Pricing Credit Derivatives in Credit Classes Frameworks

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For illustration, the KMV Corporation provides a transition ... arbitrage-free framework to price risky bonds by identifying credit classes to the rating agencies' ...

Pricing Credit Derivatives in Credit Classes Frameworks Franck Moraux, Patrick Navatte Universit de Rennes 1-IGR and CREREG-Axe Finance. 11, rue Jean Mac 35000 Rennes, France

1 Introduction

Many credit management systems, based on dierent underlying frameworks, are now available to measure and control default and credit risks1 . Homogeneous credit classes and associated transition matrix may thus be constructed within many dierent frameworks. For illustration, the KMV Corporation provides a transition matrix within a structural approach la Black-SholesMerton Crouhy-Galai-Mark 5 . Independentely from the underlying framework, a methodology based on credit classes may therefore be used to price any claim contingent on credit events among which the default. Altman and Kao 1 have early suggested the Markov chain methodology to model a credit process based on credit classes. Skinner 21 , studying a trinomial tree with both a re ecting boundary and a absorbing one for valuing corporate bonds, has in fact explored a special Markov Chain. However Jarrow-Lando-Turnbull 10 JLT hereafter have oered the rst complete arbitrage-free framework to price risky bonds by identifying credit classes to the rating agencies' ones2 . Other contributors are Kijima 13 , KijimaKomoribayashi 14 KIK hereafter and Arvanitis-Gregory-Laurent 3 AGL hereafter . This paper focuses on the ability of the Markov Chain methodology for pricing credit derivatives in credit classes framework, matching any observable risky term structures when the generator matrix is held constant or piecewised constant as in JLT, KIK, AGL . Chosen credit derivatives include those already priced in other approaches by Das 4 and Pierides 19 in the structural setting , and by Longsta-Schwartz 17 , Schnbucher 20 and Due 6 in the pure intensity framework . KIK 14 , who concentrate on the European spread put option, are thus systematically extended. The rest of this paper is organized as follow. Section 1 recalls the general framework. Section 2 derives an useful conditional probability density See the Journal of Banking and Finance special issues on credit risk modelling, Lando 15 and Cathcart-El Jahel 7 for surveys on the structural or reduced form theoretical backgrounds. 2 Overlapping w.r.t. the credit spreads, lack of reaction are the most documented problems of the rating process - see among others Weinstein 23 or HandHolthausen-Leftwich 8 .


H. Geman et al. (eds.), Mathematical Finance — Bachelier Congress 2000 © Springer-Verlag Berlin Heidelberg 2002


Franck Moraux, Patrick Navatte

function. Section 3 reviews many credit derivatives valuable in the chosen setting. Section 4 considers the calibration procedures. Section 5 is devoted to a numerical examples and a nal section concludes.

