed mauriy T T, he discouned price processes Z(, T ) and Z i (, T ) saisfy dz(, T ) = Z(, T )b(, T ) dw, where b(, T ) = σ (, T ), and where dz i (, T ) = Z i (, T )(λ i () d + b i (, T ) dw ) λ i () = a i (, T ) f(, ) + b i (, T )γ and a i (, T ) = g i (, ) α i (, T ) + 1 2 σ i (, T ) 2 b i (, T ) = σ i (, T ). Remark 1 Observe ha usually he process Z i (, T ) does no follow a maringale under he spo maringale measure P. This feaure is relaed o he fac ha i does no represen he (discouned) price of a radable securiy.

1.2 Recovery Schemes Le Y denoe he cash flow a mauriy T and le Z be he recovery process (an F-adaped process). We ake K = 2. FRTV: Fracional Recovery of Treasury Value Fixed recovery a mauriy scheme. We se Z = δb(, T ) and hus Y = 1 {τ>t } + δ 1 {τ T }. FRPV: Fracional Recovery of Par Value Fixed recovery a ime of defaul. We se Z = δ, where δ is a consan. Thus Y = 1 {τ>t } + δb 1 (τ, T ) 1 {τ T }. FRMV: Fracional Recovery of Marke Value The owner of a defaulable ZCB receives a ime of defaul a fracion of he bond s marke value jus prior o defaul. We se Z = δd(, T ), where D(, T ) is he pre-defaul value of he bond. Thus Y = 1 {τ>t } + δd(τ, T )B 1 (τ, T ) 1 {τ T }.

2 CREDIT MIGRATION PROCESS We assume ha he se of raing classes is K = {1,..., K}, where he class K corresponds o defaul. The migraion process C will be consruced as a (nonhomogeneous) condiionally Markov process on K. Moreover, he sae K will be he unique absorbing sae for his process. Le us denoe by F C he σ-field generaed by C up o ime. A process C is condiionally Markov wih respec o he reference filraion F if for arbirary s > and i, j K we have Q ( C +s = i F F C ) = Q (C +s = i F {C = j} ). The probabiliy measure Q is he exended spo maringale measure. The formula above will provide he risk-neural condiional probabiliy ha he defaulable bond is in class i a ime + s, given ha i was in he credi class C a ime. We inroduce he defaul ime τ by seing τ = inf { R + : C = K }. For any dae, we denoe by Ĉ he previous bond s raing.

3 DEFAULTABLE TERM STRUCTURE 3.1 Single Raing Class (K = 2) We assume he FRTV scheme (oher recovery schemes can also be covered, hough). Our firs goal is o derive he equaion ha is saisfied by he risk-neural inensiy of defaul ime. Inensiy of Defaul Time We inroduce he risk-neural defaul inensiy λ 1,2 as a soluion o he no-arbirage equaion (Z 1 (, T ) δz(, T ))λ 1,2 () = Z 1 (, T )λ 1 (). I is ineresing o noice ha for δ = 0 (zero recovery) we have simply λ 1,2 () = λ 1 (), [0, T ]. On he oher hand, if we ake δ > 0 hen he process λ 1,2 is sricly posiive provided ha D(, T ) > δb(, T ), [0, T ]. Recall ha we have assumed ha D(, T ) < B(, T ).

3.1.1 Credi Migraions Since K = 2, he migraion process C lives on wo saes. The sae 1 is he pre-defaul sae, and he sae 2 is he absorbing defaul sae. We may and do assume ha C 0 = 1. We posulae ha he condiional inensiy marix for he process C is given by he formula Λ = λ 1,2 () λ 1,2 () 0 0 For δ = 0, he marix Λ akes he following simple form Λ = The defaul ime τ now equals λ 1 () λ 1 () 0 0.. τ = inf { R + : C = 2 }. I is defined on an enlarged probabiliy space (Ω, F T, Q ) := (Ω ˆΩ, F T ˆF, P Q) where he probabiliy space (ˆΩ, ˆF, Q) is large enough o suppor a uni exponenial random variable, η say. Then τ = inf { R + : 0 λ 1,2(u) du η }.

