Credit risk valuation with rating transitions and partial
information
Donatien Hainaut†
Christian Yann Robert‡
September 24, 2013
†
‡
ESC Rennes Business School & CREST, France.
Email: [email protected]
Université de Lyon, Université Lyon 1, ISFA, LSAF, France.
Email: [email protected]
Abstract
This work intends to shed some light on a new use of Phase-type distributions in credit
risk, taking into account dierent ows of information without huge numerical calculations.
We consider credit migration models with partial information and study the inuence of a
decit of information on prices of credit linked securities. The transitions through the various credit classes are modeled via a continuous time Markov chain but they are not directly
observable by investors in the secondary market. We rst consider the case of one bond
issuer and study three settings of partial information. In a rst model, information about
ratings arrives at predetermined dates with delay periods. In a second model, information
arrives randomly according to an exogenous Poisson process, whereas in a third model, information arrives randomly according to an endogenous rule (transitions are observed only
when they lead the Markov chain to a class with a lower credit rating). We infer in the
three settings bonds and options prices, and we provide an explicit description of the dynamics of bond prices under real and pricing measures. We also consider the case of several
bond issuers. We rst study a model for two dierent issuers and analyze the cross eects
of decit of information and contagion on bonds prices and correlation of default times. We
then propose a model for several homogeneous issuers. Finally numerical illustrations show
the relevance of taking into account decits of information on prices of credit linked securities.
Credit migrations models; Partial and delayed information; Corporate bonds;
Credit derivatives; Phase-type distributions.
Keywords.
1
Introduction
There are two main types of models that attempt to describe default processes in the credit risk
literature:
structural and reduced form models.
Structural models use the evolution of rms
structural variables, such as asset and debt values, to determine the time of default.
In these
models, it is usually characterized as the rst hitting time of the rm's asset value (modeled as a
diusion) to a given boundary determined by the rm's liabilities. This time of default is usually a
predictable stopping time. In contrast, reduced form models do not consider the relation between
default and rm value in an explicit manner and the credit events are specied in terms of some
exogenous given jump processes. It is possible to distinguish between the reduced-form models
that are only concerned with the modeling of the time of default, called the intensity-based models,
and those with migrations between credit rating classes, called the credit migration models. In this
approach the time of default is a totally inaccessible stopping time. Reviews of credit risk models
can be found in many books, including among others Bielecki and Rutkowski (2002), Due and
Singleton (2003), Schönbucher (2003) and Lando (2004).
1
A quick look at the above description could lead one to conclude that structural and reducedform models are competing paradigms. However, an intrinsic connection between these two approaches has been pointed out in the last decade by several authors.
Actually it is possible to
show that structural models can be viewed as reduced-form models by secondary markets that
have only incomplete information. In the seminal paper, Due and Lando (2001), the rm asset
value is modeled as a continuous-time process but the market has only at discrete point times
noisy accounting reports of this value and knows the default history of the rm. In this setting,
Due and Lando establish that the default time admits an intensity that is proportional to the
derivative of the conditional density of the rm's value at the default barrier. Structural models
with incomplete information have also been studying by Kusuoka (1999), Nakagawa (2001), Jarrow and Protter (2004), Coculescu et al. (2008), Choi (2008), Frey and Schmidt (2009), Guo et
al. (2009) or Hillairet and Jiao (2012) among others.
Reduced form credit risk models with incomplete information have been less studied. Schönbucher (2004), Collin-Dufresne et al.
(2003) and Due et al.
(2006) considered models where
default intensities are driven by an unobservable factor process and the information of the secondary market is given by the default history of the portfolio of obligors, augmented by some
economic covariates. Given information about the factor process, the default times of obligors are
assumed to be conditionally independent, doubly stochastic random times. Frey and Runggaldier
(2010) and Frey and Schmidt (2012) considered general reduced form pricing models where default
intensities are driven by some factor processes which are not directly observable by investors in
secondary markets. Their information set only consists of the default history and of observation
of noisy prices for traded credit securities. Tang et al. (2011) also priced CDS in a reduced form
model in which the intensity is modulated by a Markov chain.
Our paper considers credit migration models with partial information and studies the inuence
of a decit of information on prices of credit linked securities. It diers from the previous contributions in several directions: First, instead of using a single credit rating intensity based model,
we rather work in a multiple credit ratings framework where we can look in greater details at the
changes in credit quality that may lead to default. This framework is of practical interest e.g. for
corporate bankers who use internal (or external) rating systems to price loans for non listed rms
such as SME. The transitions through the various credit classes are modeled via an homogeneous
or a piecewise homogeneous continuous time Markov chain. Compared to structural models, rating based models don't specify any dynamics for rm's asset. Second, with this framework, it is
possible by using the properties of Phase-type distributions to derive some explicit pricing formulas for the prices of risky bonds and options on these risky bonds. The inuence of a decit of
information is easily quantied without requiring intensive simulation of the credit rating. Third,
with regard to information, we consider a framework of discretely delayed ltrations as introduced
in Guo et al. (2009), i.e. no new information ow in between two consecutive observation times
as long as the issuer defaults. Compared to e.g. works of Due et al. (2001) or Choi (2008), that
consider a constant delay, our paper explores cases in which the information arrives at random
times, or at endogeneous dates.
The remainder of the paper is organized as follows. In Section 2, we consider several models
with one issuer under three discontinued ows of information.
First, we assume that investors
observe the issuer's rating only at discrete and deterministic points with a xed delay. Second,
we assume that information arrives randomly, but exogenously, according to a Poisson process.
Third, we assume that the ratings are observed at deterministic points or when transitions lead
the Markov chain to a class with a lower credit rating than the last rating known. In this way
information also arrives randomly, but according to an endogenous rule.
We infer in the three
settings bonds and options prices and we provide an explicit description of the dynamics of bond
prices under real and pricing measures. In Section 3, we consider contagion models with several
issuers. We rst study a model for two dierent issuers and analyze the cross eects of decit of
information and contagion on bonds prices and correlation of default times. We then propose a
model for several homogeneous issuers. Numerical illustrations are given in Section 4 and show
the relevance of taking into account decits of information on prices of credit linked securities. All
proofs are gathered in a last section.
2
2
Several models for one issuer's bond with three types of
partial information
In this section, we propose three models to price corporate bonds under a discontinued ow of
information. The rm that issues the bond is marked in a rating system counting
levels
1
and
L−1
L,
the issuer reaches the level
maturity
L
levels. The
correspond respectively to the highest and to the lowest credit qualities. When
it goes into bankruptcy and the bond pays
R,
the recovery rate, at
T.
(Xt ), a continuous-time, time-homogeneous
(Ω, F, (Ft )t , P ) where P denotes the real probability
measure. The Markov chain takes its values in a nite state space E = {e1 , e2 , . . . , eL } where ei
L
is the i-th vector of the standard basis of R (the j -th component of ei is the Kronecker delta,
δi,j , for each i = 1, ..., L). The dynamics of the chain (Xt ) is specied by the L × L matrix of
intensities of transition, Γ = (γi,j )
i,j=1,2,...L . The elements of Γ satisfy the following conditions
Information about the current rating is carried by
Markov chain dened on a probability space
L
X
γi,j ≥ 0 ∀ i 6= j
γi,j = 0 i = 1, ..., L.
j=1
Γ
is related to the probability of transition from rating
i
to
j,
on a period of time
∆t,
as follows
pi,j (t, t + ∆t) = pi,j (∆t) = e>
i exp (Γ∆t) ej ,
where
>
denotes the transpose operator and
exp (Γ∆t) is the exponential of the matrix Γ∆t. Given
L, the matrix of intensities can be rewritten in
that the issuer defaults when it reaches the rating
the following block form
Λ ς
0 0
Γ=
ς = (ς1 , . . . , ςL−1 )> is a L − 1 dimensional vector with non-negative components and Λ is
>
a (L − 1) × (L − 1) matrix such that ς = −Λ 1L−1 (with 1L−1 = (1, ..., 1) ). We introduce the
matrix D dened as