2 The General Framework Following JLT 10 , let it be a frictionless continuous economy with a nite horizon [0, TH ], whose uncertainty is represented by (Ω, F, (Ft )0≤t≤T , P) and where riskless and risky term structures of interest rates are available. One then assumes that rst there exists a unique equivalent martingale measure denoted P˜ and that second under this measure, the interest rate and the credit risks are non correlated. The credit process (νt )t is supposed to be correctly described by a continuous Markov chain dened on the nite set of credit status denoted S = (Si )i=1,K+1 . K depends on the number of classes retained in the discriminating procedure and while the rst state S1 describes the surest state, SK+1 stands for bond in default. Moreover S ∗ = (Si )i=1,K . The one period transition matrix associated with ν is written Q = (qij )i,j=1,K+1 , qij = P [νt+1 = j/νt = i], ∀i, j ∈ S is therefore the probability of going from a state i to an other j , 1 − qj,K+1 the survival probability. Since the (K + 1)th state is an absorbing one, qK+1,j = P [νt+1 = j/νt = K + 1] = δj=K+1 with δ the Dirac measure. Following JLT 10 , KIK 14 , AGL 3 , the credit process is assumed to be both Markovian and time homogeneous by period. Standard results then tell us that tractable Markov chains may be specied in terms of a constant Λ = (λi,j )i,j=S×S , any t−transition matrix being given by generator matrix,  1 Qt = etΛ = k≥0 k! (tΛ)k . Historical transition matrix Q cannot however be used for risk neutral pricing without correction. And the risk neutral transition matrix RNTM hereafter Q˜ must verify properties to be a transition matrix and generate a priori correct credit spreads. Expressed in term of the risk neutral generator matrix RNGM hereafter, Λ˜, these follow. First, ˜ K+1,j = 0 warranties that the default state is an absorbing state. Sec∀j : λ ond, ∀i ∈ cal S, 0x2200j0x2208S, i = j, λ˜i,j ≥ 0 ensures that the transition probabilities cannot be negative. Third, kAAA,,D λ˜i,k = −λ˜i,i assures a to  tal of transition probabilities equals to one. Finally, j≥k λ˜ i,j ≤ j≥k λ˜i+1,j implies that lower credit classes are riskier JLT 10 , lemma 2.. Additional conditions related to empirical monotonicities may be found in Kijima 13 . Assuming that bondholders face a constant writedown ω at maturity in case of an early default, the price of a risky bond is: H

p(t, T ; Θ) = p(t, T )(δ + ω P˜t [τ > T ])


where p(t, T ) is the price of a default-free zero-coupon bond paying surely a dollar at T and τ = inf[t : νt = K + 1] the default time. The credit status Θ includes both the credit class and the seniority, the recovery rate is

Pricing Credit Derivatives in Credit Classes Frameworks


is such that p(t, T ; Θ) = e−Y (t,T ;Θ)(T −t) , the yield spread over the Treasury one or credit spread is dened by h(t, T ; Θ) = − T 1−t ln(δ +ω P˜t [τ > T ]). By analogy with the interest rate theory, the instan taneous forward credit spread or local spread, h(t, T ; Θ) = T 1−t tT s(t, s) ds, can be computed as

δ = 1−ω 3 . Since the risky yield Y

s(t, T ) = −

∂ ln(δ + ω P˜t [τ > T ]). ∂T


Several remarks may be done. First, the survival probability, i.e. 1 −h(t,T ;Θ)(T −t) − δ) implies, among many things, that the survival probω (e ability may be interpreted as the price of a bond with yield h if and only if its recovery rate is null. Second, the credit margin is innite if and only if qt,T (j, K + 1) = 1 and δ = 0 roughly speaking when one loses everything surely. Third, the instantaneous forward credit spread is seen to be closely related to the conditional default time density under the martingale measure.

3 The Conditional Default Time Density Since Lando 15 and Sch nbucher 20 , local credit spreads are known to be critical for credit derivatives valuation. JLT 10 have already computed them as the dierence of the risky and risk free instantaneous forward rates. Equation 2.2, nevertheless, allows one to closely relate them to the risk neutral probability density function of the default time conditional on survival until then. Proposition 3.1. The risk neutral default time conditional density given Λ and no early default, in a credit class framework and ∀i ∈ S ∗ , is : fτt (s) =


q˜i,k (t, s)μk (s)λk,K+1 .



where μk (s) denotes the s−risk premium. Proof. Following JLT 10 , the transition probability is known to solve the ˜ ) ˜ T )Λ(T ˜ ) where Λ(T ˜ ) denotes the = Q(t, following forward equation ∂Q(t,T ∂T T −risk neutral generator. This equation denes : #K+1 


q˜i,k (t, T )μk (T )λk,j


And therefore: 3

˜ i [τ t , let's review the di erent pay-o s. The fall in credit quality is accompanied by a rise of the credit margin required by investors. This event may be hedged by an European call on the credit spread Longsta Schwartz 17 , or by an European put on the price Das 4 . Denoting Ch , Pp respectively the spread call and bond put, respective payo s are given by: Φ(Ch (t, T ; Θ)) = max[h(t, T ; Θ) − K, 0] Φ(Pp (t, T ; Θ)) = max[Kp(t, T ) − p(t, T ; Θ), 0].