Hypoheses (H) All processes and filraions may always be exended pas he horizon dae T by consancy. We se H = 1 {τ } and we denoe by H he filraion generaed by he process H: H = σ(h u : u ). In oher words, H is he filraion associaed wih he observaions of he defaul ime. I is clear ha in he presen seup G = F H. I is no difficul o check ha he hypoheses (H.1)-(H.3) hold in he presen conex (cf. Par 2). In he general case of a model wih muliple raings, he filraion H will be generaed by he migraions process C, ha is, we shall se H = σ(c u : u ). Due o he judicious consrucion of he migraion process C, he hypoheses (H.1)-(H.3) remain valid in he case of muliple raings.

3.1.2 Maringale Dynamics of a Defaulable ZCB Thanks o he consisency equaion, he process M 1,2 () := H 0 λ 1,2(u)(1 H u ) du is a maringale under Q relaive o he enlarged filraion G. Recall ha for any [0, T ] we have D(, T ) = exp ( T g(, u) du ) and ha D(, T ) is inerpreed as he pre-defaul value of a T - mauriy defaulable ZCB ha is subjec o he FRTV scheme. In oher words, D(, T ) is undersood as he value of a T - mauriy defaulable ZCB condiioned on he even: he bond has no defauled by he ime. Recall ha and Z 1 (, T ) = B 1 D(, T ) Z(, T ) = B 1 B(, T ).

Auxiliary Process Ẑ(, T ) We inroduce an auxiliary process Ẑ(, T ), [0, T ], Ẑ(, T ) = 1 {τ>} Z 1 (, T ) + δ 1 {τ } Z(, T ). I can be shown ha Ẑ(, T ) saisfies he SDE (A) dẑ(, T ) = Z 1(, T )b 1 (, T ) 1 {τ>} dw + δz(, T )b(, T ) 1 {τ } dw + (δz(, T ) Z 1 (, T )) dm 1,2 (). Noice ha Ẑ(, T ) follows a G-maringale under Q. This leads o consrucion of an arbirage free model of he defaulable erm srucure and o risk-neural represenaion for he price of he defaulable bond. We inroduce he price process hrough he following definiion. Definiion 2 The price process D C (, T ) of a T -mauriy ZCB is given by D C (, T ) = B Ẑ(, T ).

3.1.3 Risk-Neural Represenaions Proposiion 1 saisfies The price D C (, T ) of a defaulable ZCB D C (, T ) = 1 {τ>} D(, T ) + δ 1 {τ } B(, T ). D C (, T ) = 1 {C =1} exp ( T g(, u) du) +δ 1 {C =2} exp ( T f(, u) du). Moreover, he risk-neural valuaion formula holds D C (, T ) = B E Q (δb 1 T 1 {τ T } + B 1 T 1 {τ>t } G ). Furhermore D C (, T ) = B(, T ) E QT (δ 1 {τ T } + 1 {τ>t } G ) where Q T is he T -forward measure associaed wih Q. Special cases: For δ = 0, we obain D C (, T ) = 1 {τ>} D(, T ). For δ = 1, we have, as expeced, D C (, T ) = B(, T ).

Defaul-Risk-Adjused Discoun Facor The defaul-risk-adjused discoun facor equals and we se ˆB = exp ( 0 (r u + λ 1,2 (u)) du) ˆB(, T ) = ˆB E P ( We consider a bond wih FRTV. ˆB 1 T F ). Proposiion 2 We have and hus D C (, T ) = δb(, T ) + (1 δ) 1 {τ>} ˆB(, T ) D C (, T ) = B(, T ) (1 δ) ( B(, T ) 1 {τ>} ˆB(, T ) ). Inerpreaion: A decomposiion of D C (, T ) of he price of a defaulable ZCB ino is prediced pos-defaul value δb(, T ) and he pre-defaul premium D C (, T ) δb(, T ). A decomposiion D C (, T ) as he difference beween is defaul-free value B(, T ) and he expeced loss in value due o he credi risk. From he buyer s perspecive: he price D C (, T ) equals he price of he defaul-free bond minus a compensaion for he credi risk.

