Credit Risk Modeling with Random Fields

Transcription

1 Credi Risk Modeling wih Random Fields Inaugural-Disseraion zur Erlangung des Dokorgrades an den Naurwissenschaflichen Fachbereichen (Mahemaik der Jusus-Liebig-Universiä Gießen vorgeleg von Thorsen Schmid Gießen 23

2 D - 26 Dekan: Prof. Dr. Albrech Beuelspacher Guacher: Prof. Dr. Winfried Sue (Gießen Prof. Dr. Ludger Overbeck (Gießen Daum der Dispuaion:

3 Preface The demand for invesmens wih higher reurns in areas oher han he sock marke has increased enormously due o he sock marke crash in he las wo years. In exchange for an aracive yield he invesors ake a credi risk, and as a resul mehodologies for pricing and hedging credi derivaives as well as for risk managemen of credi risky asses became very imporan. The effors of he Basel Commiee is jus one of many examples which subsaniae his. In he las years he credi markes developed a a remendous speed while a he same ime he number of corporae defauls increased dramaically. I is herefore no surprising ha he demand for credi derivaives is growing rapidly. In view of his, he goal of his work is wofold. In he firs par, a survey of he credi risk lieraure is given, which offers a quick inroducion ino he area and presens he mahemaical mehods in a unifying way. Second, we propose wo new models of credi risk, focusing on differen needs. The firs model generalizes exising models using random fields in Hilber spaces. The second model uses Gaussian random fields leading o explici formulas for a number of derivaives, for which we propose wo calibraion procedures. This work is organized as follows. In Chaper 1, a survey of he credi risk lieraure is given. This includes srucural models, hazard rae models, mehods incorporaing credi raings, models for baskes of credi risky bonds, hybrid models, marke models and commercial models. In he las secion we illusrae several credi derivaives. Generally he mahemaical framework for he models is provided and some models are discussed in greaer deail. Addiionally, an explici formula for he defaul inensiy in he imperfec informaion model of Duffie and Lando (21 is derived. Chapers 2 and 3 focus on credi risk modeling using sochasic differenial equaions (SDEs in infinie dimensions. Alhough known in ineres rae heory, he applicaion of hese mehods is new o credi risk. Chaper 2 conains an inroducion o SDEs in Hilber spaces providing an Iô formula which is adequae for our purposes. In Chaper 3 a Heah- Jarrow-Moron formulaion of credi risk in infinie dimensions is given. The work of Duffie and Singleon (1999 and Bielecki and Rukowski (2 was enhanced wih alernaive recovery models and exended o infinie dimensions. These new models comprise mos of he known credi risk models and sill offer frameworks which are racable. Recen research in Özkan and Schmid (23 exends his furher o Lévy processes in infinie dimensions. I

4 II In Chaper 4, a credi risk model is presened which uses Gaussian random fields and ransfers he framework of Kennedy (1994 o credi risk. In conras o he funcional analyic approach in he previous wo chapers, he mehods used in his secion concenrae on deriving formulas for pricing and hedging. Explici expressions for he prices of several credi defaul opions are obained and an example for hedging credi derivaives is presened. Based on hese pricing formulas, wo calibraion mehodologies are provided. The firs calibraion procedure fis he model o prices of derivaives using a leas squares approach. As he daa for derivaives like credi defaul swapions is sill scarce, he second approach akes his ino accoun and in addiion uses hisorical daa. This new approach allows o calibrae perfecly o marke prices and is applicable using only a small amoun of credi derivaives daa. I am mos graeful o my supervisor, Prof. Dr. Winfried Sue, for his vial suppor. His fascinaing lecures and his way of inspiring mahemaics were a highly valuable encouragemen. Always having ime for fruiful discussions is jus one example of his coninual suppor hroughou he making of his hesis. I also warmly hank my friends and colleagues from he Sochasik-AG. Special hanks go o Sue, Charlie and Oli for spending hours and hours reading crypic noes. I wish, especially, o hank my family for heir educaion which encouraged he search for answers and helping me whenever I needed hem. Finally, I hank my deares Kirsen as she brighens my life wih her love.

5 Conens 1 Credi Risk - A Survey Inroducion Srucural Models Meron ( Longsaff and Schwarz ( Jump Models - Zhou ( Furher Srucural Models Hazard Rae Models Mahemaical Preliminaries Jarrow and Turnbull ( Duffie and Singleon ( Credi Raings Based Mehods Jarrow, Lando and Turnbull ( Lando ( Baske Models Kijima and Muromachi ( Copula Models Hybrid models Madan and Unal ( Duffie and Lando ( Marke Models wih Credi Risk Commercial Models The KMV Model ( CrediMonior Moody s CrediMerics Credi Derivaives Digial Opions Defaul Opions and Credi Defaul Swap Defaul Swapions Credi Spread Opions kh-o-defaul Opions SDEs on Hilber Spaces Preliminaries The Sochasic Inegral Covariances Iô s formula The Fubini Theorem Girsanov s Theorem III

6 CONTENTS IV 3 An Infinie Facor Model for Credi Risk An Infinie Facor HJM Exension Change of Measure Models wih Credi Risk Recovery of Marke Value Recovery of Treasury Models Using Raings Raing Based Recovery of Marke Value Raing Based Recovery of Treasury Pricing Credi Risk Modeling wih Gaussian Random Fields Preliminaries A Model wihou Credi Risk Models wih Credi Risk Zero Recovery Recovery of Treasury Value Fracional Recovery of Treasury Value Explici Pricing Formulas Defaul Digials Defaul Pu Credi Spread Opions Credi Defaul Swap and Swapion Hedging - an Example Calibraion Calibraion Using Gaussian Random Fields Calibraion Using he Karhunen-Loève Expansion A Basic Seup for Hazard Rae Models 136 B Auxiliary Calculaions 141 B.1 Normal Random Variables B.2 Boundary Crossing Probabiliies B.3 Some Inegrals B.4 Tools for Gaussian Models

7 Chaper 1 Credi Risk - A Survey 1.1 Inroducion The firs regulaions of lending and ineres were menioned in Hammurabi s Code of Laws. Hammurabi was a famous Babylonian king, who lived circa 18 BC. The mos remarkable source for his legal code is a sone slab discovered in 191 which is preserved in he Louvre, Paris. Oher cuneiform ables record a number of exbook-like ineres rae problems. For example, he cuneiform able VAT 8528 poses he following problem 1 : If I len one mina of silver a he rae of 12 shekels (1/6 of a mina per year, and I received in repaymen, one alen (6 minas and 4 minas. For how long was he money len? As long as lending is subjec o a person s employmen, here is risk of losing par of he loan, which in modern financial language would be called credi risk. A common definiion of credi risk is he following: Credi risk refers o he possibiliy ha a conracual counerpar may no be able o mee his obligaions so ha he lender faces a financial loss. The financial objec, which is subjec o credi risk, is a so-called bond. In oday s financial markes here is a vas variey of bonds raded, from Treasuries issued by differen counries or saes o bonds issued by corporaes. Generally speaking, a bond is a cerificae confirming ha is owner, he credior, has len a cerain amoun of money o a specified issuer. The len sum is called he principal or face value of he bond and has o be repaid a a fixed dae, called mauriy of he bond. Addiionally he bond offers a fixed rae of ineres and appears as an example of a fixed-income insrumen. Even if he credior has no kind of ownership righs, i is imporan o noe ha in he even of business liquidaion, bond holders have prioriy over shareholders in erms of abiliy o reclaim capial. 1 See Neugebauer (1969. Furher hisorical informaion on ineres raes in hisory may be found in chaper wo of James and Webber (2. 1

8 1.1 Inroducion 2 The risk of he bond holder o lose a cerain porion of his invesmen is he above menioned credi risk. Accordingly, he crediworhiness of he issuer is an imporan kind of informaion. Agencies like Moody s and Sandard & Poor s classify he crediworhiness of he issuers by he so-called raing. As a consequence, marke paricipans demand higher yields for lower raed bonds as a compensaion for he aken risk. The excess reurn of he corporae bond over a Treasury bond, i.e., a bond which is assumed o be free of credi risk, is called he credi spread; see Bielecki and Rukowski (22. A defaul occurs if he issuer is no able o mee his obligaions. The precise definiion of a defaul is complicaed, because i is iself negoiable; see Tavakoli (1998. Cerainly, an amoun of money is los, and he pos-defaul value of he bond, which is called recovery, significanly differs from he pre-defaul value. For his reason, spread-widening risk or changes in credi qualiy are also implied when alking abou credi risk. The occurrence of credi risk raises he demand for possibiliies o manage hem. This is when credi derivaives come ino play. They enable proecion agains differen ypes of credi risk o he effec ha cerain risk profiles are achieved. For example, credi derivaives can be used, if an invesor wans o hedge himself agains a credi risk, bu no agains ineres risk. As boh are enangled in a bond, credi derivaives provide he ailor-made possibiliy o rade his specific risk. I is imporan o disinguish beween reference risk and counerpary risk. The former refers o a conrac of wo defaul free paries, where he conrac relaes o he credi risk of some reference eniy. If, on he oher hand, over-he-couner derivaives are raded, which are in conras o exchange-raded conracs no backed by a clearinghouse or an exchange, hen each pary faces he defaul risk of is counerpary. We inroduce several classes of models of credi risk, which serve differen needs. Some ry o deermine he magniude of credi risk in a cerain produc while ohers are more suiable for he managemen of whole porfolios or for pricing derivaives. Srucural models dae back o he Nobel Prize paper of Meron (1974. They make a specific assumpion abou he capial srucure of a company, which leads o a precise specificaion when obligaions canno be fulfilled. Therefore, he probabiliy of a defaul can be deermined and furher calculaions done. A commercial implemenaion of his model is presened in Secion Conversely, hazard rae models focus on modeling he ime, a which he defaul even occurs, while he capial srucure of he company is no modeled a all. The defaul even is specified in erms of an exogenous jump process, which iself migh depend on ineres raes, credi raings, firms asses or ohers. Ofen also called reduced-form or inensiy based models, hey were firs menioned in Pye (1974. An imporan class of hazard rae models incorporae credi raings, readily available informaion on he crediworhiness of he bonds issuer. So-called hybrid models ry o combine hese ideas and incorporae boh hazard raes and he capial srucure of he company. From his perspecive hese ineresing models are relaively new in he financial lieraure and a lo of research is going on in his field.

9 1.2 Srucural Models 3 In he secion on baske models we presen wo mehods of modeling a porfolio of credi risky securiies. Baske models are mainly used o value credi derivaives wih a firs-odefaul feaure. Marke models represen he ransfer of a very successful class of ineres rae models o credi risk. They mainly cover he fac, ha yields (or bonds, respecively in he marke are available wih respec o a finie number (less han 2 of mauriy imes, and no for any mauriy as assumed by mos oher models. Quie differen are he commercial models which represen readily available sofware packages. These models show he implemenaion of several mehods handling credi risk and applicaions o large porfolios. Finally we presen cerain credi derivaives in a precise specificaion. These include credi defaul swaps and swapions, credi defaul opions, credi spread opions and opions wih a firs-o defaul feaure, and provide he basis for deriving prices in differen models. 1.2 Srucural Models The firs class of models ries o measure he credi risk of a corporae bond by relaing he firm value of he issuing company o is liabiliies. If he firm value a mauriy T is below a cerain level, he company is no able o pay back he full amoun of money, so ha a defaul even occurs Meron (1974 In his landmark paper Meron (1974 applied he framework of Black and Scholes (1973 o he pricing of a corporae bond. A corporae bond promises he repaymen F a mauriy T. Since he issuing company migh no be able o pay he full amoun of money back, he payoff is subjec o defaul risk. Le V denoe he firm s value a ime. If, a ime T, he firm s value V T is below F, he company is no able o make he promised repaymen so ha a defaul even occurs. In Meron s model i is assumed ha here are no bankrupcy coss and ha he bond holder receives he remaining V T, hus facing a financial loss. If we consider he payoff of he corporae bond in his model, we see ha i is equal o F in he case of no defaul (V T F and V T oherwise, i.e., 1 {VT >F }F + 1 {VT F }V T = F (F V T +. If we spli he single liabiliy ino smaller bonds wih face value 1, hen we can replicae he payoff of his bond by a porfolio of a riskless bond B(, T wih face value 1 (long and 1/F pus wih srike F (shor.

10 1.2 Srucural Models 4 Consequenly he price of he corporae bond a ime, which we denoe by B(, T, equals he price of he replicaing porfolio: B(, T = B(, T 1/F P (F, V,, T, σ V = e r(t 1 ( F e r(t Φ( d 2 V Φ( d 1 F = e r(t Φ(d 2 + V F Φ( d 1, (1.1 where Φ( is he cumulaive disribuion funcion of a sandard normal random variable. Furhermore, P (F, V,, T, σ V denoes he price of a European pu on he underlying V wih srike F, evaluaed a ime, when mauriy is T and he volailiy of he underlying is σ V. This price is calculaed using he Black and Scholes opion pricing formula. The consans d 1 and d 2 are d 1 = ln V F e r(t σ2 (T σ T d 2 = d 1 σ T. If he curren firm value V is far above F he pu is worh almos nohing and he price of he corporae bond equals he price of he riskless bond. If, oherwise, V approaches F he pu becomes more valuable and he price of he corporae bond reduces significanly. This is he premium he buyer receives as a compensaion for he credi risk included in he conrac. Price reducion implies a higher yield for he bond. The excess yield over he risk-free rae is direcly conneced o he crediworhiness of he bond and is called he credi spread. In his model he credi spread a ime equals see Figure 1.1. s(, T = 1 T ln B(, T e r(t ( Φ(d 2 + = 1 T ln V F e Φ( d 1, r(t The quesion of hedging he corporae bond is easily solved in his conex, as hedging formulas for he pu are readily available. To replicae he bond he hedger has o rade he risk-free bond and he firm s share simulaneously 2. This reveals he fac ha in Meron s model he corporae bond is a derivaive on he risk-free bond and he firm s share. We face he following problems wihin his model: The credi spreads for shor mauriy are close o zero if he firm value is far above F. This is in conras o observaions in he credi markes, where hese shor mauriy spreads are no negligible because even close o mauriy he bond holder 2 The hedge consiss primarily of hedging 1 F Scholes Dela-Hedging. pu and is a sraighforward consequence of he Black-

11 1.2 Srucural Models 5 5 x Figure 1.1: This plo shows he credi spread versus ime o mauriy in he range from zero o wo years. The upper line is he price of a bond issued by a company whose firm value equals wice he liabiliies while for he second he liabiliies are hree imes as high. Noe ha if mauriy is below.3 years he credi spreads approach zero. is uncerain wheher he full amoun of money will be paid back or no; cf. Wei and Guo (1991 and Jones, Mason and Rosenfeld (1984. The reason for his are he assumpions of he model, in paricular coninuiy and log-normaliy of he firm value process. On he oher hand, he inrinsic modeling of he defaul even may also be quesionable. In realiy here can be many reasons for a defaul which are no covered by his model. The model is no designed for differen bonds wih differen mauriies. Also i can happen ha no all bonds defaul a he same ime (senioriy. In pracice no all liabiliies of a firm have o be paid back a he same ime. One disinguishes beween shor-erm and long-erm liabiliies. To deermine he criical level where he company migh defaul Vasiček (1984 inroduced he defaul poin as a mixure of he level of ousandings. This concep is discussed in Secion The ineres raes are assumed o be consan. This assumpion is relaxed, for example, by Kim, Ramaswamy and Sundaresan (1993, as discussed in Secion As here are only few parameers which deermine he price of he bond, his model canno be calibraed o all raded bonds on he marke, which reveals arbirage possibiliies. Geske and Johnson (1984 exended he Meron model o coupon-bearing bonds while Shimko, Tejima and van Devener (1993 considered sochasic ineres raes using he ineres rae model proposed in Vasiček (1977. The second exension is essenially equivalen o pricing a European pu opion wih Vasiček ineres raes, where closed-form soluions are available. Of course, any oher ineres rae model can be used in his framework, like Cox, Ingersoll and Ross (1985 or Heah, Jarrow and Moron (1992.

12 1.2 Srucural Models Longsaff and Schwarz (1995 As already menioned defauls in he Meron model are resriced o happen only a mauriy, if a all. In pracice defauls may happen a any ime. Also, when a company offers more han one bond wih differen mauriies or senioriies, inconsisencies in he Meron model show up which can be solved by he following approach. Black and Cox (1976 firs used firs passage ime models in he conex of credi risk. This means ha a defaul happens a he firs ime, when he firm value falls below a prespecified level. They used a ime dependen boundary, F ( = ke γ(t, which resuled in a random defaul ime τ. Unforunaely, his framework proves o be unsaisfacory. Longsaff and Schwarz (1995 exended he Meron, respecively Black and Cox, framework wih respec o he following issues: Defaul may happen a he firs ime, denoed by τ, when he firm value V drops below a cerain level F. Ineres raes are sochasic and assumed o follow he Vasiček model. As a consequence, he firm value a defaul equals F. In he Meron model he value of he defauled bond was assumed o be V T /F which equals 1 in his conex. The recovery value of he bond is herefore assumed o be a pre-specified consan (1 w. This is he fracion of he principal he bond holder receives a mauriy. Since furher defauls are excluded in his model, he bond value a defaul equals B(τ, T = (1 wb(τ, T, where B(, T is he value of a risk-free bond mauring a T. This assumpion is ofen referred o as recovery of reasury value. In he following, we presen he model of Longsaff and Schwarz (1995 in greaer deail. The firm value is assumed o follow he sochasic differenial equaion dv ( V ( = µ( d + σ dw V (, and he spo rae is modeled according o he model of Vasiček (1977: dr( = ν(θ r( d + η dw r (. (1.2 Moreover, IE(W V (s W r ( = ρ (s for all and s. The las equaion reveals a possible correlaion beween he wo Brownian Moions W V and W r. The Vasiček model exhibis a mean-reversion behavior a level θ and easily allows for an explici represenaion of r. I is a classical model used in ineres rae heory and ofen aken as a saring poin for more sophisicaed models. A drawback of his model is he fac ha i may exhibi negaive ineres raes wih posiive probabiliy. See, for example, Brigo and Mercurio (21 and he discussions herein.

13 1.2 Srucural Models 7 For he price of he defaulable bond hey obain B LS (, T = B(, T IE QT 1 {τ>t } + (1 w1 {τ T } F = B(, T w Q T (τ > T F + (1 w. (1.3 Noe ha Q T (τ > T F is he condiional probabiliy (under he T -forward measure 3 ha he defaul does no happen before T. To he bes of our knowledge, a closed-form soluion for his probabiliy is no available 4. Neverheless here are cerain quasi-explici resuls provided by Longsaff and Schwarz (1995. See also Lehrbass (1997 for an implemenaion of he model. In he empirical invesigaion of Wei and Guo (1991, he Longsaff and Schwarz model reveals a performance worse han he Meron model. According o hese auhors his is mainly due o he exogenous characer of he recovery rae Jump Models - Zhou (1997 Anoher approach o solve he problem of shor mauriy spreads is o exend he firm value process o allow for jumps. Mason and Bhaacharya (1981 exended he Black and Cox (1976 model o a pure jump process for he firm value. The size of he jumps has a binomial disribuion. In his model here is some considerable probabiliy for he defaul o happen even jus before mauriy. Alernaively, Zhou (1997 exended he Meron model by assuming he firm value o follow a jump-diffusion process. The immediae consequence is ha defauls are no predicable. The model is formulaed direcly under an equivalen maringale measure Q, and he firm value is assumed o follow dv /V = (r λνd + σdw V ( + (Π 1dN. (1.4 (N is a Poisson process wih consan inensiy λ. The jumps are Π := U N, where U 1, U 2,... are i.i.d. and assumed o be independen of (N, (r and (W V (. Denoe ν := IE(U 1 1. Noe ha he inegral of (Π 1 dn is shorhand for Y s := s N s (Π 1 dn = (U i 1, so ha (Y is a marked poin process. I can be proved 5 ha (Y λν is a maringale so ha consequenly he discouned firm value is a maringale under he measure Q. 3 The T -forward measure is he risk neural measure which has he risk-free bond wih mauriy T as numeraire. For deails see Björk ( See discussions in Bielecki and Rukowski (22 and Goldsein ( See, for example, Brémaud (1981. i=1

14 1.2 Srucural Models 8 The ineres rae is assumed o be sochasic and follow he Vasiček model; see (1.2. The recovery rae is deermined by a deerminisic funcion w, so ha he bond holder receives ( 1 w(vτ /F a defaul. The funcion w represens he loss of he bond s value due o he reorganizaion of he firm. For w = 1 we have he zero recovery case. Zhou considers wo models. The firs, more general model, assumes ha defaul happens a he firs ime when he firm value falls below a cerain hreshold. See he previous chaper for more examples of his class of models. Since in his case no closed-form soluions are available, he auhor proposes an implemenaion via Mone-Carlo echniques. In he second, more resricive model, he auhor obains closed form soluions. For his a consan ineres rae and log-normaliy of he U i s is assumed and defaul happens only a mauriy T, when V T < F. Furhermore w is assumed o be linear, i.e., w(x = 1 w x. For w = 1 we obain he recovery srucure of he Meron model. Equaion (1.4 akes he form of a Doleans-Dade exponenial and can be explicily solved under hese assumpions, cf. Proer (1992, p. 77: We hen have he following V = V exp σ V W V ( + (r 1 N 2 σ2v λν Proposiion (Zhou. Denoe σ 2 U := Var(ln U 1 and ν := 1 + ν. Then he price of a defaulable bond in he above model equals B ZH (, T = w F V e λt ν (λ νt j e (r+λt j= j= Proof. The payoff of he bond equals j! ( ln F Φ i=1 U i. V (r σ2 V λνt j(ln ν σ2 U σ 2 V T + jσu 2 (λt j ( Φ ln F V (r 1 2 σ2 V λνt j(ln ν 1 2 σ2 U j! σ 2 V T + jσu 2 B ZH (, T = 1 {τ>t } + 1 {τ T } ( 1 w(vt /F = 1 {τ>t } + 1 {τ T } w V T F = ( V T {τ T } w F 1. To compue he presen value of he bond we consider he expecaion of he discouned payoff B ZH (, T = IE Q e r(t ( {τ T } ( w V T = e r(t 1 + IE Q (1 {VT <F } = e r(t 1 + w F IEQ ( 1 {VT <F }V T F F 1 F ( V T w F 1 F IE (1 Q {VT <F } F..

15 1.2 Srucural Models 9 Noe ha condiionally on {N T = j} we obain a log-normal disribuion for V T : IP(V T < F N T = j = IP (V exp (r 1 2 σ2v λνt + σ V W V (T N T ( = IP ln V + (r 1 2 σ2 V λνt + σ V W V (T + i=1 U i < F NT = j j ln U i < ln F =: IP(ξ j < ln F, where σ V W (T + j i=1 ln U i as a sum of independen normally disribued random variables is again normally disribued. Recall σu 2, he variance of ln U 1. As IE(ln U i = ln(1 + ν 1 2 σ2 U, we ge ( ξ j N ln V + (r 1 2 σ2 V λνt + j(ln ν 1 2 σ2 U, σ2 V T + jσ2 U =: N ( µ(j, σ 2 (j. I is an easy exercise o verify ha for ξ N (µ, σ 2 Conclude ha IE Q 1 {VT <F }V T = = i=1 IE ( ( e ξ 1 {e ξ <F } = e µ+ 1 ln F µ 2 σ2 Φ σ. σ Q(N T = jie Q (1 {VT <F }V T N T = j j= j= λt (λt j e j! = e λt V e (r λνt (λ νt j ( ln F Φ exp( µ(j + 1 ( ln F µ(j 2 σ2 (jφ σ(j σ(j j= j! V (r σ2 V λνt j(ln ν σ2 U σ 2 V T + jσu 2. We herefore obain B ZH (, T = e rt + w F V λt (1+ν e (λ νt j j= j! e (r+λt ( ln F Φ j= (λt j j! V (r σ2 V λνt j(ln ν σ2 U σ 2 V T + jσu 2 ( ln F Φ V (r 1 2 σ2 V λνt j(ln ν 1 2 σ2 U σ 2 V T + jσu 2. Noing ha he proof is complee. e rt = e (r+λt (λt j /(j!,

16 1.2 Srucural Models 1 In he case where no jumps are presen, i.e., λ =, he sum reduces o he summand wih j = so ha he bond price formula of Meron (1.1 is obained as a special case. This model feaures some properies which are also found in empirical invesigaions on credi risk: The erm srucure of he credi spreads can be upward-sloping, fla, humped or downward-sloping. The shor mauriy spreads can be significanly higher han in he Meron model. As he firm value a defaul is random, especially no equal o F as in he Longsaff and Schwarz (1995 model, he recovery is more realisic. The recovery rae is correlaed wih he firm value also jus before defaul Furher Srucural Models Kim, Ramaswamy and Sundaresan (1993 exended he firs passage ime models o also incorporae sochasic ineres raes following he model of Cox, Ingersoll and Ross (1985. In heir model here is an addiional possibiliy for a defaul o happen a mauriy. The payoff hey considered equals min(f, V. Possibly he company is no able o mee is liabiliies a mauriy bu did no face a defaul up o his ime. Nielsen, Saà-Requejo and Sana-Clara (1993 exended hese models o incorporae a sochasic defaul boundary. For he ineres rae hey used he model of Hull and Whie (199 bu were only able o obain explici formulas in he special case of he Vasiček model, cf. formula (1.2. Denoe σ 2 U := Var(ln U 1 and ν := 1 + ν. In he work of Ammann (1999 vulnerable claims are considered. These are possibly sochasic payoffs which face a counerpary risk. Counerpary risk plays a role if he buyer of a claim considers he defaul probabiliy of he seller as significan. He herefore will ask for a risk premium which compensaes for a possible loss in case of a defaul. The defaul is assumed o happen if V T < F, similar o Meron s model. In ha case he buyer of he claim X receives he fracion V T F X. Explici prices are derived for he Heah, Jarrow and Moron (1992 forward rae srucure and Meron-like firm dynamics. This secion on srucural models heavily relies on he assumpion ha he firm s value is observable or even radeable. From a pracical poin of view his seems no jusifiable as he firm s value is no radeable and even difficul o observe. This difficuly is discussed by Buffe (22 and also solved in he KMV-model; see Secion

17 1.3 Hazard Rae Models Hazard Rae Models In comparison o srucural models, inensiy based models or hazard rae models use a oally differen approach for modeling he defaul. In he srucural approach defaul occurs when he firm value falls below a cerain boundary. The hazard rae approach akes he defaul ime as an exogenous random variable and ries o model or fi is probabiliy o defaul. The main ool for his is a Poisson process wih possibly random inensiy λ, and jumps denoing he defaul evens. As in he firs passage ime models recovery is no inrinsic o his model and is ofen assumed o be a somehow deermined consan. The reason for his new approach lies in he very differen causes for defaul. Precise deerminaion as done in srucural models seems o be very difficul. Furhermore, in srucural models he calibraion o marke prices ofen causes difficulies, while inensiy based models allow for a beer fi o available marke daa. In some approaches basic ideas of hese model classes are combined, for example by Madan and Unal (1998 and Ammann (1999 where he defaul inensiy explicily depends on he firm value. These models are called hybrid models and will be discussed in Secion 1.6. As he firm value approaches a cerain boundary, inensiy increases sharply and defaul becomes very likely. So basic feaures of he srucural models are mimicked. A more involved hybrid model is presened by Duffie and Lando (21 where a firm value model wih incomplee accouning daa is considered. Basically we may disinguish hree ypes of hazard rae models. In he firs approach he defaul process is assumed o be independen of mos economic facors, someimes i is even modeled independenly from he underlying. The raing based approach incorporaes he firm s raing as his consiues readily available informaion on he company s crediworhiness. In principle one ries o model he company s way hrough differen raing classes up o a possible fall o he lowes raing class which deermines he defaul. A hird and very recen class is in he line of he famous marke models of Jamshidian (1997 and Brace, Gaarek and Musiela (1995, see Chaper Mahemaical Preliminaries In his secion we consider he modeling of he defaul process in greaer deail. The approach is mainly based on Lando (1994 and also discussed in many aricles and books like Jeanblanc (22 and Bielecki and Rukowski (22. We firs presen a brief inroducion o Cox processes. More deails can be found in Appendix A. As already menioned differen sopping imes denoing he defaul evens need o be modeled. The Poisson process is aken as a saring poin. Consan inensiy seems oo

18 1.3 Hazard Rae Models 12 resricive so one uses Cox processes, which can be considered as Poisson processes wih random inensiies 6. A special case which suis well for our purposes is he following: Consider a sochasic process λ which is adaped o some filraion G. For a Poisson process (N wih inensiy 1 independen of σ(λ s : s T se ( Ñ := N λ u du, T. (Ñ is a Cox process. Observe ha for posiive λ he process λ u du is sricly increasing and so Ñ can be viewed as a Poisson process under a random change of ime. This reveals a very powerful concep for he problems considered in credi risk. If jus one defaul ime τ is considered, his will be equal o he firs jump τ 1 of Ñ. If more defaul evens are considered, for example, ransiion o oher raing classes, furher jumps τ i are aken ino accoun. The bigger λ is, he sooner he nex jump may be expeced o occur. We obain, for any < T, IP(τ > = IE IP (τ > (λ s s = IE exp ( λ u du. Conclude ha condiionally on σ(λ s : s T he jumps are exponenially disribued wih parameer λ u du. I may be recalled ha a fundamenal assumpion o obain his is he independence of λ and N Jarrow and Turnbull ( In he work of Jarrow and Turnbull (1995 a binomial model is considered. In exension of he classical Cox., Ross and Rubinsein (1979 approach he auhors also modeled he non-defaul and he defaul sae. So for every ime period four possible saes may be aained: {up,down} {non-defaul,defaul}. They discovered an analogy o he foreignexchange markes. As he inensiy of he model is assumed o be consan we do no discuss i in greaer deail. In Jarrow and Turnbull (2 a Vasiček model for he spo rae is used and he hazard rae is explicily modeled. Correlaion of he hazard rae and spo raes are allowed. Denoe by Z and W Brownian moions under he risk neural measure Q, wih consan correlaion ρ. Z can be some economic facor, like an index or he logarihm of he firm value. 6 For a full reamen of Cox processes see Brémaud (1981 and Grandell (1997.