1 ... 0 0


D =  ... . . . ... ...  = IdL−1 0L−1
where
0
where
IdL−1
L−1
rst elements of
...
1
0
L − 1. The product DXt is then the vector of the
eL is the only absorbing state of the chain.
τ = inf {t > 0 : Xt = eL }, has a Phase-type distribution and
is the identity matrix of dimension
Xt .
We assume that the state
In this setting, the time of default,
the probability of survival is given by (see e.g. Rolski et al. (2009))
P (τ > t|F0 ) = X0> D> exp (Λ t) 1L−1 .
Let us introduce the following processes
ˆ
Hti = 1{Xt =ei } ,
Hti,j =
X
1{Xu− =ei } 1{Xu =ej }
and
Mti,j
is a
γi,j Hui du
for
i 6= j.
0
0<u≤t
It is well-known that
t∧τ
Mti,j = Hti,j −
F -martingale
under the measure
P
(see e.g.
Lemma 11.2.3 in
Bielecki and Rutkowski (2002)).
We may now identify the dynamics of the rating process
(Xt )
under the pricing measure
For the sake of clarity we assume for the three following subsections that
(Xt )
Q.
is also a time-
homogeneous Markov chain. We explain in the fourth subsection how to extend results to the case
of a piecewise time-homogeneous Markov chain.
3
Q, (Xt ) is dened by a L × L matrix of intensities of transition,
A = (ai,j )i,j=1,2,...L , that satises the same types of conditions as for Γ. Since Q is an
measure to P , the matrix A can be rewritten in the following block form
B b
A=
0 0
Under the pricing measure
denoted by
equivalent
where
b = −B 1L−1 .
Let us now introduce the costs of transition
ai,j = (1 + κi,j )γi,j ,
and such that
κi,j > −1
PL
and
j=1
for
κi,j
dened as
i 6= j,
(2.1)
ai,j = 0. By virtue of Proposition 11.2.3 in Bielecki and
Q is characterized by the Radon Nikodyn density ηt
Rutkowski (2002), the probability measure
given by
ˆ
ηt = 1 +
(0,t]
and is such that the generators of
(Xt )
L
X
ηu−
under
κi,j dMui,j
i,j=1,i6=j
P
and
Q
are respectively
default has also a Phase-type distribution under the pricing measure
survival under
Q
Q
Γ
and
A.
The time of
and the probability of
is now given by
Q(τ > t|F0 ) = X0> D> exp (B t) 1L−1 .
The use of continuous-time, time-homogeneous Markov-chain to model transitions between
credit rating has been rst proposed by Jarrow et al. (1997). Here we further exploit the properties
of Phase-type distributions to discuss the inuence of a decit of information. But note that it
is possible to weaken this assumption and generalize our result by considering extensions to take
into account memory and stochastic transitions as in Arvanitis et al. (1999).
Since their introduction by Neuts in 1975 and their generalization in a multivariate setting
by Assaf et al.
in 1983, Phase-type distributions have been used in a wide range of stochastic
modelling applications in areas as diverse as telecommunications, teletrac modelling, queuing
theory, reliability, ruin theory, healthcare systems modelling (see e.g.
Pérez-Ocón (2011)).
Fackrell (2003) (2008),
However, their use in modelling nancial risks is still recent: to the best
of our knowledge, multivariate Phase-type distribution have been considered to model default
contagion (Herbertsson and Rootzén (2008), Herbertsson (2011)) and to value synthetic CDO
tranche spreads, index CDS spreads,
k th -to-default
swap spreads and tranchelets (Herbertsson
(2008a), Herbertsson (2008b)).
2.1 A rst model for the defaultable bond with information at xed
dates
In practice, the credit quality of issuers is only assessed at discrete times, when credit rating agencies or corporate bankers assign credit ratings for issuers. If no CDS are quoted or if corporate
bonds are not traded regularly, no information on the credit worthiness is available between these
times except when the issuer defaults (which is supposed to be instantaneously observed).
For
non listed rms, the situation is worst. Most of corporate bankers use an internal rating system
and rms' credit worthiness are based on accounting gures.
Then, between dates of nancial
statements disclosures, no information is available.
To estimate the cost of this lack of information, we assume in this subsection that times of credit
rating publications or of nancial statements disclosure,
(p)
(p)
(p)
0 = t0 ≤ t1 ≤ . . . ≤ tn = T ,
are non
δ , due to the time needed for verication of the issuer's
(p)
ti = ti − δ , i ≥ 1, the times when the credit rating ratings are
random. Moreover they integrate a delay,
credit-worthiness. We denote by
really assessed. As pointed out in Guo et al. (2009), this seems to be a realistic assumption as
4
long as it can be assumed that investors observe ratings when credit rating agencies provide their
annual or quarterly reports.
If
Ht
is the indicator variable
1{τ >t} ,
equal to one if the issuer is still solvent, the information
{Gt , t ≤ T } where the sigma
(p) (p+1)
algebra Gt at time t ∈ [ti , ti+1 ) is given by Gt = σ {Xt0 , . . . , Xti , Hu , u ≤ t}.
As our purpose is to illustrate the impact of the lack of information on bond prices, we assume
available to the secondary market is represented by the ltration
that the risk free rate, denoted by
r,
is constant.
This assumption can be released without
introducing major modications in the main results of the paper.
Let us denote by
P (t, T )
T.
monetary unit at maturity
t ≤ T
the price at time
of a zero-coupon bond, that delivers one
The price of the defaultable bond is equal to the expected discounted
payo of the bond, conditionally to the available information
P (t, T ) = e−r(T −t) EQ (R + (1 − R) HT | Gt ) .
The following proposition gives an explicit expression for
Proposition 2.1.
(2.2)
P (t, T ).
(p)
Suppose that t ∈ [ti(p) , ti+1
) for some i = 0, ..., n − 1, then
P (t, T ) = Re−r(T −t) + (1 − R)e−r(T −t) γ(Xti , ti , t, T )Ht ,
where
γ(Xti , ti , t, T ) = Q (τ > T | Xti , t < τ ) =
Note that, if the issuer is still solvent at time
Xt>i D> exp (B (T − ti )) 1L−1
.
Xt>i D> exp (B (t − ti )) 1L−1
(p)
t = ti −,
(2.3)
we observe a jump in the price of
amplitude
(p)
(p)
∆P (t, T ) = (1 − R)e−r(T −t) γ(Xti , ti , ti , T ) − γ(Xti−1 , ti−1 , ti , T ) .
Remark
.
2.2
In case of default, the bond pays
R
at maturity
T
whatever the time of default. In
our time-homogeneous Markov framework, it is possible to analyze the eect on the price of the
immediate delivery of the recovered part of the nominal. Assume that, when the issuer reaches
(p)
(p)
L, it goes into bankruptcy and the bond pays R at time τ . For t ∈ [ti , ti+1 )
i = 0, ..., n − 1, the price of the zero coupon bond is given by (see Section 6.2 for the proof )
the level
P (t, T )
= Re
r(t−τ )
(1 − Ht ) + Ht R
+e−r(T −t) Ht
L−1
X
>
−1
e>
b.Xt>i exp(A(t − ti ))ej
j D (rIdL−1 − B)
j=1
Xt>i D> exp(B(t − ti ))1L−1
and
(2.4)
Xt>i D> exp (B (T − ti )) 1L−1
Xt>i D> exp (B (t − ti )) 1L−1
−R e−r(T −t) Ht
L−1
X
>
−1
e>
b.Xt>i exp(A(T − ti ))ej
j D (rIdL−1 − B)
j=1
Xt>i D> exp(B(t − ti ))1L−1
.
We now derive the dynamics of the zero coupon bond under the reduced ltration by using
the innovations approach.
(p)
Suppose that t ∈ (t(p)
i , ti+1 ) for some i = 0, ..., n − 1. The dynamics of the bond
price under Q and with partial information is given by
Proposition 2.3.
dP (t, T ) = rP (t, T )dt − (1 − R)e−r(T −t) γ(Xti , ti , t, T ) dM̂t0 ,
where (M̂t0 ) is a (G, Q)-martingale.
5
Remark
.
2.4
P and with
i = 0, ..., n − 1
It is also possible to give the dynamics of the bond price under
information by using Equation (2.1). Suppose that
dP (t, T )
=
t∈
(p) (p)
(ti , ti+1 ) for some
partial
rP (t, T )dt + (1 − R)e−r(T −t) γ(Xti , ti , t, T ) ×
L−1
X
κj,L aj,L Ht
j=1
Xt>i exp (Γ (t − ti )) ej
dt
Xt>i D> exp (Λ (t − ti )) 1L−1
−(1 − R)e−r(T −t) γ(Xti , ti , t, T ) dM̂t0,P ,
where
(M̂t0,P )
is a
(G, P )-martingale.
We nally focus on the calculation of the price of an call option on this zero coupon bond with
maturity
S≤T
and strike price
K
C(t, S, K, T ) = EQ
e−r(S−t) (P (S, T ) − K)+ |Gt
.
The interest rate being deterministic, the only factor of risk is the default risk.
(p)
Suppose that t ∈ [t(p)
i , ti+1 ) for some i = 0, ..., n − 1. The price of a call option
(p)
(p)
(p)
of exercise date S ∈ [t(p)
6= tS ) and of strike price K is given by
S , tS+1 ) (ti
Proposition 2.5.
C(t, S, K, T )
=
L−1
X
δ(Xti , ej , ti , t, tS , S) e −r(S−t) e−r(T −S) (R + (1 − R)γ(ej , tS , S, T )) − K Ht
+
j=1
+ (1 − γ(Xti , ti , t, S)Ht ) e−r(S−t) e−r(T −S) R − K
+
where γ(.) is dened by Equation (2.3) and
δ(Xti , ej , ti , t, tS , S)
=
Q (XtS = ej | Xti , t < τ ) Q (τ > S | XtS = ej , t < τ )
=
Xt>i exp (A (tS
>
Xti D> exp (B (t
− ti )) ej
e> D> exp (B (S − tS )) 1L−1 .
− ti )) 1L−1 j
2.2 A second model for the defaultable bond with information at exogenous random times
In the previous model, the credit quality was available at discrete and deterministic times. We now
assume that the rating is disclosed at random times till bankruptcy according to a homogeneous
Poisson process (if the company or country is in default, no more information is disclosed). The
Zk , k ≥ 0. The intervals of time
Uk = Zk − Zk−1 , are assumed to be indepen−1
dent and identically distributed as exponential random variables with mean ν
under the real
probability P . Moreover they are independent of the Markov chain (Xt ). The number of rating
r
disclosures at time t is denoted by Nt . By using the properties of τ (see e.g. Rolski et al. (2009)),
r
it is easily seen that the distribution of the stopped Poisson process (Nt ) under P is given by
random times of (possible) rating disclosures are denoted by
between two successive publication dates of rating,
P (Ntr = n | F0 )
= P (τ ≤ t , Nτr = n | F0 ) + P (τ > t , Ntr = n | F0 )
ˆ t
(νt)n −νt > >
(νu)n −νu > >
=
e
X0 D exp (Λ u) ςdu +
e
X0 D exp (Λ t) 1L−1 .
n!
n!
0
(Ntr ) is also a stopped homogeneous Poisson process under the pricing measure
by λ its intensity. As in the rst model, we introduce the cost of information, κP ,
We assume that
Q
and denote
such that
λ = (1 + κP )ν.
6
By virtue of Proposition 11.2.3 in Bielecki and Rutkowski (2002) and as the Poisson process
dening the dates of information disclosures is independent from the Markov chain dening the
rating, the probability measure
Q
is dened by the product of Radon Nikodyn densities
ηt
and
ζt
given respectively by
ˆ
ηt = 1 +
ηu−
(0,t]
L
X
ˆ
κi,j dMui,j ,
(0,t]
i,j=1
and is such that the generators of
ζu− κP d (Nu − νu) ,
ζt = 1 +
(Xt )
under
frequencies of information arrivals under
P
assume that there is no risk premium on
P and Q are Γ and A respectively and that the
Q are respectively ν and λ. In practise, we can
and
λ
(as done by insurers for several type of risks) and
estimate it from historical data. The information available to the market is represented by the
ltration
{Gt , t ≤ T }
that carries now information about the disclosure dates, the rating at these
Gt is then given by
n
o
Gt = σ Ntr , Z0 , Z1 , ..., ZNtr , XZ0 , . . . , XZN r , Hu , u ≤ t .
times and the occurrence of default. The sigma algebra
t
The price of the defaultable bond has the same expression as in the previous model.
Proposition 2.6.
The price at time t of a zero coupon of maturity T is
P (t, T ) = Re−r(T −t) + (1 − R)e−r(T −t) γ(XZN r , ZNtr , t, T )Ht .
t
The proof is similar to the proof of Proposition 2.1 and is left for the reader who may easily
supply the details. We now focus on the dynamics of the price of the defaultable bond.
Proposition 2.7.
is given by
dP (t, T )
The dynamics at time t of the bond price under Q and with partial information
0
r , t, T ) dM̂
= rP (t, T )dt − (1 − R)e−r(T −t) γ(XZN r , ZNt−
t
t−
r , t, T )
+(1 − R) e−r(T −t) γ(Xt , t, t, T ) − γ(XZN r , ZNt−
dNtr
t−
(2.5)
where M̂t0 and is a (G, Q)-martingale.
Remark
.
2.8
It is also possible to give the dynamics of the bond price under
P
and with partial
information
0,P
r , t, T ) dM̂
dP (t, T ) = rP (t, T )dt − (1 − R)e−r(T −t) γ(XZN r , ZNt−
t
t−
r , t, T )
+(1 − R)e−r(T −t) γ(XZN r , ZNt−
L−1
X
t−
κj,L aj,L Ht
j=1
Xt>i exp (Γ (t − ti )) ej
dt
>
Xti D> exp (Λ (t − ti )) 1L−1
r , t, T )
+(1 − R) e−r(T −t) γ(Xt , t, t, T ) − γ(XZN r , ZNt−
dNtr ,
t−
where
(M̂t0,P )
is a
(G, P )-martingale.
We may check that the average instantaneous bond return is well equal to the risk free rate.
Corollary 2.9.
The expected return of the bond is equal to the risk free rate
EQ (dP (t, T ) | Gt ) = rP (t, T )dt.
We now focus on the calculation of the price of an option on this zero coupon bond,
C(t, S, K, T ) = EQ
e−r(S−t) (P (S, T ) − K)+ |Gt
.
The interest rate being deterministic, the only factor of risk is the default risk.
7
Let us denote by z = ZNtr the date of the last rating disclosure before time
t. The price of a call option of exercise date S and of strike price K is given by the following
Proposition 2.10.
expression
C(t, S, K, T )
L−1
Xˆ S
fV (v)dv
= Ht
δ(Xz , ej , z, t, v, S) e −r(S−t) e−r(T −S) (R + (1 − R)γ(ej , v, S, T )) − K
j=1
+
t
+ (1 − γ(Xz , z, t, S)Ht ) e−r(S−t) e−r(T −S) R − K
+
where fV (v) is the density of the time of the last rating disclosure before S , V = ZNSr , in the
interval of time [t, S] given that τ > S ,
fV (v) = e−λ(S−t) δ(v − t) + λe−λ(t−v)
with δ(.) the Dirac function.
2.3 A third model for the defaultable bond with information at endogenous dates
In this last model, we assume that the rating is disclosed at discrete and deterministic times
(p)
0 = t0
(p)
≤ t1
(p)
≤ . . . ≤ tn
= T,
but also at random and endogenous times, when transitions
lead the Markov chain to a class with a lower credit rating than the last known rating.
This
is a more realistic assumption since credit rating agencies or corporate bankers not only assess
ratings when annual reports are published, but also when the credit worthiness of the rm or
of the country becomes worse than the last known rating could let to predict. This assumption
introduces a kind of asymmetry in the information arrival process:
an improvement of credit
worthiness is not immediately disclosed, contrary to worsening. Note that contrary to previous
X. without any delay, at the time of rating disclosures. We denote
0 = Z0 ≤ Z1 ≤ . . . the times of rating disclosures (deterministic times as well as at random
r
times) and by Nt the number of rating disclosures before time t. The information available to the
market is represented by the ltration {Gt , t ≤ T } where Gt is then given by
n
o
Gt = σ Ntr , t0 , Z1 , ..., ZNtr , XZ0 , . . . , XZN r , Hu , u ≤ t .
models, we observe the process
by
t
t is denoted by Rt and is equal to the scalar product of Xt and
(1, 2, . . . , L)> . The price of the defaultable bond has not the same
The rating of the issuer at time
of the
L
dimensional vector
expression as in the previous models because the length between the last rating disclosure is now
R = 1, ..., L − 1
D|R = 1{i=j≤R} 1≤i≤R,1≤j≤L .
informative. Let us introduce three families of matrices: for rating
B |R = (bl,j )1≤l≤R,1≤j≤R ,
Note that
B |L−1 = B
Proposition 2.11.
and
|R
IdL = 1{i=j≤R}
1≤i≤L,1≤j≤L
,
D|L−1 = D.
The price at time t of a zero coupon of maturity T is
P (t, T ) = Re−r(T −t) + (1 − R)e−r(T −t) γ̃(XZN r , ZNtr , t, T )Ht
t
where
γ̃(XZN r , ZNtr , t, T ) =
(2.6)
t
>
XZ>N r D|R
Z
t
Ntr
|RZN r
|RZN r
t − ZNtr
D|RZN r IdL t D> exp (B (T − t)) 1L−1
t
|R
.
Z
r
>
XZ>N r D|R
exp B Nt t − ZNtr
1RZN r
Z r
exp B
t
t
Nt
t
8
We now derive the dynamics of the zero coupon bond under the reduced ltration.
Suppose that t ∈ (ti(p) , t(p)
i+1 ) for some i = 0, ..., n − 1. The dynamics of the
bond price under Q and with partial information is given by
Proposition 2.12.
0
r , t, T ) dM̃
dP (t, T ) = rP (t, T )dt − (1 − R)e−r(T −t) γ̃(XZN r , ZNt−
t
t−