3.2 Muliple Credi Raings Case We work under he FRTV scheme. To each credi raing i = 1,..., K 1, we associae he recovery rae δ i [0, 1), where δ i is he fracion of par paid a bond s mauriy, if a bond belonging o he i h class defauls. As we shall see shorly, he noaion Ĉτ indicaes he raing of he bond jus prior o defaul. Thus, he cash flow a mauriy is X = 1 {τ>t } + δĉτ 1 {τ T }. To simplify presenaion we le K = 3 (wo differen credi classes) and we le δ i [0, 1) for i = 1, 2. The resuls carry over o he general case of K 2. 3.2.1 Credi Migraions Risk-neural inensiies of credi migraions λ 1,2 (), λ 1,3 (), λ 2,1 () and λ 2,3 () are specified by he no-arbirage condiion: λ 1,2 ()(Z 2 (, T ) Z 1 (, T )) + λ 1,3 ()(δ 1 Z(, T ) Z 1 (, T )) + λ 1 ()Z 1 (, T ) = 0, λ 2,1 ()(Z 1 (, T ) Ẑ2(, T )) + λ 2,3 ()(δ 2 Z(, T ) Z 2 (, T )) + λ 2 ()Z 2 (, T ) = 0.

If he processes λ 1,2 (), λ 1,3 (), λ 2,1 () and λ 2,3 () are nonnegaive, we consruc a migraion process C, on some enlarged probabiliy space (Ω, G, Q ), wih he condiional inensiy marix Λ() = λ 1,1 () λ 1,2 () λ 1,3 () λ 2,1 () λ 2,2 () λ 2,3 () 0 0 0 where λ i,i () = j i λ i,j () for i = 1, 2. Noice ha he ransiion inensiies λ i,j follow F-adaped sochasic processes. The defaul ime τ is given by he formula τ = inf{ R + : C = 3 }. 3.2.2 Maringale Dynamics of a Defaulable ZCB We se H i () = 1 {C =i} for i = 1, 2, and we le H i,j () represen he number of ransiions from i o j by C over he ime inerval (0, ]. I can be shown ha he process M i,j () := H i,j () 0 λ i,j(s)h i (s) ds, [0, T ], for i = 1, 2 and j i, is a maringale on he enlarged probabiliy space (Ω, G, Q ).

Auxiliary Process Ẑ(, T ) We inroduce he process SDE (A) Ẑ(, T ) as a soluion o he following dẑ(, T ) = (Z 2(, T ) Z 1 (, T )) dm 1,2 () + (Z 1 (, T ) Z 2 (, T )) dm 2,1 () + (δ 1 Z(, T ) Z 1 (, T )) dm 1,3 () + (δ 2 Z(, T ) Z 2 (, T )) dm 2,3 () + H 1 ()Z 1 (, T )b 1 (, T ) dw + H 2 ()Z 2 (, T )b 2 (, T ) dw + (δ 1 H 1,3 () + δ 2 H 2,3 ())Z(, T )b(, T ) dw, wih he iniial condiion Ẑ(0, T ) = H 1 (0)Z 1 (0, T ) + H 2 (0)Z 2 (0, T ). The process Ẑ(, T ) follows a maringale on (Ω, G, Q ), and hus Q is called he exended spo maringale measure. The proof of he nex resul employs he no-arbirage condiion. Lemma 2 For any mauriy T T, we have Ẑ(, T ) = 1 {C 3} Z C (, T ) + 1 {C =3} δĉ Z(, T ) for every [0, T ].