19 1.3 Hazard Rae Models 13 Assume he following dynamics dr = κ(θ r d + σdw, λ = a ( + a 1 (r + a 2 (Z. Noe ha λ may ake on negaive values wih posiive probabiliy. Recovery mus be modeled exogenously and he auhors use he already menioned recovery of reasury value 7. This means if defaul happens prior o mauriy of he bond, he bond holder receives a fracion (1 w of he principal a mauriy. For he value of he bond we calculae he expecaion of he discouned payoff under he risk-neural measure Q. For ease of noaion we consider =. By equaion (1.3, ( B(, T = (1 wb(, T + wie Q exp r u du 1 {τ>t }. In he model of Jarrow and Turnbull we obain B(, T = (1 wb(, T + wie Q exp( = (1 wb(, T + wie Q exp = (1 wb(, T + w exp( µ T v T. r u duq(τ T λ s : s T (r u + λ u du In he las equaion µ T and v T denoe expecaion and variance of (r u + λ u du. Under he saed assumpions his inegral is normally disribued and µ and v can be easily calculaed. The flexibiliy of he model leads o a good fi o marke daa, which is no obained by mos srucural models. Also he model incorporaes economic facors (Z Duffie and Singleon (1999 The paper by Duffie and Singleon (1999 combines wo very successful model classes in ineres rae modeling o access Credi Risk: exponenial affine models and he Heah, Jarrow and Moron (1992 mehodology. For he exponenial affine model he auhors model a vecor of hidden facors which underlie he erm srucure of ineres raes. This vecor is assumed o follow a mulidimensional Cox-Ingersoll-Ross model: dy( = K(Θ y(d + Σ diag(y( 1/2 dw(. 7 See he Longsaff and Schwarz model, Secion

20 1.3 Hazard Rae Models 14 Consequenly he componens of y are nonnegaive random numbers. Spo and hazard rae are assumed o be linear in y(: r( = δ + δ y(, λ((1 θ( = γ + γ y(. A main feaure of he exponenial affine models is ha he soluion of he above SDE can be explicily expressed in an exponenial affine form. Hence we obain deerminisic funcions a(, b( such ha ( IE exp iξ y(u du = expa(, ξ + b(, ξ y(. Thus he price of he defaulable bond can be calculaed in closed form as he value of he characerisic funcion a a proper poin. The second approach uses he well known Heah-Jarrow-Moron model of forward raes. Denoe by f(, T he forward raes deermined by he erm srucure of he defaulable bond prior o defaul 8 and by W(, T a d-dimensional sandard Brownian moion. Assume he dynamics of he forward rae o be f(, T = f(, T + µ(u, T du + σ(u, T dw(u. Similar o Heah, Jarrow and Moron (1992 he auhors specify he dynamics under he objecive measure and consider an equivalen measure Q. For arbirage-freeness i is sufficien - see he work of Harrison and Pliska ( ha all discouned price processes are maringales. Naurally his heavily relies on he recovery assumpion. Duffie and Singleon (1999 inroduced he recovery of marke value which means ha immediaely a defaul he bond loses a fracion of is value. This seup is paricularly well suied for working wih SDEs. The loss rae w is assumed o be an adaped process. Hence B(τ, T = (1 w B(τ, T. Under hese assumpions he auhors derived he following drif condiion for µ and σ: ( µ(, T = σ(, T σ(u, T du. On he oher hand, using he above menioned recovery of reasury value (cf and denoing he riskless forward rae by f(, T, he auhors obained ( µ(, T = σ(, T 8 The forward rae is by definiion f(, T = T ln B(, T. v(, T σ(u, T du + θ(λ( p(, T ( f(, T f(, T.

21 1.4 Credi Raings Based Mehods Credi Raings Based Mehods Simple hazard rae models are ofen criicized because hey do no incorporae available economic fundamenal informaion like firm value or credi raings. This secion reveals some models which incorporae hese daa. This is also a basic feaure of commercial models; see Secion 1.8. Credi raings consiue a published ranking of he credior s abiliy o mee his obligaions. Such raings are provided by independen agencies, for example Sandard & Poor s or Moody s and mosly financed by he gauged companies. The firms are raed even if hey are no willing o pay, bu for a fee hey ge deailed insigh in he resuls of he examinaions and migh reain fundamenal insighs in heir inernal divisions o idenify weaknesses. Each raing company uses a differen sysem of leers o classify he crediworhiness of he raed agencies. Sandard & Poor s, for example, describes he highes raed deb (riple-a=aaa wih he words Capaciy o pay ineres and repay principal is exremely srong. An obligaion wih he lowes raing, D, is in sae of defaul or is no believed o make paymens in ime or even during a grace period. The lower he raing, he higher is he risk ha ineres or principal paymens will no be made Jarrow, Lando and Turnbull (1997 The model proposed by Jarrow, Lando and Turnbull (1997 circumvens some disadvanages of he hihero inroduced models. Especially he use of credi raings is an aracive feaure. The movemens beween he single raing classes is modeled by a ime homogenous Markov chain, he enry ino he lowes raing class yielding a defaul. For example, if a bond is raed AAA, i is a member of he highes raing class (= class 1. If here exis K 1 raing classes, denoe by K he class of defaul. Defaul is assumed o be an absorbing sae, resrucuring afer defaul is no considered in his model. The generaor of he Markov chain is defined as λ 1 λ 12 λ 13 λ 1K λ 21 λ 2 λ 23 λ 2K Λ = λ K 1,1 λ K 1,2 λ K 1 λ K 1,K The ransiion raes for he firs raing class are in he firs row. So λ 1 = j 1 λ 1j is he rae for leaving his class, while λ 12 is he rae for downgrading o class 2 and so on. The rae for a defaul direcly from class one is λ 1K. We denoe q ij (, := IP(Raing is in class i a and in class j a, and by Q( he marix of he ransiion probabiliies q ij (,.

22 1.4 Credi Raings Based Mehods 16 The ransiion probabiliies can be compued from he inensiy marix via 9 Q( = exp(λ := id n +Λ + 1 2! (Λ ! (Λ3 +..., where id n is he n n ideniy-marix. Under he recovery of reasury assumpion 1 we obain for he price of a zero coupon bond under defaul risk r s ds 1 {τ>t } B(, T = 1 {τ>} IE e R τ rs ds δb(τ, T 1 {τ T } + e R T = 1 {τ>} IE δ1 {τ T } e R T r s ds + 1 {τ>t } e R T r s ds ( = 1 {τ>} δb(, T + IE (1 δe R T r s ds 1 {τ>t } = 1 {τ>} B(, T δ + (1 δq T (τ > T. (1.5 Q T is he T -forward measure 11. I is herefore crucial o have a model which deermines he ransiion probabiliies under his measure. While raing agencies esimae he ransiion probabiliies using hisorical observaions, i.e., under he objecive measure P, Jarrow, Lando and Turnbull (1997 propose a mehod which uses he defaulable bond prices and calculaes ransiion probabiliies under he he risk-neural measure Q. Consider he bond wih raing i and se Q T,i (τ > T he probabiliy ha he bond will no defaul unil T given i is raed i a. As i makes no sense o alk abou bond prices afer defaul, we furher on jus consider he bond price on {τ > } and ge ( B i (, T = B(, T δ + (1 δq T,i (τ > T τ >. (1.6 Jarrow, Lando and Turnbull (1997 spli he inensiy marices ino an empirical par (under P and a risk adjusmen like a marke price of risk: They assume ha he inensiies under Q T have he form UΛ and U denoes a diagonal marix where he enries are he risk adjusing facors µ i. For he ransiion probabiliies his yields ha q ij (, T is he ij h enry of he marix exp(uλ. Time homogeneiy of µ would enail exac calibraion being impossible. For he discree ime approximaion,, T is divided ino seps of lengh 1. Saring wih (1.6 one obains Q T,i (τ T τ > = 1 B i(, T δb(, T (1 δb(, T = B(, T B i (, T. (1.7 B(, T (1 δ 9 See, for example, Israel, Rosenhal and Wei (21. 1 The bond holder receives δ equivalen and riskless bonds in case of defaul. See Secion The T -forward measure is he risk neural measure which has he risk-free bond wih mauriy T as numeraire. For deails see Björk (1997.

23 1.4 Credi Raings Based Mehods 17 Denoe he empirical probabiliies from he raing agency by p ij (, T. Q T,i (τ 1 = µ i (p ik (, 1, and we obain This leads o µ i ( = QT,i (τ 1 = p ik (, 1 B(, 1 B i (, 1 p ik (, 1 B(, 1(1 δ. By his one obains (µ 1,..., µ K 1 and consequenly q ij (, 1. For he sep from o + 1 use Q T,i (τ + 1 = Q T,i (τ + 1 τ > Q T,i (τ > o ge This leads o K 1 Q T,i (τ + 1 = µ i(p i (τ + 1 τ > q ij (, = µ i (p ik (, + 1 K 1 j=1 j=1 q ij (,. µ i ( = (1.7 = Q T,i (τ + 1 K 1 j=1 q ij(, p ik (, + 1 B(, + 1 B i (, + 1 ( K 1, B(, + 1(1 δ j=1 q ij(, p ik (, + 1 and, via q ij (, + 1 = µ i (p ij (, + 1, he required probabiliies are obained. This model exends Jarrow and Turnbull (1995 using ime dependen inensiies bu sill working wih consan recovery raes. Das and Tufano (1996 propose a model which also allows for correlaion beween ineres raes and defaul inensiies. I seems problemaic ha all bonds wih he same raing auomaically have he same defaul probabiliy. In realiy his is definiely no he case. Naurally differen credi spreads occur for bonds wih he same raing. A furher resricive assumpion is he ime independence of he inensiies. The yield of a bond in his model may only change if he raing changes. Usually he marke price precedes he raings wih informaions on a possible raing change which is an imporan insigh of he KMV model; see Secion Lando (1998 The work of Lando (1998 uses a condiional Markov chain 12 o describe he raing ransiions of he bond under consideraion. All available marke informaion like ineres raes, asse values or oher company specific informaion is modeled as a sochasic process (X. This is analogous o he case wihou raings, where Lando used λ = λ(x. 12 See also Secion 11.3 in Bielecki and Rukowski (22.

24 1.4 Credi Raings Based Mehods 18 Assume ha a risk-neural maringale measure Q is already chosen. Then he arbiragefree price of a coningen claim is he condiional expecaion under his measure Q. The auhor lays ou he framework for raing ransiions where all probabiliies are already under he risk-neural measure and calibraes hem o available marke prices. As no hisorical informaion is used he probabiliy disribuion under he objecive measure is no needed. If one wans o consider risk-measures like Value-a-Risk, noe ha he objecive measure is sill required. We denoe he generaor of he condiional Markov chain (C by λ 1 (s λ 12 (s λ 13 (s λ 1K (s λ 21 (s λ 2 (s λ 23 (s λ 2K (s Λ(s = λ K 1,1 (s λ K 1,2 (s λ K 1 (s λ K 1,K (s, where for all s λ i (s = K j=1, j i λ ij (s, i = 1,..., K 1. We assume (λ ij ( o be adaped and nonnegaive processes. I is imporan for he inensiies o depend on boh ime and ineres raes. Especially for low raed companies he defaul raes vary considerably over ime 13. I was observed by Duffee (1999, e.g., ha defaul raes significanly depend on he erm srucure of ineres raes. I is cerainly bad news for companies wih high deb when ineres raes increase whereas for oher companies i migh be good news. The consrucion of (C can be done as follows. Consider a series of independen exponenial(1-disribued random variables E 11,..., E 1K, E 21,..., E 2K,... which are also independen of σ(λ(s : s and denoe he raing class of he company a he beginning of he observaion by η. Define τ η,i := inf{ : λ η,i(s ds E 1i }, i = 1,..., K and τ := min i η τ η,i, η 1 := arg min i η τ η,i. The τ η,i model he possible ransiions o oher raing classes saring from raing η. The firs ransiion o happen deermines he ransiion ha really akes place, compare Figure 1.2. The reached raing class is denoed by η 1 while τ denoes he ime a which his occurs. Analogously, he nex change in raing saring in η 1 is defined, and similarly for η i and τ i. Then, for τ i 1 < τ i, C is defined by C := η i. Defaul is assumed o be an absorbing sae of he Markov chain and we denoe he overall-ime o defaul by τ. This is he firs ime when η i = K. 13 Cf. Chaper 15 in Caouee, Almann and Narayanan (1998.

25 1.4 Credi Raings Based Mehods 19 Raing η η 1 η 2 τ τ 1 Figure 1.2: A possible realizaion of raing ransiions. The raing sars in η and drops o η 1 a τ. The nex change is a τ 1, o raing class η 2. The ransiion probabiliies P (s, for he ime inerval (s, saisfy Kolmogorov s backward differenial equaion 14 PX(s, s = Λ(s P X (s,. Consider he price of a defaulable zero recovery bond a ime, Bi (, T, which has mauriy T and is raed in class i a ime. Then we obain he following Theorem. Theorem Under he above assumpions he price of he defaulable bond equals ( B i (, T = 1 {C=i}IE exp ( r s ds (1 P X (, T i,k F. Here P X (, T i,k is he (i,k-h elemen of he marix of ransiion probabiliies for he ime inerval (, T, P X (, T. Proof. As already menioned he Markov chain is modeled under Q so ha he arbiragefree price of he bond is he following condiional expecaion: ( B i (, T = IE exp ( r s ds 1 {τ>t,c=i} F. Using condiional expecaions and he independence of E 1K and (Λ(s one concludes ( B i (, T = 1 {C=i}IE exp ( ( = 1 {C=i}E exp ( r s ds IP ( τ > T σ(λs : s T F F r s ds (1 P X (, T i,k F. 14 For non-commuaive Λ he soluion is in general no of he form P X (s, = exp Λ(u du. See Gill s and Johannsen (199 for soluions using produc inegrals.

26 1.4 Credi Raings Based Mehods 2 For he calibraion o observed credi spreads explici formulas are needed and herefore furher assumpions will be necessary. Lando chooses an Eigenvalue-represenaion of he generaor. Denoe wih A(s he marix wih enries λ 1 (s,..., λ K 1 (s, on he diagonal and zero oherwise. Assume ha Λ(s admis he represenaion Λ(s = B A(s B 1, where B is he K K-marix of he Eigenvecors of Λ(s. We conclude P X (s, = B C(s, B 1 wih exp C(s, = λ s 1(udu..... exp s λ K 1(udu 1 I is easy o see ha P X (s, saisfies he Kolmogorov-backward differenial equaion. For uniqueness, see Gill and Johannsen (199. Under hese addiional assumpions he price of he defaulable bond in Theorem simplifies considerably. Proposiion Denoing by β ij := B ij B 1 jk, he price of he defaulable bond equals B i (, T = K 1 j=1 β ij IE exp ( (λ j (u r u du F. Proof. In his seup he condiional probabiliy for a defaul when he bond is in raing class i equals Wih B ik B 1 KK = 1 we obain IP X (, T i,k = 1 {τ>} K 1 IP X (, T i,k = K 1 j=1 j=1 B ij exp( T B ij B 1 jk exp( λ j (udub 1 jk. λ j (udu. and he conclusion follows as in Using he readily available ools for hazard rae models i is now easy o consider opions which explicily depend on he credi raing or credi derivaives wih a credi rigger.

27 1.4 Credi Raings Based Mehods 21 Calibraion Assuming a Vasiček model 15 for he ineres rae we are in he posiion o use he model laid ou above for calibraion o observed credi spreads. There are no economic facors considered oher han he ineres rae and, as a consequence, λ mus be adaped o G = σ(r s : s. Furhermore, we assume wih consans γ j, κ j. λ j (s = γ j + κ j r s, j = 1,..., K 1, The dynamics of he generaor marix is Λ(s = B A(s B 1 and B has o be esimaed from hisorical daa while γ j, κ j are calibraed. The credi spread is he difference of he offered yield o he spo rae. By Theorem he bond price saisfies K 1 B i (, T = j=1 β ij IE exp Therefore, we obain for he bond s yield T T = log B i (, T = T K 1 = j=1 T = K 1 j=1 ( β ij IE exp (γ j (1 κ j r u du F. ( β ij lim IE (γ j + (κ j 1r T exp T K 1 = β ij (γ j + (κ j 1r. j=1 Hence he credi spread equals K 1 s i ( = β ij (γ j + κ j r. j=1 (γ j (1 κ j r u du F ( (γ j + κ j r u r u du F For calibraion a second relaion is needed. Lando uses he sensiiviy of he credi spreads w.r.. he spo rae: K 1 s i ( = β ij κ j. r Denoe by ŝ, dŝ he observed credi spreads and heir esimaed sensiiviies. One finally has o solve he following equaion o calibrae he model: 15 see equaion (1.2. j=1 β(γ + κr = ŝ βκ = dŝ.

28 1.5 Baske Models 22 I urns ou o be problemaic ha observed credi spreads are no always monoone wih respec o he raings. The auhor argues ha in pracice his would occur raher seldom. 1.5 Baske Models Usually here is a whole porfolio under consideraion insead of jus one single asse. Therefore he so far presened models were exended o models which may handle he behavior of a larger number of individual asses wih defaul risk, a so-called porfolio or baske. There are several approaches in he lieraure and hey can be grouped ino models which use a condiional independence concep and ohers which are based on copulas. From he firs class we presen he mehods of Kijima and Muromachi (2, which provide a pricing formula for a credi derivaive on baskes wih a firs- or second-odefaul feaure. An example is he firs-o-defaul pu, which covers he loss of he firs defauled asse in he considered porfolio, see also Secion From he second class we discuss an implemenaion based on he normal copula in Secion Besides ha, Jarrow and Yu (21 model a kind of direc ineracion beween defaul inensiies of differen companies. In heir model he defaul of a primary company has some impac on he hazard rae of a secondary company, whose income significanly depends on he primary company Kijima and Muromachi (2 Consider a porfolio of n defaulable bonds and denoe by τ i he defaul ime of he i-h bond. Le (G represen he general marke informaion (see Appendix A. Furhermore assume ha for any 1,..., n T Q(τ 1 > 1,..., τ n > n G T = Q(τ 1 > 1 G T Q(τ n > n G T, (1.8 where Q is assumed o be he unique risk neural measure. Using he represenaion via Cox processes, his yields (1.8 = exp( n i=1 i λ i (s ds. In he recovery of reasury model, he loss of bond i upon defaul equals he pre-specified consan w i := (1 δ i. So he firs-o-defaul pu is he opion which pays w i if he ih asse is he firs one o defaul before T and zero if here is no defaul. Denoe he even ha he firs defauled bond is number i by D i := {τ i T, τ j > τ i, j i}.

29 1.5 Baske Models 23 Then, using he risk neural valuaion principle, he price of he bond can be compued as he expecaion w.r.. he risk-neural measure Q and equals S F = IE exp( = r u du n w i IE exp( i=1 We obain his probabiliy using he facorizaion n w i 1 Ai i=1 IP(τ i T, τ k > τ i, k i G T {τ i = x} Wih Theorem A.1.2 we obain r u duq(a i G T. = 1 {x T } IP(τ k > x, k i G T {τ i = x} = 1 {x T } exp( k i IP(τ i T, τ k > τ i, k i G T = IE 1 {τi T } exp( k i = IE u λ i (u exp( τ i x λ k (s ds. λ k (s ds G T λ i (s ds exp( k i u λ k (s ds du u = IE λ i (u exp( k=1 We conclude for he price of he firs-o-defaul pu: n λ k (s ds du. S F = n i=1 w i IE λ i (u exp( r s ds n u k=1 λ k (s ds du. This formula simplifies considerably if w i w, as in ha case S F = wie n i=1 u λ i (u exp( ( n = wie exp( i=1 n k=1 = (1 δb(, T 1 IE T ( exp( λ k (s ds du exp( λ i (u du T T exp( r s ds n λ i (u du. i=1 r s ds

30 1.5 Baske Models 24 Using similar mehods, we deermine he swap-price, if w i is paid immediaely a defaul o he swap-holder. Se τ S F = IE exp( r u du n w i 1 Ai. i=1 Cerainly, τ r u du is no G T -measurable, so ha a sligh modificaion of he previously used mehod is necessary. We obain for he facorizaion IE x exp( r u du1 {x T } 1 {τk >x, k i} G T {τ i = x} = 1 {x T } exp x (r u + k i λ k (u du and conclude S F = n i=1 w i IE λ i (u exp u (r s + n λ k (s ds du. k=1 Similarly, he auhors provide he following price of a (firs and second-o-defaul swap, which proecs he holder agains he firs wo defauls in he porfolio: S S = n i=1 + i j δ i IE exp( (n 2 (δ i + δ j n i=1 δ i λ i (s ds B(, T IE λ k (u exp( IE λ i (u exp( n i=1 r s ds r s ds δ i n u j=1 u λ j (s ds du n λ j (s ds du j=1 Exended Vasiček implemenaion Kijima and Muromachi (2 discuss a special case of he above implemenaion. The main idea is o perform a calibraion similar o he one of Hull and Whie (199 for credi risk models. Assume for he dynamics of he hazard raes dλ i ( = ( φ i ( a i λ i ( d + σ i dw i (, i = 1,..., n, (1.9 where w i are sandard Brownian moions wih correlaion ρ ij, which is someimes saed as dw i dw j = ρ ij d. Furhermore, assume for he shor rae r dr = ( φ ( a r d + σ dw (.

31 1.5 Baske Models 25 Noe ha equaions of he ype (1.9 admi explici soluions, see Schmid (1997. From his, we ge λ i ( = λ i (e a i + φ i (se a i( s ds + σ i Using he recovery of reasure assumpion he bond price equals B i (, = δ i B(, + (1 δ i IE exp( e a i( s dw i (s. (r u + λ i (u du. Noe ha (r u + λ i (u du is normally disribued and herefore he expecaion equals he Laplace ransform of a normal random variable wih mean IE (r u + λ i (u du = ( r e a u + u (φ (se a (u s ds du ( λi (e a iu + u (φ i (se a i(u s ds du and variance Var (r u + λ i (u du = Var σ u e a (u s dz (s du + σ i u e ai(u s dw i (s du. To compue he variances i is sufficien o calculae he variances of all summands and he covariances. Seing ρ ii = 1, we have IE u 1 u 2 σ i σ j exp( a i (u 1 s 1 a j (u 2 s 2 dw j (s 2 dw i (s 1 du 2 du 1 u 1 = u 2 = = σ i σ j IE = σ i σ j IE = σ i σ j ρ ij s 1 s 2 exp( a i (u 1 s 1 a j (u 2 s 2 du 2 du 1 dw j (s 2 dw i (s 1 e a is 1 +a j s 2 1 (1 e a is 1 (1 e a js 2 dw j (s 2 dw i (s 1 a i a j e a is+a j s 1 a i a j (1 e a is (1 e a j s ds = σ iσ j ρ ij + 1 (e ai (e aj (1 e (a i+a j a i a j a i a j a i + a j =: c ij (

32 1.5 Baske Models 26 Therefore, Var σ i u e ai(u s dw i (s du = σ2 i + 2 (e ai (1 e 2ai =: v a 2 2 (. i a i 2a i Recall ha we wan o calibrae he model o he bond prices, which means calculaing φ i (s. φ (s is compued as in he risk neural case, see Hull and Whie (199. Consider 1 B(, IE exp( (r u + λ i (u du = 1 Bi (, 1 δ i B(, δ i =: γ i (, which can be obained from available prices, since δ i is assumed o be known. Noe ha γ i ( does no involve φ (s as γ i ( = exp ( λi (e a iu + u + 1 2( ci ( + v 2 (. φ i (se a i(u s ds du As we wan o solve his expression for φ i, we consider he following derivaives: ln γ i( = λ i (e a i + φ i (se a i( s ds 1 2 c i ( + v 2 ( =: g i ( Wih g i( = a i λ i (e ai + φ i ( a i e a i we conclude φ i ( = g 1 i( + a i g i ( + a i c i ( + v 2 ( 2 Hence φ i (se a is ds c i ( + v 2 ( c i ( + v 2 (. 1 a i c i ( + c i ( = σ σ i ρ i 1 e a 1 e ai + 1 e (a +a i a a a a 1 +σ σ i ρ i e a + 1 e ai a + a i e (a +a i a i a a a i = σ σ i ρ i 1 e a a + e a 1 e ai a i

33 1.5 Baske Models 27 and a i v 2 ( + v 2 ( = σi 2 = σ2 i 1 + e 2a i a i which finally leads o 1 a i 2 a i e a i 1 a i e 2a i + 2 a i e a i + 2 a i e 2a i φ i ( = g i( + a i g i ( + σ2 i (1 e 2ai e a 2a i 2 σ σ i ρ i + e a 1 e ai. a a i Using similar mehods Kijima and Muromachi (2 obain an explici formula for he firs-o-defaul swap. In Kijima (2 hese mehods are exended o pricing a credi swap on a baske, which migh incorporae a firs-o-defaul feaure Copula Models The concep of copulas is well known in saisics and probabiliy heory, and has been applied o finance quie recenly. Modeling dependen defauls using copulas can be found, for example, in Li (2 or Frey and McNeil (21. We give an ouline of Schmid and Ward (22, who apply a special copula, he normal copula, o he pricing of baske derivaives. Fix =. The goal of he model is o presen a calibraion mehod. Consider he defaul imes τ 1,..., τ n and assume for he beginning ha =. The link beween he marginals Q i ( := Q(τ i and he join disribuion is he so-called copula C( 1,..., n. Assuming coninuous marginals, U i := Q i (τ i is uniformly disribued. The join disribuion of he ransformed random imes is he copula C(u 1,..., u n := Q(U 1 u 1,..., U n u n and defines he join disribuion of he τ i s via Q(τ 1 1,..., τ n n = C ( Q 1 ( 1,..., Q n ( n. For more deailed informaion on copulas see Nelsen (1999. The choice of he copula cerainly depends on he applicaion. Schmid and Ward (22 choose he normal copula because in a Meron framework wih correlaed firm value processes such a dependence is obained, and secondary he normal copula is deermined by correlaion coefficiens which can be esimaed from daa. Assume ha (Y 1,..., Y n follows an n dimensional normal disribuion wih correlaion marix Σ = (ρ ij, where ρ ii = 1 for all i. Denoing heir join disribuion funcion by Φ n (y 1,..., y n, Σ yields he normal copula C(u 1,..., u n = Φ n ( Φ 1 (u 1,..., Φ 1 (u n.

34 1.5 Baske Models 28 For modeling purposes i is useful o noe ha seing τ i := Q 1 i (Φ(Y i, resuls in {τ 1,..., τ n } having a normal copula wih correlaion marix Σ. The above mehods enable us o calculae he join disribuion of n defaul imes, and he required correlaions can be esimaed using hisorical daa. Thus, a value a risk can be deermined. For he pricing of a derivaive wih firs-o-defaul feaure, noe ha Q(τ 1s T = 1 Q(τ 1 > T,..., τ n > T (1.1 which can be calculaed from he copula and he marginals. A more involved, bu also explici formula can be obained for a kh-o-defaul opion. For example, consider a firs-o-defaul swap, which is also discussed in Secion This is a derivaive which offers defaul proecion agains he firs defauled asse in a specified porfolio. Under he assumpion, ha all credis have he same recovery rae δ i δ, he swap pays (1 δ a τ 1s if τ 1s T. In exchange o his, he swap holder pays he premium S a imes T 1,..., T m, bu a mos unil τ 1s. As explained in Secion 1.9.2, calculaing expecaions of he discouned cash flows yields he firs-o-defaul swap premium. Thus, using equaion (1.14, we obain S 1s = (1 δie exp( τ 1s r u du1 {τ 1s T } m i=1 IE exp( i. r u du1 {τ 1s >T i } To calculae he expecaions, he disribuion of τ 1s under any forward measure is needed. Assuming, for simpliciy, independence of he defaul inensiy and he risk-free ineres rae, one obains IE exp( i r u du1 {τ 1s >T i } = B(, Ti Q(τ 1s > T i. The bond prices are readily available and he probabiliy can be calculaed via (1.1, once he copula is deermined. For he second expecaion, use IE exp( τ 1s r u du1 {τ 1s T } = B(, sie s exp( λ 1s u duλ 1s (s ds.

35 1.6 Hybrid models 29 Noe ha his expecaion can be obained via s Q(τ 1s > s = s s IE exp( = IE s exp( λ 1s u du λ 1s u duλ1s (s. Furher on, Schmid and Ward (22 derive ineresing resuls on spread widening, once a defaul occurred. For example, if one of wo srongly relaed companies defauls, i migh be likely ha he remaining one ges ino difficulies, and herefore credi spreads increase. I seems ineresing ha raders have a good inuiion on his amoun of spread widening, which also could be used as an inpu parameer o he model, which deermines he copulas. 1.6 Hybrid models Hybrid models incorporae boh preceding models, for example he firm value is modeled, and a hazard rae framework is derived wihin his model Madan and Unal (1998 The approach of Madan and Unal (1998 mimics he behavior of he Meron model in a hazard rae framework. They assume he following srucure for he defaul inensiy: λ( = ( c 2. ln V ( F B( Here V ( denoes he firm value which as in Meron s model is assumed o follow a geomeric Brownian moion. B( is he discouning facor exp( r u du and F is he amoun of ousanding liabiliies. If he firm value approaches F he defaul inensiy increases sharply and i is very likely ha he bond defauls. As defauls can happen a any ime his model is much more flexible han he Meron model. Unlike in Longsaff and Schwarz s model, he defaul can even happen when he firm value is far above F, hough wih low probabiliy. The auhors also consider parameer esimaion in heir model. A closed form soluion for he bond price is no available and for calculaing he prices of derivaives numerical mehods need o be used. Furher hybrid models of his ype can be found in Ammann (1999 or Bielecki and Rukowski (22.

36 1.6 Hybrid models Duffie and Lando (2 The model of Duffie and Lando (21 accouns for he fac ha bond holders receive only imperfec informaion on he issuer s asses. The approach sars wih a srucural model for he firm value and assumes ha he bond holder obains observaions on he firm value disurbed and only a discree ime poins, which leads o a hazard rae model. Afer presening he framework proposed by he auhors we derive he hazard rae explicily. Suppose he firm value can be modeled by a geomeric Brownian moion, as in he Meron framework, i.e., V = V exp((µ σ2 2 + σw =: V exp(m + σw. The firm is operaed by equiy owners, which have complee informaion on he firm s asses, represened by F = σ(v s : s. The firs sep is o deermine he opimal liquidaion policy. Assume ha he drif of he firm value is smaller han he risk-free ineres rae, µ < r, and furher on, he firm generaes cash flow a he rae δv for some consan δ >. Then he presen value of he firm s fuure cash flow is finie, respecively IE e r(s δv s ds F = δv e (µ r(s ds = δv r µ. If µ r he presen value of he firm s fuure cash flow is infinie. This case poses several problems and an opimal exercise policy like he one deermined in equaion (1.11 below is no available. Neverheless, one could assume ha equiy owners liquidae he firm a he firs ime when he firm value falls below a cerain boundary, hus, assuming direcly ha (1.11 holds. If he equiy holders choose o liquidae he firm, a fracion α, 1 of he asses is los because of liquidaion coss. The ousanding deb D has o be paid o he deb-holders, if possible, and he remaining value goes o he equiy holders, ha is min(d, (1 α δv r µ max(, (1 α δv r µ D debholders equiy. If he deb akes he form of a consol bond, meaning ha he coupons are paid coninuously a rae C > and he ax benefi herefore yields he consan rae θc, we conclude for he iniial value of equiy, according o a cerain liquidaion policy represened by a (F -sopping ime τ, ha τ F (V, C, τ = IE e r( δv + (θ 1C d + e rτ max(, (1 αδv τ r µ D.