 r
r , t, T )
+(1 − R) e−r(T −t) γ̃(Xt , t, t, T ) − γ̃(XZN r , ZNt−
dNt −
t−

X

aRt ,j Ht dt
j>RZN r
t
where (M̃t0 ) and Ntr −
´tP
j>RZN r
0
aRu ,j Hu du are (G, Q)-martingales.
u
The pricing of options in this model is more complex than in the previous models because the
density of the time of the last rating disclosure before the exercise date of the option is intricate.
However, it is possible to price an option when the exercise date of the option,
a deterministic date of information disclosure
(p)
ti
for some
S,
corresponds to
i = 1, ..., n.
Let us denote by z = ZNtr , the date of the last rating disclosure before time t.
Assume that there exists i = 1, ..., n such that S = ti(p) . The price of a call option of exercise date
S ≥ t and of strike price K is given by the following expression
Proposition 2.13.
C(t, S, K, T )
=
L−1
X
δ̃(ej , Xz , z, t, S) e −r(S−t) e−r(T −S) (R + (1 − R)γ̃(ej , S, S, T )) − K Ht
+
j=1
+ (1 − γ̃(Xz , z, t, S)Ht ) e−r(S−t) e−r(T −S) R − K
+
where γ̃(.) is dened by Equation (2.6) and
δ̃(ej , Xz , z, t, S)
=
Q ( Xs = ej | Gt )
=
X
e>
l
l≤Rz
>
exp B |Rz (t − z) D|Rz ej
Xz> D|R
z
exp (A (S − t)) ej
.
> exp B |Rz (t − z) 1
Xz> D|R
Rz
z
2.4 An extension of the rst model to a piecewise time-homogeneous
rating process
Lando (2004) mentions that a better calibration to market prices is obtained when the the costs of
transitions
κi,j , dening the transition matrix under the risk neutral measure (see equation (2.1)),
are piecewise constant. In this subsection we release partly the assumption of time-homogeneity
and consider the case where
(Xt )
is a piecewise time-homogeneous Markov chain as in Jarrow
et al. (1997). For the reader interested in the calibration of piecewise time-homogeneous rating
processes, we refer to Chapter 6 of Lando (2004). We present an extension of results about bonds
and options pricing, when the information arrives at non random dates.
The current rating is now carried by
chain. The dynamics of
(Xt )
(Xt ), a continuous-time, piecewise time-homogeneous Markov
(Ω, F, Q) is specied by the L × L intensity
on the probability space
matrices
A
(k)
=
B (k)
0
9
b(k)
0
on the time interval
(k − 1)∆
and
k∆
((k − 1)∆, k∆]
for
k = 1, 2, ...
∆ > 0.
and
The transition matrix between
is then given by
exp A(k) ∆
=
exp B (k) ∆
0
We may check that the same matrix between
exp A(k) ∆ exp A(k+1) ∆ =
exp B (k) ∆ exp B (k+1) ∆
0
1L−1 − exp B (k) ∆ 1L−1
.
1
(k − 1)∆
=
X0> D>
j−1
Y
(k + 1)∆
is equal to
1L−1 − exp B (k) ∆ exp B (k+1) ∆ 1L−1
.
1
Then, we infer that the probability of survival till time
Q(τ > t | F0 )
and
t ∈ ((j − 1)∆, j∆]
is given by
exp B (k) ∆ exp B (j) (t − (j − 1)∆) 1L−1
k=1
=: X0> D> Σ(0,t) 1L−1 .
For times
t0 ∈ ((j1 − 1)∆, j1 ∆]
Σ(t0 ,t)
=
and
t ∈ ((j2 − 1)∆, j2 ∆]
such that
t0 < t,
we let
2 −1
jY
exp B (j1 ) ((j1 ∆ − t0 )
exp B (k) ∆ exp B (j2 ) (t − (j2 − 1)∆) .
k=j1 +1
This operator will be used in bonds and options pricing formulas.
As in Section 2.1, we assume that times of credit rating publications,
T,
are non random. If
Ht
is the indicator variable
1{τ >t} ,
(p)
(p)
(p)
0 = t0 ≤ t1 ≤ . . . ≤ tn =
equal to one if the issuer is still solvent,
{Gt , t ≤ T }
= σ {Xt0 , . . . , Xti , Hu , u ≤ t}.
the information available to the secondary market is represented by the ltration
(p) (p+1)
where the sigma algebra Gt at time t ∈ [ti , ti+1 ) is given by Gt
Then, based on previous results, we infer the following corollaries.
Corollary 2.14.
(p)
Suppose that t ∈ [t(p)
i , ti+1 ) for some i = 0, ..., n − 1, then
P (t, T ) = Re−r(T −t) + (1 − R)e−r(T −t) γ(Xti , ti , t, T )Ht ,
where
γ(Xti , ti , t, T ) = Q (τ > T | Xti , t < τ ) =
Xt>i D> Σ(ti ,T ) 1L−1
.
Xt>i D> Σ(ti ,t) 1L−1
(2.7)
Similarly, we price the impact of a lack of information on the price of a call option.
(p)
Suppose that t ∈ [t(p)
i , ti+1 ) for some i = 0, ..., n − 1. The price of a call option
(p)
(p)
(p)
of exercise date S ∈ [t(p)
6= tS ) and of strike price K is given by
S , tS+1 ) (ti
Corollary 2.15.
C(t, S, K, T )
=
L−1
X
δ(Xti , ej , ti , t, tS , S) e −r(S−t) e−r(T −S) (R + (1 − R)γ(ej , tS , S, T )) − K Ht
+
j=1
+ (1 − γ(Xti , ti , t, S)Ht ) e−r(S−t) e−r(T −S) R − K
+
where γ(.) is dened by Equation (2.7),
δ(Xti , ej , ti , t, tS , S)
=
Xt>i Θ(ti ,tS ) ej
e> D> Σ(tS ,S) 1L−1 ,
Xt>i D> Σ(ti ,t) 1L−1 j
and for times t0 ∈ ((j1 − 1)∆, j1 ∆] and t ∈ ((j2 − 1)∆, j2 ∆] such that t0 < t,
Θ(t0 ,t)
=
exp A(j1 ) (j1 ∆ − t0 )
jY
2 −1
exp A(k) ∆ exp A(j2 ) (t − (j2 − 1)∆) .
k=j1 +1
10
3
Information and contagion between issuers
Pricing of multinames credit derivatives, such as recently studied by Herbertsson (2011) or Giesecke
et al.
(2011) requires to take into account dependence of default events.
First we study the
inuence of a lack of information and contagion for two obligors. In this model, each rm has its
own matrix of transition probabilities and default events modify them. This framework is then
considered to several homogeneous obligors.
3.1 A model with contagion for two issuers
In this section, we extend the single name framework developed previously to introduce a contagion
eect between two rms. More precisely, we consider two issuers
Y
and
Z
such that the proba-
bilities of bankruptcy of each one is aected by the default of the other actor. Both issuers are
L levels. Information about current ratings is still carried by a
(Xt ), dened on (Ω, F, (Ft )t , P ). There exists exactly L2 combinations
For this reason, (Xt ) takes now its values in a wider state space E = {e1 , e2 , . . . , eL2 }
L2
Y
Z
set of unit basis vector of R . We will denote by Rt and Rt the ratings of Y and Z
marked in a rating system counting
continuous Markov chain
of ratings.
that is the
t
at time
- if
- if
and we adopt the following convention
Xt = ej
j = 1, ..., (L − 1)2 , then the
j
RtY =
(L − 1)
ratings are equal to
RtZ = j −
j = (L − 1)2 + 1, ..., (L − 1)2 + L − 1,
default) and the rating of Z is given by
Xt = ej
in
for
for
j
(L − 1);
(L − 1)
then the rating of
Y
is equal to
L (Y
and is
RtZ = j − (L − 1)2 ;
- if
Xt = ej
for
j = (L − 1)2 + L, ..., (L − 1)2 + 2L − 2,
Y is given by
then the rating of
Z
is equal to
L (Z
is in
default) and the rating of
RtY = j − L(L − 1);
- if
Xt = ej
for
j = L2 ,
then both ratings are equal to
L.
Y and Z are in default are respectively given by
ΘY =
ej ∈ E|j = (L − 1)2 + 1, ..., (L − 1)2 + L − 1 and j = L2
ΘZ =
ej ∈ E|j = (L − 1)2 + L, ..., L2 .
Y
We denote by τ
= inf t ≥ 0|Xt ∈ ΘY and τ Z = inf t ≥ 0|Xt ∈ ΘZ the times of rms default.
Q
τ Y > 0, τ Z > 0 =
Of course we assume that at t = 0, both issuers are not in default such that P
1. The indicator variables, HtY = 1{t<τ Y } and HtZ = 1{t<τ Z } , are equal to one if issuers are still
The sets of states in which
solvent.
Θ = (ΘY )c ∩ (ΘZ )c , ΘY D = ΘY ∩ (ΘZ )c , ΘZD = (ΘY )c ∩ ΘZ and ΘY Z =
Θ ∩ Θ . Θ is the set of states where both issuers are still solvent, ΘY D (resp. ΘZD ) is the set of
YZ
= {eL2 } is the state
states where Y (resp. Z ) has defaulted but Z (resp. Y ) is still solvent, Θ
We also dene
Y
Z
where both rms have defaulted.
The pricing of bonds is done under a risk neutral measure
dened by a
L2 ×L2
Q, under which the chain (Xt ) is
A = (ai,j )i,j=1,2,...L . The matrix
matrix of intensities of transition denoted by
of intensities of transitions takes the following form

B (1,1)
 0(L−1),(L−1)2
A=
 0(L−1),(L−1)2
01,(L−1)2
B (1,2)
B (2,2)
0(L−1),(L−1)
01,(L−1)
11
B (1,3)
0(L−1),(L−1)
B (3,3)
01,(L−1)

b(1)
b(2) 

b(3) 
0
B (i,j) are submatrices of A such that B (1,1) = A|Θ→Θ , B (1,2) = A|Θ→ΘY D , B (1,3) =
A|Θ→ΘZD , B (2,2) = A|ΘY D →ΘY D , B (3,3) = A|ΘZD →ΘZD and b(j) are the following vectors
where
b(1)
=
A|Θ→ΘY Z = −(B (1,1) + B (1,2) + B (1,3) )1(L−1)2 ,
b(2)
=
A|ΘY D →ΘY Z = −B (2,2) 1(L−1) ,
b(3)
=
A|ΘZD →ΘY Z = −B (3,3) 1(L−1) .
We provide in Section 4.4 a detailed example in which the scale of ratings counts three levels so
as to visualize the structure of the matrix
following submatrices of
B (1,1)