Symmetrically, the rise in the credit quality can be hedged either by a European put on the credit spread KIK 14 or by an European call on the risky bond price whose payo s are respectively : Φ(Ph (t, T ; Θ)) = max[K − h(t, T ; Θ), 0], Φ(Cp (t, T ; Θ)) = max[p(t, T ; Θ) − Kp(t, T ), 0].

Following standard arbitrage arguments, these contracts are priced by discounting the payo s. The market risk and the credit risk being not correlated each other, option prices written on credit margin are ˜


Hh (t , t, T, Θ) = EtP [βt βt−1 Φ(Hh (t, T, Θ))] = p(t , t)EtP [Φ(Hh (t, T, Θ))]

where H stands for C , P

call, put and then nowadays values are ˜

p(0, t)E0P [Φ(Hh (t, T, Θ))] = p(0, t)

K+1  j=1

q0,t (i, j)Φ(Hh (t, T, Θj ))



Franck Moraux, Patrick Navatte

where q0,t (i, j) = (etΛ )ij . Option prices involving risky bonds are given by: ˜

Hp (t , t, T, Θ) = EtP [βt βt−1 Φ(Hp (t, T, Θ))]

˜ ˜ = EtP [βt βt−1 p(t, T )]EtP [max[(−1)1H=C (K − δ − ω P˜t (τ > T )), 0]] ˜ = p(t , T )EtP [max[(−1)1H=C (K − δ − ω P˜t (τ > T )), 0]].

In particular, nowadays values are: p(0, T )


q0,t (i, j) max[(−1)1H=C (K − δ − ωQSt,T (j)), 0]


where q0,t (i, j) = (etΛ )ij and QSt,T (j) = 1 − qt,T (j, K + 1). Let's point out that these credit derivatives do not involve the same information regarding the riskless term structure. Furthermore, strike prices appear critical whereas they would have been endogenously computed in other framework Das 4 and Pierides 19

. 4.3

Credit Derivatives and Credit Class Events

Credit derivatives hedging against a credit class change tend to be more important as the credit portfolio management becomes a risk classes management5 . These contingent contracts include both digital contracts delivering a unit in case of a change in credit class and credit options whose strike prices are xed at a suitable level. These latter contracts may however be shown not to be valuable in our framework because they need border credit spread between credit classes unattainable in a constant generator framework. An European credit class digital put option, digecc , delivers a unit at maturity if the change of credit class occurs during the life of the contract. Its payo is given at any date t by Φt (digecc (0, T, Θ)) = 1{τ ≤T }∩{t=T } (t) where τcc denotes the time the credit status changes. Its American version, digacc , pays the unit as soon as the event occurs its payo is therefore Φt (digacc (0, T, Θ)) = 1{τ ≤T }∩{t=τ } (t). Once the proper transition matrix dened, these contracts are simple generalisations of previous default digitals. For ease of presentation, let's consider the special case of a credit class digital option paying one unit if the creditworthiness changes to the BB one or less. The relevant transition matrix for pricing is dened on S = (Si )i=1,BB and considers the BB-rated class as an absorbing state. More precisely, one needs the squared matrix Q dened by cc


with qi,j

= qi,j for qBB,j = δBB (j). 5




Q = (qij )i,j=1,BB  i, j < BB , qi,BB = k≥BB qi,k

speculative classes are often non desired.

for all i < BB and

Pricing Credit Derivatives in Credit Classes Frameworks


Then, thanks to the independence assumption, the respective prices are: ˜

digecc (0, T, Θ) = E P [βT−1 ΦT (digecc (0, T, Θ))] = p(0, T )q0,T (i, BB) ˜

digacc (0, T, Θ) = E P [βτ−1 Φτ (digacc (0, T, Θ))]

T  λk,BB p(0, s))qi,k (0, s) ds = k∈S


Formal proofs are similar to the default digitals ones. Other credit class digitals are valued along the same lines by associating the proper transition matrix.