37 1.6 Hybrid models 31 As he equiy owners will choose he liquidaion policy maximizing he iniial value of equiy, his leads o he opimizaion problem S = sup F (V, C, τ, τ T where T is he se of all (F -sopping imes. The opimal sraegy, as shown by Leland and Tof (1996, is given by τ(v B = inf{ : V V B }, (1.11 wih a cerain level V B which can be deermined by solving a Hamilon-Jacobi-Bellman differenial equaion 16. For convenional parameers, he auhors are able o show ha V B = V B (C = (1 θcγ(r µ, γ = m + m 2 + 2rσ 2. r(1 + γδ σ 2 Turning o he bond holder s perspecive, we noice ha hey receive informaion on he firm value jus a seleced imes 1, 2,.... This is modeled by a noisy observaion of V i, i.e., insead of observing V i he marke paricipans observe 17 Ṽ i := V i exp(z i σ2 Z 2. The Z i are assumed o be independen normally disribued random variables wih variance σ 2 Z and being independen of (W s s. If we assume for simpliciy ha equiy is no raded on he public marke, he informaion available o he bond holder is H = σ(ṽ 1,..., Ṽ n, 1 {τ s} : 1,..., n and s. In his framework he probabiliy for no defaul unil T equals 1 {τ>} IP(τ > T H = 1 {τ>} IP( inf s (,T V s > V B H. Fix and denoe by k he las n which is smaller han or equal o. Because (W is a Markov process, i is sufficien o condiion on a smaller σ algebra, and herefore 1 {τ>} IP(τ < T H = 1 {τ>} IP ( V inf s (,T exp(m(s + σ(b s B > V B Ṽk, 1 {τ } = 1 {τ>} IP ( inf m(s + σ(b s B > ln V B Ṽk, 1 {τ }, (1.12 s (,T V 16 For a deailed reamen of opimizaion problems in he financial conex, see Korn and Korn (1999, Chaper V. 17 Duffie and Lando (21 use Ṽ i := V i exp(z i insead. This is equivalen in erms of informaion, bu seems counerinuiive as in ha case he expecaion of Ṽ i is no V i.

38 1.6 Hybrid models 32 where ln V B V = ln V B ln V k m( k σ(b B k = ln V B ln Ṽ k + Z k σ2 Z 2 m( k σ(b B k. Applying Lemma B.1.1 wih X 1 := Z k N (, σ 2 Z and X 2 := ln V k m k N (, σ 2 k yields he represenaion k Z k = σ2 Z (ln Ṽ k m k + σ2 Z 2 σz σ Zσ ξ, σ2 k σ 2 Z + σ 2 k where ξ has a sandard normal disribuion and is independen of Ṽ k. We obain he decomposiion ln V B σ = ln V Ṽ Z 2 k ( σz σ Zσ k ξ σ(b σ2 k σ 2 B Z + σ 2 k + M(, k k where we se = ln Ṽ k σ 2 k σ 2 Z + σ2 k + k σ Zσ ξ σ(b σ 2 B Z + σ 2 k + M(, k, k M(, k = ln V B + σ2 Z ( σ2 Z m 2 k σz 2 + σ2 Z σ2 k 2 m( k = ln V B m + σ2 k σ 2 Z + σ2 k ( σ 4 Z 2σ 2 k σ2 Z 2 + m k This decomposiion of V ino independen random variables will enable us o calculae he desired probabiliy. Consider (1.12 = 1 {τ>} IP ( inf m(s + σ(b s B > s (,T k σ 2 k ln Ṽ k σz σ Zσ ξ σ(b σ2 k σ 2 B Z + σ 2 k + M(, k Ṽk, 1 {τ } k = 1 {τ>} IE IP ( inf m(s + σ(b s B > η Ṽk, 1 {τ }, B B k, ξ Ṽk, 1 {τ } s (,T. wih σ 2 k η := ln Ṽ k σz σ Zσ k ξ σ(b σ2 k σ 2 B Z + σ 2 k + M(, k. k Because η is measurable w.r.. B := σ(ṽ k, 1 {τ }, B B k, ξ and B s B is independen of B, we can apply equaion (B.2. Recall ha equaion (B.2 yields for c < IP ( inf m(s + σ(b s B > c s (,T = Φ ( m(t c σ T e 2cm/σ 2 Φ ( m(t + c σ T

39 1.7 Marke Models wih Credi Risk 33 and zero oherwise. Therefore we obain ( (1.12 = 1 {τ>} IE 1 {η<} Φ ( m(t η σ T e 2ηm/σ2 Φ ( m(t + η σ T Furhermore, η is (condiionally on Ṽ k normally disribued wih mean ln Ṽ k Ṽ k. (1.13 σ 4 Z 2σ 2 k σ2 Z 2 σ 2 k ln σz M(, k = ln V B m + σ 2 Ṽ k + m k + k σ2 k σz 2 + =: µ η ( k σ2 k and deerminisic variance σz 2 σ2 k σz σ 2 ( k = σ 2 σ4 2 k σ2 k σz 2 + =: σ σ2 η. 2 k I is easy o check ha k implies σ 2 η. One of he main asserions of Duffie and Lando (21 is ha his imperfec informaion model resuls in a hazard rae model wih a cerain hazard rae λ. Our objecive is o compue his hazard rae explicily. The calculaions are posponed o he appendix and in Lemma B.3.2 we come up wih he hazard rae λ = 1 {τ>} T T = ln = 1 {τ>} 1 2 2π ( µη σ 2 IP(τ > T H σ 2 η + m exp ( µ2 η. 2ση Marke Models wih Credi Risk Schönbucher (2 discusses he framework for a defaulable marke model. The difference beween he marke models and he coninuous mauriy models is ha marke models rely only on a finie number of bonds, whereas coninuous mauriy models assume a coninuiy of bonds raded in he marke, ha is bonds for all mauriies in a cerain range. As a maer of fac, many imporan variables are no available in marke models as, for example, he shor rae or coninuously derived forward raes, which form he basis for he seing in Heah, Jarrow and Moron (1992. Inroducions o marke models wihou defaul risk can be found for example in Brace, Gaarek and Musiela (1995, Rebonao (1996 or Brigo and Mercurio (21. Assume we are given a collecion of selemen daes T 1 < < T K, he enor srucure, which denoes he mauriies of all raded bonds. Denoe by B k ( := B(, T k he riskless bonds raded in he marke. The discree forward rae for he inerval T k, T k+1 is defined as 1 ( Bk ( F (, T k, T k+1 =: F k ( = T k+1 T k B k+1 ( 1.

40 1.7 Marke Models wih Credi Risk 34 The defaulable zero coupon bond is denoed by B(, T k. As a saring poin for modeling, i is assumed ha his is a zero recovery bond, i.e., a defaul he value of he bond falls o zero. Pu B k ( = B(, T k = 1 {τ>} B(, Tk. The defaul risk facor is denoed by D k ( := B k ( B k (. If here exiss an equivalen maringale measure Q we have D k ( = 1 B k ( IEQ exp( k = B k( B k ( IET k 1{τ>Tk } F ( = Q T k τ > Tk F r u du1 {τ>tk } F where Q T k denoes he Tk -forward measure 18 and IE T k he expecaion w.r.. his measure. So D k ( denoes he probabiliy ha, under he forward measure, he bond survives ime T k. Remark In a recovery of reasury model 19 he defaulable bond is modeled as a sum of zero recovery bond B (, T and a risk-free bond B k ( = wb k ( + (1 wb k(. We immediaely conclude ha in his case ( D k ( = wq T k τ > Tk F + (1 w. Define H(, T k, T k+1 := H k ( = 1 ( Dk ( T k+1 T k D k+1 ( 1. To simplify he noaion we wrie B 1 for B 1 ( (similarly for F, D, H and T j+1 T j = δ j. This leads o he following decomposiion B k = B k 1 1 j=1 B j+1 B j = B k 1 B j+1 B j B j+1 1 B j+1 B j B j j=1 k 1 = D 1 j=1 k 1 D j+1 B j+1 B 1 D j B j=1 j k 1 ( 1 = D 1 B δj H j (1 1 + δ j F j. j=1 18 The T k -forward measure is he risk neural measure which has he risk-free bond wih mauriy T k as numeraire. For deails see Björk ( See Secion

41 1.7 Marke Models wih Credi Risk 35 The discree forward raes of he defaulable bond are spli ino a risk-free par and a risky par which is represened by he discree-enor hazard rae H. Defining he credi spread we immediaely obain S k ( = S(, T k, T k+1 := F k ( F k (, S k ( = 1 ( Bk 1 1 ( Bk 1 δ k B k+1 δ k B k+1 = B k B k+1 1 δ k ( Bk B k+1 B k+1 B k 1 = (1 + δ k F k H k. The main moivaion for marke models was o reproduce Black-like formulas for prices of caps and swapions. This was paricularly possible in he so-called LIBOR-marke models. The basic assumpion in hese models is ha he discree forward rae has a log-normal disribuion. There are also oher models, see, for example, Andersen and Andreasen (2. Schönbucher (2 concenraes on LIBOR-like models and assumes df k ( F k ( ds k ( S k ( = µ F k ( d + σf k dw( = µ S k ( d + σ S k dw(. Here W denoes a N-dimensional sandard Brownian moion, whereas σ k are consan vecors and µ k are adaped processes. Alernaively, also he dynamics of H could be specified and he dynamics of S derived. Since H k = S k /(1 + δ k F k, we obain dh k ( = 1 (1 + δ (1 + δ k F k 2 k F k S k (µ S k ( d + σ S k dw S k δ k F k (µ F k ( d + σ F k dw S k δ k F k σ S k σ F k d S k + (1 + δ k F k 3 δ2 kfk 2 σk F σk F d S k =... d + σ S k δ kf k σ F k dw 1 + δ k F k 1 + δ k F k =: H k ( µ H k ( d + σ H k ( dw. Noe ha σ H k is no a consan, bu an adaped process wih σ H k ( = σs k δ kf k ( 1 + δ k F k ( σf k.

42 1.8 Commercial Models 36 Using Iô s formula we obain for he dynamics of he defaulable forward raes d F k ( = ds k ( + df k ( + d < S k, F k > = S k µ S k + F k µ F k + S k F k σ S k σ F k d + ( S k σ S k + F k σ F k dw =: Fk ( µ Fk ( d + σ Fk ( dw. The main reason for he populariy of he marke models lies in he agreemen beween he model and well-esablished marke formulas for basic derivaive producs. Therefore he model is usually calibraed o acual marke daa and aferwards used, for example, o price more complicaed derivaives. For his reason he dynamics are direcly modeled under he risk-neural measure, or even more convenienly, under he T k -forward measures. In search of somehing analogous for marke models wih credi risk, he T k -survival measure urns up naurally. I is he measure under which he defaulable bond B k ( becomes a numeraire. The T k -survival measure Q k is defined by he densiy L k := exp( k r s ds1 {τ>tk } B k ( = d Q k dq. Noe ha he densiy has Q-expecaion 1 bu becomes zero if he defaul happens before T k. In view of his, Qk is no equivalen o Q bu only absoluely coninuous w.r.. Q. A his poin differen changes of measures can be obained. Changes from he survival o he forward measure and he analogy of he spo LIBOR measure in a credi risk conex are also discussed in Schönbucher (2. Finally, consider an F T -measurable claim X T, which is paid only when τ > T. Assuming zero recovery, hen his claim can be valued by he following resul, see Bielecki and Rukowski (22: S = B(, T Ēk( XT F. Here Ēk denoes he expecaion wih respec o Q k. 1.8 Commercial Models The models presened in his secion, he so-called commercial models, are quie differen from he models presened up o now. These models were developed by several companies and are widely acceped in pracice. They all offer an implemened sofware, bu he complee procedure of his implemenaion is published only for some models.

43 1.8 Commercial Models The KMV Model ( CrediMonior The procedure of KMV is based on Meron s approach (see Secion and combines i wih hisorical informaion via a saisical procedure. KMV do no publish he exac procedure implemened in heir sofware bu he following illusraive example may be considered o be very close o heir approach. In Meron s model he firm value of he company was assumed o be observable. In realiy his is unforunaely no he case. Usually shares of a company are raded bu he real firm value is even difficul o esimae for inernals. Using he raded shares as an esimae of he unknown firm value daes back o Modigliani and Miller, see Caouee, Almann and Narayanan (1998, p. 142 p.p. for more informaion. The share is viewed as a call opion on he firm value, where he exercise price is he level of he company s deb. Wih he dynamics chosen as in Meron s model and denoing by D he deb level a ime T, he value of he shares E corresponds o he Black-Scholes formula where he consans d 1, d 2 are E = V Φ(d 1 De r(t Φ(d 2, d 1 = ln V De r(t σ2 (T σ T d 2 = d 1 σ T. Invering his relaion resuls in he firm value. Also an esimae for he volailiy of he share resuls in an esimae of he firm s value. KMV found ha in general firms do no defaul when heir asse value reaches he book value of heir oal liabiliies. This is due o he long-erm naure of some of heir liabiliies which provides some breahing space. The defaul poin herefore lies somewhere in beween he oal liabiliies and he shor-erm (or curren liabiliies. For his reason se defaul poin := shor-erm deb + 5% long-erm deb. In he nex sep hey calculae he disance-o-defaul DD = firm value defaul poin firm value vola of firm value. Finally KMV obains he defaul probabiliy from daa on hisorical defaul and bankrupcy frequencies including over 25, company-years of daa and over 4,7 incidens of bankrupcy 2. 2 See Crosbie and Bohn (21 for furher informaion.

44 1.8 Commercial Models Moody s Besides Meron s approach, which is ofen saed as coningen claims analysis (CCA, here are saisical approaches, pioneered by Alman (1968, which predic defaul evens using marke informaion and accouning variables via economeric mehods. Moody s public firm risk model bridges beween hese models and is herefore named a hybrid model. The procedure, as described in Sobehar and Klein (2, uses a varian of Meron s CCA as well as raing informaion (if available, cerain repored accouning informaion and some macroeconomic variables o represen he sae of he economy and of specific indusries hrough logisic regression. On his basis hey provide a one-year esimaed defaul probabiliy (EDP CrediMerics CrediMerics was originally developed by J.P. Morgan and belongs o RiskMerics Group since The procedure is oally published o clarify he model and he used daa are provided in he Inerne. The arge of CrediMerics is he valuaion of a whole porfolio. This includes differen asses and derivaives like loans, bonds, commimens o lend, financial leers-of-credi, receivables and marke driven insrumens like swaps, forwards and opions. The deerminaion of he acual price of he porfolio proceeds in hree seps. Firs he probabiliy of a defaul is deermined, second he probabiliy of changes in raing (which direcly resuls in a differen price and hird he deerminaion of he changes in value which are evoked by eiher a defaul or a change in raing. For he hree seps cerain inpus are needed. They can be obained by hisorical esimaion or are observable in he marke 21 : Transiion marices - ransiion probabiliies for changes in raing, Recovery raes in defaul - ordered by senioriy, counries and secors, Risk-free yield curve, Credi spreads - for all mauriies and raings. The ransiion marices are also provided by Moody s and Sandard & Poor s and herefore have o be lised separaely (Moody s raes in eigh and Sandard & Poor s in 18 classes. In our example we consider he Table 1.1. Observe ha here are some unusual figures in his able. For example, he probabiliy ha a company raed CCC is raed AAA afer one year equals.22 %. This seems o be unusually high in comparison o he oher enries. As here are few CCC raings his 21 See

45 1.8 Commercial Models 39 Table 1.1: The able displays he ransiion probabilies (in % for he ime horizon of 1 year. Raing (now Raing in 1 year AAA AA A BBB BB B CCC D AAA AA A BBB BB B CCC seems o be a consequence of an excepional even. Also criical is ha he probabiliy o defaul for a company raed AAA or AA equals zero. For sure here is a small bu posiive probabiliy ha such an even may happen. A his poin smoohing algorihms are recommended o obain a ransiion-marix which is well suied for furher calculaions; see Gupon, Finger and Bhaia (1997, p For he second se of daa, recovery raes are esimaed on a hisorical basis. Usually his informaion is provided by raing agencies. There are some sudies on recovery raes, and we discuss an example of Asarnow and Edwards (1995. CrediMerics hough uses jus mean and sandard deviaion. The use of a bea disribuion is discussed bu no implemened. Figure 1.3: Recovery Raes

46 1.8 Commercial Models 4 The senioriy of he bond cerainly has a significan influence on he recovery rae. Table 1.2 illusraes his. Table 1.2: Senioriy mean (% SD (% Senior Secured Senior Unsecured Senior Subordinaed Subordinaed Junior Subordinaed CrediMerics also uses he acual erm srucure of ineres raes and observable credi spreads. As he arge is he valuaion of bonds in a year s horizon no only defaul informaion should be used bu also price changes due o raing changes. One needs o answer he quesion Wha will be he value of a bond raed XXX in a year?. This is done by calculaing sripped forward raes wih respec o he raing. Sripping is he procedure o calculae zero coupon prices from a se of bonds offering coupons. Assume for now ha he curren credi spreads do no change. The risk-free erm srucure provides forward raes and he curren credi spreads are added o obain he fuure (defaulable forward-raes. We show he full procedure in he conex of an example. We face he problem o price a BBB-raed senior unsecured bond wih mauriy 5Y and annual coupons of 6%. Face value is 1 USD. As described above one srips he bond prices o obain he defaulable forward zero coupon curve. We wan o explain his procedure in greaer deail using he figures in Table 1.3. Assume he bond has raing A a he end of he year. The forward value hen becomes F V = 6 + The oher forward values are % + 6 ( % ( % ( % 4 = Raing AAA AA A BBB BB B CCC Forward Value($ The resuls may be found in Table 1.4.

47 1.8 Commercial Models 41 Table 1.3: Caegory 1Y 2Y 3Y 4Y (in % AAA AA A BBB BB B CCC The value a defaul is assumed o be he mean of hisorical recovery values for senior unsecured deb. In he above calculaion we followed he CrediMerics Technical Documen. For he sandard deviaion hey do no include he esimaed sandard deviaion of he recovery raes. If his is incorporaed (SD for senior unsecured deb = 25.45%, see he able on he previous page one obains a sandard deviaion of 1.11 which is considerably higher. Table 1.4: Sae in 1Y Prob. (% Forward Value (F V F V 2 AAA AA A BBB BB B CCC Defaul mean/ SD:

48 1.9 Credi Derivaives Credi Derivaives In his secion we inroduce several ypes of derivaives ha relae o credi risk. Unless explicily menioned, we assume ha he proecion seller has no defaul risk. In realiy, srong correlaions beween proecion seller and underlying prove o be quie dangerous. The proecion seller migh defaul shorly afer he underlying and he proecion becomes worhless. Addiionally o he derivaives presened in his secion, here exis so-called vulnerable opions. These are derivaives whose wrier may defaul, hus facing a counerpary risk. They are considered, for example, in Ammann (1999 or Bielecki and Rukowski (22. We do no consider derivaives on large baskes like collaeralized deb obligaions or ohers. See Blum, Overbeck and Wagner (23 for more informaion. A credi defaul swap or a credi defaul opion is an exchange of a fee for a coningen paymen if a credi defaul even occurs. The fee is usually called defaul swap premium. The difference beween swap and opion is deermined by he way he fee is paid. If he fee is paid up-fron, he agreemen is called opion, while if he fee is paid over ime, i is called swap 22. The defaul even is no a precise noion. Quie conrary, he even, which riggers he paymen, is negoiable. I could be a cerain level of spread widening, occurrence of publicly available informaion of failure o pay or an even, ha he parners can agree upon. See Das (1998 for examples of credi derivaives and he underlying conracs. No surprisingly, erms of documenaion risk or legal risk arise in he conex of credi risk. If he payoff is some predeermined consan, he derivaive is called digial, for example defaul digial pu or defaul digial swap. There are also opions on a baske which have specific feaures. For example, a firso-defaul swap is based on a baske of underlyings, where he proecion seller agrees o cover he exposure of he firs eniy riggering a defaul even. The firs-o-defaul srucure is similar o a collaeralized bond or loan obligaion. Usually here are bonds or loans wih similar credi raings in he baske, because oherwise he weakes credi would dominae he derivaive s behavior. Like in he ineres rae case, here are opions wih early exercise possibiliy, called American, credi derivaives wih knock-in/ou feaures, opions direcly on he credi spreads or leveraged credi defaul srucures, see Tavakoli (1998. Also reduced loss credi defaul opions are menioned herein, which yields a way o reduce he cos of defaul proecion. In his conrac he proecion buyer sill akes a fixed percenage of he loss on a defaul even, while he furher loss is covered by he proecion seller. 22 See, for example, Tavakoli (1998, p.61 p.p..

49 1.9 Credi Derivaives Digial Opions In he case of a digial swap or opion he paymen, which is exchanged if he defaul even occurs wihin he lifeime of he opion, is fixed. Assume, for simpliciy, ha he payoff equals 1. There are wo possibiliies for he ime, when he payoff is exchanged, eiher a mauriy T of he opion or direcly a defaul τ: (i If he payoff akes place a mauriy, he price of he opion (usually called pu a ime, if here was no defaul before, equals 23 1 {τ>} P d (, T = 1 {τ>} IE exp( = 1 {τ>} B(, T Q T r u du1 {τ T } τ T. Remark The payoff of he digial defaul pu in his case is similar o he payoff of he zero recovery bond. In fac, if we denoe he defaulable bond wih zero recovery and mauriy T by B (, T, we obain T 1 {τ>} P d (, T = 1 {τ>} IE exp( = 1 {τ>} B(, T B (, T. r u du(1 1 {τ>t } So, once he price of he zero recovery bond is known, he price of he defaul pu can be easily calculaed. Economically spoken, as a defaulable pu and a zero recovery bond wih same mauriies guaranee he payoff 1, heir price mus be equal o he price of a risk-free bond, which is B(, T. (ii If he payoff is done a defaul, Theorem A.1.3 yields for, T τ 1 {τ>} P d (, T = 1 {τ>} IE exp( = 1 {τ>} IE = 1 {τ>} exp( r u du1 {τ T } s (r u + λ u duλ s ds s B(, s IE s exp( λ u duλ s ds.

50 1.9 Credi Derivaives 44 B(τ, T B(τ, T τ T P (, T Figure 1.4: Cash flows for a defaul pu. Defaul occurs a τ before he opion expires. The payoff is agreed o be he difference o an equivalen defaul-free bond, which is denoed by B(τ, T B(τ, T. The price of he defaul pu is denoed by P (, T and is paid iniially a Defaul Opions and Credi Defaul Swap To clarify he paymens aking place for a defaul opion or a credi defaul swap, consider figures 1.4 and 1.5. In he case of he defaul opion, he proecion buyer pays a fee upfron, which equals he price of he opion. For he credi defaul swap (CDS he premium S is paid a ime poins T 1,..., T n unil eiher mauriy of he conrac or defaul. There are wo srucural possibiliies for he defaul paymen Difference o par. If a defaul even occurs, he proecion seller has eiher o pay he par value (which we always assume o be 1 in exchange for he defauled bond, or pay he par value minus he pos-defaul price of he underlying bond. The payoff is equivalen o 1 B(τ, T, if τ T. 2. Difference o an equivalen bond. The payoff in he case ha a defaul even occurs is he value of an equivalen, defaul-free bond minus he marke value of he defauled bond. In his case he payoff equals B(τ, T B(τ, T, if τ T. In he case of a coupon bond, here is usually a proecion of he principal, and possibly of he accrued ineres. The firs sep in pricing he defaulable swap is he pricing of he defaulable opion wih he same payoff. The price of he opion, denoed by P (, T, yields he discouned value 23 For convenience we wrie IE ( for IE Q ( F and IE T ( for E QT ( F, when Q T is he T-forward measure. 24 See, for example, Das (1998, p. 63.

51 1.9 Credi Derivaives 45 1 B(τ, T S τ T Figure 1.5: Cash flows for a credi defaul swap. Defaul occurs a τ before he opion expires. The payoff is agreed o be he difference o par, 1 B(τ, T. The defaul swap spread, S, is paid regularly a imes T 1,..., T 4 (unil defaul. of he payoff a ime. The premium S is paid a imes T 1,..., T n, bu a mos unil a defaul even occurs. Denoing he price of a zero recovery bond by B (, T, his yields P (, T = n S B (, T i. i=1 Consequenly, he swap premium can be obained, once he price of he defaulable opion and he zero recovery bond prices are known, as S( = P (, T n i=1 B (, T i. (1.14 For example, if we assume recovery of reasury for he defaulable bond, we have τ P (, T = IE exp( which can be expressed using he defaul digial pu as P (, T = (1 δp d (, T. r u du (1 δ1 {<τ T }, As already menioned, his ges slighly more difficul if he underlying is a coupon bond, see Schmid (22 for deails Defaul Swapions A credi defaul swapion offers he righ, bu no he obligaion, o buy or sell a credi defaul swap a a fuure ime poin T for a pre-specified swap premium K. The conrac is knocked ou if a defaul of he reference eniy occurs before T. We refer o a credi defaul swap call (CDS call if he assigned righ is o buy a credi defaul swap and oherwise o a credi defaul pu (CDS pu. Credi defaul swapions are no ye sandard insrumens

52 1.9 Credi Derivaives 46 which are liquidly raded, bu, for example, Hull and Whie (22 repor ha a marke for such conracs is developing. Denoing he enor srucure of he underlying swap by T = {T 1,..., T n } and he price of he CDS call a ime by C S (, T, T, we obain for he payoff of he CDS call a mauriy T T 1 CS(T, T, T = + n S(T K B (T, T i 1 {τ>t }. S(T is he swap premium a ime T. For simpliciy we se he day-coun fracion o one 25. If he swap offers he replacemen of he difference o an equivalen defaul-free bond in he case of a defaul, he swap rae equals We conclude for he price of he CDS call C S (, T, T = IE exp( i=1 S(T = B(T, T n B(T, T n n. i=1 B (T, T i ( r u du B(T, T n B(T, T n K n i=1 +1{τ>T B (T, T i }. Oherwise, if difference o par is considered, he swap price depends on he recovery. In a recovery of reasury model, he swap rae, as shown in he previous secion, equals S(T = (1 δ P d(t, T n n i=1 B (T, T i. This yields ha he price of he CDS call can be compued via C S (, T, T = IE exp( Credi Spread Opions ( r u du (1 δ P d (T, T n K n i=1 + B (T, T i. A credi spread opion is an opion which depends on he credi spread, ha is he difference beween he yield of he underlying defaulable bond and he yield of a reference bond, which is usually assumed o be defaul-free. For example, a credi spread call wih srike (yield K a mauriy T has he payoff ( B(T, T e K(T T +, B(T, T where T > T is he mauriy of he underlying defaulable bond. 25 For a discussion on he differen day-coun fracions, see James and Webber (2, p. 51 p.p.. Wih arbirary day-coun fracion i we would have o consider n i=1 ib (T, T i.

53 1.9 Credi Derivaives 47 Thus he call is in he money if he yield of he defaulable bond is higher han he yield of he riskless bond plus he srike (yield K. We use coninuous compounding 26 of he yield rae, and noe ha his represens an annual yield, if he ime scale is denoed in eniies of 1 year. Schmid (22 discusses credi spread opions wih a knock-ou feaure. In his case a credi spread call opion wih mauriy T on an underlying defaulable bond wih mauriy T and srike K, knocked ou a defaul, has he payoff 1 {τ>t } ( B(T, T e K(T T B(T, T +. In conras o he opion-specific payoff, a credi spread swap wih srike K and mauriy T has he payoff B(T, T e K(T T B(T, T. To replicae he payoff of he credi spread swap, he seller buys a porfolio a ime, which consiss of he defaulable bond wih mauriy T and sells exp K(T T risk free bonds wih mauriy T. A replicaing argumen yields he value a ime of he above payoff o be B(, T B(, T exp K(T T. Consequenly, he credi spread swap premium, which has o be paid a imes T 1,..., T n, equals S( = B(, T e K(T T B(, T n i=1 B(, T. i If he credi spread swap is knocked ou a defaul of he underlying, he premium relaes o zero recovery bonds B (, T, which promise he par value, 1, if he reference bond B(, T did no defaul unil is mauriy T and zero oherwise. Then he premium equals S( = B(, T e K(T T B(, T n. i=1 B (, T i kh-o-defaul Opions Derivaives wih a kh-o-defaul feaure are quie common in he marke. For example, a firs-o-defaul pu covers he loss of he firs defauled asse in a considered porfolio. These ypes of producs offer a cheaper proecion agains losses, if one considers more han k asses o defaul in a cerain ime inerval as unlikely, and herefore offer ailormade credi risk profiles, which may be used o redisribue credi risk or release regulaory capial. 26 The relaion o he discree ime value of money concep is he following. The discouning facor for a ime period of T years are 1 (1 + y nt = e K T, if he yield y is paid n imes a year. This yields he relaion y = (ln K 1 n.

54 1.9 Credi Derivaives 48 Once a price for a kh-o-defaul pu is obained, he premium of a kh-o-defaul swap can be calculaed via formula (1.14. See Secion 1.5 for applicaions, where we already obained he following formula for he premium of a firs-o-defaul swap τ 1s (1 δie exp( r u du1 {τ S 1s ( = 1 1s T } {τ 1s >} m i=1 IE i. exp( r u du1 {τ 1s >T i }

55 Chaper 2 SDEs on Hilber Spaces This secion develops some heory of sochasic processes on Hilber spaces, in paricular, he Iô - calculus for such processes. A deailed reamen may be found in he work of Da Prao and Zabczyk (1992. We adap heir mehodology o our framework and provide an inroducion o sochasic analysis on Hilber spaces. The Iô - formula is exended from real-valued funcions o funcions which have values in a Hilber space. Noe ha here is a similar exension in Filipović (21, sec using a differen proof. For he ools from analysis and funcional analysis we refer o Dieudonné (1969, Yosida (1971 or Werner (2. The erm srucure of ineres raes and is evoluion may be described by he se of forward raes {f(s, : s }. If we fix he ime s, he forward rae curve x f(s, x appears as an elemen of a funcional space. Therefore, a sochasic process (f(s s which iself akes values in a funcional space may well serve as a model for he forward raes. To formulae he dynamics of he forward raes, we develop some mehodology for Wiener processes in funcional spaces. As poined ou by Yor (1974, here are fundamenal problems defining he sochasic inegral of a Banach space valued process, while Hilber spaces are more suiable. This leads us o sochasic processes wih values in Hilber spaces. For echnical reasons we always consider a finie ime horizon T IR, i.e., invesigae he process (f(s s,t. 2.1 Preliminaries Consider a separable Hilber space H wih an inner produc <, >. The space of linear, coninuous mappings from H ino iself is denoed by L(H. Noe ha L(H is a Banach space and for D L(H and h H we ofen wrie D h insead of D(h. The Borel σ-algebra B(H is he σ-algebra induced from he norm of H. We sar by defining normaliy for probabiliy measures in H. 49

56 2.1 Preliminaries 5 Definiion A probabiliy measure µ on (H, B(H is said o be Gaussian, if and only if for any h H here exis p, q IR wih q, such ha µ{x H, < h, x > A} = N (p, q(a, A B(IR. Here, N (p, q denoes he Gaussian measure on (IR, B(IR wih mean p and variance q. Second, we generalize he concep of mean and variance. Definiion For a Gaussian measure µ on (H, B(H he elemen m H such ha < h, x > µ(dx =< m, h > h H H is called he mean of µ. The symmeric, nonnegaive operaor D L(H wih < h 1, x > < h 2, x > µ(dx < m, h 1 > < m, h 2 > =< Dh 1, h 2 > h 1, h 2 H H is called he covariance operaor of µ. Here D is symmeric in he sense, ha < Dh 1, h 2 >=< Dh 2, h 1 >. As for IR n, mean and covariance operaor uniquely deermine µ. For a random variable wih disribuion µ we wrie ξ N (m, D and Cov(ξ := D. The connecion wih Definiion is he following. For h H and p, q IR such ha we have p =< m, h > and q =< Dh, h >. µ{x H, < h, x > A} = N (p, q(a, Remark If H = IR n, a measure is Gaussian, iff he characerisic funcion akes he form ϕ(λ = expiλ m 1 2 λ Σ λ, λ IR n, for appropriae m IR n and Σ IR n IR n. The appearing erms equal λ m = < λ, x > µ(dx and λ Σ λ = λ ( (x m (x m µ(dx λ = < λ, x >< λ, x > µ(dx < λ, m > 2.