A
in a simple setting.
We also denote
B
and
b
the
A
B =  0(L−1),(L−1)2
0(L−1),(L−1)2
B (1,2)
B (2,2)
B (1,3)
0(L−1),(L−1)

0(L−1),(L−1)  ,
B (3,3)

b(1)
b =  b(2) 
b(3)

G(1) (resp. G(2) ) be a diagonal L2 − 1 × L2 − 1 matrix whose i-th diagonal element for
c
c
i = 1, ..., L2 − 1 equals 1 if ei ∈ ΘY (resp. ΘZ ) and is 0 otherwise. Finally to characterize
the dynamics of ratings of Y or Z when the other issuer has defaulted, we introduce the following
and let
notation
D(2)
=
0(L−1),(L−1)2
Id(L−1)
0(L−1),L
D(3)
=
0(L−1),L(L−1)
Id(L−1)
0(L−1),1
and
A(2) =
B (2,2)
01,(L−1)
b(2)
0
A(3) =
,
B (3,3)
01,(L−1)
b(3)
0
.
We now assume that the credit quality of issuers is only assessed at discrete (and non random)
0 = t0 ≤ t1 ≤ . . . ≤ tn = T . The information available to the market is represented
{Gt , t ≤ T } where the sigma algebra Gt at time t ∈ [ti , ti+1 ) is now dened
Gt = σ Xt0 , . . . , Xti , HuY , HuZ , u ≤ t .
times
by
the ltration
by
Proposition 3.1. Suppose that t ∈ [ti , ti+1 ) for some i = 0, ..., n − 1, then the price of bonds of
maturity T issued by Y and Z are given by the following expressions:
P Y (t, T ) = Re−r(T −t) + (1 − R)e−r(T −t) Q(τ Y > T | Gt ),
P Z (t, T ) = Re−r(T −t) + (1 − R)e−r(T −t) Q(τ Z > T | Gt ),
where
Q τ Y > T | Gt
=
γ Y (Xti , ti , t, T )HtY HtZ + γ Y S (Xti , ti , t, τZ , T )HtY (1 − HtZ )
Q τ Z > T | Gt
=
γ Z (Xti , ti , t, T )HtZ HtY + γ ZS (Xti , ti , t, τY , T )HtZ (1 − HtY )
and
γ Y (Xti , ti , t, T )
γ Z (Xti , ti , t, T )
=
Q(τ Y > T | Xti , t < τ Z , t < τ Y )
=
Xt>i D> exp(B(t − ti ))G(2) exp(B(T − t))G(1) 1L2 −1
,
Xt>i D> exp(B(t − ti ))G(2) G(1) 1L2 −1
(3.1)
= Q(τ Z > T | Xti , t < τ Z , t < τ Y )
=
Xt>i D> exp(B(t − ti ))G(1) exp(B(T − ti ))G(2) 1L2 −1
,
Xt>i D> exp(B(t − ti ))G(1) G(2) 1L2 −1
12
(3.2)
γ Y S (Xti , ti , t, τZ , T ) = Q(τ Y > T | Xti , t ≥ τ Z = tz , t < τ Y )
P
>
>
(3) >
) exp B (3,3) (T − tz ) 1L−1
ej ∈ΘZD Xti exp (A(tz − ti )) ej ej (D
,
= P
>
>
(3) )> exp B (3,3) (t − t ) 1
z
L−1
ej ∈ΘZD Xti exp (A(tz − ti )) ej ej (D
γ ZS (Xti , ti , t, τy , T ) = Q(τ Z > T | Xti , t < τ Z , t > τ Y = ty )
P
(2) >
>
>
) exp B (2,2) (T − ty ) 1L−1
ej ∈ΘY D Xti exp (A(ty − ti )) ej ej (D
= P
.
>
>
(2) )> exp B (2,2) (t − t ) 1
y
L−1
ej ∈ΘY D Xti exp (A(ty − ti )) ej ej (D
(3.3)
(3.4)
In this setting, we are also able to calculate the correlation between default events.
Suppose that t ∈ [ti , ti+1 ) for some i = 0, ..., n − 1, then the correlation between
default events are provided by
Proposition 3.2.
Cor 1{τY <T } , 1{τZ <T } | Xti , τ Z > t, τ Y > t = p
γ Y,Z − γ Y γ Z
γ Y γ Z (1 − γ Y ) (1 − γ Z )
where γ Y = γ Y (Xti , ti , t, T ) and γ Z = γ Z (Xti , ti , t, T ) are dened by Equation (3.1) and (3.2),
and
γ Y,Z = γ Y,Z (Xti , ti , t, T ) =
Remark
.
3.3
Xt>i D> exp(B(T − ti ))G(1) G(2) 1L2 −1
.
Xt>i D> exp(B(t − ti ))G(1) G(2) 1L2 −1
The pricing of options in the contagion model is much more complex than in model
with a single entity. However, it is possible to price an option on a basket of two zero coupon bonds
with weights
(ωY , ωZ )
when both actors issue zero coupon bonds of same maturity and when the
S , corresponds to a date of information disclosure tS . Suppose that
t ∈ [ti , ti+1 ) for some i = 0, ..., n − 1. The price of a call option of exercise date S and of strike
price K is given by the next expression (see Section (6.13))
exercise date of the option,
C(t, S, K, T ) =
X
Q(XtS = ei | Gt )e−r(S−t) (ωY + ωZ )R + (1 − R)ωY γ Y + (1 − R)ωZ γ Z − K +
ei ∈Θ
ˆ
X
+
Q(XtS = ei | Gt )e
X
ˆ
Q(XtS = ei | Gt )e−r(S−t)
t
ei ∈ΘY D
X
+
ei
(ωY + ωZ )R + (1 − R)ωY γ Y S (tz ) − K + fZD (tz )dtz
t
ei ∈ΘZD
+
tS
−r(S−t)
tS
(ωY + ωZ )R + (1 − R)ωZ γ ZS (ty ) − K + fY D (ty )dty
Q(XtS = ei | Gt )e−r(S−t) [(ωY + ωZ )R − K]+
(3.5)
∈ΘY Z
γ Y = γ Y (ei , tS , tS , T ), γ Z = γ Z (ei , tS , tS , T ), γ Y S (tz ) = γ Y S (ei , tS , tS , tz , T ), γ ZS (ty ) =
(ei , tS , tS , ty, T ),
where
γ
ZS
Xt>i exp (A(tS − ti )) ej
>
ek ∈Θ Xti exp (A(t − ti )) ek
Q(XtS = ei | Gt ) = P
and
fZD (u)
= fτz |t<τz ≤tS (u)
=
fY D (ty )
Xt>i D> exp(B(t − ti ))G(1) exp(B(u − t))BG(2) 1L2 −1
Xt>i D> exp(B(t − ti ))G(1) exp(B(tS − t))G(2) 1 − Xt>i D> exp(B(t − ti ))G(1) G(2) 1L2 −1
= fτy |t<τy ≤tS (ty )
=
Xt>i D> exp(B(t − ti ))G(2) exp(B(u − t))BG(1) 1L2 −1
Xt>i D> exp(B(t − ti ))G(2) exp(B(tS − t))G(1) 1 − Xt>i D> exp(B(t − ti ))G(2) G(1) 1L2 −1
13
The contagion model such as presented is rather dicult to extend to more than two counterparties in practice due to a too large number of states, except if, for a given number of observed
bankruptcies, all surviving rms have the same matrix of transition probabilities.
This case is
developed in the next subsection.
3.2 A model with contagion for several homogeneous issuers
Pricing of CDOs requires to model the dependence of default events between a large number of
rms. We adapt the model of the previous subsection to introduce a contagion eect between a
large number of obligors and study the cross eect of contagion and lack of information on survival
probabilities.
We consider
m homogeneous rms marked in the same rating system counting L levels.
The in-
formation about the number of rms per rating is carried by a continuous-time, time-homogeneous
Markov chain
(Xt ),
dened on
(Ω, F, (Ft )t , P ).
N (m, n)
If no rm is in default, there are
numbers of rms in each rating. If
n ≥ m,
n
.
m
Let, for
=
N (m, L + m − 2) possible combinations for characterizing the
d rms are closed, there are only Nd = N (m − d, L + m − d − 2)
possible combinations for characterizing the numbers of rms in each rating. Then, there exist
exactly
N=
m
X
Nd = N (m, L + m − 1)
d=0
(Xt ) takes now its values in a state space E = {e1 , e2 , . . . , eN }
RN . N may be huge: e.g. for 125 rms and 3 ratings, we
possible combinations. For this reason,
that is the set of unit basis vector of
have
N = 8001.
However the transition matrix is sparse and the model stays tractable by choos-
ing appropriately
L
and
m,
as shown by the work of Herbertsson (2008b): he has successfully
implemented a similar model with only two ratings (alive/default) and for a state space counting
215 = 32768
states.
Pd
Pd+1
k=0 Nk + 1, ...,
k=0 Nk , d rms have
defaulted. To limit the number of parameters, we assume that only one rm can change of rating
We rank states of
E
such that when
Xt = ej
for
j=
or go to bankruptcy at a given time. This implies that a state
ej
is attainable from a state
ei ,
only if the numbers of combinations of the corresponding ratings dier of one unit. The matrix of
intensities of transition takes the following form:

A





= 




(0,0)
(0,1)
BN0 N0
0N1 N0
0N2 .N0
BN0 N1
(1,1)
BN1 N1
0N2 .N1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
01N0
01N1
01N2
...
0N0 N2
(1,2)
BN1 N2
(2,2)
BN2 N2
.
.
.
0N0 N3
0N1 N3
(2,3)
BN2 N3
...
...
...
0N0 1
0N1 1
0N2 1
..
..
.
.
.
.
.
(m−1,m−1)
BNm−1 Nm−1
...
(m−1,m)
BNm−1 1
0