5 Implementation Any objective transition matrix must be calibrated before risk neutral pricing. JLT 10 , KIK 14 and AGL 3 have proposed di erent procedures to estimate credit risk premia.

5.1 A Comparison of Calibration Procedures Apart minor di erences, all of them have suggested risk premia as solutions of a squared errors minimization between observed and theoretical prices6 . Let's denote Λ = (λi,j )i,j∈S×S the historical generator, Λ˜ = (λ˜i,j )i,j∈S×S the risk neutral generator matrix or RNGM, I the identity matrix, e the square zero matrix whose last column is the unity and pˆ the observed bond prices. The ˜ recursive relation Q˜ 0,t+1 = Q˜ 0,t Q˜ t,t+1 = Q˜ 0,t eΛ(t) must hold for any horizon and, as a result, two distinct assumptions concerning the RNTM over a period are possible. First, following AGL 3 , a unique RNGM may be considered. Any rated bond prices available on the market are used and the generator matrix is designed to match simultaneously, any risky term structures and therefore any horizon. Formally, the RNGM Λ˜ is solution of the following least square optimization: ⎡

Λ˜ = arg min ⎣ ¯ Λ


¯ i,j − λi,j ]2 + (βi,j )−1 [λ


¯ 2⎦ [ˆ pn (0, T ; Θn ) − p(t, T ; Λ)]

i∈S n=1

subject to conditions warrantying a well-dened risk neutral generator recalled in the rst section. β stands for some subjective weights. Second, along the lines of JLT 10 and KIK 14 , the RNGM may be assumed as a deterministic function of the historical generator and the horizon 6

To warranty some consistent value, JLT 10 have constrained solutions to positive ones, AGL 3 have prevented them to be far from the historical one and KIK 14 have bounded them.


Franck Moraux, Patrick Navatte

˜ = U (t)Λ. Both recursive procedures identify U (t) to Π(t) = considered - Λ(t) (πi,j (t))i,j∈S×S the diagonal matrix that contains the K transition matrix

risk premia and the unity. Both also assume proportional adjustments between the one-period RNTM and the historical one, as exposed in Table 1. Risk premia are supposed invariant either for any attainable class from the credit class considered JLT 10  or for any attainable class except the default class KIK 14 . As demonstrated by KIK 14 , arbitrage conditions must be also veri ed. It is noteworthy that they have excluded the JLT 10 'unconstrained' procedure in most empirical expriments conducted7 . Other operational details are given in appendix, we now turn to the numerical examples.

Table 1: Horizon-dependent Risk Premia for the Transition Matrix Recall that Π(t) = (πi,j (t))

, Π(0) = I . Conditions are ∀i ∈ S ∗ Alternative Procedures KIK 14 JLT 10


Proportional Adjustment [Q˜ t,t+1 − I] = Π(t)[Q − I] [Q˜ t,t+1 − e] = Π(t)[Q − e] Conditions πi (t) ∈ ]0, 1/ (1 − qi,i )[ πi (t) ∈ ]0, 1/ (1 − qi,K+1 )[

5.2 Numerical Illustrations of Credit Derivatives Valuation An investor who fears the credit quality to fall may consider many dierent credit derivatives among which calls on credit spread, puts on the risky bond price with a strike price sensible to interest rates and default digital put options. In the following, observable term structures of interest rates, plotted in Figure 1, are used. Based on data reported in KIK 14 160597, they are interpolated with a standard modi ed Nielsen-Siegel's procedure 18 . This is consistent with Helwege-Turner 9 who nd that risky bonds better than B have upward-sloping credit yield curves8 . For calibration, the S&P 's average one-period transition matrix9 has been chosen. Finally, subjective weights in the AGL 3 procedure have been chosen to match at the best the observed credit risky term structure. Figure 2 rst compares American digital put option prices to their European counterparts. While the European digital put option has been shown JLT's proportional adjustment may imply a division by zero avoided by imposing a striict postive value for any qj,K+1 even j = 1. In the KIK procedure, the rst computed risk premia are well dened iif the write down faced by the bondholder is such that ω > 1 − pˆ(0, T ; Θi )/ˆp(0, T ). This restriction is avoided by assuming below that the recovery rate is null. 8 For the riskiest exposure, a strictly downward sloping term structure is nevertheless dened as a + be−ct with a,b some strict positive values. It allows the AGL calibration procedure. See the discussion below. 9 available in Standard & Poor's Special Report  22 , Table 8 or from the authors. 7

