STOCHASTIC METHODS IN CREDIT RISK MODELLING, VALUATION AND HEDGING

Transcription

1 STOCHASTIC METHODS IN CREDIT RISK MODELLING, VALUATION AND HEDGING Tomasz R. Bielecki Deparmen of Mahemaics Norheasern Illinois Universiy, Chicago, USA (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

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

3 4 Defaulable Coupon Bond 5 Credi Derivaives 5.1 Defaul Swap 5.2 Toal Reurn Swap 6 Defaulable Lévy Term Srucures

4 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, M. Kijima and K. Komoribayashi (1998): A Markov chain model for valuing credi risk derivaives. Journal of Derivaives 6, Fall, D. Lando (2000): Some elemens of raing-based credi risk modeling. In: Advanced Fixed-Income Valuaion Tools, J. Wiley, Chicheser, pp 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

5 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.

6 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 }.

7 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 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 ).

8 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 ).]

9 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 ].

10 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 ) 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.

11 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 }.

12 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.

13 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 ).

14 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 () 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 }.

15 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.

16 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 ).

17 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 ).

18 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 ).

19 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.

20 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 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.

21 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 () 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 } 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 ).

22 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 ].

23 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.

24 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.

25 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)

26 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().

27 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() 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.

28 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.

29 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.

30 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.

31 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).