 
B

=
0




b
0
(d,d)
is the submatrix of transition of size Nd × Nd , between states of E in which d rms
(d,d+1)
are closed. BN N
is the submatrix of transition of size Nd × Nd+1 , between states with d closed
d d+1
rms, to states in which d + 1 rms are in default.
where
BNd Nd
Let us denote by
I base
the transition matrix of one single rm without any contagion eect under
the risk neutral measure
Q,
I base =
B base
0
14
bbase
0
.
The elements
(i, j)
(d,d)
BNd Nd
of
for
i = 1, ..., Nd
and
j = 1, ..., Nd
are related to this matrix as
(d,d)
follows. If ePd
is not attainable from ePd
, then BN N (i, j) = 0. If ePd
is
d d
k=0 Nk +j
k=0 Nk +i
k=0 Nk +j
P
attainable from e d
, let t(i, j) be the rating for which the number of rms with this rating
k=0 Nk +i
f (i, j) be the rating for which the number of rms with this
n(i, j) be the number of rms which had the rating t(i, j), then
has increased of one unit,
decreased of one unit,
(d,d)
BNd Nd (i, j)
where
θd
rating has
= θd × n(i, j) × Bfbase
(i,j),t(i,j)
is a constant introducing a contagion eect when
d
rms are in default. If
θd
is above
or below one, the intensities of transitions are higher or lower than initial ones, and rms' ratings
respectively change at a quicker or lower rate.
(i, j)
(d,d+1)
for i = 1, ..., Nd and j = 1, ..., Nd−1 are build as follows. If
(d,d+1)
P
is not attainable from e d
, then BN N
(i, j) = 0. If ePd+1 Nk +j is attainable
d d+1
k=0 Nk +i
k=0 Nk +j
k=0
P
from e d
N +i , we have with the same notation as previously
The elements
of
BNd Nd+1 (i, j)
ePd+1
k=0
k
(d,d+1)
BNd Nd+1 (i, j)
= θd × n(i, j) × bbase
f (i,j) .
As in the previous section, the credit quality of issuers is only assessed at discrete (and non ran-
0 = t0 ≤ t1 ≤ . . . ≤ tn = T . The information available to the market is represented
{Gt , t ≤ T } where the sigma algebra Gt at time t ∈ [ti , ti+1 ) is now dened by
Gt = σ {Xt0 , . . . , Xti , Hu , u ≤ t} with Hu = 1{τm >u} .
dom) times
by the ltration
∆d the set of
τd = inf {t > 0 : Xt ∈ ∆d } the
E
Let us denote by
states of
by
stopping time when
in which at least
d
d
rms are in bankruptcy and
bankruptcy are observed.
Let, for k ≥ 1, Tk > Tk−1 > ... > T1 > 0 and dk > dk−1 > ... > d1 > 0.
Suppose that t ∈ [ti , ti+1 ) for some i = 0, ..., n − 1 and that τd1 > ti , then the joint probability of
(τd1 , ..., τdk ) is given by
Proposition 3.4.
Q(τd1 > T1 , τd2 > T2 , ... , τdk > Tk | Gt ) =
Xt>i D> exp (B (T1 − ti )) G(1) exp (B (T2 − T1 )) G(2) . . . exp (B (Tk − Tk−1 )) G(k) 1N −1
Xt>i D> exp (B (t − ti )) G(1) 1N −1
where D = (IdN −1 0N −1 ) and G(j) is a diagonal (N − 1) × (N − 1) matrix whose ith diagonal
element for i = 1, ..., N − 1 equals 1 if i ∈ ∆C
dj and is 0 otherwise.
Based on this last proposition, it is possible to price credit derivatives on a basket of names,
but we don't develop the CDO pricing in this work (see Herbertsson (2008b) for further details).
4
Numerical applications
4.1 Model with predetermined times of information disclosure
The purpose of this section is to illustrate numerically the inuence of information discontinuities
on prices, in all models previously studied. It is well known that historical transition probabilities
provided by rating agencies such as Moody's or S&P are not adapted for pricing purposes (see e.g.
Lando 2004, chapter 6). So as to work with a realistic rating system, we instead t a transition
matrix with eight levels (AAA, AA, A, BBB, BB, B, CCC and D) to default probabilities of
Vodafone, Alstom and Arcelor Mittal, for time horizons from 1 to 10 years. These probabilities
are bootstrapped from EUR CDS quotes, retrieved from Bloomberg on the 5th of September 2013,
and presented in Table 4.1 for some maturities.
15
CDS quotes
S&P Rating
1
2
3
5
7
Vodafone
A(-)
16.30
27.70
39.90
72.60
100.80
114.60
Alstom
BBB
38.50
78.10
117.70
193.70
250.20
272.20
Arcelor Mittal
BB(+)
105.70
159.60
243.80
399.10
476.40
511.50
1
2
3
5
7
10
0.46%
0.70%
0.98%
1.52%
1.92%
2.34%
Eur swap rates 17/5/11
10
Survival probabilities
S&P Rating
1
2
3
5
7
10
Vodafone
A(-)
0.00
0.01
0.02
0.06
0.12
0.20
Alstom
BBB
0.01
0.03
0.06
0.26
0.30
0.47
Arcelor Mittal
BB(+)
0.02
0.05
0.12
0.33
0.57
0.86
Table 4.1: CDS quotes. Eur swap rates and bootstrapped probabilities of default per maturities.
We now explain how we calibrated the transition matrix coecients under the risk neutral
measure. To limit the number of parameters, we assume that rms can only be up/downgraded
to the closest upper/lower rating, or go to bankruptcy. More precisely, we assume that the block
B of the transition matrix A, is tridiagonal, i.e. that there exist parameters η1,d , η1,D ,
η2,D , ...,η7,u , η7,D such that A is equal to
 P
−1
−1
−1
− j=d,D (1 + eη1,j )
(1 + eη1,d )
0
... (1 + eη1,D )
P

−1
−1
−1
−1
(1 + eη2,u )
− j=u,d,D (1 + eη2,j )
(1 + eη2,d )
... (1 + eη2,D )


..

−1
−1
.

0
(1 + eη3,u )
... (1 + eη3,D )

.
.

..
..
.
.

.
.
.
.


.
.
.
−1
..
.
..

... (1 + eη7,D )
.
0
0
0
0
where
bmax
η2,u , η2,d
,






 × bmax





is a positive constant. As criterion of optimization, we minimize the quadratic sum of
errors between modeled and real probabilities of default. Table 4.2 contains the matrix
A obtained
by this method and used in numerical applications.
AAA
AA
A
BBB
BB
B
CCC
D
AAA
-2.9519
2.9506
0
0
0
0
0
0.0013
AA
11.7497
-12.1536
0.4011
0
0
0
0
0.0029
A
0
6.1599
-9.0208
2.8610
0
0
0
0.0000
BBB
0
0
0.5346
-1.1117
0.5612
0
0
0.0159
BB
0
0
0
0.0000
-0.4269
0.4269
0
0.0000
B
0
0
0
0
0.0000
-0.4227
0.4227
0.0000
CCC
0
0
0
0
0
0.0000
-0.4206
0.4206
D
0
0
0
0
0
0
0
0
Table 4.2: Matrix A obtained by calibration to CDS quotes of Vodafone, Alstom, Arcelor Mittal.
Figure 4.1, that compares modelled and bootstrapped probabilities of bankruptcy, reveals that
the quality of the calibration is rather satisfactory. We use this transition matrix in the remainder
of the section.
16
Vodafone
Alstom
0.2
Arcelor Mittal
0.9
0.5
0.18
0.45
Mod.
Mkt
0.16
0.8
Mod.
Mkt
0.4
0.14
0.35
0.12
0.3
0.1
0.25
0.08
0.2
0.06
0.15
0.04
0.1
0.02
0.05
Mod.
Mkt
0.7
0.6
0.5
0.4
0
0
5
time
10
0
0.3
0.2
0.1
0
5
time
0
10
0
5
time
10
Figure 4.1: Modelled and bootstrapped probabilities of bankruptcy.
We try now to understand how the decit of information inuences prices of bonds and options.
For this purpose, we compare zero coupon bonds prices for each rating and for maturities that
range from 1 to 20 years. In our rst set of tests, we assume that the rating known at time zero
has been updated for the last time, a half year ago (t0
to
r = 2%.
= −δ = −0.5)
and the interest rate is set
A credit spread is retrieved by inverting the formula:
P (t0 =−0.5) (0, T ) = e(r+spread(t0 =−0.5))T .
(4.1)
In a second set of tests, bond prices are computed under the assumption that information about
ratings is available at the date of calculation (which is equivalent to set
are in this case noted
spread(t0 = 0).
t0 = 0).
Credit spreads
The part of the credit spread related to the decit of
information is then calculated as the following dierence:
spread attributed to information
= spread(t0 = −0.5) − spread(t0 = 0)
The evolution of the part of spreads attributable to the information delay is displayed in the left
graph of Figure 4.1, for bonds issued by A, BB and BBB issuers. We observe that the decit of
information does not have the same impact on every rating. In particular, the higher is the rating,
the lower is the importance of the delay.
The part of spreads attributable to the information
increases for term maturities and next decreases for maturities longer than 5 years. Finally, we
have tested the inuence of the delay between the last disclosure of information and the date of
bonds issuance.
The right graph of Figure 4.1 presents the part of spreads attributable to the
information delay, for dierent time lag:
t0 = −1.5, −1, −0.5 and for BB bonds.
For this category
of bonds, the higher is the delay of disclosure, the higher is the cost of this lack of information.
We have priced call options on a zero coupon bond of maturity 10 years and of rating BB.
0.65
t0 = −4 to −1 years and tS = 4
The right graph of Figure 4.3 presents options prices for t0 = 0 years and tS = 2 to 3 years.
The recovery rate is still null. The exercise date is set to
to
0.90.
years.
S=5
years and strikes ranges from
The left graph of Figure 4.3 presents options prices for
We clearly see in both cases that the higher is the decit of information, the higher should be
options prices.
4.2 Model with information at exogenous random dates
In this section, we test the model in which the information about the real rating of the issuer is
published at random times. As in the previous paragraph, we work with the matrix of transition
17
Part of the credit spread related to the partial information
Influence of the delay of information on the spread
2
0.8
0.7
t0=−0.5
1.5
t0=−1
0.6
t0=−1.5
0.5
%
%
1
0.4
0.5
0.3
0.2
0
0.1
A
BB
BBB
−0.5
0
5
10
15
0
20
0
5
10
15
20
Figure 4.2: Inuence of the delay of information on bonds credit spreads (y -axis), per maturity
(x-axis).
−7
2.5
x 10
−7
Influence of the delay of information on option prices
2
x 10
Influence of the delay of information on option prices
1.8
2
1.6
tS=2
t0=−4
1.4
t0=−3
tS=3
tS=4
t0=−2
t0=−1
1.5
tS=5
1.2
1
1
0.8
0.6
0.5
0.4
0.2
0
0.65
0.7
0.75
0.8
0.85
0
0.65
0.9
0.7
0.75
0.8
0.85
0.9
Figure 4.3: Inuence of the delay of information on 5y options prices (y -axis) for dierent strikes
(x-axis), underlying: 10y bond.
18
Influence of the frequency of disclosure, rating A.
0.08
0.07
0.06
λ=1
λ=2
λ=3
0.05
0.04
0.03
0.02
0.01
0
0.8
0.85
0.9
0.95
Figure 4.4: Inuence of disclosure frequencies on 5y options prices (y -axis), for dierent strikes
(x-axis), underlying: 10y bond.
Evolution of prices
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0
2
4
6
8
10
Figure 4.5: Evolution of 10y bond price over time (x-axis), in three scenarios.
rates of Table 4.2. The formula of bond prices being identical to the one of the previous model, we
limit our analyze to option prices. Figure 4.4 presents prices of call options on a zero coupon bond
of maturity 10 years and of ratings A. The recovery rate is still null. The exercise date is set to
S = 5 years and strikes
λ = 1, 2, 3 years. A
to
range from
0.65
to
0.90.
The frequency of information disclosure is equal
low frequency of information release rises the prices, whatever the strike
prices. In Figure 4.5, we display three simulated trajectories of a bond price (maturity 10 years
and rating A) under the risk neutral measure. These trajectories are computed by discretization
of equation (2.5). The jumps are related to the arrival of a new information and the amplitude of
jumps depends on the new rating disclosed.
4.3 Model with information at endogenous random dates
In this section, we assess numerically the credit spread of bonds, when the information is disclosed
at xed times and when a credit worsening occurs. As mentioned earlier, this assumption introduces asymmetry in the information arrival process. We still work with the matrix of transition
rates of Table 4.2. Figure 4.6 presents the credit spreads, such as dened by (4.1), for a BBB zero
coupon bond. The risk free rate is 2% and two scenarios are considered: when the last rating has
been published one and two years ago. We also reported on the gure, the spreads of BBB bond,
when the information is not endogenous. We observe that spreads are lower when the information
is endogenous. This observation can be justied as follows: when the information is fully exoge-
19
8
7
6
BBB z=−2 endo.
BBB z=−2 exo.
BBB z=−1 endo.
BBB z=−1 exo.
%
5
4
3
2
1
0
Figure 4.6:
0
2
4
6
8
10
12
14
16
18
20
Comparison of credit spreads (y -axis) with and without endogeneity for dierent
maturities (x-axis).
nous, the probability that an obligor has in fact a lower rating than the last one reported, is not
null. It is not the case when the information is endogenous: in case of worsening of the economic
situation of a rm, the rating is adjusted and disclosed immediately. The fact that the rm has
not been downgraded since one or two years, gives us an indication that the rm has the same or
a higher rating. Similar conclusions can be drawn for options prices: the endogeneity attenuates
the impact of the delay of information.
4.4 Model with contagion for two issuers
So as to present explicitly the matrix
count three levels and that the issuers
A, we work under the assumption that the scale of ratings
Y and Z cannot default at the same time. Furthermore, we
assume that, when both issuers are in activity, matrix of intensities, ruling the marginal evolution
of ratings, are identical and such that the one year probabilities of transition are:

γ11
eA = exp  γ21
0
γ12
γ22
0

 
γ13
0.05 0.90 0.05
γ23  =  0.10 0.80 0.10 
0
0
1
0
If
Z
α
strictly is bigger than one, the probabilities that
Y are multiplied by a positive constant α. If
Y changes of rating or goes to bankruptcy are
then increased. The parameter α allows us to model the contagion eect of the default of Z on
the rating of Y . In case of bankruptcy of Y , the transition intensities of Z are multiplied by a
positive constant β . If β diers from α, the contagion eect is asymmetric.The transition matrix
A of the system is in this case equal to:
is in default, then the transition intensities of
Xt
e1
e2
e3
e4
e5
e6
e7
e8
e9
(RY , RZ )
(1, 1)
(1, 2)
(2, 1)
(2, 2)
(3, 1)
(3, 2)
(1, 3)
(2, 3)
(3, 3)
(1, 1)
2γ11
γ21
γ21
0
0
0
0
0
0
(1, 2)
γ12
γ11 + γ22
0
γ21
0
0
0
0
0
In the following tests, we have set
(2, 1)
γ12
0
γ22 + γ11
γ21
0
0
0
0
0
(2, 2)
0
γ12
γ12
2γ22
0
0
0
0
0
α = 0.8 and β = 1.2.
(3, 1)
γ13
0
γ23
0
αγ11
αγ21
0
0
0
(1, 3)
γ13
γ23
0
0
0
0
βγ11
βγ21
0
(2, 3)
0
0
γ13
γ23
0
0
βγ12
βγ22
0
(3, 3)
0
0
0
0
αγ13
αγ23
βγ13
βγ23
0
2% while
−0.5. The
The risk free rate is still equal to
the recovery rate is null. When required, the default times
20
(3, 2)
0
γ13
0
γ23
αγ12
αγ22
0
0
0
tz
and
ty
have been set to
Contagion effect of Z on Y
Contagion effect of Y on Z
1.4
0
1.2
−0.2
1
−0.4
∆ Spread Y RY=1, RZ=1
∆ Spread Y RY=1, RZ=2
0.8
−0.6
∆ Spread Z RY=1, RZ=1
%
%
∆ Spread Y RY=1, RZ=3
0.6
∆ Spread Z RY=2, RZ=1
−0.8
∆ Spread Y RY=3, RZ=1
0.4
−1
0.2
−1.2
0
0
5
10
15
−1.4
20
0
5
10
15
20
Figure 4.7: Contagion eect on bond spreads (y -axis) per maturity (x-axis).
0.08
0.06
ρ
0.04
0.02
t0=0 RY=RZ=1
t0=−3 RY=RZ=1
0
t0=0 RY=RZ=2
t0=−3 RY=RZ=2
−0.02
−0.04
0.5
1
1.5
2
β
2.5
Figure 4.8: Inuence of the delay of disclosure and of
last disclosure is at the date of the issuance,
t0 = 0 .
3
β (x-axis)
on correlations (y -axis).
Figure 4.7 exhibits the dierence between
credit spreads computed with and without contagion (situation in which
we could expect, for the chosen values of
α
and
β,
if
Y
α = 1 and β = 1). As
Z , the probability of
collapses before
bankruptcy of this last issuer decreases and the spreads follow the same trend. In the opposite
case, the default of
Z
directly raises the probability of default of
Y
and increases the spreads.
In Figure 4.8, we plot the correlation between the one year default times (1{τY <T } and
1{τZ <T } ),
α is set to 1, while β varies from 0.5 to 3. Two couples of rating
are considered: (RY , RZ ) = (1, 1) and (2, 2). The graph reveals that the rating directly aects
the level of correlation of default times. When β = 1, default times are independent. For lower or
higher β 's, the correlation is respectively negative or positive. We also observe that the decit of
for dierent levels of contagion.
information directly inuences the correlation. If the last rating disclosure has been made 3 years
ago, the correlation is higher (for
(RY , RZ ) = (1, 1))
or lower (for
(RY , RZ ) = (2, 2))
than in case
of a recent disclosure. Figure 4.9 illustrates the inuence of the information delay on the price
of a basket option for dierent strikes. The exercise date is set to 5 years and the maturity of
both bonds is 10 years. The risk free and recovery rates are respectively 2% and 0%. The ratings
are set to
(RY , RZ ) = (2, 2)
and the weights of the basket are
section, a lack of recent information raises the option prices.
21
ωY = ωZ = 50%.
As in previous
Influence of the delay of information on basket option prices
0.06
0.05
0.04
t0=−4
t0=−3
0.03
t0=−2
t0=−1
0.02
0.01
0
0.65
0.7
0.75
0.8
Figure 4.9: Inuence of the delay of disclosure on 5y basket option prices (y -axis) for dierent
strikes (x-axis).
5
Conclusions
This paper explores a new use of Phase-type distributions in credit risk under dierent ows
of incomplete information.
In particular, it details the inuence of discontinuity and delay of
information on prices of credit linked securities, in a credit rating system.
In the rst model
considered, information about the rating arrives at predetermined dates. In this framework, we
infer prices of zero coupon bonds, their dynamics in continuous time, and prices of call options.
Numerical applications reveal that a decit of information raises most of the time credit spreads
and option prices but the impact varies considerably across ratings. In the second model studied,
information is disclosed at random times. This assumption is more realistic and aects mainly the
dynamics of bonds and option prices. In the third model rating is disclosed at non random dates or
when the credit worthiness of the rm or of the country becomes worse than the last known rating
could let to predict.
This asymmetry in the information arrival process reduces the impact of
information delay on credit spreads. Finally, we propose a contagion model for two issuers, when
information ows at xed dates and an extension to multiple obligors.
Numerical applications
show us that a decit of information aects either positively or negatively the correlation between
default times.
6
Proofs
6.1 Proof of Proposition 2.1
Equation (2.2) can be developed as follows
P (t, T )
= e−r(T −t) EQ (R + (1 − R) HT | Xt0 , . . . , Xti , Hu , u ≤ t)
= Re−r(T −t) + (1 − R)e−r(T −t) EQ (HT |Xt0 , ...Xti , Ht = 1) Ht ,
and according to the Bayes' rule
EQ (HT |Xt0 , ...Xti , Ht = 1)
= Q (τ > T | Xti , t < τ )
Q(τ > T | Xti , ti < τ )
=
Q(τ > t | Xti , ti < τ )
Xt>i D> exp (B (T − ti )) 1L−1
=
.
Xt>i D> exp (B (t − ti )) 1L−1
Note that the second relation is also shown in theorem 13 in Guo et al. (2009).
22
6.2 Proof for Equation 2.4
The price of the defaultable bond is given by
P (t, T )
=
=
=
EQ e−r(T −t) 1{τ >T } + e−r(τ −t) R 1{τ ≤T } | Gt
REQ e−r(τ −t) | Gt + e−r(T −t) EQ 1 − R e−(τ −T ) HT | Gt
REQ e−r(τ −t) | Gt + e−r(T −t) EQ (HT | Gt ) − Re−r(T −t) EQ e−(τ −T ) HT | Gt .
The rst expectation can be developed as follows
EQ e−r(τ −t) | Gt
= EQ e−r(τ −t) | Xt0 , . . . , Xti , Hu , u ≤ t
= er(t−τ ) (1 − Ht ) + EQ e−r(τ −t) | Xti , Ht = 1 Ht
= e
The Laplace transform of
τ
r(t−τ )
EQ e−r(τ −t) 1{τ >t} | Xti , τ > ti
(1 − Ht ) +
Ht
Q (τ > t | Xti , τ > ti )
is given by (see e.g. Rolski et al. (2009))
−1
EQ e−ω(τ −t) | Xt = Xt> D> (ωIL−1 − B) b
and therefore
EQ e−r(τ −t) 1{τ >t} | Xti , τ > ti
=
L−1
X
EQ e−r(τ −t) 1{Xt =ej } | Xti , τ > ti
j=1
=
L−1
X
EQ e−r(τ −t) |Xt = ej P Q (Xt = ej |Xti , τ > ti )
j=1
=
L−1
X
>
−1
e>
bXt>i exp(A(t − ti ))ej .
j D (rIL−1 − B)
j=1
Finally, the expectation can be rewritten as follows
L−1
>
−1
X e>
bXt>i exp(A(t − ti ))ej
j D (rIL−1 − B)
Ht .
EQ e−r(τ −t) | Gt = er(t−τ ) (1 − Ht ) +
Xt>i D> exp(B(t − ti ))1(L−1)
j=1
The second expectation has already been calculated in the proof of Proposition 2.1, and is equal
to
EQ (HT | Gt ) =
Xt>i D> exp (B (T − ti )) 1L−1
Ht .
Xt>i D> exp (B (t − ti )) 1L−1
The last expectation is calculated as follows:
EQ e−(τ −T ) HT | Gt
= EQ e−r(τ −T ) 1{τ >T } | Xti , Ht = 1 Ht
= Ht
L−1
X
EQ e−r(τ −T ) 1{XT =ej } | Xti , Ht = 1
j=1
= Ht
L−1
X
EQ e−r(τ −T ) |XT = ej Q (XT = ej |Xti , τ > t)
j=1
=
Ht
= Ht
L−1
X
Q (X = e |X , t < τ )
T
j
ti
i
EQ e−r(τ −T ) |XT = ej
Q(τ
>
t
|
X
,
t
<
τ)
t
i
i
j=1
L−1
X
>
−1
e>
b
j D (rIL−1 − B)
j=1
23
Xt>i exp(A(T − ti ))ej
.
Xt>i D> exp (B (t − ti )) 1L−1
6.3 Proof of Proposition 2.3
By Proposition 2.1, we get that for
dP (t, T )
(p)
(p)
t ∈ (ti , ti+1 )
∂
γ(Xti , ti , t, T ) Ht dt
∂t
+(1 − R)e−r(T −t) γ(Xti , ti , t, T ) dHt .
rP (t, T )dt + (1 − R)e−r(T −t)
=
γ,
First, note that, according to the denition of
∂
γ(Xti , ti , t, T )
∂t
Xt>i D>
its derivative is equal to
exp (B (T − ti )) 1
> >
2 Xti D (exp (B (t − ti )) B) 1L−1
exp (B (t − ti )) 1
=
−
=
−γ(Xti , ti , t, T )
Xt>i D>
Xt>i D> (exp (B (t − ti )) B) 1L−1
.
Xt>i D> exp (B (t − ti )) 1L−1
Ht
Second, let us characterize the martingale representation of
Recall that the rating
(1, 2, . . . , L)> .
Rt
is equal to the scalar product of
Xt
under the market ltration
and of the
Ht0 = 1{τ ≤t} is a stopped jump process
ˆ t
0
Ht =
aRu ,L Hu du + Mt0
We know that
(6.1)
L
G.
dimensional vector
that can be written as
0
0
where Mt is a real martingale on the enlarged ltration
and Rutkowski (2002)).
(F, Q)
(see e.g. Lemma 11.2.3 in Bielecki
Let us now recall that:
1. For every
(F, Q)
martingale
Mt ,
2. For any progressively measurable process
Q
ˆ
t
(G, Q)
with
t
EQ (ψu | Gu ) du,
−
0
is a
ψt
ˆ
ψu du | Gt
E
E (Mt | Gt ) is a (G, Q) martingale.
´
T
EQ 0 |ψu | du < ∞, the dierence
its optional projection
martingale.
We can then infer that
ˆ
M̂t0 = EQ Ht0 | Gt −
ˆ
t
EQ (aRu ,L Hu | Gu ) du = Ht0 −
0
is a
(G, Q)
t ≤ T,
0
t
EQ (aRu ,L | Gu ) Hu du.