Pricing Credit Derivatives in Credit Classes Frameworks


Term structures of interest rates from the risk free to the CC risky one. Term structurs of interest rates are based on observable data reported in KIK 1998 and interpolated with the Nielsen-Siegel 1987 modi ed procedure Fig. 1.

proportional to the default probability, the American digital put option may be computed as  

T /Δt

diga (0, T, Θ) =

˜ k,K+1 (jΔt)Δt p(0, jΔt)˜ qi,k (0, jΔt)λ

j=1 k∈S ∗

where T is the expiration date. The supervision frequency, Δt, of the American digital put option is chosen daily and p(0, jΔt) = e−N S(jΔt) where N S stands for the modied Nielsen-Siegel function. Let's precise that λ˜ k,K+1 (jΔt) = ln q ˜ k,K+1 for AGL 3 and λ ˜ k,K+1 (jΔt) ≈ q λ k,K+1 q −1 for KIK 14 and JLT 10 where q is the calibrated risk neutral matrix over the suitable one year period As expected, the price of an American digital is always greater than its European equivalent and riskier is the exposure, more costly is the insurance contract. This graph also illustrates the pricing dierence induced by the risk neutral calibration procedure. k,k


European vs. American default digital put option prices for two underlying exposures rated AA B

Fig. 2.


Franck Moraux, Patrick Navatte

Figure 3 nally plots credit spread call options and risky bonds put options hedging credit exposures rated from AAA to B and maturing 10 years later to be secured at a BBB-rated level. It then appears that these two insurance contracts behave quite similarly suggesting a parity relation. This latter has already been pointed by Pierides 19 10 .

Credit derivatives hedging against a fall in the credit quality of a 10-maturing bond below a BBB level. Underlying exposures are rated from AAA to B

Fig. 3.


The Constant Generator Case

In fact, it can be shown that a single constant risk neutral generator matrix la AGL 3 cannot, by construction, handle with any shapes of risky term structures. This claim may be illustrated by Figure 4 that provides the credit spreads implied by the constant RNGM after risk neutral calibration on strictly increasing term structures of Figure 1. The CCC-rated credit spread is seen to be constant w.r.t. the maturity. This property of constant risk neutral generator matrix may be formally proved as follows. Proposition 5.1. A Markov Chain process with a constant generator well

dened w.r.t. credit risk cannot generate, by construction, any strictly increasing tem structure of credit spread for the last credit class before default. Moreover, the lowest possible at CCC-term structure of implied credit spreads is directly given by − ln(˜qCCC,CCC ) = −λ˜CCC,CCC = λ˜CCC,D . Proof. First, let's recall that the CCC-rated class is, in our setting, the last class before default. Let's also point out that if Q = (qij )i,j=1,K+1 is such that qCCC,j > 0 for any j < CCC then the probability of being upgraded at the next instant is not null and as a result the credit margin is decreasing with respect to the horizon. qCCC,j are therefore null ∀j < CCC and it then remains qCCC,CCC = 0 and qCCC,D = 0. Recalling furthermore that T −t the default state is absorbing, one has P˜t [τ > T ] = qCCC,CCC . As a result, 10

and may considered in the present framework as well available upon request .

Pricing Credit Derivatives in Credit Classes Frameworks


Term structures of credit spreads implied by a single subjective generator matrix. Term structures are based on observable data plotted in Figure 1.