57 2.1 Preliminaries 51 Coming back o linear operaors on H, we inroduce an elemenary bu imporan ool, he race. Definiion Consider an orhonormal basis {e k : k IN} of H. For any linear operaor D on H we define he race of D hrough r D := < De j, e j >, j=1 if he above series converges absoluely, and se he race equal o infiniy oherwise. The race is independen of he chosen basis. If < De j, e j > <, he operaor D is called race-class. We denoe he Banach space of race-class operaors by L 1 (H and is norm by 1, compare Da Prao and Zabczyk (1992, Appendix C. I can be shown ha he covariance operaor of a Gaussian probabiliy measure is a race class operaor, see Da Prao and Zabczyk (1992, Proposiion Noe ha for posiive D race-class already follows from r D <. In our applicaions H will be a space of funcions h : IR IR. For a sochasic process (X(s s which akes values in H we se X(s, := X(s(. Tha is, if we consider he process of forward raes (f(s, s, f(s, represens he forward rae a ime s wih mauriy, while f(s represens he whole erm srucure a ime s. Definiion For a symmeric, nonnegaive race-class operaor D L(H he H-valued process (X(s s is called a D-Wiener process if (i X( =, (ii X has coninuous rajecories, (iii X has independen incremens, (iv he disribuion of X(s 2 X(s 1 is a Gaussian measure on H wih mean and covariance operaor (s 2 s 1 D. If he considered probabiliy space admis a filraion (F s s saisfying he usual condiions 1, X(s is F s -measurable and X(s 2 X(s 1 is independen of F s1 for all s 2 > s 1, we say ha X is a D-Wiener process wih respec o (F s s. Propery (iv specifies he covariance srucure of (X s s. The covariance operaor of a cerain incremen, say X s2 X s1, may be decomposed ino a facor which depends only on ime, namely s 2 s 1, and an operaor D. The operaor D refers o he covariance in he Hilber space. For s 1 = and s 2 = s one migh hink of Cov ( X(s, 1, X(s, 2, so he second facor describes he covariance w.r.. he mauriy ( 1, 2 respecively. 1 This means, ha F conains all IP-null ses and he filraion is righ-coninuous, which is called he usual augmenaion of (F, see Revuz and Yor (1994.

58 2.1 Preliminaries 52 Compare o he case H = IR n. A Brownian moion wih covariance funcion (s Σ = (s AA is obained from a Brownian moion (B s s wih independen componens via (AB s s. As his procedure can no be ransferred o he infinie dimensional case, he covariance operaor always needs o be specified explicily, and his is why we speak of a D-Wiener process. We now develop he Eigenvalue expansion of a H-valued random variable ξ. This will be crucial if we consider linear operaors on ξ = X(s. Observe ha for any orhonormal basis {e k : k IN} of H and f H we have f = k < f, e k > e k, where he Fourier-coefficiens < f, e k > are real-valued random variables. For D L(H and f, g H we obain < Df, g > = < D k e k < f, e k >, l e l < g, e l >> = k,l < f, e k > < De k, e l > < g, e l >. I is ineresing o find a basis which simplifies he above expression. Assume ha D is a covariance operaor, in paricular, D is nonnegaive and race-class. The Eigenvecors of D form a complee orhonormal sysem {e k : k IN} while he Eigenvalues λ k form a bounded sequence of nonnegaive real numbers, such ha 2 D e k = λ k e k. (2.1 Because D is a race-class operaor, we have k λ k < and obain < Df, g > = k λ k < e k, f > < e k, g >. If we consider ξ = X(s, where (X(s is a D-Wiener process, his Eigenvalue expansion gives a useful represenaion. Proposiion Consider a D-Wiener process (X(s s and denoe by {e k : k IN} he Eigenvecors of D. Define β k (s :=< X(s, e k >. 1 Then, for λ k >, λk β k (s are muually independen Brownian moions. Moreover, we have he decomposiion X(s = β k (se k, (2.2 k=1 and he series in (2.2 converges in L 2 (Ω, A, IP. 2 See Da Prao and Zabczyk (1992, p.86. Noe ha he sysem {e k } cerainly depends on D. In he following we always refer o his paricular {e k } wihou sressing he dependence on D.

59 2.1 Preliminaries 53 Proof. Clearly β k is a coninuous, cenered Gaussian process. Then βk (s β l ( IE = 1 IE < X(s, e k >< X(, e l >, λk λ l λk λ l and because (X(s s has independen incremens, we obain IE < X(s, e k >< X(, e l > = IE < X(s, e k >< X(s, e l > According o iem (iv of Definiion 2.1.5, he disribuion of X(s is Gaussian wih mean and covariance operaor (s D. Therefore IE < X(s, e k >< X(s, e l > = < x, e k >< x, e l > µ Xs (dx =< (s D e k, e l >. and βk (s β l ( IE λk λ l = 1 λk λ l (s < De k, e l > = 1 λk λ l (s λ k δ kl = (s δ kl, where δ kl equals one if k = l and zero oherwise. We conclude ha he β k s are independen Brownian moions 3. Furhermore, yields IE(< X(s, e k >< X(s, e l > = s < De k, e l >= sλ k δ kl, IE m β k (se k 2 = s k=n and, because D is a race-class operaor, k λ k <. So he β k (se k form a Cauchysequence and (2.2 converges in L 2. m k=n λ k The above resul also yields a possibiliy o consruc a Wiener process from a series of independen Brownian moions (W k : For any orhonormal basis {e k : k IN} and posiive λ k such ha k λ k <, (2.1 deermines a covariance operaor, say D. Then a D-Wiener process is obained by puing X(s := λk W k (se k. k=1 Because of his, X is ofen called infinie dimensional Brownian moion. 3 See, for example, Remark in Karazas and Shreve (1988.

60 2.2 The Sochasic Inegral The Sochasic Inegral In his secion we aim o define he sochasic inegral wih respec o a D-Wiener process on a Hilber Space H. In he case where H = IR n, he inegral w.r.. an n-dimensional Brownian moion (B(s s wih covariance Marix Σ (see page 52 of a IR n IR n - valued sochasic process (σ(s s is well known and denoed by σ(s db s. Noe ha he marix σ(s is a linear mapping IR n IR n, operaing on B(s. Imiaing his, we consider inegrands which ake values in he space of linear funcions from H H. As before, for Φ L(H and f H we wrie Φ f for Φ(f. Definiion Consider he Hilber Space H and a process (Φ(s s,t, which akes values in L(H. Φ(s is called elemenary, if here exis = < 1 < < n = T and Φ k L(H, measurable wih respec o F k, such ha Φ( = and Φ( = Φ k for ( k, k+1, k =,..., n 1. In his case we define he sochasic inegral for, T by n 1 Φ(s dx(s := Φ k (X( k+1 X( k. k= Noe ha he sochasic inegral is iself a sochasic process which has values in H. Sochasic inegrals prove o be a powerful concep o describe he behavior of maringales. Denoe he norm 4 on H by. A sochasic process (X(s s,t wih E X(s < for all s, T is called a maringale w.r.. he filraion (F s s, iff IE( X F s = X s, for all s < T. Usually one considers filraions of he ype F s = σ(x : s. As a consequence of is independen incremens, a D-Wiener process (X(s s,t w.r.. (F s s is a maringale. This leads o he quesion, under which circumsances his propery is inheried by he sochasic inegral. The following proposiion considers such a case. 4 The norm in a Hilber space is induced by he inner produc, such ha h :=< h, h > 1 2 for h H.

61 2.2 The Sochasic Inegral 55 Proposiion For an elemenary sochasic process (Φ(s s,t wih values in L(H we have 5 Φ T := IE ( T r(φ(sd (Φ(sD 2 2 ds = ( IE Φ(s dx(s Furhermore, if Φ T <, hen he sochasic inegral Φ(s dx(s is a square-inegrable maringale for all T. Proof. Enhance he pariion by =: m. Seing k X = X( k+1 X( k we obain IE m 1 Φ(s dx(s 2 = IE = IE ( m 1 k= =: (1 + (2. k= Φ k k X 2 ( Φ k k X 2 + IE 2 m 1 i,j=,i<j < Φ i i X, Φ j j X > Considering he firs erm, noe ha, using expansion (2.2, we may wrie k X = X( k+1 X( k = k β i e i, where k β i = β i ( k+1 β i ( k and 1 λi β i are muually independen Brownian moions. This leads o (1 = = = m 1 k= m 1 IE < Φ k k X, Φ k k X > IE < Φ k k= m 1 k= ( IE i,j=1 k β i e i, Φ k i=1 i=1 k β j e j > j=1 Fk IE k β i k β j < Φ k e i, Φ k e j > As he 1 λk β k are independen Brownian moions, we have. Fk IE ( k β i k β j λi λj = IE k ( β i k ( β j λi λj = ( k+1 k λ i δ ij. (2.3 5 Using posiiviy and he Eigenvalue expansion of D, we define D 1 2 (x := k λk < x, e k > e k, see Werner (2, p Furhermore, for T L(H we denoe is Hilber space adjoin by T, see Werner (2, p. 28. Tha is, a, b H yield < T a, b >=< a, T b >.

62 2.2 The Sochasic Inegral 56 We conclude (1 = = m 1 k= m 1 ( IE ( k+1 k λ i < Φ k e i, Φ k e i > i=1 ( IE < Φ k D 1 2 ei, Φ k D 1 2 ei > k= i=1 ( = IE r (Φ(sD (Φ(sD 2 ds. ( k+1 k Analogously we obain (2 = 2 m 1 i,j=1,i<j ( IE < Φ i i X, Φ j j X > =. For he maringale propery we enhance he pariion furher by s =: m. Then s = m < = m so ha IE Φ(u dx(u F s = = s s Φ(u dx(u + IE Φ(u dx(u, Φ j j X F s m 1 j= m because of independen incremens and zero means of (X(s s,t. As a nex sep we wan o exend he sochasic inegral o more general funcions Φ. Therefore we look for a class of processes which can be approximaed by elemenary funcions, such ha a he same ime he maringale propery of he inegral is preserved. I urns ou ha he proper class is formed by cerain Hilber-Schmid operaors. Firs, consider he space H := D 1 2 (H, which, endowed wih he inner produc 6 < u, v > := k 1 < u, e k >< v, e k >=< D 1 2 u, D 1 2 v > λ k is a Hilber space. For an orhonormal basis {e k : k IN} of H, seing e k := D 1 2 e k yields an orhonormal basis {e k : k IN} of H. 6 Similar o D 1 2, we define D 1 2 (x := k=1 1 λk < x, e k > e k.

63 2.2 The Sochasic Inegral 57 Then, denoe by L 2 (H, H he space of all Hilber-Schmid operaors from H ino H, ha is, linear operaors T, wih < T e k, T e k >2 < k=1 for an orhonormal basis {e k : k IN} of H. Noe ha he inner produc < S, T > 2 := < Se k, T e k > k=1 induces he norm T 2 := ( k=1 T e k 2 1 2, and L 2 (H, H is again a Hilber space. See, for example, Werner (2, p. 268 p.p.. Wih he above noaions we define Φ T also for non-elemenary processes wih values in L 2 (H, H as Φ T := IE 1 Φ(s ds Because e k = D 1 2 e k we obain = IE k=1 < Φ(se k, Φ(se k > ds 1 2. Φ T = = = IE IE IE k=1 k=1 < Φ(sD 1 2 ek, Φ(sD ek > ds ( < Φ(sD Φ(sD 2 e k, e k > ds r ( (Φ(sD 1 2 (Φ(sD 1 2 ds 1 2. We hen have he following Lemma For a predicable process (Φ(s s,t wih values in L 2 (H, H and Φ T <, here exiss a sequence of elemenary processes Φ n, such ha Φ Φ n T as n. Proof. See Da Prao and Zabczyk (1992, Lemma 4.7.

64 2.3 Covariances 58 Now we define for predicable (Φ(s s,t wih values in L 2 (H, H and Φ T < Φ(s dx(s := lim Φ n (s dx(s. (2.4 n I can be shown ha his sochasic inegral is well-defined and furhermore a maringale if Φ T < (see Proposiion So we finally found he class of suiable inegrands which ensure ha he sochasic inegrals inheri he maringale propery, namely predicable processes wih values in L 2 (H, H which saisfy Φ T <. Finally he sochasic inegral may be, by use of a localizaion procedure, exended o sochasically inegrable processes, i.e., processes, for which IP ( T Φ(s 2 2 ds < = 1. Noe ha, in doing so, he maringale propery is los, bu sochasic inegrals sill remain local maringales. For a full reamen see Da Prao and Zabczyk (1992, p. 94 p.p Covariances In his secion we consider covariances of he previously defined sochasic inegrals. The following definiion is in analogy o Definiion Definiion For wo H-valued random variables X i wih mean m i, i = 1, 2, he symmeric operaor D L(H, such ha IE < X 1, f >< X 2, g > < m 1, f >< m 2, g >=< Df, g > is called he covariance of X 1 and X 2 and denoed by Cov(X 1, X 2. Proposiion Assume (Φ 1 (s s,t, (Φ 2 (s s,t are predicable processes wih values in L 2 (H, H, Φ 1 T < and Φ 2 T <. Then for all, T IE Φ i (s dx(s =, IE Φ i (s dx(s 2 <, i = 1, 2 and he covariance operaor equals for all, s, T ( Cov s Φ 1 (u dx(u, Φ 2 (v dx(v = = IE s ( Φ2 (ud 1 2 ( Φ1 (ud 1 2 du.

65 2.3 Covariances 59 Furhermore, IE < s Φ 1 (u dx(u, Φ 2 (u dx(u > = IE s r(φ 2 (ud 1 2 (Φ1 (ud 1 2 du. (2.5 Proof. As (Φ 2 (ud 1 2 and (Φ 1 (ud 1 2 are L 2 (H valued processes, he process (Φ 2 (ud 1 2 (Φ 1 (ud 1 2 akes values 7 in L 1 (H. Da Prao and Zabczyk (1992, p. 12 obain he inequaliy IE which ensures exisence of he inegral. (Φ 2 (ud 1 2 (Φ1 (ud du Φ 1 T Φ 2 T, (2.6 Furher on, consider elemenary processes Φ 1 and Φ 2. We proceed similarly o he proof of Proposiion Assume w.l.o.g. ha s and enhance he pariion by and s a he poins m and m, say. Then ( IE < Φ 1 (u dx(u, a >< s Φ 2 (vdx(v, b > (2.7 As for i j = m i,j=,i j m + + i= j= m+1 m ( i= IE ( IE < Φ 1 ( i i X, a > < Φ 2 ( j j X, b > m ( IE < Φ 1 ( i i X, a > < Φ 2 ( j j X, b > < Φ 1 ( i i X, a > < Φ 2 ( i i X, b > ( IE < Φ 1 ( i i X, a > < Φ 2 ( j j X, b > =. he firs wo sums vanish. Furhermore, i X N (, ( i+1 i D and we obain ( IE < Φ 1 ( i i X, a > < Φ 2 ( i i X, b > ( = IE < Φ 1 ( i = IE j,k=1 = IE j,k=1 i β j e j, a > < Φ 2 ( i i β k e k, b > j=1 k=1 < Φ 1 ( i e j, a >< Φ 2 ( i e k, b > IE ( i β j i β k F i < Φ 1 ( i e j, a >< Φ 2 ( i e k, b > δ jk λ k ( i+1 i = IE < Φ 1 ( i D 1 2 ek, a >< Φ 2 ( i D 1 2 ek, b > k=1 7 Here, L 1 (H is he Banach space of all race-class operaors in L(H, see Page 51. ( i+1 i. (2.8

66 2.4 Iô s formula 6 This yields s (2.7 = < IE (Φ 2 (ud 1/2 (Φ 1 (ud 1/2 du a, b >. So he conclusion holds for elemenary processes. Wih he bound (2.6 he general conclusion follows from an appropriae approximaion hrough elemenary processes. Observe ha equaion (2.8 yields (2.7 = m i= IE < (Φ 2 ( i D 1 2 (Φ1 ( i D 1 2 a, b > ( i+1 i and (2.5 follows immediaely. 2.4 Iô s formula The formula of Iô (1946 yields he chain rule for funcions of diffusion processes. In comparison o he fundamenal heorem of calculus here appears an unexpeced second erm. As he formula mainly relies on he Taylor formula his is a resul of he non vanishing second-order erm and leads o ineresing probabilisic inerpreaions. The reason for is appearance is due o infinie variaion of he Brownian moion. Ineresingly, here is a close analogue o processes in Hilber spaces which is derived in his chaper. We only cie he Taylor formula for Hilber spaces. A deailed reamen may be found in Dieudonné (1969. Consider a Hilber space H and an open subse A H. I may be recalled ha, if he derivaive of a coninuous mapping f : A H denoed by Df exiss, i is a coninuous and linear mapping form H ino H and herefore an elemen of he Banach space L(H. Furhermore, if he second derivaive D 2 f exiss, i is an elemen of L(H; L(H and a symmeric 8 mapping. The space L(H; L(H can be idenified 9 wih he space of coninuous bilinear mappings of H H ino H, denoed by L(H, H; H. As a resul of he mean value heorem we obain Taylor s formula: Theorem Assume f is a wice coninuously differeniable mapping of A ino H. If x + θ A for x, H and all θ, 1, we have where ζ is an elemen of, 1. f(x + = f(x + Df(x D2 f(x + ζ (,, 8 D 2 f is symmeric in he sense ha D 2 f (f, g = D 2 f (g, f. 9 By h (s, (h s.

67 2.4 Iô s formula 61 For he Iô-formula on Hilber spaces we consider a D-Wiener process (X(s s on H and a predicable process (Φ(s s wih values in L 2 (H, H, such ha Φ T <. Then he sochasic process S( = S( + Φ(s dx(s (2.9 is a square-inegrable maringale, as already menioned in he previous secion. Theorem For an open subse A of he Hilber space H, le f : A H be a funcion, whose firs and second derivaive is uniformly coninuous on bounded subses of A. For (S(,T, as in (2.9 we have for all, T IP-a.s. f(s( = f(s( + Df(S(u ds(u + Noe ha he firs inegral equals λ k D 2 f(s(u (Φ(u e k, Φ(u e k du. k=1 Df(S(u Φ(u dx(u. Proof. By a localizaion procedure we can resric ourselves o bounded (X(s s,t and (Φ(s s,t, see Da Prao and Zabczyk (1992, p. 16. Furher on, consider a pariion Π = {, 1,..., n } of, wih = < 1 < < n and denoe is mesh by Π := max 1 i n ( i i 1. Using he Taylor formula on Banach spaces and wriing ( (2 for (,, we obain n 1 f(s f(s = f(s( j+1 f(s( j = = j= n 1 Df(S( j (S( j+1 S( j D2 f( S j (S( j+1 S( j (2 j= n 1 Df(S( j j S D2 f(s( j ( j S (2 j= + 1 D 2 f( 2 S j D 2 f(s( j ( j S (2 = I + II + III, where we se S j := S( j + ζ j (S( j+1 S( j wih ζ j = ζ j (ω, 1. Considering I, we inend o approximae Y s := Df(S(s by he elemenary process n 1 Ys n := Df(S(1 {} (s + Df(S( j 1 (j, j+1 (s. j=

68 2.4 Iô s formula 62 Indeed, uniform coninuiy of he derivaive and he bounded convergence heorem yield Y Y n T = IE as he mesh of he pariion ends o zero. Y (u Y n (u 2 2 du Then, by Definiion (2.4, we have IP-a.s. for Π, I Df(S(s ds(s. The hird summand, III, converges o IP-a.s. for Π, because of coninuiy of he derivaive, using a similar argumen. Consider he second erm, II. We calculae he condiional expecaion of he summands IE (D 2 f(s( j ( j S (2 Fj = IE (D 2 f(s( j (Φ( j j X (2 Fj. (2.1 Using he Eigenvalue expansion of X, we obain (2.1 = (2.3 = k,l=1 Fj IE ( j β k j β l D 2 f(s( j (Φ( j e k, Φ( j e l ( j+1 j λ k D 2 f(s( j (Φ( j e k, Φ( j e k k=1 =: ( j+1 j J( j. (2.11 To show L 2 -convergence i suffices o prove ha he following expecaion converges o zero: n 1 ( 2 IE D 2 f(s( j ( j S (2 ( j+1 j J( j j= = n 1 j= ( IE D 2 f(s( j ( j S (2 2 (j+1 j 2 IE J( j 2. (2.12 In he las sep we used he fac ha he summands are independen from each oher and, due o (2.11, have zero mean. If we expand he firs summand via (2.2 and denoe D 2 f(s( j (Φ( j e k, Φ( j e l =: ξ j k,l

69 2.5 The Fubini Theorem 63 we ge n 1 2 IE D 2 f(s( j ( j S (2 j= = = n 1 { IE j= n 1 j= k,m=1 k,l,m,n=1 } Fj IE j β k j β l j β m j β n ξ j k,l ξj m,n λ k λ m ( j+1 j 2 IE(ξ j k,k ξj m,m + n 1 j= k=1 λ 3 k( j+1 j 2 IE ( (ξ j k,k 2. Second momens of ξ j k,l are bounded for any j, k, l because D 2 f iself is bounded by assumpion. So he las sum converges o zero as Π. For he second summand of (2.12 we conclude n 1 2 ( j+1 j 2 IE λ k D 2 f(s( j (Φ( j e k, Φ( j e k j= = = k=1 n 1 ( j+1 j 2 IE λ k ξ j k,k j= j= k=1 k,m=1 which also converges o zero as sup j ( j+1 j. n 1 ( j+1 j 2 λ k λ m IE ξ j k,k m,m ξj, Up o now we obained convergence in L 2. Considering a subsequence of {Π (n } n=1 yields he desired IP-a.s. convergence, c.f. Karazas and Shreve (1988, p The Iô-formula can be exended o processes of he ype S( = S( + µ(s ds + Φ(s dx(s, where (µ(s s,t is an adaped, H-valued process. Also he funcion f migh be imedependen. See Da Prao and Zabczyk (1992, p The Fubini Theorem The Fubini heorem is jus saed for convenience. For a proof, see Da Prao and Zabczyk (1992, p. 19 p.p.. Le (E, E denoe a measurable space and µ be a finie, posiive measure on (E, E. Furhermore, consider a predicable, measurable mapping 1 Φ(, ω, x : (, T ω E L 2 (H, H. 1 For deails on measurabiliy and predicabiliy in his case, see Da Prao and Zabczyk (1992, p. 19.

70 2.6 Girsanov s Theorem 64 Theorem Assume ha Φ(, ω, x T µ(dx <, for IP-almos all ω. Then i follows ha E Φ(, x dx( µ(dx = Φ(, x µ(dx dx(, IP-a.s. E E 2.6 Girsanov s Theorem Recall ha we already defined he Hilber space H = D 1 2 (H wih inner produc <, > and he induced norm by on page 56. Theorem Consider predicable process (µ(s s,t wih values in H and se Φ(s( := < µ(s, >. Assume ha ( IE exp Φ(s dx(s 1 2 µ(s 2 ds = 1, where (X(s s,t is a D-Wiener process under he measure P. Then he process X( := X( µ(s ds,, T is a D-Wiener process under he measure P, defined by d P := exp T Φ(s dx(s 1 µ(s 2 2 ds dp. Noe ha < µ(s, > is a linear mapping from H ino IR, hus Φ(s L(H, IR. This requires a slighly more general definiion of he sochasic inegral as obained up o now. Neverheless, his is achieved analogously and he reader is referred o Da Prao and Zabczyk (1992, p. 29, where he heorem is proved. The process ( X(,T is a so-called D-Wiener process wih drif (µ s s,t. The Girsanov heorem shows ha one obains a D-Wiener process wih zero drif under he equivalen measure P Under cerain circumsances, he Girsanov heorem already describes all equivalen measures, see Bogachev (1991.

71 Chaper 3 An Infinie Facor Model for Credi Risk Modeling credi risk may sar wihin he framework of Heah, Jarrow and Moron (1992 (henceforh HJM and hen be exended o credi risk. There are several ways o do his, and in he nex wo secions we presen an approach in a framework due o Duffie and Singleon (1999. We sar by formulaing he exension of he HJM framework o sochasic differenial equaions on Hilber spaces. Our presenaion uses he parameerizaion due o Musiela (1993, see also Bagchi and Kumar (2 or Filipović (21. In Secion 3.3 we presen an approach based on credi raings. We use a Markov model in combinaion wih wo differen recovery srucures. For a raing based recovery of marke value approach wih finiely many facors, see Acharya, Das and Sundaram (2, and for a raing based recovery of reasury value approach, see Bielecki and Rukowski (2. We exend boh models using SDEs on Hilber spaces. Furhermore, recen research in Özkan and Schmid (23 exends his o Lévy processes in infinie dimensions. The arbirage-free condiions are presened in a fashion which clarifies he connecion beween he defaulable spo rae, defaul inensiy and he recovery srucure. A his poin he quesion naurally arises, why o consider infinie dimensional models for he erm srucure of ineres raes. Tradiionally he infinie number of forward raes in a erm srucure model are defined via a diffusion driven by a finie number of Brownian moions. This choice enables analyical racabiliy and is usually jusified wih a view owards he empirical fac ha he firs hree principal componens describe 95% of he observed variance. However, as poined ou in Con (21, dealing wih ineres rae derivaives ypically involves expecaions of non-linear funcions of he forward rae curve. Therefore, a model which migh explain he variance of he forward rae quie well may sill lack some principal componens which have a non-negligible effec on he flucuaions of such derivaives. Anoher argumen owards infinie dimensional models arises in Chaper 4, namely ha a calibraion based on such a model may show beer numerical resuls and may help o avoid frequen re-calibraions, while analyical racabiliy is preserved. From now on we always consider he objecive measure P and a measure Q which is equivalen o P. The following heorems offer condiions, under which all discouned 65

72 3.1 An Infinie Facor HJM Exension 66 bond prices are maringales under Q. Then Q is called an equivalen maringale measure and, as shown by Björk, di Masi, Kabanov and Runggaldier (1997, he marke is free of arbirage. 3.1 An Infinie Facor HJM Exension To develop our model wih credi risk in infinie dimensions, we firs discuss he mehodology in he case wihou credi risk. Kennedy (1994 gives an ineres rae formulaion wih Gaussian random fields. This approach was exended o more general models using SDEs on Hilber spaces by Goldsein (1997, Sana-Clara and Sornee (1997 and Bagchi and Kumar (2. The framework we presen includes he firs wo and is a special case of he las. We derive he analogue of he drif condiion of Heah, Jarrow and Moron (1992 in an infinie dimensional seing. Saring wih a model under some measure Q, we derive a condiion under which Q is a maringale measure. The idea of he HJM approach is o model he dynamics of he forward raes iself raher han o model he dynamics of he insananeous ineres rae and hen derive he dynamics of he forward raes. The forward raes have a one-o-one correspondence o bond prices, which in he coninuous-ime case amouns o B(, T = exp f(, u du. Usually, he forward rae is modeled via an n-dimensional Brownian moion as df(, T = α(, T d + σ(, T dw, T, (3.1 where α(, T IR and σ(, T IR n form predicable processes. Noicing ha he forward-rae curve a ime, denoed by f(, :, T IR, is a funcion (of T, one could model a sochasic process f( which iself akes values in a funcional space. So he quesion arises which funcional space o choose. Usually here are forward raes up o a maximum ime-o-mauriy in he marke, say T. Consequenly, on can express he forward raes as f(, + x :, T, T IR. This leads o he so-called Musiela parameerizaion 1. One considers r (x := f(, + x, where he sochasic process (r,t akes values in a funcional space IR,T. Someimes we use r o denoe he spo rae, r (. 1 See Musiela (1993.

73 3.1 An Infinie Facor HJM Exension 67 I urns ou ha i is appropriae o consider sochasic differenial equaions on Hilber spaces. Throughou his chaper H sands for a separable Hilber space, and our inenion is o use a space of real-valued funcions on an inerval, T. A differen approach owards modeling he forward raes uses Gaussian random fields and is presened in Chaper 4. Firs we have o resae equaion (3.1 in erms of r (x. equivalen o (se x := T r (x = f(, T = f(, T + = r ( + x + α(u, T du + α(u, + x du + σ(u, T dw u Noe ha his equaion is σ(u, + x dw u. Le {S( IR + } denoe he semigroup of righ shifs, defined by S(g(x = g(x +, for any funcion g : IR + IR. This enables us o obain a consisen formulaion wihin a funcional seing by r (x = S(r (x + r = S(r + S(α(u, x du + S(α(u du + S(σ(u, x dw u S(σ(u dw u, where r, α(u and σ(u are iself elemens of H. In his formulaion he shif operaor arises naurally, as forward raes wih fixed mauriy correspond o forward raes wih decreasing ime-o-mauriy, see also Figure 3.1. In his secion we will generalize he inegral wih respec o W o Wiener processes on he Hilber space H. Consider sochasic processes α :, T Ω H and σ :, T Ω L(H; H, boh predicable w.r.. (F, saisfying IP( α(s ds < = 1 and σ T <. Furher on, assume ha (X( is a D-Wiener process as defined in he preceding chaper. Assume he forward rae dynamics o follow r = S(r + S(α(u du + S( σ(u dx(u. (3.2 Noe ha r akes values in H, so i represens he whole forward-rae curve, oherwise denoed by f(, + x. For α we could explicily wrie α(, x while his is no possible for σ. Sill, even if he index x does no appear direcly, i is no obsolee. As he las inegral is an elemen of H for all we can wrie i eiher as σ(u dx(u =: I( H (3.3

74 3.1 An Infinie Facor HJM Exension 68 T x + 1 x 1 Figure 3.1: The shif occurring in he Musiela parameerizaion: denoes curren ime, while T denoes mauriy. The forward rae f( 1, x relaes naurally o he forward rae f(, 1 + x = S( 1 f(, x. This leads o he shif erms in equaion (3.2. or direcly as I(, x. The shif operaor herefore yields S(I(, x = I(, + x. Using he Eigenvalue expansion of X, see equaion (2.2, we have he decomposiion X(u = β k (ue k, k=1 1 where βk (u denoe are independen, sandard Brownian moions on he real line. σ k (u, v := ( σ(u e k (v, and ge, in he above noaion, he following Then we Theorem Se α (u, T := α(u, v dv and u σ k (u, T := σ u k(u, v dv. Then all discouned bond prices are maringales iff α(, T = λ k σk(, T σ k (, T, T, T, + T. (3.4 k=1 Equaion (3.4 is ofen referred o as he drif condiion. Noe ha he drif condiion derived by Heah, Jarrow and Moron (1992 is he special case corresponding o λ k = 1 for k = 1 and zero oherwise. Inuiively, he drif condiion means ha, once he volailiy (and dependence srucure is specified, he dynamics under he arbirage-free measure is fixed. As a change of measure does no change he volailiy srucure, his could be esimaed using hisorical daa. A differen approach o obain he volailiy srucure uses a calibraion o marke prices, as discussed in deail in Secion 4.5.