(6.2)
0
martingale. Moreover we have
EQ (aRt ,L | Gt )
=
=
EQ (aRt ,L | Xt0 , ...Xti , Ht = 1) Ht
L−1
X
aj,L Ht Q (Xt = ej | Xti , t < τ ) ,
j=1
and the probability that the issuer has the rating
Q (Xt = ej | Xti , t < τ ) =
j
at time
t
is calculated by the Bayes' rule
X >i exp (A (t − ti )) ej
Q (Xt = ej | Xti , ti < τ )
= > t>
.
Q (τ > t | Xti , ti < τ )
Xti D exp (B (t − ti )) 1L−1
Combining equations (6.2) and (6.1) allows us to rewrite the dynamics of the zero coupon bond
price as follows
dP (t, T )
=
rP (t, T )dt − (1 − R)e−r(T −t) γ(Xti , ti , t, T )
−(1 − R)e−r(T −t) γ(Xti , ti , t, T )
L−1
X
aj,L
j=1
−(1 − R)e
−r(T −t)
γ(Xti , ti , t, T ) dM̂t0 .
24
Xt>i D> (exp (B (t − ti )) B) 1L−1
Ht dt
Xt>i D> exp (B (t − ti )) 1L−1
Xt>i exp (A (t − ti )) ej
Ht dt
Xt>i D> exp (B (t − ti )) 1L−1
Third, it remains to prove that
Xt>i D> (exp (B (t − ti )) B) 1L−1 = −
L−1
X
aj,L Xt>i exp (A (t − ti )) ej
(6.3)
j=1
to conclude. According to the denition of
exp(At) =
Then, for
j 6= L,
A,
exp(tB)
0
we know that (see Rolski et al. (2009)) that
1L−1 − exp(tB)1L−1
1
.
we have that
Xt>i exp (A (t − ti )) ej = Xt>i D> exp (B (t − ti )) D ej .
By construction, the vector
b = −B 1L−1 .
b
is the vector with components
aj,L for j = 1...L − 1
and it satises
Therefore
−
L−1
X
aj,L Xt>i exp (A (t − ti )) ej
= −Xt>i D> exp (B (t − ti )) b
j=1
= Xt>i D> (exp (B (t − ti )) B) 1L−1 ,
which ends the proof.
6.4 Proof of Proposition 2.5
According to Proposition 2.1, the price of the call can be rewritten as follows
C(t, S, K, T )
=
=
=
EQ e−r(S−t) (P (S, T ) − K)+ |Gt
−r(S−t) Q
−r(T −S)
e
E
e
(R + (1 − R)γ(XtS , tS , S, T )HS ) − K |Gt
+
−r(S−t) Q
−r(T −S)
e
E
e
[R + (1 − R)γ(XtS , tS , S, T )HS ] − K |Xt0 , ...Xti , Ht = 1 Ht
+
−r(S−t)
−r(T −S)
+ (1 − Ht ) e
e
R−K
+
and then
=
C(t, S, K, T )
Xt0 , ...Xti , Ht = 1 Ht
e−r(S−t) EQ HS e−r(T −S) (R + (1 − R)γ(XtS , tS , S, T )) − K
+ + 1 − EQ (HS |Xt0 , ...Xti , t < τ ) Ht e−r(S−t) e−r(T −S) R − K .
(6.4)
+
The expectation in the second term of equation (6.4) is equal to
E (HS |Xt0 , ...Xti , Ht = 1) = Q (τ > S | Xti , t < τ, ti < τ ) = γ(Xti , ti , t, S).
So as to lighten future calculations, we use the notation
g(XtS ) = e−r(T −S) (R + (1 − R)γ(XtS , tS , S, T )) − K .
+
25
The expectation in the rst term of Equation (6.4) becomes then
HS
= EQ
=
e−r(T −S) (R + (1 − R)γ(XtS , tS , S, T )) − K
1{XS 6=eL } g(XtS ) | Xt0 , ...Xti , t < τ
EQ
L−1
X
Xt0 , ...Xti , Ht = 1
+
g(ej )Q (τ > S, XtS = ej | Xt0 , ...Xti , t < τ )
j=1
=
L−1
X
g(ej )Q (τ > S | XtS = ej , t < τ ) Q (XtS = ej | Xti , t < τ ) .
j=1
Given that the default state is absorbing, we have for
Q (XtS = ej | Xti , t < τ )
=
=
j 6= L
that
Q (XtS = ej , τ > t | Xti , ti < τ )
Q (τ > t | Xti , ti < τ )
Xt>i exp (A (tS − ti )) ej
Xt>i D> exp (B (t − ti )) 1L−1
and
>
Q (τ > S | XtS = ej , t < τ ) = e>
j D exp (B (S − tS )) 1L−1 .
6.5 Proof of Proposition 2.7
The proof is similar to the proof of Proposition 2.3, excepted that we have now a jump component
related to the disclosure of information. If the last date of rating disclosure before
then according to the bond price formula, the dierential of
dP (t, T )
P (t, T )
t
is
r
ZNt−
= z,
is given by
∂
γ(Xz , z, t, T ) Ht dt
∂t
+(1 − R)e−r(T −t) γ(Xz , z, t, T ) dHt
= rP (t, T )dt + (1 − R)e−r(T −t)
+(1 − R)e−r(T −t) (γ(Xt , t, t, T ) − γ(Xz , z, t, T )) dNtr
The optional projection of
G -adapted
Ht
on
Gt
(6.5)
is built as in Proposition 2.3. To conclude note that
and is therefore equal to its optional projection
(Nt )
is
E (Nt | Gt ).
6.6 Proof of Corollary 2.9
Let
r
ZNt−
=z
be the last date of rating disclosure before
0
that M̂t and is a
(G, Q)-martingale,
t.
As
E (dNtr | Gt ) = λHt dt
and given
we obtain from Equation (2.5) that
EQ (dP (t, T ) | Gt )
= rP (t, T )dt + (1 − R) e−r(T −t) (E (γ(Xt , t, t, T ) | Gt ) − γ(Xz , z, t, T )) λHt dt
with
E (γ(Xt , t, t, T ) | Gt )
= Ht E (γ(Xt , t, t, T ) | X0 , ...Xz , t < τ )
= Ht
L−1
X
γ(ej , t, t, T )Q(Xt = ej | Xz , t < τ )
j=1
= Ht
L−1
X
γ(ej , t, t, T )
j=1
26
Q (Xt = ej | Xz , z < τ )
.
Q (τ > t | Xz , z < τ )
(6.6)
On another side, we have
>
γ(ej , t, t, T ) = e>
j D exp (B (T − t)) 1L−1 = Q(τ > T |Xt = ej , t < τ ).
Equation (6.6) can then be rewritten as follows
E (γ(Xt , t, t, T ) | Gt )
L−1
X
= Ht
Q(τ > T |Xt = ej , t < τ )
j=1
Since
(Xt )
Q (Xt = ej | Xz , z < τ )
.
Q (τ > t | Xz , z < τ )
is a Markov process, we have the relation
Q (τ > T |Xz , z < τ ) =
L−1
X
Q(τ > T |Xt = ej , t < τ )Q (Xt = ej | Xz , z < τ )
j=1
that allows us to infer that
L−1
X
Q (Xt = ej | Xz , z < τ ) γ(ej , t, t, T ) =
j=1
Q(τ > T |Xz , z < τ )
= γ(Xz , z, t, T )
Q (τ > t | Xz , z < τ )
and that
E (γ(Xt , t, t, T ) | X0 , ...Xz , t < τ ) = γ(Xz , z, t, T ).
6.7 Proof of Proposition 2.10
Remember that
V
is the date of the last rating disclosure, before time
S , V = ZNSr .
According to
the proof of Proposition (2.5) (see Equation (6.4)), the price of the call can be rewritten as follows
C(t, S, K, T )
= e−r(S−t) EQ HS e−r(T −S) (R + (1 − R)γ(XV , V, S, T )) − K |X0 , ...Xz , t < τ Ht
+
+ 1 − EQ (HS |X0 , ...Xz , t < τ ) Ht e−r(S−t) e−r(T −S) R − K .
(6.7)
+
First note that
E (HS |X0 , ...Xz , t < τ ) = γ(Xz , z, t, S).
So as to lighten future calculations, we denote
g(XV , V ) = e−r(T −S) (R + (1 − R)γ(XV , V, S, T )) − K .
+
The expectation in the rst term of equation (6.7) becomes then
EQ (HS g(XV , V ) | X0 , ..., Xz , t < τ )
= EQ 1{XS 6=eL } g(XV , V ) | X0 , ...Xz , t < τ
=
L−1
X
EQ (Q (XV = ej , τ > S | Xz , t < τ, V ) g(ej , V ))
(6.8)
j=1
We note
fV (v)
the density of the last rating disclosure before time
have occurred, the times
Z1 ...Zn
S.
If
n
information disclosures
are unordered random variables, distributed as iid uniform (see
27
Rolski et al. 2009, p157). Then,
fV (v)
that is dened for
v ≤ S,
can be written as the following
sum
fV (v)
P (NS−t = 0) δ(v − t) +
=
∞
X
P (NS−t = k)k
k=1
∞
X
(v − t)
k−1
(S − t)
k
k−1
k
(S − t) k −λ(S−t) (v − t)
λ e
k
(k − 1)!
(S − t)
k=1
!
∞
k−1
X
((v − t) λ)
δ(v − t) + λ
(k − 1)!
k=1
δ(v − t) + λe−λ(t−v)
=
P (NS−t = 0) δ(v − t) +
=
e−λ(S−t)
=
e−λ(S−t)
The expectation in Equation (6.8) can then be rewritten as follows
EQ (Q (XV = ej , τ > S | Xz , t < τ, V ) g(ej , V ))
ˆ S
=
Q (Xv = ej , τ > S | Xz , t < τ, V = v) g(ej , v) fV (v) dv
t
ˆ
S
=
δ(Xz , ej , z, t, v, S)g(Xv , v) fV (v) dv.
t
6.8 Proof of Proposition 2.11
As in the proof of Proposition (2.1), we have
P (t, T ) = Re−r(T −t) + (1 − R)e−r(T −t) EQ (HT |Gt ) Ht ,
where
= Q τ > T | XZN r , Ru ≤ RZN r , ∀u ∈ [ZNtr , t]
EQ (HT |Gt )
t
=
t
Q(τ > T, Ru ≤ RZN r , ∀u ∈ [ZNtr , t]| XZN r , ZNtr < τ )
t
t
Q(Ru ≤ RZN r , ∀u ∈ [ZNtr , t] | XZN r , ZNtr < τ )
t
=
t
Q(τ > T, Ru ≤ RZN r , ∀u ∈ [ZNtr , t]| XZN r , ZNtr < τ )
t
t
|R
ZN r
>
t
r
XZ>N r D|R
exp
B
t
−
Z
1RZN r
Nt
Z r
t
Nt
t
As the numerator of this last equation is equal to
Q(τ > T, Ru ≤ RZN r , ∀u ∈ [ZNtr , t]| XZN r , ZNtr < τ )
t
t
= Q(τ > T, τ > ZNtr , Ru ≤ RZN r , ∀u ∈ [ZNtr , t]| XZN r )Q τ > ZNtr | XZN r
t
t
t
X
=
Q (τ > T | Xt = ej ) Q Xt = ej , Ru ≤ RZN r , ∀u ∈ [ZNtr , t]| XZN r
t
t
j≤RZN r
t
=
X
>
>
>
e>
j D exp (B (T − t)) 1L−1 XZN r D|RZ
t
j≤RZN r
Ntr
|R
Z r
exp B Nt
t − ZNtr
D|RZN r ej
t
>
= XZ>N r D|R
Z
t
Ntr
|R
Z r
exp B Nt
t − ZNtr
|RZN r
D|RZN r IdL
t
we get the result.
28
t
D> exp (B (T − t)) 1L−1
t
6.9 Proof of Proposition 2.12
By Proposition 2.11, we get that for
dP (t, T )
(p)
(p)
t ∈ (ti , ti+1 )
∂
γ̃(XZN r , ZNtr , t, T ) Ht dt
t
∂t
r , t, T ) dHt
+(1 − R)e−r(T −t) γ̃(XZN r , ZNt−
t−
−r(T −t)
r , t, T )
γ̃(Xt , t, t, T ) − γ̃(XZN r , ZNt−
dNtr
+(1 − R) e
= rP (t, T )dt + (1 − R)e−r(T −t)
(6.9)
t−
To simplify further calculations, let us denote:
>
= D|R
Z
G0
G1 =
B
|R
Z r
exp B Nt
Ntr
|RZN r
|RZN r
D|RZN r IdL
t
t
t
,
|RZN r
D> − D|RZN r IdL
t
D> B
.
t
First, note that, according to the denition of
∂
γ̃(XZN r , ZNtr , t, T )
t
∂t
t − ZNtr
γ̃ ,
its derivative is equal to
h
1
|RZ r
−γ̃(XZN r , ZNtr , t, T )XZ>N r G0 B Nt 1RZN r
>
0
t
t
t
XZN r G 1RZN r
t
t
i
+XZ>N r G0 G1 exp (B (T − t)) 1L−1 .
=
t
Second, as in the proof of Proposition 2.3,we can then infer that
ˆ
ˆ
t
Mt0 = EQ Ht0 | Gt −
EQ (aRu ,L | Gu ) Hu du.
0
is a
(G, Q)
0
Ht0 = 1{τ ≤t} .
martingale, where
t
EQ (aRu ,L Hu | Gu ) du = Ht0 −
Moreover we have that
R ZN r
EQ (aRt ,L | Gt )
Xt
=
aj,L Q Xt = ej | XZN r , Ru ≤ RZN r , ∀u ∈ [ZNtr , t] Ht
t
t
j=1
XZ>N r G0 D|RZN r
t
=
PRZNtr
j=1
t
aj,L ej
Ht .
XZ>N r G0 1RZN r
t
t
We can rewrite (6.9) as follows
0
r , t, T ) dM
dP (t, T ) = rP (t, T )dt − (1 − R)e−r(T −t) γ̃(XZN r , ZNt−
t
t−
r , t, T )
+(1 − R)e−r(T −t) γ̃(Xt , t, t, T ) − γ̃(XZN r , ZNt−
dNtr
t−
−r(T −t)
−(1 − R)e
γ̃(XZN r , ZNtr , t, T )
t
1
t