Price of a Defaulable ZCB We inroduce he price process of a T -mauriy defaulable ZCB by seing D C (, T ) = B Ẑ(, T ) for any [0, T ]. In view of Lemma 2, he price of a defaulable ZCB equals D C (, T ) = 1 {C 3} D C (, T ) + 1 {C =3} δĉ B(, T ) wih some iniial condiion C 0 {1, 2}. An analogous formula can be esablished for an arbirary number K of raing classes, namely, D C (, T ) = 1 {C K} D C (, T ) + 1 {C =K} δĉ B(, T ). Properies of D C (, T ): D C (, T ) follows a (Q, G)-maringale, when discouned by he savings accoun. In conras o he condiional price processes D i (, T ), he process D C (, T ) admis disconinuiies, associaed wih changes in credi qualiy. I represens he price process of a radable securiy: he defaulable ZCB of mauriy T.

3.2.3 Risk-Neural Represenaions Recall ha δ i [0, 1) is he recovery rae for a bond which is in he i h raing class prior o defaul. Proposiion 3 defaulable ZCB equals The price process D C (, T ) of a T -mauriy D C (, T ) = 1 {C 3} B(, T ) exp ( T + 1 {C =3} δĉ B(, T ) s C (, u) du) where s i (, u) = g i (, u) f(, u) is he i h credi spread. Proposiion 4 The price process D C (, T ) saisfies he riskneural valuaion formula D C (, T ) = B E Q (δĉt B 1 T 1 {τ T } + B 1 T 1 {τ>t } G ). I is also clear ha D C (, T ) = B(, T ) E QT (δĉt 1 {τ T } + 1 {τ>t } G ) where Q T sands for he T -forward measure associaed wih he exended spo maringale measure Q.

3.3 Saisical Probabiliy We shall now change, using a suiable generalizaion of Girsanov s heorem, he measure Q o he equivalen probabiliy measure Q. In he financial inerpreaion, he probabiliy measure Q will play he role of he saisical probabiliy. I is hus naural o posulae ha he resricion of Q o he original probabiliy space Ω necessarily coincide wih he saisical probabiliy P for he defaul-free marke. Condiion (L): We se dq dq = L T, Q -a.s., where he Q -local posiive maringale L is given by he formula dl = L γ dw + L dm, L 0 = 1, and he Q -local maringale M equals dm = i j = i j κ i,j () dm i,j () for some processes κ i,j > 1. κ i,j () (dh i,j () λ i,j ()H i () d)

3.3.1 Prices for Marke and Credi Risks For any i j we denoe by κ i,j > 1 an arbirary nonnegaive F-predicable process such ha T 0 (κ i,j () + 1)λ i,j () d <, Q -a.s. We assume ha E Q (L T ) = 1, so ha he probabiliy measure Q is well defined on (Ω, G T ). Financial inerpreaions: The process γ corresponds o he marke price of ineres rae risk. Processes κ i,j represen he marke prices of credi risk. Le us define processes λ Q i,j by seing for i j and λ Q i,j() = (κ i,j () + 1)λ i,j () λ Q i,i() = j i λ Q i,j().

3.3.2 Saisical Defaul Inensiies Proposiion 5 Under an equivalen probabiliy Q, given by Condiion (L), he process C is a condiionally Markov process. The marix of condiional inensiies of C under Q equals Λ Q = λ Q 1,1()... λ Q 1,K()..... λ Q K 1,1()... λ Q K 1,K() 0... 0 If he marke price for he credi risk depends only on he curren raing i (and no on he raing j afer jump), so ha. κ i,j = κ i,i =: κ i for every j i hen Λ Q = Φ Λ, where Φ = diag [φ i ()] wih φ i () = κ i ()+1 is he diagonal marix (see, e.g., Jarrow, Lando and Turnbull (1997). Imporan issues: Valuaion of defaulable coupon-bonds. Modelling of correlaed defauls (dependen migraions). Valuaion and hedging of credi derivaives. Calibraion o liquid insrumens.