75 3.1 An Infinie Facor HJM Exension 69 Forward raes observed in he marke have a ime-o-mauriy of up o 2 years or more, while he ime horizon for credi derivaives is relaively small. This implies for our model ha T > T, which plays a role, for example, in he drif condiion. Denoe he measure under which he above dynamics akes place by Q. If his measure is equivalen o he objecive measure P and he drif-condiion is saisfied, hen he marke is free of arbirage. Compleeness follows if he equivalen maringale measure is unique. Condiions under which his holds rue in he above seing are o he bes of our knowledge no ye available. Proof of Theorem In he Musiela parameerizaion, he bond price equals B(, T = exp( T r (v dv. Seing y(, T := r (v dv, we need o derive is dynamics. Using he noaion of he sochasic inegral via I(, v, see (3.3, we wrie y(, T = T r (v + dv T α(u, v + du dv T I(, v + dv. Wih y(, T = r (v dv we have ha T r (v + dv = y(, T + r (v dv T r (v + dv = y(, T + r (v dv. As we would like o apply Iô s formula o prove he maringale propery we need o have some dynamics of y w.r.. dx, which requires inerchanging he inegraion. Wih he aid of he Eigenvalue expansion we have I( = = = σ(u dx(u ( σ(u d β k (ue k k=1 k=1 σ(u e k dβ k (u. The las equaliy holds because σ T <, see Da Prao and Zabczyk (1992, p. 99. Boh I( and σ(u e k are elemens of H and may be wrien as I(, v and (σ(u e k (v,

76 3.1 An Infinie Facor HJM Exension 7 respecively. We obain he represenaion I(, v = k=1 σ k (u, v dβ k (u. (3.5 The inegrabiliy condiion σ T < allows us o use he sochasic Fubini Theorem 2.5.1, see Filipović (21. This yields T I(, v + dv = = T k=1 k=1 T Applying he obained represenaion leads o σ k (u, v + dβ k (u dv σ k (u, v + dv dβ k (u. y(, T = y(, T + r (v dv T α(u, v + du dv (3.6 k=1 T σ k (u, v + dv dβ k (u. Using he sandard Fubini heorem we can inerchange he order of he α-inegral. We wan o inroduce he spo rae, r ( ino he above formula. By is dynamics (3.2 we obain r v ( dv = v r (v + v α(u, v du + σ(u dx(u(v dv = r (v dv + v α(u, v du dv + v σ(u dx(u(v dv. Again using he decomposiion (3.5 and Fubini s heorem yields r v ( dv = = v r (v dv + α(u, v du dv + r (v dv + u α(u, v dv du + v k=1 k=1 u σ k (u, vdβ k (u dv σ k (u, v dv dβ k (u.

77 3.1 An Infinie Facor HJM Exension 71 Noe ha r (vdv also appears in (3.6. Thus, we obain y(, T = y(, T + k=1 u = y(, T + where we used α (u, T = u r v ( dv α(u, v dv du σ k (u, v dv dβ k (u r u ( du α (u, T du k=1 α(u, v dv du k=1 u σ k (u, T dβ k(u, α(u, v dv and σ k (u, T = u σ k (u, v dv dβ k (u σ k(u, v dv. To apply he Iô - formula we look for a represenaion in a more funcional analyic way. Define an operaor Φ :, T Ω L(H; H, by Φ(u f( := u σ(u f(v dv. Then Φ(u dx(u(t = = = Φ(u ek (T dβk (u k=1 k=1 u σ(u e k (v dv dβ k (u σk(u, T dβ k (u. k=1 Seing µ(u, := r u ( α (u, we obain y( = y( + µ(u du Φ(u dx(u. This is he represenaion of y ha we were looking for. The second sep is o derive he dynamics of he bond price B(, T = exp(y(, T. To apply Iô s formula, we define F : A H, g( exp(g(.

78 3.1 An Infinie Facor HJM Exension 72 Here A is chosen in a way, such ha exp(g(, defined by x exp(g(x, for all x IR, is again an elemen of H. Then we have B(, = F (y(( or B( = F (y(, respecively. We compue he firs and second derivaive of F. Firs, define for f, g H he produc of f and g by (f g( := f( g( and wrie g k := g g. } {{ } k imes Then F (g( = exp(g( = g( k k=1. The derivaive of g 2 is k! D(g 2 (x = 2x id, where id is he ideniy on H. This is rue, because for x, x IR and herefore g 2 (x g 2 (x 2x (x x = x x x x 2x (x x g 2 (x g 2 (x 2x (x x lim x x x x = (x x 2 = lim x x (x x 2 x x x x lim x x x x x x =. The derivaive of g n is easily obained by inducion and we may conclude as well as DF (g = F (g id D 2 F (g = F (g id id. Applying Iô s formula yields db( = DF (B(u µ( d Φ( dx( + 1 λ k D 2 F (B( (Φ( e k, Φ( e k d 2 k=1 = B( µ( d Φ( dx( + 1 λ k B( (Φ( e k (Φ( e k d. 2 k=1 Evaluaing B(, a mauriy T leads o db(, T = B(, T (r ( α (, T d σk (, T dβ k( k=1 λ k σk (, T 2 d. (3.7 k=1

79 3.1 An Infinie Facor HJM Exension 73 Define he discouning process D := exp( r u du. Noe ha as D is differeniable, i is of finie variaion. Applying he common Iô-formula 2 o he discouned bond price herefore yields dd B(, T = ( r D B(, T d + D db(, T = (r D B(, T ( r α (, T k=1 λ k σk(, T 2 d k=1 σk(, T dβ k (. (3.8 Noe ha we sress he dependence on (r ( r, which is in his case equal o. In he case wih credi risk we consider r ( insead of r ( and his erm will no vanish. Consequenly he discouned bond price is a maringale under σ T <, iff α (, T = 1 λ k σ 2 k(, T 2 d, T, + T. k=1 Using he definiions of α and σ we obain α(, u du = 1 2 k=1 λ k σ k (, u du 2 du d. Taking he parial derivaive w.r.. T, we ge α(, T = λ k σk (, T σ k(, T. k= Change of Measure Up o now we considered he model under a measure Q and obained condiions, under which Q is a maringale measure. In fac, he observed dynamics akes place under he objecive measure P, and we have o perform a change of measure o obain he riskneural dynamics, which is necessary for pricing and hedging. The main ools for doing so is he Girsanov Theorem Once he drif condiion is obained, he procedure for obaining Q is similar hroughou all models. Observe ha he dynamics remains he same under all measures, jus he properies of he considered processes change. In paricular, if (X(s s,t is a D-Wiener process under P, X(s := X(s s µ(u du 2 As D is of finie variaion, his equals he produc rule, compare Revuz and Yor (1994, p. 199 p.p..

80 3.1 An Infinie Facor HJM Exension 74 is a D-Wiener process under Q, if dq := exp Φ(s dx(s 1 2 µ(s 2 ds dp and Φ(s( := < µ(s, >. We obain he following Proposiion If here exiss a predicable process (µ(s s,t which saisfies he condiions for Theorem and σ( µ( (T = α(, T λ k σk (, T σ k(, T, for all, T and T, +T, hen he measure Q as defined above is an equivalen maringale measure. Proof. The dynamics of he forward raes equal r = S(r + = S(r + = S(r + =: S(r + k=1 S(α(u du + S( S(α(u du + S( σ(u dx(u σ(u d ( u X(u S( α(u σ(u µ(u du + S( S( α(u du + S( σ(u d X(u. µ(v dv σ(u d X(u Girsanov s heorem yields ha ( X(s s is a D-Wiener process under Q. Therefore, if he drif condiion for ( α(s is saisfied, Q is an equivalen maringale measure. The drif condiion reveals α(, T = α(, T σ( µ( (T = λ k σk(, T σ k (, T. k=1 Thus, he change o he risk-neural measure Q is possible and he marke is free of arbirage. If credi risk is incorporaed in his seing, he change of measure furhermore resuls in a change of he inensiy. This is also rue for he raings model of Secion 3.3, cf. Bielecki and Rukowski (22, Secions 4.4 and 7.2.

81 3.2 Models wih Credi Risk Models wih Credi Risk A his poin we add defaul risk o our model. In he HJM framework wih finie dimension his was firs considered by Duffie and Singleon (1999. In he following we exend heir resuls o infinie dimensions. Consider a hazard-rae model, ha is, for a given filraion (G of general marke informaion, he defaul ime τ admis an inensiy (λ which is adaped o (G. For deails see Appendix A. As previously, we consider a separable Hilber space H, whose elemens are inended o be funcions f :, T IR. The following assumpion is basic for he nex wo secions and summarizes he infinie dimensional seing for he defaulable forward raes. Assumpion (A1: Le ᾱ :, T Ω H and σ :, T Ω L(H; H be sochasic processes, which are predicable w.r.. (F and saisfy IP( ᾱ(s ds < = 1 and σ T <. Furhermore, assume ha he defaulable forward rae follows r = S( r + S(ᾱ(u du + S( σ(u d X(u, where ( X(s s,t is a D-Wiener process Recovery of Marke Value For mehods using SDEs he recovery of marke value model is paricularly well suied. In his model he dynamics before a defaul occurs is modeled analogously o he risk-free case. If a defaul occurs, say a τ, he bond loses a random fracion q τ of is pre-defaul value, where (q s s,t is a predicable process wih values in, 1. The remaining value is insananeously paid o he bond holder, and herefore no more subjec o defaul risk. The dynamics of he defaulable bond unil a defaul occurs is modeled by specifying he dynamics of he defaulable forward raes, denoed by r (x. Hence, 1 {τ>} B(, T = 1{τ>} exp( T r (u du. If he bond defauls wihin is lifeime is value a defaul is assumed o become 1 {τ T } B(τ, T = 1{τ T } (1 q τ B(τ, T. In conras o oher recovery models he value of he bond immediaely before defaul has some influence on he repaymen, which seems reasonable.

82 3.2 Models wih Credi Risk 76 The value of (1 q τ B(τ, T is immediaely available o he bond owner a defaul and no more subjec o any risk. Therefore, he value of he defaulable bond can be represened by B(, T = 1 {τ>} exp( T r (u du + 1 {τ } exp( r u du(1 q τ B(τ, T. τ Now we can sae he following Theorem Assume ha ᾱ(s, x is coninuous in s for any x, T and assumpion (A1 holds. Under he recovery of marke value model, discouned bond prices are maringales, iff he following wo condiions are saisfied on {τ > }: (i For any, T, T + T (ii For any, T ᾱ(, T = λ k σ k (, T σ k(, T. (3.9 k=1 r ( = r ( + q λ. (3.1 Proof. If we denoe he discouning facor by D = exp( r u du, he discouned gains process G(, T := D B(, T equals G(, T = 1 {τ>} D B(, T + 1{τ } exp r u du + = 1 {τ>} D B(, T + 1{τ } D τ (1 q τ B(τ, T = 1 {τ>} D B(, T + D s (1 q s B(s, T dλ s. τ r u du (1 q τ B(τ, T For he las represenaion we se Λ s := 1 {τ s}. The -dynamics of G(, T becomes dg(, T = d(1 Λ D B(, T + (1 q D B(, T dλ =: (1 + (2. Taking ino accoun ha Λ is of finie variaion he firs summand equals (1 = dλ D B(, T + (1 Λ dd B(, T. The compuaion of he discouned bond s dynamics is analogous o he risk-free case. Using formula (3.8 wih λ k, β k, respecively, we obain { dd B(, T = D B(, T ( r ( r ᾱ (, T + 1 λ k σ k 2 (, T 2 d k=1 } σ k (, T d β k (. (3.11 k=1

83 3.2 Models wih Credi Risk 77 r s ( is coninuous in s, because ᾱ(s, is coninuous by assumpion and X(s by definiion. Therefore, on {τ > }, we have B(, T = B(, T. By definiion of (λ s s, we have ha Λ s s τ λ s ds is a H-maringale, which implies ha d M := dλ 1 { τ} λ d = dλ (1 Λ λ d is he differenial of a H-maringale. See Bielecki and Rukowski (22, Lemma This leads o dg(, T = D B(, T + (1 q D B(, T dλ { +(1 Λ D B(, T ( r ( r ᾱ (, T + 1 λ k σ 2 k(, T 2 d k=1 } σ k (, T d β k ( = D B(, T { q d M k=1 σ k (, T d β k ( k=1 +(1 Λ q λ + r ( r ᾱ (, T k=1 } λ k σ k (, T 2 d. Hence he d-erm represens he drif. As (G(, T is a maringale, iff he drif is zero, i is a maringale, iff 1 {τ>} q λ + r ( r ᾱ (, T λ k σ k(, T 2 = T. (3.12 Noe ha his is needed only for τ. This is due o he assumpion ha he recovery value is insananeously paid o he bond holder and herefore here is no risky dynamics afer defaul. Consequenly, equaion (3.12 is rue under (3.9 and (3.1. For he converse, since his equaion mus hold for any τ T and he -erms equal zero if T = we obain (3.9 and hen (3.1. k=1 Remark If one prefers a drif condiion which does no depend on a paricular realizaion of τ, he equivalency in Theorem migh be dropped. Tha is, if condiions (3.9 and (3.1 hold rue for any, T and T, + T, discouned bond prices are maringales, because equaion (3.12 holds. Noe ha he converse follows only on {τ > }. The underlying measure is a maringale measure iff condiions (3.9 and (3.1 are saisfied, which implies ha he marke is free of arbirage. We are no able o conclude ha he marke is complee, because here is, o our bes knowledge, no uniqueness resul available ye. If his would be rue, resuls of Björk, di Masi, Kabanov and Runggaldier (1997 could be used o show approximae compleeness.

84 3.2 Models wih Credi Risk 78 Some simple Models wih infinie Facors In his secion we discuss some simple models in he above presened framework. Assuming ha σ(s :, T L(H; H is deerminisic immediaely resuls in a Gaussian model. In analogy o Vargiolu (2 a hisorical esimaion of he covariance srucure using he Karhunen-Loève 3 decomposiion is possible. The procedure requires wo seps. Firs, he covariance operaor is esimaed using hisorical daa. In he second sep he firs Eigenvecors /values are obained, say up o a number N. This resuls in a N-facor HJM model which is used as an approximaion of he infinie facor model. Le us consider he procedure in furher deail. Wih Proposiion he covariance operaor of r( becomes Var( r( = ( σ(sd 1 2 ( σ(sd 1 2 ds. Assuming we consider a ime inerval which is small enough so ha variaions of σ(s do no play a significan role, one could use 4 as an esimaor of D n ( n n 1 := 1 n where = n (or 1, respecively n/2. n r( i r( i i=1 ( σ(d 1 2 ( σ(d 1 2, Similar o Secion focusing on he error of a finie dimensional approximaion raher han pre-specifying he dimension naurally involves he Karhunen-Loève decomposiion in he following way. The firs n Eigenvalues and Eigenvecors of D n ( n can be obained as follows. Fix k H and define k n+1 := D n ( n k n. Then k n+1 iself is an elemen of H. Vargiolu (2 shows ha k n e 1 and k n+1 k n λ 1, as n. Using D 1 := D n ( n λ 1 e 1 e 1, and applying he procedure o D 1 yields e 2 and λ 2 and so on. 3 See, for example, Bogachev (1991, p. 55 p.p., Da Prao and Zabczyk (1992, p. 99 p.p. or Adler ( Here denoes he ensor produc of elemens of H. The decomposiion of a linear operaor D ino is Eigenvecors e k and Eigenvalues λ k hen can be wrien in he form D = k=1 λ k e k e k. See Reed and Simon (1974.

85 3.2 Models wih Credi Risk 79 The number of Eigenvecors, n, will be chosen such ha he desired precision is obained. Finally, we approximae ( σ(d 1 2 n k=1 λ 1 2 k e k, and his represens he approximaing n-facor classical HJM model. Con (21 also inroduces a quie simple model using sochasic processes in Hilber spaces, and shows ha cerain saisical feaures of he erm srucure of ineres raes, which were observed in empirical sudies, can be reproduced. In paricular, he model capures imperfec correlaion beween mauriies, mean reversion and he srucure of principal componens of erm srucure deformaions Recovery of Treasury There are differen models of recovery, as already discussed in Chaper 1. An alernaive o he recovery of marke value is he recovery of reasury formulaion, see Secion In his model, he defaul enails a reducion of he face value by a pre-specified consan. The reduced face value, denoed by δ, is assumed o be no more subjec o defaul risk and is paid o he bond holder a mauriy T. This is cerainly equivalen o paying δb(τ, T immediaely a defaul. Therefore he value of he defaulable bond in his model is B(, T = 1 {τ>} exp( T r (u du + 1 {τ } δb(, T, T. Theorem Assume a recovery of reasury model and he riskless bond marke o be arbirage-free. Under assumpion (A1, discouned defaulable bond prices are maringales, iff on {τ > } for any, T, T, + T and condiion (3.9 holds. ( B(, T r ( = r + λ 1 δ B(, T Defaul always yields loss of money, so a sensible choice of he model s recovery should imply δb(, T < B(, T, so ha he promised ineres of he defaulable bond r ( exceeds han he risk-free ineres rae, r. Proof. Wih he noaion of he previous proof, he discouned gains process in his model becomes G(, T = D B(, T = (1 Λ D exp( T r (u du + Λ δd B(, T

86 3.3 Models Using Raings 8 wih dynamics dg(, T = (1 Λ d(d exp( T r (u du D exp( T r (u dudλ +Λ δd(d B(, T + δd B(, T dλ. Taking ino accoun ha on {τ > }, exp( r (u du = B(, T, he value of d(d B(, T is given in equaion (3.11. This yields dg(, T = (1 Λ D B(, T {( r ( r ᾱ (, T k=1 D exp( T + Λ δd(d B(, T σ k (, T d β } k ( λ k σk (, T 2 d k=1 r (u du + δd B(, T d M + (1 Λ λ d = (1 Λ D B(, T { r ( r λ ᾱ (, T δd B(, T (1 Λ λ d + d M, } λ k σk (, T 2 d where we denoe he sum of all maringale erms by M. Noe ha D B(, T is a maringale, as we assumed he riskless bond marke o be free of arbirage. Consequenly he drif of (G(, T is zero, iff on {τ > } for all T = B(, { T r ( r λ ᾱ (, T k=1 } λ k σk(, T 2 + δb(, T λ k=1 = r ( r λ + δλ B(, T B(, T ᾱ (, T λ k σk (, T 2. k=1 Similar argumens as for Theorem yield he desired resul. 3.3 Models Using Raings As raings are readily available and a widely used ool in markes subjec o credi risk, a model should be capable of using his informaion. In his secion we lay ou he framework for a model in infinie dimensions ha incorporaes differen raing classes. We presen wo alernaive recovery srucures wih recovery levels dependen on he pre-defaul raing.

87 3.3 Models Using Raings 81 The basic assumpion of he nex wo secions describes he behavior of he defaulable forward raes wih respec o he curren raing. Assumpion (A2. Assume ha here are K 1 raings, where 1 denoes he highes raing and K 1 he lowes, while K is associaed wih defaul. Denoing by K = {1,..., K 1} he se of possible raings and puing K = K {K}, we assume ha he raing i forward rae saisfies for, T r i = S(r i + S(α i (u du + S( σ i (u dx i (u, where (X i (,T is a D i -Wiener process. Furhermore, α i :, T Ω H and σ i :, T Ω L(H; H are sochasic processes, which are predicable w.r.. (F and saisfy IP( ᾱ i (s ds < = 1 and σ i T <, for all i K. To exclude arbirage we furhermore assume ha r K 1 (x > > r 1 (x > r (x x, T. This corresponds o he fac ha higher raed bonds are more expensive han lower raed ones. If his would no be he case he raing of he bond would seem o be wrong. This could happen because of speculaive behavior or when he raing is delayed by some oher effecs and is no modeled here. The above relaion could be saed equivalenly by he condiion ha he iner-raing spreads mus be posiive, see Acharya, Das and Sundaram (2. The process which describes he curren raing of he bond, (C 1 (, akes values in K and is assumed o be a Markov process a his sae. Inuiively, his means ha he hisory of raings for his paricular bond does no influence he price nor defaul risk of he bond, only he curren raing does 5. We denoe by C 2 ( he previous raing before C 1 (. If here were no changes in raing up o ime we se C 2 ( = C 1 (. The defaul τ occurs a he firs ime, when he sae K is reached, τ := inf{ : C 1 ( = K}. Denoe he condiional infiniesimal generaor of C 1 given G under he measure Q by λ 11 ( λ 12 ( λ 13 ( λ 1K ( λ 21 ( λ 22 ( λ 23 ( λ 2K ( Λ = λ K 1,1 ( λ K 1,2 ( λ K 1,K 1 ( λ K 1,K ( Each (λ ij ( is a (G -adaped process saisfying he condiion λ ii ( = λ ij (, for all. (3.13 i,j K,j i 5 Noe ha his assumpion enables us o obain easier formulas, bu migh no be fulfilled in realiy. For example, if he bond has been downgraded, i is empirically observed ha furher downgradings are more likely han upgradings.

88 3.3 Models Using Raings 82 We sae he following proposiion which is proved, for example, in Bielecki and Rukowski (22, Prop Proposiion For any funcion f : K IR he following process is a maringale: M( = f(c 1 ( =: f(c 1 ( j=1 K λ C 1 (u,jf(j du (Λf(C 1 (u du. (3.14 For he raing ransiion o he defaul sae, using equaion (11.51 of Bielecki and Rukowski (22, we immediaely conclude Proposiion The process (M i ( is a maringale for any i K: M i ( = 1 {C 2 (=i,c 1 (=K} λ ik (u1 {C 1 (u=i} du. ( Raing Based Recovery of Marke Value Assume he raing i recovery rae (q i ( o be a nonnegaive sochasic process which is predicable w.r.. (F for all i K. In exension o Secion we model he defaulable bond wih raing ransiions for all, T and T, + T by B(, T = 1 {C1 ( K} exp( + 1 {C 1 (=K}q C2 ( τ T r C1 ( (u du B(τ, T exp( τ r u du. (3.16 We call his recovery modeling raing based recovery of marke value. This may be compared o he case wihou raings in Secion The advanages of he recovery of marke value model carry hrough o his model. A his poin we can compue he defaulable forward rae, he forward rae offered by he bond B(, T. Seing x := T we obain r (x = T ln B(, T = = 1 B(, T T 1 B(, T T B(, T 1 {C 1 ( K} exp( +1 {C 1 (=K}q C2 ( τ T r C1 ( (u du B(τ, T exp( τ r u du.

89 3.3 Models Using Raings 83 Compuing he derivaive yields r (x = 1 1 {C B(, T 1 ( K} exp( = 1 {C 1 ( K}r C1 ( +1 {C 1 (=K}q C2 ( τ T exp( r C1 ( (u du r C1 ( (T τ r u du exp( (x + 1 {C 1 (=K}r C2 ( τ (x +. T τ r C2 ( τ (u du r C2 ( τ (T τ Ineresingly, his expression does no depend on he differen recovery raes, which is due o he fac ha he forward raes describe he behavior of relaive price changes. So he defaulable forward rae equals he forward rae wih respec o he bond s raing. If he bond defauled, he forward rae curve remains saic, as here is no furher movemen excep he risk-free ineres. Denoe B i (, T = exp( T r i (u du. Theorem Assume ha (A2 and (3.16 hold under he measure Q. Then discouned defaulable bond prices are maringales under Q iff he following wo condiions are saisfied on {τ > }: (i For, T, T, + T, r C1 ( ( = r + (1 q C1 ( + (ii For, T, T, + T, K 1 j=1,j C 1 ( λ C 1 (,K( 1 Bj (, T B C1 ( (, T λ C 1 (,j(. (3.17 α C1 ( (, T = k=1 λ C1 ( k σ C1 ( k (, T σ C1 ( k (, T. (3.18 Under he condiions of he above heorem and, if Q is equivalen o he objecive measure P, Q is an equivalen maringale measure and so he marke is free of arbirage. Proof. Using equaion (3.16, we deermine he discouned gains process K 1 G(, T = D i=1 + D τ 1 {C 1 (=K} 1 {C 1 (=i}b i (, T K 1 i=1 1 {C 2 (=i}q i τ B(τ, T.

90 3.3 Models Using Raings 84 Noe ha he indicaors have finie variaion, jus like (D, and herefore Iô s formula yields he dynamics For he las erm, dg(, T = + K 1 i=1 1 {C 1 (=i}d(d B i (, T K 1 D B i (, T d1 {C 1 (=i} i=1 + d ( K 1 1 {C 1 (=K,C (=i} 2 q i τ B(τ, T Dτ. i=1 q i τ B(τ, T D τ d1 {C 1 (=K,C 2 (=i} = q i Bi (, T D d1 {C 1 (=K,C 2 (=i}, as he indicaor changes only a = τ. Furhermore, because of coninuiy of he forward raes, B(τ, T = B C 2 (τ (τ, T. Using (3.14 wih f i (x = 1 {x=i} for i K, we have d1 {C 1 (=i} = d( M i ( + = d( M i ( + j=1 K λ C 1 (u,jf i (j du λ C 1 (u,i du = d M i ( + λ C 1 (,i d. (3.19 Analogously o he defaul-free case (see 3.11 he dynamics of each i-raed bond for, T and T, + T can be expressed as 6 (r d(d B i (, T = D B i i (, T ( r α i (, T k=1 k=1 λ i k σi k (, T 2 d σ i k (, T dβi k (. (3.2 6 As before, we use he abbreviaions α i (, T = α i (, u du σk i (, T = σ i ( e k (u du.

91 3.3 Models Using Raings 85 Use (3.15 o obain dg(, T = + + = K 1 i=1 K 1 i=1 {( 1 {C 1 (=i}d B i (, T r i ( r α i (, T k=1 } σk i (, T dβk( i D B i (, T d M i ( + λ C 1 (,i( d K 1 q i B i (, T D dm i ( + λ i,k (1 {C 1 (=i} d i=1 K 1 { ( D B i (, T 1 {C 1 (=i} r i ( r + q i λ i,k( i=1 +d M, α i (, T k=1 k=1 } λ i kσk i (, T 2 + λ C 1 (,i( d λ i k σi k (, T 2 d where we denoed he sum of he maringale pars by M. The d-erm yields he drif, and G(, T is a maringale, iff he drif is zero. We spli he drif ino wo pars. The firs par consiss of 1 {C 1 (=i} α i (, T k=1 λ i k σi k (, T 2, i K, which is equal o zero (see equaion (3.12, iff on {C 1 ( = i} α i (, T = k=1 λ i k σi k (, T σi k (, T. Hence, condiion (3.18 follows. The second par yields = 1 {C 1 ( K} = r C1 ( K 1 + j=1 { B C1 ( (, T r C1 ( + K 1 j=1 } B j (, T λ C 1 (,j( ( r + q C1 ( λ C 1 (,K( ( r + q C1 ( λ C 1 (,K( B j (, T B C1 ( (, T λ C 1 (,j(, on {C 1 ( K}. (3.21

92 3.3 Models Using Raings 86 Using equaion (3.13 leads o K 1 j=1 B j (, T B C1 ( (, T λ C 1 (,j( = = = = K 1 j=1,j C 1 ( K 1 j=1,j C 1 ( K 1 j=1,j C 1 ( Finally we obain, on {C 1 ( K}, B j (, T B C1 ( (, T λ C 1 (,j( + λ C 1 (,C 1 (( B j (, T B C1 ( (, T λ C 1 (,j( (3.21 r C1 ( ( = r + (1 q C1 ( K j=1,j C 1 ( λ C 1 (,j( B j (, T B C1 ( (, T 1 λ C 1 (,j( λ C 1 (,K(. + K 1 j=1,j C 1 ( λ C 1 (,K( 1 Bj (, T B C1 ( (, T λ C 1 (,j(. Remark Again, if one prefers a drif condiion no depending on a paricular realizaion of (C 1 (, equivalency in Theorem canno be obained, see Remark In his case we require he above equaions o be saisfied for any i K, which leads o he following condiions: (i For, T, T, + T and i K r i ( = r + (1 q i λ i,k ( + (ii For, T, T, + T and i K, K 1 j=1,j i 1 Bj (, T λ B i i,j (. (3.22 (, T α i (, T = k=1 λ i k σi k (, T σi k (, T. ( Raing Based Recovery of Treasury Anoher way o model recovery is based on he recovery of reasury model developed by Bielecki and Rukowski (2. We adap heir framework bu exend heir model by considering infinie dimensional Wiener processes.

93 3.3 Models Using Raings 87 Wih he noaions of he previous secion, he defaulable bond wih raing ransiions is modeled for, T and T, + T by B(, T = 1 {C 1 ( K} exp( T r C1 ( (u du + 1 {C 1 (=K}δ C 2 (B(, T. (3.24 The raing i-recovery rae δ i is assumed o be consan. This recovery modeling is referred o as raing based recovery of reasury. Compuing he defaulable forward rae in his model yields r (x = 1 {C 1 ( K}r C1 ( (x + 1 {C 1 (=K}r (x. This is similar o he raing based recovery of marke value seing, and, of course, differences appear jus for he behavior afer defaul. In his model he defaulable forward rae afer defaul equals he defaul-free rae. Anyway, some par of he invesed money is los. Theorem Assume ha (A2 and (3.24 holds under he measure Q. Then discouned defaulable bond prices are maringales under Q, iff for, T, T, + T on {τ > } r C1 ( ( = r + K 1 j=1,j C 1 ( λ C 1 (,j( ( 1 Bj (, T B C1 ( (, T + λ C 1 (,K( ( B(, T 1 δ C 1 ( B C1 ( (, T (3.25 and condiion (3.18 holds. A his poin suiable parameers should ensure ha he sum in equaion (3.25 is posiive. The las erm is posiive for δb(, T < B(, T, as already noed in he case of recovery of reasury wihou raings. Proof. Using he noaion of Theorem 3.3.3, he dynamics of B(, T becomes d B(, T = + K 1 i=1 i=1 1 {C 1 (=i}db i (, T + B i (, T d1 {C 1 (=i} K 1 1 {C 1 (=K,C 2 (=i}δ i db(, T + B(, T δ i d1 {C 1 (=K,C 2 (=i}. Noe ha he differenials of he indicaors are 1 or 1 when a jump occurs and zero oherwise.