Fig. 4.

the term structure of implied credit spreads given by − T1 ln P˜0 [τ > T ] is − ln qCCC,CCC . At last, equalities are obtained using eq. 28, JLT 1997

where q˜i,K+1 = λ˜i,K+1 ( q˜ln q˜−1 ). Indeed, since q˜i,i = 1 − q˜i,K+1 for CCC , one has q˜i,i = eλ˜ . The risk neutral calibration suggested in AGL 3 therefore fails to match risky term structures whose CCC one is strictly increasing as that reported in KIK 14 . This is not the case of KIK 14 's recursive procedure. i,i



6 Conclusion This paper has questioned the ability of the Markov Chain methodology with a constant or piecewised constant generator matrix for valuing credit derivatives, matching the observable risky term structures and generating implied credit spreads. This setting has been shown capable to price many credit derivatives. However since it generates nite and deterministic credit spreads and cannot handle with correlation between exposures, nor spread risk credit derivatives nor those involving a basket of credit exposures can be considered. Moreover it has been demonstrated that the constant risk neutral generator matrix cannot match any shapes of risky term structures. In particular, no strictly increasing CCC-term structure of credit spread, as the one reported in Kijima-Komoribayashi 14 , can be implied by the framework.

7 Appendix 7.1

Operational Notes

Eective estimation procedures depend on the available information. Let's assume available a set of available risky zero-coupon bond prices


Franck Moraux, Patrick Navatte

as numerous as the number of credit class and maturities. Since the theoretical pricing equation holds ∀T > 0,∀i ∈ S one has pˆ(0, T ; Θi ) = p(0, T )(δ + ω P˜i [τ > T ]) or equivalently (ˆ p(0, T, Θi ))i∈S,T

qi,K+1 (T ) = a + bˆ p(0, T ; Θi ), ∀i ∈ S,

with a = 1 + δω−1 , b = −[ωp(0, T )]−1 . Since, on the other hand, ∀T : qi,K+1 (T ) =

a(i, j)πj (T ), ∀i ∈ S


with a(i, j) = q˜ij (T −1)qj,K+1 , one has a system of (K+1) equations with (K+ 1) unknowns. At each horizon there exists a unique set of risk premia if and only if rst the previous (˜qi,j (T − 1))i,j∈S and therefore (πj (T − 1))j∈S  have been previously calculated and second the resulting (a(i, j))i,j∈S×S matrix is inversible. In this case, however, nothing warranties that this one-to-one relation is costless in particular w.r.t. the stripping procedure. More general recursive least square minimization procedure subject to conditions ensuring suitable properties appears then reasonable to provide risk premia. Finally, to compute the RNGM Λ˜ from the RNTM Q˜ , a couple of numerical approximations are available, A rst one, presented in JLT 10 Eq. 28, may be applied to I + Π(t)(Q − I). It holds as soon as 'more than one transition per year' is a low probability event see Appendix A, 517-520.. A second one is related to an analytical approximation suggested in JLT 10 and exploits ˜ and that U is identied to Π . both Q˜ t,t+1 ≈ I + Λ(t) 7.2

Default Swaps Valuation

As already claimed, digital default options allow one to price many other credit derivatives among which the default swaps. Default swaps are insurance contracts paying, at the default time and against an annuity premium, an amount covering the incured loss  6 . Two default swaps with constant fee may be considered. − The default digital swap is an exchange contract of a regular fee paid until default against a unit amount paid at default. Pricing this contract is equivalent to valuing the fee noted x. Along Schnbucher's lines 20 , the constant and continuous fee rate is given by:

x 0


[1 − qi,K+1 (0, t)] dt =

 k∈S ∗




p(0, s)qi,k (0, s) ds

− The default swap, dsw, is an exchange contract of a regular fee paid until default against an amount paid at default equaling the resulting lost value. Schnbucher 20 and Due 6 have demonstrated that it is duplicated by a portfolio containing a long position in a default free oating rate note