4 Defaulable Coupon Bond Consider a defaulable coupon bond wih he face value F ha maures a ime T and promises o pay coupons c i a imes T 1 <... < T n < T. The coupon paymens are only made prior o defaul, and he recovery paymen, proporional o he face value, is made a mauriy T. The migraion process C may depend on boh he mauriy T and on recovery raes. We wrie C = C (δ, T ), where δ = (δ 1,..., δ K ), and D C(δ,T ) (, T ) insead of D C (, T ). We consider a coupon bond as a porfolio of: (i) defaulable coupons = defaulable zero-coupon bonds wih zero recovery, (ii) defaulable face value = defaulable zero-coupon bond wih recovery δ. The arbirage price of a defaulable coupon bond hus equals D c (, T ) := n i=1 c id C(0,Ti )(, T i ) + F D C(δ,T ) (, T ) wih he convenion ha D C(0,Ti )(, T i ) = 0 for > T i.

5 Credi Derivaives 5.1 Defaul Swap The coningen paymen is riggered by he even {C = K}. I is seled a ime τ = inf { < T : C = K } and equals Z τ = (1 δĉt B(τ, T )). Noice he dependence of Z τ on he iniial raing C 0 hrough he defaul ime τ and he recovery rae δĉt. Consider wo cases: (i) he buyer pays a lump sum a conrac s incepion (defaul opion), (ii) he buyer pays an annuiy (defaul swap). In case (i), he value a ime of a defaul opion equals S = B E Q (B 1 τ (1 δĉt B(τ, T )) 1 {<τ T } G ). In case (ii), he annuiy κ can be found from S 0 = κ E Q ( T i=1 B 1 i 1 {i <τ}). Boh S 0 and κ depend on he iniial raing C 0.

5.2 Toal Rae of Reurn Swap As a reference asse we ake he coupon bond wih he promised cash flows c i a imes T i. Suppose he conrac mauriy is ˆT T. In addiion, suppose ha he reference rae paymens (he annuiy paymens) are made by he invesor a fixed scheduled imes i ˆT, i = 1, 2,..., m. The owner of a oal rae of reurn swap is eniled no only o all coupon paymens during he life of he conrac, bu also o he change in he value of he underlying bond. By convenion, we assume ha he defaul even occurs when C (δ, T ) = K. According o his convenion, he reference rae κ o be paid by he invesor saisfies E Q n i=1 c ib 1 T i 1 {Ti ˆT } + E Q ( B 1 τ (D c (τ, T ) D c (0, T )) ) where = κ E Q m i=1 B 1 i 1 {Ci (δ,t ) K} τ = inf { 0 : C (δ, T ) = K } ˆT.

6 Defaulable Lévy Term Srucures Eberlein and Özkian (2003) generalize he model by Bielecki and Rukowski (2000) o he case of erm srucures driven by Lévy processes. They assume ha under he measure P he dynamics of insananeous forward raes are df(, T ) = 2 A(, T )d 2 Σ(, T ) dl and dg i (, T ) = 2 A i (, T )d 2 Σ i (, T ) dl (i), where L and L (i) are Lévy processes wih canonical decomposiions L = b + cw + 0 R d px(µ L ν L )(ds, dx), L (i) = b i + c i W + 0 R d p i x(µ L ν L )(ds, dx), and where 2 is he derivaive wih respec o he T variable. In view of he above dynamics, Eberlein and Özkian (2003) appropriaely modify he consrucion by Bielecki and Rukowski (2000). In paricular, he dynamics under he counerpar of measure Q of he discouned processes Z and Z i are dz(, T ) = Z(, T ) ( β(, T )d W + R d ψ(x,, T )( µ ν)(d, dx) ), dz i (, T ) = Z i (, T )( α i (, T ) + β i (, T )d W + R d ψ i (x,, T )( µ ν)(d, dx)). Proposiions 3 and 4 remain valid in he se-up of Eberlein and Özkian (2003).