94 3.3 Models Using Raings 88 Using (3.19, we have K 1 i=1 B i (, T d1 {C 1 (=i} = K 1 i=1 B i (, T (d M i ( + λ C 1 (,i d. Furhermore, use (3.15 and (3.2 o obain d B(, T = K 1 i=1 K 1 i=1 (r 1 {C 1 (=i}b i i (, T ( αi (, T k=1 B i (, T λ C 1 (,i(d + d M i K 1 1 {C 1 (=K,C 2 (=i}δ i db(, T i=1 i=1 σ i k (, T dβi k ( K 1 δ i B(, T λ i,k (1 {C 1 (=i} d + dm i. k=1 λ i k σi k (, T 2 d Separaing he drif and maringale pars, his leads o d B(, T = K 1 i=1 1 {C 1 (=i}b i (, T r i ( αi (, T k=1 K 1 B i (, T λ C 1 (,i(d + 1 {C 1 (=K,C 2 (=i}δ i db(, T i=1 K 1 δ i B(, T λ i,k (1 {C 1 (=i} d i=1 K 1 1 {C 1 (=i}b i (, T i=1 k=1 σ i k (, T dβi k ( K 1 B i (, T d M i ( + δ i B(, T dm i (. i=1 λ i k σi k (, T 2 d

95 3.3 Models Using Raings 89 If we denoe he discouning facor by D he discouned bond price equals d(d B(, T = ( r D B(, T d + D d B(, T K 1 = r D i=1 1 {C 1 (=i}b i (, T + K 1 i=1 { K 1 + D 1 {C 1 (=i}b i (, T r( i α i (, T K 1 i=1 i=1 i=1 1 {C 1 (=K,C 2 (=i}δ i B(, T d k=1 B i (, T λ C 1 (,i(d + 1 {C 1 (=K,C 2 (=i}δ i db(, T K 1 } δ i B(, T λ i,k (1 {C 1 (=i} d + d M, λ i kσ i k (, T 2 d where we added he maringale pars up o d M. As he discouned risk-free bond is a maringale by assumpion, we conclude ha d(d B(, T = r D B(, T d + D db(, T is a maringale and so he 1 {C 1 (=K,C 2 (=i}-erms sum up o a maringale. We have d(d B(, T { K 1 = D + d M, i=1 K {C 1 (=i}b i (, T r + r( i α i (, T B i (, T λ C 1 (,i(d + i=1 denoing he maringale par by M. K 1 Therefore, D B(, T is a maringale, iff on {C 1 ( K} = B C1 ( (, T r + r C1 ( ( α C1 ( (, T K 1 j=1 i=1 B j (, T λ C 1 (,j( + δ C 1 (B(, T λ C 1 (,K. Furher on we consider he drif on {τ > }. This leads o = B C1 ( (, T r C1 ( ( r α C1 ( (, T δ C 1 (B(, T λ C 1 (,K + K 1 j=1 k=1 } δ i B(, T λ i,k (1 {C 1 (=i} d k=1 k=1 λ i kσ i k (, T 2 d λ C1 ( k σ C1 ( k (, T 2 λ C1 ( k σ C1 ( k (, T 2 B j (, T λ C 1 (,j(. (3.26

96 3.3 Models Using Raings 9 Again, we spli he above condiion in wo pars. The firs par consiss of α C1 ( (, T k=1 λ C1 ( k σ C1 ( k (, T 2, which is equal o zero (see equaion (3.12, iff α C1 ( (, T = k=1 λ C1 ( k σ C1 ( k (, T σ C1 ( k (, T. We consider he second par on {C 1 ( = i} wih i K. This yields K 1 = r i ( r B(, T + δ i B i (, T λ B j (, T i,k + B i (, T λ i,j(. j=1 Using equaion (3.13 and denoing q i,j (, T = { B j (, T /B i (, T δ i B(, T /B i (, T j K j = K leads o K r i ( = r q i,j (, T λ i,j ( j=1 K = r j=1,j i K = r + j=1,j i q i,j (, T λ i,j ( q ii (, T ( K λ i,j ( λ i,j ( ( 1 q i,j (, T, j=,j i and we conclude (3.25. Remark Similar o Remark 3.2.2, we obain he following condiion, which does no depend on C 1 bu also implies an arbirage-free marke ogeher wih (3.23: r i ( = r + K 1 j=1,j i λ i,j ( ( 1 Bj (, T B i (, T + λ i,k ( ( B(, T 1 δ i, i K. (3.27 B i (, T I seems naural ha condiion (3.1 exends o he raing model. Equaion (3.27 represens he relaionship under no-arbirage beween he ineres offered by a bond raed i, he likelihood of raing changes wih heir consequences o he bond s price, as well as wih defaul and recovery.

97 3.4 Pricing 91 An equivalen bu more concise version of (3.26 is obained on {C 1 ( = i} by seing a i (, T := r + r i ( αi (, T k=1 λ i k σi k (, T 2. Recall ha i K. Subsiuing λ ii ( = K j=1,j i λ ij(, we obain = B i (, T a i (, T + K 1 j=1 = B i (, T a i (, T + δ i B(, T λ i,k ( + B j (, T λ i,j ( + δ i B(, T λ i,k ( K 1 j=1,j i and hence obain he equivalen represenaion of (3.26, = B i (, T a i (, T + K 1 j=1,j i B j (, T λ i,j ( B i (, T K j=1,j i λ i,j ( ( B j (, T B i (, T λ i,j ( + ( δ i B(, T B i (, T λ i,k (. The firs par of his expression relaes o he drif of he bond iself, while he oher pars refer o he possible changes ino a differen raing class. A change of he raing immediaely enails a change of he bond s price. These are muliplied wih he rae, ha such a change may happen. See also Proposiion Noe ha D T B(T, T = exp( r u du 1 {τ>t } + δ C 2 (T 1 {τ T }. As we have shown ha he discouned bond price is a maringale his leads o B(, T = 1 D IE Q D T B(, T F = IE Q exp( r u du (1 {τ>t } + δ C 2 (T 1 {τ T } F. This is ofen saed as he bond price equals he condiional expecaion of he discouned payoff, which proves o be rue in our seing as well. 3.4 Pricing Pricing in credi risky models is usually done via compuaion of he expecaion of he discouned coningen claim, see for example Lando (1994, Duffie and Singleon (1999 or Bielecki and Rukowski (22. We presen a series of examples where we are able o obain closed form soluions in Secion 4.4.

98 3.4 Pricing 92 Saring wih a model for he defaul-free and defaulable forward rae, one uses equaion (3.1 wih a specific assumpion on he recovery srucure o obain a model for he defaul inensiy λ. Depending on he model i sill migh be difficul o evaluae he expecaion. There are wo main possibiliies, eiher o simplify he model o compue he expecaion explicily or oherwise o use Mone-Carlo mehods.

99 Chaper 4 Credi Risk Modeling wih Gaussian Random Fields In his chaper we use Gaussian random fields o model ineres raes and credi risk. Afer inroducing he basic erminology for Gaussian random fields, we presen he ineres rae model of Kennedy (1994, and a he same ime simplify some proofs. Secion 4.3 exends his model o also incorporae credi risk in differen recovery siuaions, while Secion 4.4 presens several explici pricing formulas and a hedging scheme of an opion on a defaulable bond under zero recovery. In Secion 4.5 we discuss wo calibraion mehodologies. The use of random fields in credi risk modeling seems new, and he approach using Gaussian random fields has he advanage of admiing explici pricing and hedging formulas. As shown in Pang (1998, he explici formulaion and flexibiliy of Gaussian random fields proves o be advanageous for calibraion issues in he ineres rae conex. 4.1 Preliminaries A random field is a sochasic process indexed by vecors. As in he univariae case, is disribuion is uniquely deermined by is finie dimensional disribuions (fidis. I is called Gaussian, if all fidis are Gaussian. For our purpose we only consider random fields in, T, T. More general versions can be found in Adler (1981. A deailed reamen of Gaussian measures on Banach spaces may be found in Bogachev (1991. Definiion A sochasic process (X(s, s,,t is called a Gaussian random field, if for all (s i, i, i = 1,..., n and n 1 he vecor admis a Gaussian law in IR n. (X(s 1, 1,..., X(s n, n A Gaussian random field may be fully described by is expecaion and covariance func- 93

100 ion 1 µ(s, := IEX(s, 4.1 Preliminaries 94 dependen s independen s 1 s 2 s 3 s 4 s Figure 4.1: The consequences of equaion (4.1: Incremens like X(s 4, X(s 3, and X(s 2, s X(s 1, s are assumed o be independen, while X(s 4, u X(s 3, and X(s 2, s X(s 1, s may sill be dependen (for u. c(s,, u, v := IE (X(s, µ(s, (X(u, v µ(u, v. Condiions on he covariance funcion imply a cerain smoohness of a random field, for example coninuiy. Lemma A Gaussian random field (X(s, s,,1 wih zero mean and coninuous covariance funcion has a.s. coninuous sample funcions, if here exis < C < and ε > such ha for all s 1, s 2, 1, 2, 1 IE X(s 1, s 2 X( 1, 2 2 For a proof see Adler (1981, p. 6. C log s 1+ε. Remark Assume, for example, ha (X(s, s,,t has zero mean and he covariance funcion IE(X(s 1, 1 X(s 2, 2 = c(s 1 s 2, 1, 2, (4.1 where c(s, 1, 2 is a deerminisic funcion. Then (X(s, s,,t has independen incremens in he s-direcion, as for s 4 > s 3 s 2 > s 1, s,, T, we have IE (X(s 2, s X(s 1, s(x(s 4, X(s 3, = c(s 2, s, c(s 2, s, + c(s 1, s, c(s 1, s, =. 1 Mean and covariance funcion are sufficien o describe he Gaussian random field, see Bogachev (1991, p. 52 p.p..

101 4.2 A Model wihou Credi Risk 95 Coming back o he seing in Hilber spaces, we may compue he covariance operaor D(s 1 := Cov(X(s 1 in many cases. Consider he case where H = L 2 (λ, he space of square-inegrable funcions w.r.. a measure λ. Then Fubini s heorem allows o inerchange expecaion and he inner produc, and he definiion of D yields IE(< X(s 1, f >< X(s 1, g > = IE X(s 1, xf(xλ(dx X(s 1, yg(yλ(dy = =! = < Df, g >. IE(X(s 1, xx(s 1, yf(xg(yλ(dxλ(dy c(s 1, x, yf(xλ(dxg(yλ(dy Therefore, Cov(X(s is he linear mapping D(s wih D(s : H H, f c(s, 1, f( 1 λ(d 1, (4.2 if he inegral exiss. Noe ha x c(s 1, 1, xf( 1 λ(d 1 is a funcion iself, bu D(s is no necessarily a race-class operaor, cf. Da Prao and Zabczyk (1992, Secion A Model wihou Credi Risk Before considering bonds wih defaul risk, we presen he framework wihou credi risk: he ineres rae case. This secion follows he approach of Kennedy (1994, while simplifying some proofs. The forward raes are modeled by a Gaussian random field and we obain a drif condiion under which he model is arbirage-free. In his secion we always consider a finie ime horizon T and a maximum ime-o-mauriy T, so ha an overall ime horizon T := T + T seems appropriae. The considered marke herefore consiss of bonds B(, T where, T and T, + T. Basic o his secion is he following Assumpion (B1: Le (X(s, s,, T be a coninuous, zero-mean Gaussian random field whose covariance funcion can be represened by a funcion c : IR 3 IR such ha Cov ( X s1, 1, X s2, 2 = c(s1 s 2, 1, 2. Also, c(, 1, 2 = (which refers o a deerminisic iniial erm srucure. Noe ha c is symmeric in he sense ha c(, 1, 2 = c(, 2, 1. The informaion available a ime is described by he σ-algebra F = σ ( X u,v : u, v u, u + T.

102 4.2 A Model wihou Credi Risk 96 This reveals a basic fac for forward raes, namely, as o f(, T, he wo indices and T are reaed differenly. The index represens he calendar ime, while T denoes mauriy. For a cerain ime, he whole ineres rae curve is known, ha is, all {f(, T : T, + T } are assumed o be observable in he marke a ime. Usually, his informaion is only available for a discree enor srucure T 1,..., T n, which is a basic moivaion o consider marke models. On he oher hand, one can eiher inerpolae hem, using splines or some parameric families, which is discussed in Filipović (21, or view he discree observaions as a parial informaion of he whole, bu unknown erm srucure. We ake his las viewpoin and, neverheless, model he whole erm srucure. Laer on, in he calibraion process, we accoun for he discree observaions by an approximaion argumen. Take µ(s, : IR 2 IR o be a coninuous funcion. The T -forward rae a ime is hen modeled hrough f(, T = µ(, T + X(, T. (4.3 This also specifies he dynamic of he bonds, since B(, T = exp( f(, u du. Remark The Gaussian HJM model is a special case of his. For a deerminisic drif µ(, T and volailiy σ(, T and df(, T = µ(, T d + σ(, T dw, he covariance funcion of he forward raes becomes Cov(f(s 1, 1, f(s 2, 2 = IE s1 σ(u, 1 dw u s 2 σ(v, 2 dw v = s 1 s 2 σ(u, 1 σ(u, 2 du c(s 1 s 2, 1, 2. Remark The model of Hull and Whie (199 can be formulaed in his framework. The spo rae hen saisfies dr = φ a r d + σ dw. Hull and Whie (199 showed ha he bond price can be expressed in an exponenialaffine form: B(, T = α(, T exp( β(, T r,

103 4.2 A Model wihou Credi Risk 97 where α is a deerminisic funcion, and β(s, = β(, β(, s. β(, s/ s I was shown by Schmid (1997, using ime ransformaions of Brownian moions, ha for appropriae deerminisic funcions f, g, τ he disribuion of r has he following form: where r( L = f( + g(w (τ(, τ( = As f(, T = / T ln B(, T, we obain Hence, as τ is increasing, σ(u 2 du. β(, u/ u Cov(f(s 1, 1, f(s 2, 2 = β(s 1, 1 1 β(s 2, 2 2 Cov(r s1, r s2. Insering he definiion of β and τ leads o Cov(r s1, r s2 = g(s 1 g(s 2 τ(s 1 s 2. Cov(f(s 1, 1, f(s 2, 2 = β(s 1, 1 1 β(s 2, 2 2 s 1 s 2 σ 2 u du. β(, u/ u In his chaper we always consider he objecive measure P and a measure Q which is equivalen o P. Assume (B1 o hold under Q. Girsanov s Theorem (see B.4.1 may be used o show ha (B1 also holds under P. Furhermore, he following heorems offer condiions, so ha all discouned bond prices are maringales under Q. Then Q is called an equivalen maringale measure and, as shown by Harrison and Kreps (1979, he marke is free of arbirage. Theorem (Kennedy Under he assumpions (B1, he measure Q is an equivalen maringale measure iff for all, T and T, + T µ(, T = µ(, T + c( v, v, T dv. (4.4 Proof. If all discouned bond prices are maringales, hen Q is an equivalen maringale measure. This is equivalen o IE (e R ru du B(, T F s = e R s ru du B(s, T s T s + T.

104 4.2 A Model wihou Credi Risk 98 We ge ( s exp r u du ( f(s, u du = IE exp r u du Fs f(, u du ( 1 = IE exp s r u du f(, u du + Fs f(s, u du ( = IE exp s f(u, u f(s, u du s Fs f(, u f(s, u du (4.5 s =: IE ( e A B Fs = IE(e (A+B. The las equaion holds because of Remark In our Gaussian seup he forward raes are normally disribued. Then also he inegrals of he forward raes and A and B are normally disribued. The above expecaion can herefore be calculaed using he Laplace-ransform of (A + B. By he definiion of he forward raes, (4.3, we obain IE(A + B = µ(u, u µ(s, u du + µ(, u µ(s, u du. s The variances equal ( Var(A = Var X(u, u X(s, u du = = s s s ( Cov X(u, u X(s, u, X(v, v X(s, v du dv c(u v, u, v c(s, u, v du dv and s s Var(B = c(, u, v c(s, u, v du dv, while he covariance of A and B becomes Cov(A, B = c(u, u, v c(s, u, v c(s, u, v + c(s, u, v dv du = s c(u, u, v c(s, u, v dv du. s

105 4.3 Models wih Credi Risk 99 We herefore obain for he variance of A + B Var(A + B = c(u v, u, v c(s, u, v du dv + c(, u, v c(s, u, v du dv s s +2 c(u, u, v c(s, u, v dv du = s c ( (u v, u, v c(s, u, v dv du = 2 s s u u=s v=s c ( (u v, u, v c(s, u, v dv du, } {{ } =v by he symmery of c. Equaion (4.5 requires IEexp (A + B = exp IE(A + B Var(A + B being equal o one. Therefore he exponen needs o be zero, which is equivalen o u c(v, u, v c(s, u, v dv du u=s v=s = = µ(u, u µ(s, u du µ(, u µ(s, u du s s µ(u, u µ(s, u du. Seing s = and and aking he parial derivaive wih respec o T, he following drif-condiion is obained µ(, T = µ(, T + c( v, v, T dv. 4.3 Models wih Credi Risk In his secion we consider a marke which has wo ypes of bonds. In conras o he riskless bonds, we denoe he price of a bond incorporaing a cerain defaul risk by B(, T. We model he forward raes of he riskless bonds as in he preceding chaper and he forward raes of he defaulable bonds similarly.

106 4.3 Models wih Credi Risk 1 Assumpion (B2: Assume ( X(s, s,, T is a zero-mean, coninuous Gaussian random field wih covariance funcion Cov ( Xs1, 1, X s2, 2 = c(s1 s 2, 1, 2, where c(, 1, 2 =. Furher on, assume ha incremens of he ype X(s 2, X(s 1, and X(s 2, X(s 1, for s 1 s 2 T are independen of G s1 = σ ( X(s,, X(s, : s s 1, s, s + T. The defaulable forward rae is modeled wih a deerminisic funcion µ(s, : IR 2 IR by for all, T and T, + T. f(, T := µ(, T + X(, T, (4.6 In he considered hazard rae framework he defaul inensiy (λ is assumed o be a nonnegaive (G -adaped process, see Appendix A. G can be inerpreed as he available informaion a ime Zero Recovery In a marke wih credi risk he dynamics of he bond relae o several facors. The riskfree ineres rae has cerainly a fundamenal influence on he behavior of he defaulable bond. Besides ha, he crediworhiness of he bond plays an imporan role. Crediworhiness is represened by he probabiliy of a defaul, respecively he defaul inensiy. The hird componen is he price of he bond afer defaul, named recovery. In his firs approach we consider he case of zero recovery, ha is, he case where he value of he bond afer defaul is zero. Hence only risk-free ineres and defaul inensiy remain o be considered. Theorem Assume (B2 and consider a measure Q which is equivalen o he objecive measure. If he defaulable forward raes ake he form (4.6 under Q and λ s ds < a.s., hen Q is an equivalen maringale measure iff he following wo condiions hold on {τ > }: (i For all T f(, = r + λ, (4.7 (ii For all, T and T, + T µ(, T = µ(, T + c( v, v, T dv. (4.8

107 4.3 Models wih Credi Risk 11 Proof. The case of risk-free bonds was already examined in he previous secion. Consider he risky bonds. For s T and all T s, s + T, we show 2 IE (e R ru du B(,! T F s = e R s ru du B(s, T, which is equivalen o IE exp ( r u du 1 {τ>} exp ( f(, u du Fs = 1 {τ>s} exp (! = exp ( s s r u du ( IE exp( r u du 1 {τ>s} exp ( s (r u + λ u du f(s, u du. f(, u du F s The firs equaliy follows using Lemma A.1.2. Condiion (4.7 implies, on {τ > s}, ( IE exp( ( IE exp( ( IE exp( s s s r u du s ( f(, u du F s = exp f(u, u f(s, u du f(u, u f(s, u du s f(s, u du f(, u f(s, u du F s = 1 f(, u f(s, u du = 1. (4.9 This is exacly (4.5, wih defaulable forward raes f(, T raher han f(, T. So we obain analogously he drif condiion (4.8 for he defaul case. For he converse, if Q is already an equivalen maringale measure, he price of he bond is he expecaion of is discouned payoff under Q, i.e., B(, T = IE exp( = 1 {τ>} IE exp( r u du1 {τ>t } F = 1 {τ>} B(, T IE T exp( = 1 {τ>} exp( r u + λ u du F λ u du F f(, u duie T exp( λ u du F. 2 All expecaions are wih respec o his equivalen maringale measure, if no saed oherwise. IE T denoes he expecaion w.r.. he T -forward measure, see page 16.

108 4.3 Models wih Credi Risk 12 This yields he defaulable spo rae (on {τ > s} f(, = ( ln 1 {τ>} IE exp( (r u + λ u du F T T = ( IE exp( T (r u + λ u dur T + λ T F = ( IE exp( T (r u + λ u du F T = = r + λ. Using his wih he assumpion ha Q is a maringale measure leads o (4.9 which implies (4.8 as in he risk-free case. We emphasize ha no assumpion on he dynamics of he risk-free ineres rae is needed excep (4.7. Neverheless, his equaion srongly connecs f, f and λ. The credi spread s(, T is he difference beween defaulable and risk-free rae, and we obain s(, T = f(, T f(, T = T ln IE ( 1 {τ>} exp( = T ln 1 {τ>} B(, T IE T ( exp( = T ln ( IET exp( An explici Model for he Inensiy (r u + λ u du F f(, T λ u du F f(, T λ u du F. (4.1 We examine an example in greaer deail. Jarrow and Turnbull (2 use he following model for dynamics of he inensiy under Q: λ = a ( + a 1 (W ( + a 2 (r. (W ( is a Brownian Moion which could represen he log-reurns of he asse value of a company or of an index. We assume (W ( o be independen of (r, he calculaions for correlaed processes being analogous. This model suggess a specific srucure for (λ, while up o now we considered primarily he forward raes. Neverheless, saring wih a random field model for he risk-free ineres rae and he assumpions on (λ, we will derive f and show ha i fis well in he above presened defaulable random field model.

109 4.3 Models wih Credi Risk 13 Remark Noe ha in his model he nonnegaiviy assumpion for (λ is violaed. Neverheless, his model can sill be seen as an approximaion of he model wih defaul inensiy ( (λ + if he probabiliy of he defaul inensiy being negaive is small, which should be rue for a suiable choice of he parameers. This is also a common problem in Gaussian ineres models, which admi negaive ineres raes wih posiive probabiliy, see Rogers (1995. To calculae f(, T via (4.1 we sar wih 3 IE T exp( λ u du T F = exp( T a (u du IE exp( a 1 (uw (u du T IE T exp( a 2 (ur u du. (4.11 This holds, because W λ is independen of r. The las facor becomes IE T T exp( a 2 (ur u du T = B(, T 1 IE exp( (1 + a 2 (ur u du = exp( T µ(, u + X(, u du IE exp( (1 + a 2 (ur u du. The exponens are normally disribued. To calculae he Laplace ransform we need heir expecaions and variances, which equal for he firs erm in (4.11 T IE a 1 (uw (u du = a 1 (uie(w (u F du = W a 1 (u du and T Var a 1 (uw (u du = a 1 (ua 1 (vie ( (W (u W (W (v W ( dudv = a 1 (ua 1 (v(u v dudv. For he las erm in (4.11 we obain T IE (a 2 (u + 1r u du = (a 2 (u + 1µ(u, u du + (a 2 (u + 1IE ( X(u, u du = (a 2 (u + 1 ( µ(u, u + X(, u du 3 For convenience we wrie IE ( for IE Q ( F, IE T ( for E QT ( F and Var ( for Var Q ( F.

110 4.3 Models wih Credi Risk 14 and T Var (a 2 (u + 1r u du = 2 = 2 This yields u u (4.11 = exp (a 2 (u + 1(a 2 (v + 1IE (X(u, u X(, u(x(v, v X(, v dv du (a 2 (u + 1(a 2 (v + 1 ( c(v, u, v c(, u, v dv du. a (u du a 1 (uw ( du + u a 1 (ua 1 (vv dudv 2 ( T a 1 (u du 2 + µ(, u + X(, u (a 2 (u + 1 ( µ(u, u + X(, u du + u For he credi spread we herefore obain (a 2 (u + 1(a 2 (v + 1 ( c(v, u, v c(, u, v dv du. s(, T = a (T + W (a 1 (T a 1 (T a 1 (uu du + a 1 (T µ(, T + (a 2 (T + 1µ(T, T + a 2 (T X(, T (a 2 (T + 1 Direcly from he drif condiion (4.4 we ge µ(t, T µ(, T = (a 2 (u + 1 ( c(u, u, T c(, u, T du. a 1 (u du ( c(u, u, T c(, u, T du, (4.12 which implies µ(, T + (a 2 (T + 1 µ(t, T = µ(, T + (a 2 (T + 1 µ(, T (a 2 (u + 1 ( c(u, u, T c(, u, T du a 2 (u ( c(u, u, T c(, u, T du = a 2 (T µ(, T (a 2 (T + 1 a 2 (u ( c(u, u, T c(, u, T du.

111 4.3 Models wih Credi Risk 15 We conclude s(, T = a (T + W (a 1 (T +a 2 (T µ(, T (a 2 (T + 1 a 1 (T a 1 (uu du + a 1 (T =: µ(, T + W (a 1 (T + a 2 (T f(, T, µ(, T being a deerminisic funcion. Since a 2 (u ( c(u, u, T c(, u, T du f(, T = f(, T + s(, T, a 1 (u du + a 2 (T X(, T he defaulable forward rae akes he form (4.6 wih a Gaussian random field X, which has he covariance funcion Cov(X s1, 1, X s2, 2 = a 1 ( 1 a 1 ( 2 (s 1 s 2 + (1 + a 2 ( 1 (1 + a 2 ( 2 c(s 1 s 2, 1, 2. So his model fis well ino our seup Recovery of Treasury Value In his recovery model he bondholder receives he payoff 1 {τ>t } + (1 w1 {τ T } a mauriy. I is useful o spli he bond s value ino wo pars, a zero recovery bond and a riskless bond B(, T T V = w B (, T + (1 wb(, T. For no-arbirage we need for all s T and T s, s + T IE s exp( r u du ( w B (, T + (1 wb(, T = w B (s, T + (1 wb(s, T s ( w IE s exp( r u du B (, T B (s, T ( +(1 w IE s exp( s s r u dub(, T B(s, T =. This condiion is cerainly valid if for boh riskless and zero recovery bonds he noarbirage condiion is valid. If we examine he Gaussian random field seup (cf. 4.2 and we ge ha he marke is free of arbirage, if for all, T and T, + T µ(, T = µ(, T + c( v, v, T dv,

112 4.3 Models wih Credi Risk 16 and µ(, T = µ(, T + c( v, v, T dv. For he converse, use ha he defaul-free bond is raded in he marke. Therefore a marke free of arbirage implies ( IE s exp( s r u dub(, T B(s, T =. Thus he discouned zero recovery bond has o be a maringale and we conclude ha he drif condiions hold Fracional Recovery of Treasury Value The hisory of a defaulable bond possibly admis several defaul evens. Evenually a coupon can no be paid, while he company is no necessarily forced o defaul. Also raing migraion is someimes referred o as a defaul even, and can be incorporaed in he following, more general model. Evidenly a model wih muliple defaul evens is needed, while his yields ha a defaul even no necessarily leads o defaul of he bond. Moreover, such a defaul even could be an upgrading, where he value of he bond increases. Assumpion (B3: In he fracional recovery of reasury value model he bond is modeled for all, T, T, + T as B F R (, T = Q( exp( f(, u du, wih Q( := τ i (1 L τi, where he loss process (L akes values in (, 1 and is adaped o (G. The τ 1, τ 2,... are he jump imes of a Cox Process, cf. Appendix A. In his seing we define he defaulable forward raes by f(, T := T ln B F R (, T. Q( As before 4 we model he defaulable forward rae by a Gaussian random field X wih covariance funcion c under Q, i.e., f(, T = µ(, T + X(, T. 4 See Assumpion (B2.

113 4.3 Models wih Credi Risk 17 Theorem Under assumpion (B3, discouned bond prices are maringales iff for all, T f(, = r + λ L (4.13 and drif condiion (4.8 holds. In his case he marke is arbirage-free. Proof. All discouned bond prices are maringales iff for all s T and T s, s + T IE s exp( r u du B(, s T = exp( r u du B(s, T, which is equivalen o 1 = IE s exp = IE s s<τ i A.1.4 = IE s exp s ( ru f(s, u du Q( Q(s exp (1 L τi exp λ u L u du s ( f(, u f(s, u du T ( ( f(u, u f(s, u du f(, u f(s, u du T ( ( f(u, u f(s, u du f(, u f(s, u du. Using (4.13 we obain s s 1 = IE s exp and analogously o (4.5, his is rue under (4.8. s ( f(u, u f(s, u du, For he converse, noe ha if discouned bond prices are maringales, B F R (, T = IE exp( = Q(B(, T IE T which yields for he defaulable spo rae r = f(, = T = r + λ L. ln T = τ i r u du Q(T F exp( T ( (1 L τi IE exp λ u du, (r u + λ u L u du Similar o he preceding proofs we obain he conclusion.

114 4.3 Models wih Credi Risk 18 Remark If in he previous heorem (L is admied o become equal o 1, he value of he bond may drop o zero. In his case he defaulable dynamics vanish and no condiions are needed o ensure an arbirage-free marke. Thus, for equivalency in Theorem 4.3.3, he drif condiions need o be saisfied on {Q( > } only. We also conclude for he credi spread in his model: s(, T = ln B(, T ln B(, T T = T T ( ln Q( + ln B(, T + ln IE T exp( = T T IET exp( λ u L u du. λ u L u du ln B(, T

115 4.4 Explici Pricing Formulas Explici Pricing Formulas This secion provides explici pricing formulas for cerain credi derivaives inroduced already in Secion 1.9. They provide he basis for he calibraion mehods developed laer on. Throughou his secion, we say wihin he noaion of Secion 4.2 and Defaul Digials A basic derivaive based on a credi risky underlying is he defaul digial pu. I promises a fixed payoff, say 1, if a defaul occurred before mauriy, and zero oherwise. We focus on he derivaive where he payoff is seled a mauriy. I may be recalled ha he defaul digial pu wih payoff a mauriy is inrinsically relaed o he zero recovery bond, as P d (, T + B (, T = B(, T. Assumpion (C1: Assume ha boh risk-free and defaulable forward raes admi a represenaion via Gaussian random fields and he drif-condiions (4.4 and (4.8 are saisfied. Furher on, assume ha he considered defaulable bond admis a fracional recovery of marke value 5 wih posiive, deerminisic loss funcion L and (4.13 holds. If assumpion (C1 holds, Theorems and yield ha he marke is free of arbirage. Furhermore, we deduce from (4.13 ha λ = f(, f(, L. (4.14 Insead of defining he dynamics of f(, T and λ and hen deriving f(, T, we wan o propose he dynamics of f(, T and f(, T and invesigae he consequences for λ. This reflecs he fac ha λ is no observable in he marke, while he forward raes are. Therefore, we use equaion (4.14 as a saring poin for his secion. This immediaely has some consequences. By definiion, (λ is assumed o be a nonnegaive process. Quie conrary, equaion (4.14 sugges λ o have a normal disribuion, which has a posiive probabiliy o be negaive. Thus, (λ mus be deerminisic. We wan o relax his rigid assumpion and raher ake (4.14 as a definiion for he sochasic process (λ. Because L is deerminisic, λ urns ou o be a Gaussian random field. Following Remark 4.3.2, for appropriae parameers, (λ migh be negaive wih jus a small probabiliy and herefore can be used as an approximaion of he rue defaul inensiy. 5 See Secion

116 4.4 Explici Pricing Formulas 11 For ease of noaion we wrie f(u insead of f(u, u and similarly µ(u, µ(u, X(u and X(u. Furhermore, se :=. In he following calculaions we will need a measure for ineracion beween he risk-free and he defaulable forward rae. For his, we se for s, 1, 2, T ς(s, 1, 2 := Cov ( f(s, 1, f(s, 2 = Cov ( X(s, 1, X(s, 2. Noe ha ς(s, 1, 2 is no necessarily symmeric in 1 and 2. Furhermore, assumpion (A2 immediaely yields Cov ( X(s1, 1, X(s 2, 2 = ς(s 1 s 2, 1, 2. Frequenly, we will consider erms similar o and herefore se r + λ = r (1 1 L + f 1 L, l := ( 1 1 L. Lemma Under (C1, he price of he zero recovery bond is for T, T B (, T = exp u ( lu µ(, u + µ(, u T du + 2 L u u ( l u c(v, u, v + l v c(v, u, v dv du. L v L u Proof. We obain for he price of he zero recovery bond B (, T = IE ( exp( The exponen s expecaion becomes = IE ( exp r u + λ u du (r u l u + f u L u du. l u L v ς(v, v, u dv du IE l u f(u + f u du L u = (4.8 = u ( l u µ(u + µ(u L u du ( l u µ(, u + µ(, u L u du ( l u c(v, u, v + 1 c(v, u, v dv du L u

117 4.4 Explici Pricing Formulas 111 and is variance equals Var l u X(u + X u du L u = The bond price hus equals { B (, T = exp l u l v c(u v, u, v + 1 c(u v, u, v + 2 l u ς(u v, v, u dv du. L u L v L v u { = exp + 2 ( l u µ(, u + u u µ(, u L u du ( l u c(v, u, v + 1 c(v, u, v dv du L u ( l u l v c(v, u, v + 1 c(v, u, v + 2 l } u ς(v, v, u dv du L u L v L v ( l u µ(, u + u µ(, u L u du ( lu c(v, u, v + l v c(v, u, v dv du L v L u } l u ς(v, v, u dv du. L v A a firs glance, i seems confusing ha he loss rae L appears in he price of a zero recovery bond. Noe ha his follows, because we ake equaion (4.14 as a definiion of he hazard rae. If he price of he zero recovery bond is available, he following formula allows o calibrae he loss rae L. Denoing he forward rae of he zero recovery bond by f (, T, we have f( = r + λ L = r + (f ( f(l L = f( f( f ( f(. The zero recovery bond is a basis for evaluaing more complicaed derivaives. I herefore will prove useful o obain some auxiliary resuls.