Pricing Credit Derivatives in Credit Classes Frameworks


and a short position in default risky one, this latter being obtained with the help of a digital default put option. The price of a default swap is given by: dsw = f rn(0, T ) + c

 k∈S ∗




p(0, s)qi,k (0, s) ds − 1

where f rn denotes the price of a par default free oating rate note. These formulae are valid for any model of term structure of interest rates. Acknowledgements. We thank participants at the BFS Congress - Paris 2000 for comments and stimulating discussion, in particular T. Bielecki, H. Geman, JN. Hugonnier, M. Jeanblanc, JL. Prigent, O. Renault, O. Scaillet, P. Sch nbucher and L. Schloegl. We are especially indebted to P. Sch nbucher for suggesting digital contracts contingent to a change in credit class. S. Aboura, M. Bellalah, F. Quittard-Pinon are also acknowledged.

References 1. E. Altman, D. Kao : The Implications of Corporate Bond Ratings Drift , Financial Analyst Journal, May-June, 1992, 64-75. 2. E. Altman, V. Kishore : Almost Everything You Wanted to Know About Recoveries on Defaulted Bonds. Financial Analyst Journal Nov 1996 5764 3. A. Arvanitis, J. Gregory, J.P. Laurent : Building Models for Credit Spreads. Journal of Derivatives Spring 1999 2743 4. S. Das : Credit Risk Derivatives. Journal of Derivatives 2 1995 723 5. M. Crouhy, D. Galai, R. Mark : A Comparative Analysis of Current Credit Risk Models. Journal of Banking & Finance 24 12 2000 59117 6. D. Due : Credit Swap Valuation. Financial Analyst Journal Jan 1999 7387 7. L. Cathcart, L. El Jahel : Valuation of Defaultable Bonds. Journal of Fixed Incomes 1998 6578 8. J. Hand, R. Holthausen, R. Leftwich : The Eect of a Bond Rating Announcements on Bond and Stock Price. Journal of Finance 47 1992 733750 9. J. Helwege, C. Turner : The Slope of the Credit Yield Curve for SpeculativeGrade Issuers. Journal of Finance 54 5, 1999 18691884 10. R. Jarrow R., D. Lando, S. Turnbull : A Markov Model for the Term Structure of Credit Risk Spread. Review of Financial Studies 10 1997 481523 11. R. Jarrow, S. Turnbull : Pricing Derivatives on Financial Securities subject to Credit Risk, Journal of Finance 50 1995 5386 12. Journal of Banking & Finance. Special issues 22 1011, 24 12 13. M. Kijima : Monotonicities in Markov Chain Model for Valuing Corporate Bonds subject to Credit Risk. Mathematical Finance 8 1998 229247 14. M. Kijima, K. Komoribayashi : A Markov Chain Model for Valuing Credit Risk Derivatives. Journal of Derivatives Fall 1998 97108. 15. D. Lando : Modeling Bonds and Derivatives with Default Risk, in M. Dempster and S. Pliska eds. Mathematics of Derivatives Securities Cambridge University Press 1997a 369393 16. D. Lando : On Cox processes and credit risky bonds. Review of Derivatives Research 2 23 1997b 99120


Franck Moraux, Patrick Navatte

17. F. Longsta, E. Schwartz : Valuing Credit Derivatives. Journal of Fixed Income 5 1995 25 32 18. C. Nielsen, A. Siegel : Parsimonious modeling of Yield Curves. Journal of Business 60 4 1987 473-489 19. Y. Pierides : The Pricing of Credit Risk Derivatives. Journal of Economic Dynamics and Control 21 1997 1579 1611. 20. P. Schnbucher : Pricing Credit Risk Derivatives. Working paper University of Bonn 1998 21. F. Skinner : A trinomial model of bonds with default risk. Financial Analyst Journal 50 2 1994 73 78 22. Standard & Poor's Special Reports. 1998 23. M. Weinstein : The eect of a rating change announcement on bond price. Journal of Financial Economics 5 1977 329 350

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