 >
0 |RZ r
>
0
XZN r G B Nt 1RZN r + XZN r G D|RZN r
t
−(1 − R)e−r(T −t)
t
t
XZ>N r G0
×
1RZN r
t


RZN r
t
X


aj,L ej  Ht dt

XZ>N r G0
t
j=1
1
G exp (B (T − t)) 1L−1
t
XZ>N r G0 1RZN r
t
29
t
Ht dt .
(6.10)
Now, we note that
1RZN r + XZ>N r G0 D|RZN r

t
X


aj,L ej 




R ZN r
XZ>N r G0 B
|RZN r
t
t
t
t
t
j=1

t
X


aj,L ej 

R ZN r
 |RZ r
= XZ>N r G0 B Nt 1RZN r + D|RZN r
t
t
t
j=1

RZN r

= XZ>N r D> exp B t − ZNtr
t
 X 
D|RZN r −

t
j=1

L−1
X
t

 
aj,l  ej  ,
(6.11)
l=RZN r +1
t
and that



R ZN r
 X
E 
Q

aRt ,j Ht dt | Gt 
X 

=
j=1
j>RZN r

L−1
X
t

aj,l 
XZ>N r G0 D|RZN r ej
t
l=RZN r +1
t
t
XZ>N r G0 1RZN r
t
t
XZ>N r G0 D|RZN r
t
=
PRZNtr
j=1
t
XZ>N r G0
t
PL−1
Ht dt
t
l=RZN
aj,l ej
r +1
t
Ht dt.(6.12)
1RZN r
t
Moreover, we can prove the following relations,


X

EQ γ̃(Xt , t, t, T )

aRt ,j Ht dt | Gt 
(6.13)
j>RZN r
t


X

= EQ Xt> D> exp (B (T − t)) 1L−1

aRt ,j Ht dt | Gt 
j>RZN r
t
RZN r
=
L−1
X
Xt
>
al,j e>
j D exp (B (T − t)) 1L−1
XZ>N r G0 D|RZN r el
l=1 j=RZN r +1
XZ>N r G0 D|RZN r
=
t
t
t
t
t
XZ>N r G0 1RZN r
PRZN r
t
j=1
t
PL−1
l=RZN r +1
aj,l el
>
e>
j D
Ht dt
t
exp (B (T − t)) 1L−1
t
Ht dt
XZ>N r G0 1RZN r
t
t
and
XZ>N r G0 G1 exp (B (T − t)) 1L−1 =
t


R ZN r
L−1
t
X

 X
XZ>N r G0 D|RZN r 

t
t
j=1



>
aj,l el  e>
j D  exp (B (T − t)) 1L−1 .
(6.14)
l=RZN r +1
t
If we insert (6.11) (6.12) (6.13) (6.14) in (6.10), we can infer the existence a
(M̃t0 )
(G, Q)
martingale
such that
0
r , t, T ) dM̃
dP (t, T ) = rP (t, T )dt − (1 − R)e−r(T −t) γ̃(XZN r , ZNt−
t
t−

 r
r , t, T )
+(1 − R) e−r(T −t) γ̃(Xt , t, t, T ) − γ̃(XZN r , ZNt−
dNt −

X

aRt ,j Ht dt .
t−
j>RZN r
t
30
6.10 Proof of Proposition 2.13
According to the proof of Proposition (2.5) (see Equation (6.4)), the price of the call can be
rewritten as follows
C(t, S, K, T )
−r(S−t) Q
−r(T −S)
= e
E
HS e
(R + (1 − R)γ̃(XS , S, S, T )) − K |Gt Ht
+
+ 1 − EQ (HS |Gt ) Ht e−r(S−t) e−r(T −S) R − K .
+
First note that
E (HS |Gt ) = γ̃(Xz , z, t, S).
So as to lighten future calculations, we use the notation
g(XtS ) = e−r(T −S) (R + (1 − R)γ̃(XS , S, S, T )) − K .
+
The expectation in the rst term of Equation (6.4) becomes then
EQ
=
HS
e−r(T −S) (R + (1 − R)γ̃(XS , S, S, T )) − K
Gt
+
X
L−1
g(ej )Q ( Xs = ej | Gt )
EQ 1{XS 6=eL } g(XS ) | Gt =
j=1
=
L−1
X
X
g(ej )
j=1
Q ( Xs = ej | Xt = el ) Q (Xt = el | Gt ) .
l≤RZN r
t
=
L−1
X
j=1
X
g(ej )
e>
l exp (A (S − t)) ej
l≤RZN r
>
XZ>N r D|R
Z
t
>
XZ>N r D|R
Z
t
t
Ntr
|R
Z r
exp B Nt t − ZNtr
D|RZN r ej
t
|R
.
ZN r
t
r
t
−
Z
exp
B
1
N
R
Z
r
t
N
r
Nt
t
6.11 Proof of Proposition 3.1
We use the result of Assaf et al. (1983) that provides an analytical formula of the joint probability
of survival of both issuers and the Bayes' rule to prove that
Q τ Y > T | Xt0 , ..., Xti , τ Z > t, τ Y > t
If
=
Q τ Y > T, τ Z > t | Xti , τ Y > ti , τ Z > ti
Q (τ Z > t, τ Y > t | Xti , τ Y > ti , τ Z > ti )
=
Xt>i D> exp(B(t − ti ))G(2) exp(B(T − t))G(1) 1L2 −1
.
Xt>i D> exp(B(t − ti ))G(2) G(1) 1L2 −1
ti ≤ τ Z = tz ≤ t
Q τ Y > T | X0 , ..., Xti , τ Z = tz ≤ t, τ Y > t
= Q τ Y > T | Xti , Xtz ∈ ΘZD , τ Y > t
Q τ Y > T, Xtz ∈ ΘZD | Xti
=
Q (Xtz ∈ ΘZD , τ Y > t | Xti )
31
We can conclude by noticing that
Q τ Y > T, Xtz ∈ ΘZD | Xti , τ Y > ti , τ Z > ti
X
Q (Xtz = ej | Xti ) Q τ Y > T | Xtz = ej
=
ej ∈ΘZD
=
X
(3) >
Xt>i exp (A(tz − ti ) ej e>
) exp B (3,3) (T − tz ) 1L−1
j (D
ej ∈ΘZD
and that
Q Xtz ∈ ΘZD , τ Y > t | Xti , τ Y > ti , τ Z > ti
X
=
Q (Xtz = ej | Xti ) Q τ Y > t | Xtz = ej
ej ∈ΘZD
=
X
(3) >
Xt>i exp (A(tz − ti ) ej e>
) exp B (3,3) (t − tz ) 1L−1 .
j (D
ej ∈ΘZD
6.12 Proof of Proposition 3.2
First note that
Cov( 1{τY <T } , 1{τZ <T } | Xti , τ Z > t, τ Y > tt
=
Cov(1{τY >T } , 1{τZ >T } | Xti , τ Z > t, τ Y > tt )
= Q(τ Y > T, τ Z > T | Xti , τ Z > t, τ Y > t)
−Q τ Y > T | Xti , τ Z > t, τ Y > t Q τ Z > T | Xti , τ Z > t, τ Y > t
By using the result of Assaf et al. (1983)
Q τ Y > T, τ Z > T | Xti , τ Z > t, τ Y > t
=
Xt>i D> exp(B(T − ti ))G(1) G(2) 1L2 −1
Q(τ Y > T, τ Z > T | Xti )
=
Q(τ Z > t, τ Y > t | Xti )
Xt>i D> exp(B(t − ti ))G(1) G(2) 1L2 −1
= γ Y,Z .
Finally, we have that
Q(τ Y
>
T | Xti , τ Z > t, τ Y > t) = γ Y (Xti , ti , t, T )
Q(τ Z
>
T | Xti , τ Z > t, τ Y > t) = γ Z (Xti , ti , t, T )
and
V ar(1{τY <T } | Xti , τ Z > t, τ Y > t)
= Q(τ Y < T | Xti , τ Z > t, τ Y > t)
× 1 − Q(τ Y < T | Xti , τ Z > t, τ Y > t)
= γ Y (Xti , ti , t, T ) 1 − γ Y (Xti , ti , t, T ) .
6.13 Proof for Equation 3.5
The relation (3.5) is the expectation of the option payo.
conditionally to the available information at time
Q(XtS = ei | Xti , τ Z > t, τ Y > t)
t
Q(XtS = ei | Xti )
ek ∈Θ Q(Xt = ek | Xti )
=
P
=
P
32
The probability of being in state
can be developed as follows:
Xt>i exp (A(tS − ti )) ej
.
>
ek ∈Θ Xti exp (A(t − ti )) ek
i
And the density of the time of default of
Z , conditionally to the fact that Z
has gone to bankruptcy
between time is equal to the following ratio:
fτz |t<τz ≤tS (u) =
where
Ft<τz (u)
u ∈ [t, tS ].
is the cumulative distribution of the survival time of
Ft<τz (u)
=
=
=
and where
ft<τz (u)
Ft<τz (tS )
ft<τY (u)
1 − Q τ Z > u | Xti , τ Z > t, τ Y
Q τ Y > t, τ Z > u | Xti
1−
Q (τ Y > t, τ Z > t | Xti )
1−
Z,
>t
Xt>i D> exp(B(t − ti ))G(1) exp(B(u − t))G(2) 1L2 −1
Xt>i D> exp(B(t − ti ))G(1) G(2) 1L2 −1
is obtained by dierentiation:
ft<τz (u) = −
Xt>i D> exp(B(t − ti ))G(1) exp(B(u − t))BG(2) 1L2 −1
.
Xt>i D> exp(B(t − ti ))G(1) G(2) 1L2 −1
6.14 Proof of Proposition 3.4
According to the Bayes' rule, we have that
Q(τd1 > T1 , ..., τdk > Tk | Gt )
=
Q(τd1 > T1 , ..., τdk > Tk | Xti , ti < τd1 )
Q(τd1 > t| Xti , ti < τd1 )
and the numerator and denominator are determined by the result of Assaf et al. (1983, p.694):
Q(τd1 > T1 , ..., τdk > Tk | Xti , ti < τd1 ) =
Xt>i D> exp (B (T1 − ti )) G(1) exp (B (T2 − T1 )) G(2) . . . exp (B (Tk − Tk−1 )) G(k) 1N −1
and
Q(τd1 > t| Xti , ti < τd1 ) = Xt>i D> exp (B (t − ti )) G(1) 1N −1 .
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