118 4.4 Explici Pricing Formulas 112 We define he zero recovery measure Q by dq = exp( r u + λ u du dq. (4.15 B (, T Then Q is equivalen o Q. Noe ha we suppress he dependence of T in he noaion. The following lemma gives a represenaion of f (, T in erms of X(, T and X(, T. Lemma Under (C1, he forward rae offered by he zero recovery bond can be represened for, T, T, + T as f (, T = µ (, T + l T X(, T + X(, T L T, (4.16 µ (, T := l T µ(, T + µ(, T + L T { LT 1 c(u, u, T c(, u, T (ς(u, u, T ς(, u, T L u L T } L u 1 c(u, u, T c(, u, T (ς(u, T, u ς(, T, u L u L T Furhermore, f (, T has independen incremens in he firs coordinae and he following covariance funcion for s, T and 1, 2 s, s + T : Proof. By definiion, c (s, 1, 2 = l 1 l 2 c(s, 1, 2 + c(s, 1, 2 L 1 L 2 + l 1 L 2 ς(s, 2, 1 + l 2 L 1 ς(s, 1, 2. f (, T = T ln B (, T = T 1 B (, T IE exp( r u + λ u du(r T + λ T = IE ( rt + λ T = IE ( l T r T + f(t, L T where IE denoes expecaion wih respec o he zero recovery measure Q. du. As IE (r T = µ(t, T + X(, T + IE ( X(T, T X(, T,

119 4.4 Explici Pricing Formulas 113 we can use Lemma B.4.1 o compue IE ( X(T, T X(, T and obain IE (r T 4.8 = µ(, T + ( c(u, u, T c(, u, T du + X(, T Furhermore, + Cov X(T, T X(, T, = f(, T + IE ( f(t = µ(, T + X(, T + ( ru + λ u du ( c(u, u, T c(, u, T (1 l u 1 ς(u, u, T ς(, u, T du. L u ( l u c(u, u, T c(, u, T l u ς(u, T, u ς(, T, u du. The asserion abou he covariance funcion remains o be shown. Acually, Cov(f (s 1, 1, f (s 2, 2 = IE ( (l 1 X(s 1, 1 + X(s 1, 1 L 1 (l 2 X(s 2, 2 + X(s 2, 2 L 2 = l 1 l 2 c(s 1 s 2, 1, 2 + c(s 1 s 2, 1, 2 L 1 L 2 + l 1 L 2 ς(s 1 s 2, 2, 1 + l 2 L 1 ς(s 1 s 2, 1, Defaul Pu In his secion we consider a defaul pu wih knock-ou feaure. The pu is knocked ou if a defaul occurs before mauriy of he conrac, which means ha he promised payoff is paid only if here was no defaul unil mauriy of he conrac. So his pu proecs agains marke risk bu no agains he loss in case of a defaul. Denoing he price of a (knock ou defaul pu wih mauriy T on a defaulable bond wih mauriy T by P k (, T, T, he risk neural valuaion principle yields for all T < T T. P k (, T, T = IE exp( r u du ( K B(T, T + 1 {τ>t },

120 4.4 Explici Pricing Formulas 114 Furher on, denoe by B k (, T, T a knock-ou conrac on he defaulable bond, which delivers he defaulable bond wih mauriy T a ime T, if no defaul happened unil T and zero oherwise. This derivaive seems a bi synheic, bu if boh defaul pu and defaul call wih knock-ou are raded, i can be replicaed by he following combinaion of pu and call: C k (, T, T P k (, T, T = IE exp( = IE exp( r u du ( B(T, T K + (K B(T, T + 1 {τ>t } r u du B(T, T 1 {τ>t }. If he knock-ou bond is no available one can use expression (4.17 for a sill explici, bu more complicaed pricing formula. Theorem The price of a defaul pu wih mauriy T, T on a defaulable bond wih mauriy T (T,, which is knocked ou if defaul occurs before T equals wih deerminisic erms P k (, T, T = B (, T KΦ( d 2 B k (, T, T Φ( d 1, σ(t, T := T c(t, u, v du dv, T T µ(t, T := ln B(, T B(, T + T T d 2 := µ(t, T ln K, σ(t, T d 1 := d 2 + σ(t, T. Proof. The price of he pu equals P k (, T, T = IE exp( l v c(t, u, v ς(t, u, v dv du σ(t, T, r u + λ u du ( K exp( T (K = B (, T IE exp( f(t, u du +. T T f(t, u du + From Lemma B.4.1, f(t, u du is normally disribued under Q, wih he same vari- T

121 4.4 Explici Pricing Formulas 115 ance as under Q bu expecaion IE ( ( f(t, u du = IE ( f(t, u du + Cov f(t, u du, (r u + λ u du T = T µ(t, u du T T ( 1 L v c(t, u, v + l v ς(t, u, v dv du. T T Formula (B.5 reveals he opion pricing formula 6 P k (, T, T = B (, T KΦ( d 2 e σ(t,t 2 µ(t,t 2 Φ( d 1, where µ(t, T = µ(t, u du T ( 1 L v c(t, u, v + l v ς(t, u, v dv du, T T he compuaion of he oher parameers being sraighforward. Furher on, we have B (, T e σ(t,t 2 µ(t,t ( = B (, T IE exp ( (r u + λ u du f(t, u du ( = B (, T IE exp( r u du B(T, T 1 {τ>t } T and he proof is finished. = B k (, T, T If he use of B k (, T, T seems inappropriae, he following is useful: B k (, T, T = B (, T e σ(t,t 2 µ(t,t = B (, T B(, T B(, T exp( T T l v c(t, u, v ς(t, u, v dv du. (4.17 Using he pu-call represenaion of B k (, T, T he alernaive represenaion respecively is obained. P k (, T, T = B (, T KΦ( d 2 C k (, T, T Φ( d 1, Φ(d 1 Φ(d 1 P k (, T, T + Φ( d 1 C k (, T, T = B (, T KΦ( d 2 6 Noe ha we apply B.5 wih respec o Q, seing ξ 2.

122 4.4 Explici Pricing Formulas Credi Spread Opions The pricing of credi spread opions can be done in a more or less similar fashion. Consider a pu on he credi spread of a defaulable bond which is knocked ou a defaul, i.e., a derivaive, which proecs agains spread widening risk, bu no defaul risk. Theorem Under assumpion (C1, he price of he (knock ou credi spread pu wih mauriy T, T on a defaulable bond wih mauriy T (T, equals P k CS (, T, T = B k (, T, T Φ ( d 1 KB (, T Φ ( d 2, wih he abbreviaions µ 1 := u µ(, u µ(, u + ( c(v T, v, u c(v T, v, u dv du, σ 1 := T T c(u v, u, v ς(t, u, v ς(t, v, u + c(u v, u, v dv du, σ 2 := + 2 T T T T l 1 (u, T l 1 (v, T c(u v, u, v dv du + l 1 (u, T ς(u v, v, u dv du, L v c(u v, u, v L u L v dv du ρ := + T T T l 1 (u, T ς(u T, v, u c(u T, v, u dv du 1 L u c(u T, u, v ς(u T, u, v dv du, d 2 := µ 1 ln K σ 1 + ρσ 2, d 1 := d 2 + σ 1, l 1 (u, T := { l u if u T, 1 if u > T..

123 4.4 Explici Pricing Formulas 117 Proof. The price of he credi spread pu equals Denoe PCS k (, T, T = IE exp( ξ 1 := ξ 2 := = IE exp T ( exp +1{τ>T r u du( B(T, T KB(T, T } (r u + λ u du ( f(t, u f(t, u du, T (r u + λ u du T f(t, u du ( f(t, u f(t, u du K f(t, u du +. T = IE(ξ 2 l 1 (u, T X(u T, u du X(u du. L u Then Lemma B.4.3 applies. We compue µ 1 = IE(ξ 1 = = T ( µ(t, u µ(t, u du µ(, u µ(, u du + u ( c(v T, v, u c(v T, v, u dv du, T while he oher parameers are obained direcly from heir definiion. Noe ha ( IE exp (r u + λ u du f(t, u du = B k (, T, T T and IE ( exp (r u + λ u du = B (, T. Thus, Lemma B.4.3 yields he desired conclusion.

124 4.4 Explici Pricing Formulas Credi Defaul Swap and Swapion In his secion we consider he pricing of a credi defaul swapion, in paricular he price of a so-called CDS call. I may be recalled from Secion ha his derivaive is a call on he swap premium, which is knocked ou if a defaul of he underlying eniy occurs before mauriy. I was also shown, ha, if he swap offers he replacemen of he difference o an equivalen risk-free bond on defaul, he swap rae is S(T = B(T, T n B(T, T n n. i=1 B (T, T i The pricing of he credi defaul swap herefore mainly relies on he pricing of he zerorecovery bond. Therefore, Lemma immediaely leads o a price of he credi defaul swap. Thus, he price of he CDS call equals CS k (, T, T = IE exp( = IE exp( ( r u du B(T, T n B(T, T n K ( n r u + λ u du B(T, T n exp( n i=1 +1{τ>T B (T, T i } f(t, u du K n exp( i=1 T i T + f (T, u du. (4.18 Usually he final repaymen, represened by B(T, T n, dominaes he coupon paymens. This jusifies he following Assumpion (C2: For he considered mauriy T, T and he enor srucure T < T 1 < < T n T assume ha he random variable exp ( n T f(t, u du + K n exp ( i i=1 T f (T, u du (4.19 can be approximaed by a log-normal random variable, denoed by B(T, T 1,..., T n. Furhermore, we assume ha discouned zero recovery bonds are maringales, i.e., drif condiion (4.8 is saisfied for µ ( and c (, respecively. If he drif condiion for zero recovery bonds is no saisfied, he following mehods can be applied in a similar fashion and i is sill possible o obain explici pricing formulas. Under assumpion (C2 he pricing of he credi defaul swapion is very similar o he pricing of a credi spread call, where he underlying is B(T, T 1..., T n. Denoe he mean and variance of B(T, T 1,..., T n by m and σ 2. This leads o

125 4.4 Explici Pricing Formulas 119 Lemma Under assumpion (C2 he mean and variance of B(T, T 1,..., T n equal m = B(, T n B(, T exp( + K σ 2 = K 2 n i=1 n T T B (, T i B (, T exp( n A ij + K i,j=1 c(v, u, v dv du i T T n B i + C, i=1 c (v, v, u dv du, where A ij, B i and C have he explici expressions (4.2, (4.21 and (4.22, respecively. Proof. According o assumpion (C2, B(T, T 1,..., T n = e ξ, wih normally disribued ξ. We wan o mach he momens and hereby compue mean and variance of ξ. Noe ha m = expieξ + 1 Var ξ 2 and herefore IEξ = ln m 1 Var ξ. This leads o 2 σ 2 = exp2ieξ + Var ξ ( exp(var ξ 1 Thus, = m 2 exp(var ξ 1. ( ξ N ln m 1 + σ2 m, ln ( σ 2 m Furher on, we derive explici expressions for m and σ 2. Firs, n m = IE exp( T f(t, u du + K n exp( i=1 i T f (T, u du. wih n IE exp( T f(t, u du = exp n = B(, T n B(, T exp( T µ(t, u du n T T n T n T T c(v, u, v dv du c(t, u, v dv du

126 4.4 Explici Pricing Formulas 12 and, if he drif condiion for zero recovery bonds is also saisfied, IE exp( i T f (T, u du So he asserion on m follows. = B (, T i B (, T exp( i T T c (v, v, u dv du. The variance of B(T, T 1,..., T n hus becomes σ 2 = Var K which may be simplified o n exp( i=1 n σ 2 = K 2 Cov ( exp( +2K i,j=1 i T i n Cov ( exp( i=1 + Var ( n exp( T n f (T, u du + exp( T i T f (T, u du, exp( f(t, u du. T j n f (T, u du, exp( T T f(t, u du f (T, u du f(t, u du Therefore σ 2 has he form K 2 n i,j=1 A ij + 2K n i=1 B i + C and, using equaion (B.8, we obain A ij = B (, T i B (, T j B (, T 2 exp( { exp Similarly we obain ( i T j T T B i = B(, T n B (, T i B(, T B (, T exp( and, finally, C = B(, Tn B(, T 2 T n exp( 2 i T T c (v, v, u dv du l u l v c(t, u, v + l u L v ς(t, v, u j T T + l v ς(t, u, v + 1 c(t, u, v dv du 1 L u L u L v T n T T c(v, u, v dv du c(v, u, v dv du exp( i T T n T n T T c (v, v, u dv du }. (4.2 c (v, u, v dv du (4.21 c(t, u, v du dv 1. (4.22

127 4.4 Explici Pricing Formulas 121 We inroduce an auxiliary produc, which we call he convering bond, B C (, T, T. I is used as an abbreviaion in he pricing formula for he swapion, and an explici formula for is price is available. The convering bond is a derivaive which pays 1 a mauriy T if no defaul occurred unil T < T. Thus, i behaves like a zero recovery bond unil T and is convered ino a defaul-free bond a T, if no defaul occurred so far. Denoing B C (, T, T := IE exp( λ u du r u du we obain he following Lemma The price of he convering bond a ime = equals for T < T T B C (, T, T = B(, T exp Proof. Seing = we have IE exp( λ u du f(, u f(, u du r u du = B(, T IE exp( = B(, T exp exp( L u 1 ( lv c(u v, u, v + (1 + 1 c(u v, u, v L u L v 1 L u L v ς(u v, u, v + ς(u v, v, u du dv ς(u v, u, v c(u v, u, v L u λ u du exp Cov( µ(u, u µ(u, u L u du λ u du, dv du. r u du 1 L u L v c(u v, u, v + c(u v, u, v ς(u v, u, v c(u v, u, v L u ς(u v, u, v ς(u v, v, u du dv dv du.

128 4.4 Explici Pricing Formulas 122 Using he drif condiions we may conclude { B C (, T, T = B(, T exp f(, u f(, u du L u 1 lv c(u v, u, v + (1 + 1 c(u v, u, v L u L v 1 L u L v ς(u v, u, v + ς(u v, v, u du dv ς(u v, u, v c(u v, u, v L u } dv du. Under furher assumpions his formula can be simplified considerably. For example, if L u L exp f(, u f(, u L u du = B(, T 1 L. B(, T Recalling he definiions of m and σ 2 from Lemma 4.4.5, we obain Theorem Under he assumpions (C1 and (C2 he price of a CDS call equals S C (, T, T n = B C (, T, T n Φ( d 2 B k (, T, T n + K n B (, T n Φ( d 1, i=1 wih deerminisic µ 1 := m + B(, T B(, T n + σ 1 := ln σ 2 m n u T n T n T T B(, Tn + ln B(, T exp + K n i=1 c(v, u, v dv du, c(t, u, v du dv m + σ2 2 n T B (, T i B (, T exp T i T c(v, u, v dv du T i T n T T c (v, u, v dv du l T c(t, u, v + n T n T T ς(t, u, v dv du ς(t, u, v L T dv du,

129 4.4 Explici Pricing Formulas 123 σ 2 := + 2 n T n n T l 2 (u, T l 2 (v, T c(u v, u, v dv du + l 2 (u, T ς(u v, v, u dv du, L v d 2 := µ 1 ln K + ρσ 2, σ 1 d 1 := d 2 + σ 1, { l u for u T, l 2 (u, T := 1 for u > T. c(u v, u, u L u L v dv du In equaion (B.6 an explici expression is given for ρ := Cov ln B(T, T 1,..., T n T, r u + λ u du ln B(T, T n. B(T, T n Proof. Using he abbreviaion B C (, T, T n, see Lemma 4.4.6, we obain IE exp( r u + λ u du ( B(T, T n + K n B (T, T i = B C (, T, T n + K i=1 n B (, T i. i=1 The discouned payoffs of he swapion were already derived in equaion (4.18. Seing ξ 1 = ln B(T, T 1,..., T n B(T, T n ξ 2 = r u + λ u du + n f(t, u du T we can use formula (B.5 o compue he expecaion of he discouned payoffs. We need o derive he consans, hus µ 1 = IE ln B(T, T 1,..., T n B(T, T n = m + n µ(, u + u c(v T, u, v dv du T

130 4.4 Explici Pricing Formulas 124 The variances are σ 1 = Var ln B(T, T 1,..., T n B(T, T n = ln σ 2 m n T n T T c(t, u, v du dv ( + Cov ln B(T, T 1,..., T n, n T f(t, u du and = ln σ 2 m n T n T T B(, Tn + ln B(, T exp +K n i=1 m + σ2 2 σ 2 = Var = n T n n T 2 n c(t, u, v du dv n T B (, T i B (, T exp T i T c(v, u, v dv du T i T n T T l 2 (u, T X(u T, u du + c (v, u, v dv du l T c(t, u, v + X(u L u l 2 (u, T l 2 (v, T c(u v T, u, v dv du + l 2 (u, T ς(u v, v, u dv du. L v The asserion now follows using Lemma B.4.4. n T n T T ς(t, u, v dv du ς(t, u, v L T du dv du c(u v, u, v L u L v dv du If one prefers o use he replacemen of he difference o par insead of he difference o an equivalen risk-free bond, pricing formulas are obained proceeding similarly Hedging - an Example Blanche-Scallie and Jeanblanc (21 inroduced a hedging mehodology for derivaives on underlyings which bear credi risk. Their approach concenraes on derivaives which

131 4.4 Explici Pricing Formulas 125 promise a riskless coningen claim X T if no defaul occurred before T. In his framework i is essenial ha X T can be replicaed on he riskless marke. The above considered derivaives ofen incorporae a similar knock-ou feaure, bu in mos cases he payoff canno be replicaed on he riskless marke. For example, consider a call on a zero recovery bond wih srike K, offering a T ( B (T, T K +. If a defaul occurred before T, he call is worhless. This is very similar o a knock-ou feaure, bu he call also can become worhless if he value of B (T, T drops below K. Firs, we derive an explici pricing formula for he call opion and aferwards sugges a hedging scheme. Theorem The price of a call wih mauriy T, T on a zero recovery bond wih srike K and mauriy T (T, T equals wih C(, T, T = B (, T Φ(d 1 K B (, T Φ(d 2, (4.23 σ 2 (T, T := T T T c (T, u, v dv du, B(,T ln KB(,T d 2 := σ(t, T 1 2 σ(t, T, d 1 := d 2 + σ(t, T. Proof. The risk-neural valuaion principle yields C(, T, T = IE ( exp( = IE ( exp( KIE ( exp( r u du B (T, T K + r u du1 {B (T,T >K}B (T, T r u du1 {B (T,T >K} =: (1 (2.

132 4.4 Explici Pricing Formulas 126 Observe ha {B (T, T > K} {τ > T }. Hence (2 = K IE ( exp( = K IE ( exp r u du1 {B (T,T >K,τ>T } (r u + λ u du1 R { T T f (T,u du> ln K} T = K B (, T Q ( f (T, u du > ln K, T where Q denoes he zero recovery measure, see (4.15. According o Girsanov s Theorem B.4.1, he inegral f (T, u du is normally disribued under Q wih he same variance T as under Q, namely σ 2 (T, T := Var T f (T, u du = T c (T, u, v dv du. T T T Proceeding similarly we obain for (1: (1 = IE ( exp = B (, T IE ( exp( (r u + λ u du exp( Applying Lemma B.4.2 we may conclude (1 = B (, T IE ( exp( = B (, T Q ( Finally, noe ha T T T T f (T, u du1 R { T T f (T,u du>ln K} f (T, u du1 R { T T f (T,u du>ln K}. f (T, u du Q ( f (T, u du > ln K σ 2 (T, T. T f (T, u du > ln K σ 2 (T, T d 2 = ln K + µ(t, T + σ 2 (T, T σ(t, T B(,T ln KB(,T = σ(t, T σ(t, T. The explici pricing formula for he defaul bond opion admis an explici derivaion of he hedging sraegy.

133 4.4 Explici Pricing Formulas 127 The price of he opion (4.23 depends on wo differen securiies, B (, T and B (, T. Naurally he hedge consiss in rading in hese wo asses. Observe ha he call price is a coninuous funcion of B (, T and B (, T unil defaul. Therefore, he dela-hedging mehodology can be applied. For he firs par of he hedge we have 1 (s := C(, T, T B (, T = Φ(d 1 + B d 1 (, T φ(d 1 B (, T d 2 KB (, T φ(d 2 B (, T. Wriing σ for σ(s,, T, we obain which yields d 1/2 B (, T = φ(d 1 = 1 B(, T σ, 1 exp d d 2σ + σ 2 2π 2 = φ(d 2 K B (, T B (, T, = φ(d2 exp( σd 2 σ2 2 1 (s = Φ(d 1 + φ(d 2 B (, T KB (, T B (, T σ B (, T KB (, T = Φ(d 1. (4.24 Similarly, we obain for he hedge w.r.. B(s, : 2 (s = B (, T d 1 φ(d 1 B (, T KΦ(d 2 KB d 2 (, T φ(d2 B (, T = KΦ(d 2. Alogeher, he hedge is perfec because up o he jump of he underlying, we have dc( B(s, T, B(s,, K = 1 (sd B(s, T + 2 (sd B(s, and a he disconinuiy τ he value of he call and he value of he hedging porfolio boh jump o zero. Noe he analogy o he Delas in he Black-Scholes formula 7. 7 See, for example Hull (1993.

134 4.5 Calibraion Calibraion In his chaper we presen wo approaches how o calibrae a Gaussian random field model o marke daa. This is moivaed by he resuls of Pang (1998, who shows ha in he ineres rae case he calibraion of a random field model in comparison o a n-facor HJM model permis more sabiliy over ime and frequen re-calibraion can be avoided. This is due o he differen approaches specifying he number of significan facors. In n-facor models, n is pre-specified by some reasoning and hen he calibraion is carried ou. In conras, in random field models, n is specified during he calibraion, such ha he error of he n-dimensional approximaion does no exceed a cerain level. Thus, he laer mehod allows choosing n depending on he daa and he required precision. If we wan o avoid assuming a parameric covariance srucure as in Kennedy (1997, a relaively large daa se needs o be available. We herefore assume ha prices of credi defaul swaps and swapions are accessible. Nowadays hese opions are no ye raded liquidly, bu as he credi marke is increasing rapidly, i is jus a quesion of ime unil hey will be available Calibraion Using Gaussian Random Fields As in Pang (1998 we make he following assumpions: (i We assume ha he riskless model is already calibraed. (ii The covariance funcions saisfy c(s, 1, 2 = ς(s, 1, 2 = s s ḡ( 1 u, 2 u du, g( 1 u, 2 u du. (iii Furhermore, he surfaces ḡ : IR 2 IR and g : IR 2 IR are piecewise riangular: For nodes {u 1,..., u m } any (u i, u i, (u i+1, u i, (u i+1, u i+1 or (u i, u i, (u i, u i+1, (u i+1, u i+1 define he corners of he surfaces riangles. The second assumpion yields saionary volailiy facors, while he hird assumpion allows for quick calibraion of he covariance funcion. The {u 1,..., u m } do no necessarily coincide wih he enor srucure, denoed by {T 1,..., T n }. For example, in Pang (1998 he i are muliples of.25 while he enor srucure is {1, 2, 3, 5, 7, 1}. If we wan o ensure ha he erm srucure of forward raes is coninuous or smooh, we would have o assume coninuiy and boundedness of he covariance funcion (respecively heir second derivaive, which is violaed by he second assumpion. Neverheless,

135 4.5 Calibraion 129 rounding he edges yields lile differences in derivaives prices and suiable regulariy of he forward raes. For he calibraion we would use daa of a cerain ime period, say some weeks or a monh, and use sandard opimizaion sofware o minimize, for example S-Plus wih he funcion nlmin, he residual sum of squared differences beween he calculaed prices and marke prices. In his procedure, calculaing model prices is done in wo seps. Firs, deermine c(s, 1, 2 and c(s, 1, 2 on he basis of g(u, v and ḡ(u, v for u, v {u 1,..., u m }, 1, 2 {T 1,..., T n } and every considered daa ime s {s 1,..., s p }. For he second sep, he prices of he considered derivaives are compued using he c(s, 1, 2 and c(s, 1, 2 deermined in he firs sep Calibraion Using he Karhunen-Loève Expansion An alernaive calibraion mehod uses he Karhunen-Loève decomposiion 8 inspired by Vargiolu (2. The approach presened in his secion incorporaes a mixure beween hisorical esimaion and calibraion o acual marke daa. This has he advanage ha on one side he procedure profis from useful hisorical informaion, while on he oher side he requiremens of raders, ha a model should calibrae perfecly o marke prices, is fulfilled. The daa problem in calibraion issues of credi risk models has been addressed in several papers, for example Schönbucher (22. I herefore seems beneficial ha he proposed procedure parsimoniously uses he available daa. Cenral parameers of our Gaussian model are he covariance funcions. Wih a view owards applicaions, he flexibiliy provided by he model encouners he problem ha he daa for calibraion issues is sill scarce. In he following we presen an inermediary soluion which serves boh needs. Consider he covariance funcion c(s, 1, 2, where we se s =. Then c( can be decomposed ino c(, 1, 2 = λ k e k ( 1 e k ( 2, k using an orhonormal basis {e k : k IN} of L 2 (µ, he Hilber space of funcions f : IR IR which are square inegrable w.r.. a suiable measure µ. For our applicaion a cerain period of mauriies will be of ineres, for example he inerval, T, and we choose for µ he Lebesgue measure. Noe ha, o deermine he covariance funcion, one has o specify boh he {e k : k IN} and he {λ k : k IN}. For he former, we use hisorical informaion, while he laer are obained via calibraion. 8 See, for example, Bogachev (1991, p. 55 p.p., Da Prao and Zabczyk (1992, p. 99 p.p. or Adler (1981.

136 4.5 Calibraion 13 The firs sep is o esimae he covariance funcion using a se of hisorical daa. Consider a small ime inerval, so ha saionariy of he considered random fields in his ime inerval may be assumed. The hisorical daa consiss of observaions of f(s, a a se of ime poins T := {(s i, j : 1 i n 1, 1 j n 2 }. Following Hall, Fisher and Hoffmann (1994 we sugges an esimaor based on kernel mehods. For he poins a = (s 1, 1 and b = (s 2, 2 we define he covariance esimaor by X(ci X X(d j X c(a, b := c i,d j T K( a c i, b d j h h i,j T K( a c i h, b d, j h where K(c, d is a symmeric kernel Observe ha he sum is over all ime poins in T, labeled c i and d j, respecively. Esimaion of he covariance funcion ρ( 1, 2 a a cerain ime s is hus obained by considering s 1 = s 2 = s. The following second sep is opional, bu ensures ha he esimaor is posiive definie, hus a covariance funcion iself. This yields increased performance for he eigenvecor decomposiion below. We inver he characerisic funcion of our esimaor, ϕ(λ := exp(iλ ρ( d for λ IR 2. Because he esimaor is symmeric, we have ϕ(λ = cos(λ ρ( d. IR 2 Following Bochner s heorem, we need ϕ(λ o ensure ha ρ is a covariance funcion, hus we use he posiive par of ϕ(λ in he inversion of he Fourier ransform and sugges he following esimaor of he covariance funcion ˆρ( = 1 (2π 2 cos(λ ϕ(λ + dλ. Figure 4.2 shows he resul of he covariance esimaion on a se of U.S. Treasury daa using hisorical daa of 4 weeks. The implemenaion uses a Gaussian kernel and he covariance esimaor is ploed for mauriies of 3 monhs o 3 years. Afer obaining an esimaor for he covariance funcion, we can calculae is Eigenfuncions up o a required precision. Vargiolu (2 presens a recursive scheme o obain he Eigenfuncions from he covariance operaor 9. We apply he procedure o our seing wihin Gaussian random fields. The eigenvecor decomposiion is done applying he Mises-Geiringer ieraion procedure o our seing, cf. Ruishauser (1976 for he applicaion o IR n. Fix k L 2 (µ and define 1 k n+1 ( := ˆρ(, k n ( d. 9 As already discussed in Chaper 3, on page Compare o equaion (4.2.

137 4.5 Calibraion 131 Figure 4.2: Esimaed covariance funcion for U.S. Treasury daa (May 22. esimaion uses a Gaussian kernel and shows mauriies of 3, 6,..., 36 monhs. The Then k n+1 iself is an elemen of L 2 (µ. The resuls of Vargiolu (2 can be used o show ha k n k n+1 e 1 and k n λ 1, as n. As hese Eigenvecors need no be normed, we inroduce he normed eigenvecors ē k. Using ˆρ 1 ( 1, 2 := ˆρ( 1, 2 λ 1 ē 1 ( 1 ē 1 ( 2, and applying he procedure o ˆρ 1 yields e 2 and λ 2 and so on. For he applicaion one migh wan o obain he covariance funcion on a cerain grid, hus readily available implemenaions for marices may be used afer a suiable ransformaion 11. Figure 4.3 shows he calculaed Eigenvecors for he U.S. Treasury daa. The firs wo Eigenvecors show significan Eigenvalues ( and.569, while he remaining Eigenvalues are of much smaller magniude. In his example i herefore urns ou o be sufficien o use he firs wo Eigenvecors only. More generally, assume ha we already have deermined he firs N Eigenfuncions. Then we use he following covariance funcion for he calibraion: ˆρ(λ 1,..., λ N, 1, 2 := N λ k e k ( 1 e k ( 2. k=1 11 Noe ha he scalar produc in IR n and L 2 (µ is differen.

138 4.5 Calibraion Eigenvecors Figure 4.3: Esimaed, normed Eigenvecors of he covariance funcion in Figure 4.2. The firs wo Eigenvecors correspond o he Eigenvalues and.569, respecively, while he furher are of magniude As before, a sandard sofware package can be used o exrac he λ 1,..., λ N from observable derivaives prices by a leas-squares approach. Noe ha, in comparison o he previously presened model, a much smaller se of derivaives can be used for he calibraion. The implemenaion of his las sep using credi derivaives daa is subjec o fuure research. Neverheless, we already analyzed some bond daa and esimaed he covariance funcions and he Eigenvecors / values. Take for example he daa from Greece Treasury bonds. The esimaion resuls may be found in Figures 4.5 and 4.6. Firs, noe ha he variance for bonds wih small mauriies is higher han for bonds wih large mauriies. Second, for he period June o Augus 21 negaive correlaions for bonds wih small versus bonds wih large mauriies were observed. This reflecs a movemen in opposie direcion as o ineres raes in his period. Taking a closer look a he Eigenvecors reveals he componens of he covariance funcion. The firs Eigenvecor generaes more or less he shape of he covariance funcions. The already menioned effec, ha larger mauriies relae o smaller variances, may be observed here as well. Noe ha he scale of he z-axis changes (Max.1 o.48 wih he firs Eigenvalue (.3318 o The second Eigenvecor covers he wriggly srucure of he covariance funcion. Noe ha his is in srong relaion wih he Eigenvalues.

139 4.5 Calibraion Eigenvecors Figure 4.4: Esimaed covariance funcion and normed Eigenvecors for U.S. Treasury daa, July - Sepember 21. The firs wo Eigenvalues are ,.1186, he remaining ones being of much smaller magniude (1 9.

140 4.5 Calibraion 134 Jun-Aug 1 Jun-Aug 2 Mar-May 3 Figure 4.5: Esimaed Covariance funcion for Greece Treasury daa. The plos are based on 4 observaions of Bonds wih mauriies 3, 5, 7, 1, 15, 2, 3 years. The covariance funcion is ploed for mauriies T = 3, 6, 9,..., 24 years.

141 4.5 Calibraion Jun-Aug Eigenvecors Jun-Aug Eigenvecors Mar-May Eigenvecors Figure 4.6: Esimaed Eigenfuncions for Greece Treasury daa. For he firs plo Eigenvalues are.3318,.9484,.558,.2444,.1.66 (he ohers being of magniude For he second.8929,.233,.88127,.6616 (ohers: 1 5 and for he hird.1371,.25,.8344,.822 (ohers: 1 5.

142 Appendix A Basic Seup for Hazard Rae Models A deailed reamen of proofs and mehods wihin he hazard rae framework can be found, among ohers, in Lando (1994, Bielecki and Rukowski (22, Jeanblanc and Rukowski (2 or Jeanblanc (22. Consider a probabiliy space (Ω, A, Q, endowed wih a filraion (G. The probabiliy measure Q will represen a risk-neural measure, which is fundamenal in pricing coningen claims. The filraion (G represens he general marke informaion, which could include informaion on cerain indices, ineres raes and so on. Inroducing defaul risk ino he model, we consider a defaul ime τ, which is a posiive random variable on (Ω, A, Q. The associaed jump process 1 {τ } induces he defaul informaion represened by H := σ(1 {τ s} : s. Therefore, he oal informaion available a ime is F = H G. In hazard-rae models, one uses a specific ype of process for 1 {τ }, namely Cox processes. As some models incorporae more han on jump, we aim a defining a jump process, which jumps a imes τ 1, τ 2,... and se τ := τ 1, if jus one defaul even is of ineres. Consider a Poisson process (Ñ wih inensiy 1, which is independen of G for all, and a nonnegaive, nondecreasing and righ-coninuous process (Λ adaped o (G. We hen obain a Cox process hrough a (random ime change of he process Ñ by seing1 If (Λ( admis he represenaion N := ÑΛ(. Λ( := hen (λ( is called he inensiy of N. λ(u du, The defaul ime is represened by he firs jump of (N, so ha τ := τ 1 = inf{s : N s = 1}, 1 For a deailed reamen on Cox processes see Grandell (

143 Appendix A. Basic Seup for Hazard Rae Models 137 while he n-h jump is τ n := inf{s : N s = n}. I is easy o deduce he following Lemma A.1.1. For a Cox process (N wih inensiy (λ and τ being he firs jump of (N, we have Q(τ > G = exp( λ u du. For pricing a defaulable bond he following Theorem, firs menioned in Lando (1994, is indispensable: Theorem A.1.2. For a G T -measurable random variable X T, and τ as well as (N defined as in he preceding Lemma, we have ha IE(X T 1 {τ>t } F = 1 {τ>} IE ( exp( = 1 {τ>} IE ( exp( λ u dux T F λ u dux T G. Proof. Using he definiion of he Cox process as given above yields IE(X T 1 {τ>t } F = 1 {τ>} IE X T IE ( 1 {τ>t } F G T F = 1 {τ>} IE X T IE ( 1 { Ñ(Λ T Ñ(Λ=} F G T F. By definiion of F we have σ(f G T = σ(h G T. Furhermore, since a Poisson process has independen incremens, we have, condiionally on G T and on {τ > }, ha Ñ(Λ T Ñ(λ is independen of H. Therefore and we may conclude IE ( 1 {Ñ(ΛT Ñ(Λ =} H G T = IE ( exp (Λ T Λ G T IE(X T 1 {τ>t } F = 1 {τ>} IE X T exp( λ u du G. Using he arbirage-free pricing principle, which yields ha he fair price of a coningen claim is he expecaion of he discouned payoff under an equivalen maringale measure, we obain he following formula for he defaulable bond B(, T : B(, T = IE ( exp( = 1 {τ>} IE ( exp r u du1 {τ>t } F (r u + λ u du G.

144 Appendix A. Basic Seup for Hazard Rae Models 138 Valuing non-european claims or using differen conceps of recovery, one will find he following heorem useful (see Lando (1994: Theorem A.1.3. For a Cox process (N wih inensiy (λ and τ being he firs jump of (N and a sochasic process (Y s s, we have (i IE T τ exp( s r u duy s ds F = 1 {τ>} IE exp s (r u + λ u duy s ds G. (ii τ IE exp( r u duy τ 1 F { τ T } = 1 {τ>} IE exp s (r u + λ u duy s λ s ds G. The firs formula allows for pricing a payoff sream, which is coninuously paid unil T and sopped a τ. The second formula prices he random payoff Y τ, which is paid a defaul. Proof. For (i, observe ha IE T τ exp( s r u duy s ds F = = s IE 1 {s τ} exp( s IE exp( r u duy s F ds r u duy s IE ( 1 {s τ} F G T F. As s > we have {s τ} = { < s τ}. Hence he inner expecaion can be represened via (Ñ(, so ha 1 {τ>} IE ( 1 {s τ} F G T = 1{τ>} IE ( 1 {s τ} H G T = IE ( 1 {τ>} 1 { Ñ(Λ s 1} H G T. On {τ > } we have ha Ñ(Λ =, because no jump occurred before. This yields 1 {τ>} IE ( 1 {s τ} F G T = 1{τ>} IE ( 1 { Ñ(Λ s Ñ(Λ 1} H G T and (i follows. s = 1 {τ>} exp (Λ s Λ = 1 {τ>} exp( λ u du Asserion (ii of he heorem is covered by Bielecki and Rukowski (22, Prop

145 Appendix A. Basic Seup for Hazard Rae Models 139 If several defaul evens are under consideraion, one uses he following Theorem A.1.4. If (L is a process which is adaped o (G we have under he assumpions of A.1.2 N T N IE (1 L τi X T F = (1 L τi IE exp( L u λ u du X T F. i=1 i=1 Proof. As (L is an adaped process, we have ( N T GT IE IE (1 L τi X T H F i=1 = N i=1 (1 L τi IE IE ( N T i=n +1 (1 L τi X T GT H F. (A.1 Consider he inner expecaion k ( IE 1 {NT N =k} = k N +k (1 L τi G T H X T (A.2 i=n +1 ( N +k IE 1 {NT N =k}ie i=n +1 GT (1 L τi G T H σ(n T H X T. The condiional disribuion of he τ i s can be replaced by an uncondiional one 2, because Here, he η i are i.i.d. wih densiy L ( τ N+1,..., τ NT NT N = k = L ( η 1:n,..., η k:n. λ u λ u du on (, T. Because he order wihin he produc can be inerchanged, i is possible o swich back o he η i. The inner expecaion (A.2 hen becomes ( k IE (1 L ηi G T H i=1 = = (1 IE ( L η1 GT H k ( 1 L u λ u k. λ w dw du 2 See Rolski, Schmidli, Schmid and Teugels (1999, p.52. The η i:n denoe he order saisics of η i, ha is he η i are ordered, such ha η 1:n η 2:n η n:n.

146 Appendix A. Basic Seup for Hazard Rae Models 14 Therefore we may conclude (A.2 = X T k = X T exp( = exp( IE 1 {NT N =k} GT H 1 λ u du k L u λ u du X T. ( L u λ u du k 1 k! L u λ u λ w dw du λ u k λ w dw du k

147 Appendix B Auxiliary Calculaions B.1 Normal Random Variables Consider wo independen normally disribued random variables X 1 and X 2 wih zero mean and variances σ 2 1 and σ 2 2. The following lemma may be used o deermine he disribuion of X 1 condiionally on X 1 + X 2. Lemma B.1.1. There exiss ξ N (, 1, which is independen of X 1 + X 2, and X 1 = σ2 1 σ1 2 + (X 1 + X 2 + σ 1σ 2 ξ. σ2 2 σ σ2 2 Proof. We define σ 2 σ ξ := X 1 1 X 2. σ 1 σ σ2 2 σ 2 σ σ2 2 Then ξ is normally disribued wih expecaion zero and variance 1. I remains o show ha ξ is independen of X 1 + X 2. This follows from Cov ( ( σ 2 σ ξ, X 1 + X 2 = IE 1 ( X 1 X 2 σ 1 σ σ2 2 σ 2 σ 2 X1 + X σ2 2 = σ 2 σ 1 σ σ 2 2 σ1 2 σ 1 σ 2 σ 2 σ σ2 2 2 =. B.2 Boundary Crossing Probabiliies We have he following (see, e.g., Pechl (1996 Theorem B.2.1. For a sandard Brownian moion (B s s, consans b and m, we have for b < IP( inf ms + B s b = Φ ( b m + e 2bm Φ ( b + m. <s 141

148 Appendix B. Boundary Crossing Probabiliies 142 Proof. By he reflecion principle 1 we conclude for b < IP( inf s B s < b = 2IP(B < b = 2Φ ( b. (B.1 Consider a probabiliy measure P, defined by dp = exp ( mb m2 2 dp = exp( mb + m2 dp, 2 wih B := m + B. P is equivalen o P, and he Girsanov heorem yields ha B is a Brownian moion under P. We conclude P ( inf s B s < b = 1 {inf s Bs <b} exp(mb m2 2 dp = exp(mb m2 2 P inf s B s < b B dp. The condiional probabiliy equals one for B b. For B > b, a resul for condiional expecaions yields The numeraor equals P inf B s < b B = x = lim h 1 lim h h P ( inf s B s b, B 1 h 1 B x,x+h {inf s Bs <b} dp 1 P. h (B x, x + h x, x + h = x P ( inf s B s b, B > x = x Φ( 2b x = 1 (x 2b2 exp(, 2π 2 where we again used he reflecion principle 2. For he denominaor we obain x Φ( x = 1 2π exp( x2 2, so ha for x > b P inf Bs < b B = x = exp( 4b2 4xb. 2 1 See, for example, Karazas and Shreve ( To conclude, ha (x > b P ( inf B s < b, B > x = Φ ( 2b x.

149 Appendix B. Some Inegrals 143 Subsiuing his leads o P ( inf s B s < b = = 1 2π 1 2π b b exp( exp( (x m2 dx + 2 (x m2 dx + 2 b = Φ ( b m + e 2bm 1 Φ ( b m = Φ ( b m + e 2bm Φ ( b + m. b exp ( mx m2 2 4b2 4xb 2 exp ( (x 2b m2 2 x2 dx 2 + 2bm dx Noe ha for m = we obain he special case (B.1. We conclude ha, for c <, IP ( inf {m(s + σ(b s B } > c s (,T = IP ( inf {ms + σb s } > c s (,T = IP ( inf {m s (,T σ s + B s} > c σ = 1 IP ( inf s (,T { m σ s + B s} < c σ = 1 Φ ( c m(t σ + e 2cm/σ 2 Φ ( c + m(t T σ T = Φ ( m(t c σ e 2cm/σ 2 Φ ( m(t + c T σ. (B.2 T For c > his probabiliy equals zero, because inf{... }. B.3 Some Inegrals Lemma B.3.1. (i (ii x exp ( x exp ( (x a2 2b (x a2 2b ( a 2 ( a dx = b exp + a 2πbΦ b 2b (x d2 bc dx = 2c b + c exp ( bc(a d2 + (ac + db 2 2bc(b + c + ac + db 2πbc exp ( (b + c 3 2 (a d2 ( ac + db Φ 2(b + c bc(b + c

150 Appendix B. Some Inegrals 144 Proof. We have x exp (x a2 2b dx = b +a x a b (x a2 exp dx 2b (x a2 exp dx 2b = b exp( a2 2b + a 1 2πb 2πb = b exp( a2 2b + a 2πbΦ ( a b. (x a2 exp dx 2b For (ii, we have x exp ( (x a2 2b = = (i = exp ( = (x d2 dx 2c x exp ( x2 (b + c 2(ac + dbx + a 2 c + bd 2 dx 2bc x exp ( ( x ac+db 2 b+c 2bc/(b + c + ac + db 2πbc Φ ( (b + c 3 2 (a d2 dx 2(b + c (a d2 bc 2(b + c b + c exp ( ac + db bc(b + c (ac + db2 2bc(b + c bc b + c exp ( bc(a d2 + (ac + db 2 2bc(b + c + ac + db 2πbc exp ( (b + c 3 2 (a d2 ( ac + db Φ. 2(b + c bc(b + c The following lemma is an auxiliary resul for Secion Wih he noaion herein, we have Lemma B.3.2. The defaul inensiy λ equals λ = m 2π 8σ exp ( (ln V B + (ln Ṽ k m k + σ2 Z 2 (σ2 Z 2 σ2 Z +σ2 k ( k σ 2 Z +σ2 k 1 m2.

151 Appendix B. Some Inegrals 145 Proof. All condiional expecaions are wih respec o H, so we wrie IE for IE(... H. Noe ha, by definiion of a defaul inensiy, Since λ = 1 {τ>} ln IP(τ > T H. T T = T ln IP (τ > T = IP T (τ > T IP (τ > T and 1 {τ>} IP (τ > = 1 we jus need o compue he numeraor. numeraor equals T IE ( (m(t η 1 {η<} Φ σ e 2ηm/σ 2 Φ ( m(t + η T σ T ( = IE 1 {η<} ϕ ( m(t η σ ( m T 2σ T + η 2σ(T 3 2 = e 2ηm/σ2 ϕ ( m(t + η σ ( m T 2σ T η 2σ(T 3 2 Using (1.13, he 1 ( (m(t η IE 2σ(T 3 η1 {η<} ϕ 2 σ + e 2ηm/σ 2 ϕ ( m(t + η T σ T m + 2σ T IE ( (m(t η 1 {η<} ϕ σ e 2ηm/σ 2 ϕ ( m(t + η T σ T = 1 1 2σ(T 3 2 2πσ η exp ( x exp ( (x µ η 2 2ση 2 (m(t x2 (2mx + exp 2σ 2 (T σ 2 (m(t + x2 dx (B.3 2σ 2 (T m + 2σ 1 T 2πσ η exp ( exp ( (x µ η 2 2ση 2 (m(t x2 (2mx exp 2σ 2 (T σ 2 (m(t + x2 dx. (B.4 2σ 2 (T Observe ha he expression in (B.4 equals zero, so we concenrae on he remaining one. As exp ( (m(t x2 (2mx (m(t + x2 + exp 2σ 2 (T σ 2 2σ 2 (T = 2 exp ( (m(t x2 2σ 2 (T

152 Appendix B. Some Inegrals 146 we may conclude, using Lemma B.3.1, ha (B.3 = = 1 x exp (x µ η 2 2ση 2 (x m(t 2 dx 2σ 2 (T 2σσ η π(t 3 2 { 1 σ 2 η σ 2 (T 2σσ η π(t 3 2 ση 2 + σ 2 (T exp σ2 η σ2 (µ η m(t 2 + (µ η σ 2 (T + m(t ση 2 2 2ση 2σ2 (T (ση 2 + σ2 (T + 2πσ 2 ση 2(T µ ησ 2 (T + m(t ση 2 (ση 2 + exp (µ η m(t 2 σ2 (T 3 2 2(ση 2 + σ 2 (T ( Φ µ ησ 2 (T + m(t ση 2 } σ η σ (T (ση 2 + σ2 (T =: I + II. We firs show, ha I equals zero when we se T =. We have as 3 T. c 1 I = T (σ 2 η + σ 2 (T exp µ 2 η (T (ση 2 + σ2 (T µ η m + σησ 2 2 (ση 2 + σ 2 (T + (T c 3 ση 2 + σ 2 (T, For II we obain 2πσ 2 ση 2 (T (T (µ η σ 2 + mση 2 II = 2πσσ η (T 3 2 (ση 2 + σ2 (T 3 2 exp (µ η m(t 2 ( 2(ση 2 + Φ µ η σ 2 + mσ 2 η T σ2 (T σ η σ ση 2 + σ2 (T 1 ( σ 2 µη + m exp ( µ2 η 1 2π ση 2 2ση 2 2. Wih he definiions of µ η and σ η we obain 4 λ = m 2π 8σ exp ( (ln V B + (ln Ṽ k m k + σ2 Z 2 (σ2 Z 2 σ2 Z +σ2 k ( k σ 2 Z +σ2 k 1 m2. 3 See Heuser (1991, p. 289 for lim τ 1 τ exp(cτ 1 =, for c <. 4 Recall, ha m = µ σ2 2.

153 Appendix B. Tools for Gaussian Models 147 B.4 Tools for Gaussian Models The following lemma is a simple version of Girsanov s Theorem Lemma B.4.1. Assume he wo random variables ξ and η are joinly normally disribued under a probabiliy measure Q. Then d Q := eξ IE(e ξ dq defines a measure equivalen o Q. η is normally disribued under Q wih Proof. The definiion of Q yields for λ IR Ẽ(η = IE(η + Cov(η, ξ, Var(η = Var(η. Ẽ ( exp(λη = IE ( e ξ which immediaely yields he desired resul. = IE(e ξ eλη exp IEξ + λieη + 1 λ2 Var ξ + λ Cov(η, ξ + Var η 2 2 exp(ie(ξ + 1 Var(ξ, 2 The following expecaion is essenial for he derivaion of he Black-Scholes formula: Lemma B.4.2. For a normally disribued random variable ξ wih variance σ 2, we have IE ( e ξ 1 {ξ>a} = IE ( e ξ IP ( ξ > a σ 2. Proof. Assume IE(ξ =, he saemen wih nonzero mean being an easy exension. Then IE(e ξ 1 {ξ>a} = 1 2πσ 2 = exp σ2 2 a a expx x2 2σ 2 dx 1 exp (x σ2 dx 2πσ 2 2σ 2 = exp σ2 2 IP(ξ + σ2 > a. Furhermore, we have Lemma B.4.3. For i = 1, 2 le ξ i be joinly normal wih expecaion µ i, variance σi 2 correlaion ρ, respecively. Then IE ( e ( ξ 2 e ξ 1 K + = IE e ξ 1 +ξ 2 ( µ1 ln K Φ + ρσ 2 + σ 1 σ 1 KIE ( e ( ξ µ1 ln K 2 Φ + ρσ 2. σ 1 and

154 Appendix B. Tools for Gaussian Models 148 Proof. Firs, IE ( e ( ξ 2 e ξ 1 K + = IE ( e ξ 1+ξ 2 ( 1 {ξ1 >ln K} KIE e ξ 2 1 {ξ1 >ln K} =: (1 + (2. We use he decomposiion ξ 2 = µ 2 + σ 2ρ σ 1 (ξ 1 µ 1 + σ 2 1 ρ2 ξ, where ξ is sandard normally disribued and independen of ξ 1. This yields for he firs erm (1 = e µ 1+µ 2 IE exp ( (ξ 1 µ 1 (1 + σ 2ρ + σ 2 1 ρ2 ξ 1 {ξ1 >ln K} σ 1 = e µ 1+µ 2 + σ 2 2 (1 ρ2 2 IE exp((ξ 1 µ 1 (1 + σ 2ρ 1 σ {(ξ1 µ 1 (1+ σ 2 ρ >(ln K µ σ 1 (1+ σ 2 ρ } 1 1 σ. 1 Applying Lemma B.4.2, we obain (1 = exp µ 1 + µ 2 + σ2 2 (1 ρ2 2 + (σ 1 + σ 2 ρ 2 2 ( IP (ξ 1 µ 1 σ 1 + σ 2 ρ > (ln K µ 1 σ 1 + σ 2 ρ σ 1 = exp µ 1 + µ 2 + σ ρσ 1σ 2 + σ2 2 ( µ1 ln K Φ 2 σ 1 σ 1 (σ 1 + σ 2 ρ 2 + σ 1 + σ 2 ρ. For he second erm we have IE(e ξ 2 1 {ξ1 >ln K} = IE exp ( µ 2 + σ 2ρ (ξ 1 µ 1 + σ 2 1 ρ2 ξ 1 {ξ1 >ln K} σ 1 = exp ( µ 2 + σ2 2 (1 ρ2 IE e σ 2 ρ (ξ σ 1 µ σ 2 { 2 ρ (ξ σ 1 µ 1 > σ 2 ρ 1 σ 1 }(ln K µ 1 B.4.2 = exp ( µ 2 + σ2 2 2 (σ 2 ρ IP (ξ 1 µ 1 > (ln K µ 1 σ 2ρ ρ 2 σ2 2 σ 1 σ 1 = exp ( µ 2 + σ2 ( 2 µ1 ln K Φ + ρσ 2. 2 σ 1 Usually he formula in Lemma B.4.3 is abbreviaed as and we immediaely obain IE e ξ 1+ξ 2 Φ(d1 KIE ( e ξ 2 Φ(d 2 IE ( e ξ 2 ( K e ξ 1 + = KIE ( e ξ 2 Φ( d2 IE e ξ 1+ξ 2 Φ( d1. (B.5

155 Appendix B. Tools for Gaussian Models 149 We use he noaion of Secion I may be recalled ha he mean and variance of B(T, T 1,..., T n was denoed by m and σ 2 and The compuaion of is done in he following l 2 (u, T = { l u for u T, 1 for u > T.. ρ := Cov ln B(T, T 1,..., T n T, r u + λ u du ln B(T, T n B(T, T n Lemma B.4.4. Under he assumpion (B2 we have B(, Tn ρ = ln B(, T exp + K n i=1 c(v, u, v dv du B (, T i B (, T exp B(, Tn ln B(, T exp( + K + n i=1 n T n T i T n T n T B (, T i B (, T exp i T T n T T c (v, u, v dv du l 2 (v, T l u c(v T, u, v + T i T (1 + 1 L v c(v, u, v dv du T l 2 (v, T c(v T, u, v dv du Proof. By he definiion of ρ, c (v, u, v dv du n T T ρ = Cov ln B(T, T 1,..., T n ln B(T, T n, = Cov ln B(T, T 1,..., T n, + n T n T n l 2 (v, T c(v T, u, v dv du l 2 (v, T ς(v T, u, v dv du i T T ς(v T, u, v L u dv du c(v, u, v + l uς(v, v, u dv du L u L v L v ς(u v, v, u L v dv du. (B.6 l 2 (v, T X(v dv n T T r u + λ u du + n T X(v L v dv f(t, u du ς(u v, v, u L v dv du. (B.7

156 Appendix B. Tools for Gaussian Models 15 We compue he covariances separaely. Observe, ha for wo joinly normally disribued random variables ξ 1 and ξ 2, which is equivalen o Cov(e ξ 1, e ξ 2 = IE(e ξ 1+ξ 2 IE(e ξ 1 IE(e ξ 2 = IE(e ξ 1 IE(e ξ 2 e Cov(ξ 1,ξ 2 1, (B.8 Cov(ξ 1, ξ 2 = ln 1 + Cov(eξ 1, e ξ 2 IE(e ξ 1 IE(e ξ 2 = ln IE(e ξ 1+ξ 2 lnie(e ξ 1 IE(e ξ 2. Firs, consider Cov ln B(T, T 1,..., T n, n l 2 (v, T X(v dv B(T, T 1..., T n = ln IE exp( n l 2 (v, T X(v dv ln IE( B(T, T 1,..., T n ln IE ( exp( n Assumpion (B2 leads o B(T, T 1,..., T n IE exp( n l 2 (v, T X(v dv where he expecaions are B(T, T n IE exp( n l 2 (v, T X(v dv = IE exp( = exp l 2 (v, T X(v dv. B(T, T n = IE exp( n l 2 (v, T X(v dv n B (T, T i +K IE n T n T n T n n T i=1 f(t, u du + µ(, u + exp( n l 2 (v, T X(v dv n l 2 (v, T X(v dv c(v, u, v dv du l 2 (u, T l 2 (v, T c(u v, u, v dv du l 2 (v, ς(t v, u, v dv du., T

157 Appendix B. Tools for Gaussian Models 151 For he compuaion of he second expecaion noe ha Var i T l u X(T, u + X(T, u L u du = i T i T T and, using he drif condiion for he zero recovery bond, i µ (T, u du i T i c (T, u, v dv du c (T, u, v dv du, T T T i = µ (, u + c (v, u, v dv du. So, using (4.16, we obain B (T, T i IE exp( n l 2 (v, T X(v dv i = exp µ (, u + Conclude Cov ln B(T, T 1,..., T n, n = ln exp +K n i= T n T n i T n T T c (v, u, v dv du l 2 (u, T l 2 (v, T c(u v, u, v dv du l u l 2 (v, T c(t, u, v + l 2(v, T ς(t v, u, v L u l 2 (v, T X(v dv n T exp m + σ2. 2 f(, u + i T i T n T f (, u + Consider he second covariance in (B.7, c(v, u, v dv du n T T c (v, u, v dv du l u l 2 (v, T c(t, u, v + l 2(v, T ς(t v, u, v L u Cov ln B(T, T 1,..., T n, X(u L u du. dv du. l 2 (v, T ς(t v, u, v dv du dv du

158 Appendix B. Tools for Gaussian Models 152 We need he following wo expecaions: IE T B(T, Tn exp( X(v L v = exp dv n µ(, u du n T (1 + 1 L v c(v, u, v dv du T T c(u v, u, v L u L v dv du and IE B (T, T i exp( = exp X(v L v i T i T T dv µ (, u + c(u v, u, v L u L v c (u v, u, v dv du dv du c(v, u, v + l uς(v, v, u dv du L u L v L v. Thus, Cov ln B(T, T 1,..., T n, = ln exp ( +K X(v, v L v dv n T n exp i=1 f(, u du i T n T T f (, u + i T c(v, u, v + l uς(v, v, u dv du L u L v L v T m + σ2. 2 (1 + 1 L v c(v, u, v dv du c (v, u, v dv du Finally, puing he above equaions ogeher yields he desired resul.

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165 Index D-Wiener process, 51 L 2 (H, H, 56 baske models, 22 calibraion, 128 CDS, 44 price, 118 CDS call, 45 price, 118 collaeralized deb obligaions, 42 commercial models, 36 CrediMerics, 38 KMV, 37 Moody s, 38 convering bond, 121 counerpary risk, 2 covariance operaor, 5 Cox process, 11, 136 credi defaul swap... see CDS, 44 credi derivaives, 2, 42 credi spread, 2, 4, 35, 12 credi spread call, 46 credi spread opions, 46 price, 116 credi spread swap, 47 CrediMerics, 38 defaul, 2 kh-o-defaul swap, 47 opions, 42 poin, 37 pu, 44 swap, 42, 44 defaul digials price, 19 defaul poin, 5 defaul pu price, 113 defaul swapions, 45 defaulable HJM model, 75 defaulable raing model, 8 disance-o-defaul, 37 drif condiion, 14, 68 Duffie, 13, 3 Duffie and Lando, 3 Duffie and Singleon, 13 equivalen maringale measure, 66, 97 explici pricing formulas, 19 face value, 1 firs passage ime models, 6 firs-o-defaul swap, 28, 47 forward measure, 7 Fubini heorem, 63 Gaussian random field, 93 defaulable model, 99 definiion, 93 riskless model, 95 Girsanov Theorem, 64 hazard rae models, 11 baske, 22 Duffie and Singleon, 13 hybrid, 29 Jarrow and Turnbull, 12 raings, 15 hedging, 4 defaulable bond opion, 126 HJM model, 66 defaulable, 75 hybrid models, 11, 29 Duffie and Lando, 3 infinie facors defaulable HJM model, 75 defaulable raing model, 8 HJM model, 66 inensiy based models...see hazard rae models, 11 Iô s formula,

166 INDEX 16 Jarrow and Turnbull, 12 Jarrow, Lando and Turnbull, 15 Kennedy, 97 KMV, 37 kh-o-defaul-swap, 47 Lando, 11, 15, 17, 3 Longsaff and Schwarz, 6 marke models, 33 Schönbucher, 33 maringale, 54 mauriy, 1 mean, 5 Meron, 3 Moody s, 38 Longsaff and Schwarz, 6 Meron, 3 ohers, 1 Zhou, 7 survival measure, 36 race-class, 51 vulnerable claims, 1 vulnerable opions, 42 Wiener process, 51 zero recovery Gaussian random fields, 1 zero recovery measure, 112 Zhou, 7 normal copula, 27 pricing defaulable bond opion, 125 explici formulas, 19 infinie facors, 91 random field, 93...see Gaussian random field, 93 raing, 2 raing based approach, 11 raings, 15 Jarrow, Lando and Turnbull, 15 Lando, 17 calibraion, 21 recovery, 2 recovery of marke value, 14, 75 raing based, 82 recovery of reasury value, 6, 13, 14, 79 fracional, 16 Gaussian random fields, 15 raing based, 86 recovery raes, 39 reduced form models...see hazard rae models, 11 reference risk, 2 senioriy, 5 shor mauriy spreads, 4, 7 sochasic inegral definiion, 54 maringale propery, 54, 58 srucural models, 2

167 Erklärung Hiermi erkläre ich, dass ich die Arbei selbsändig verfass und nur die angegebenen Hilfsmiel verwende habe. Gießen, den