1
On Cox pro cesses and credit risky securities
David Lando
Department of Op erations Research
University of Cop enhagen
Universitetsparken 5
DK-2100 Cop enhagen .
Denmark
phone: +45 35 32 06 83
fax: +45 35 32 06 78
e-mail: [email protected]
First version: March 1994
This revision March 31, 1998
1
Previous title: On Cox pro cesses and credit risky b onds. This pap er is a
revised and extended version of parts of Chapter 3 of my Ph.D. thesis. I have
received useful comments from my committee memb ers Rob ert Jarrow, Rick Dur-
rett, Sid Resnick and Marty Wells, as well as from Michael Hauch, Brian Huge,
Martin Jacobsen, Monique Jeanblanc, Kristian Miltersen, Lasse Heje Pedersen,
Torb en Sk deb erg and Stuart Turnbull and from participants at seminars at the
1993 FMA Do ctoral seminar in Toronto, Merrill Lynch, Aarhus University, Odense
University, University of Cop enhagen, Sto ckholm Scho ol of Economics, University
of Lund, New York University, Erasmus University, the 1994 Risk conference in
Frankfurt, IBC conference on Credit derivatives in London 1996, the 1996 confer-
ence on Mathematical Finance in Aarhus. All errors are of course myown.
1 Intro duction
The aim of this pap er is to illustrate howCox pro cesses - also known as doubly
sto chastic Poisson pro cesses - provide a useful framework for mo deling prices
of nancial instruments in which credit risk is a signi cant factor. Both the
case where credit risk enters b ecause of the risk of counterparty default and
the case where some measure of credit risk such as a credit spread is used as
an underlying variable in a derivative contract can be analyzed within our
framework.
The key contribution of the approach is the simplicityby which it allows
credit risk mo deling when there is dep endence b etween the default-free term
structure and the default characteristics of rms. As an illustration of the
metho d we present a generalized version of a Markovian mo del presented in
Jarrow, Lando and Turnbull [13]. This generalization allows the intensities
which control the default intensities and the transitions between ratings to
dep end on state variables which simultaneously govern the evolution of the
term structure and p ossibly other economic variables of interest. This mo del
can be used to analyze a variety of contracts in which credit ratings and
default are part of the contractual setup. Two imp ortant examples are de-
fault protection contracts which provide insurance against default of a rm
and so-called 'credit triggers' in swap contracts which typically demand the
settlement of a contract in the event of downgrading of one of the parties
to the contract. By imp osing some additional structure on the mo del we
derive a new class of mo dels which in a sp ecial setup b ecome 'sums of ane
term structure mo dels'. These mo dels simultaneously mo del term structures
for di erent ratings and allow spreads to uctuate randomly even in p eri-
ods where the rating of the defaultable rm do es not change. Such mo dels
are useful for managers of corp orate b ond p ortfolios and risky government
debt who are constrained by credit lines setting upp er limits on the amount
of b onds held b elow a certain rating category. Given such lines the event of
a downgrading may force the manager to either sell o part of the p osition
or try to o -load some credit exp osure in credit derivatives markets.
The mo deling framework is similar to that of Jarrow and Turnbull [14],
Jarrow, Lando and Turnbull [13], Lando [16], Madan and Unal [19], Artzner
and Delbaen [1], Due and Singleton [7], [8], and Due, Schro der and Ski-
adas [9] in that default is mo deled through a random intensity of the default
time. However, this pap er disp enses with the indep endence assumption em-
ployed in Jarrow and Turnbull [14], Jarrow, Lando and Turnbull [13], Madan 1
and Unal [19]. In this resp ect our approach resembles that of the works of
Due and Singleton [7],[8], and Due, Schro der and Skiadas [9]. The ap-
proach in this pap er is not based on the recursive metho ds employed there.
This sacri ces some generality but the structure of Cox pro cesses pro duces
a class of mo dels which are simple to deriveyet seems rich enough to handle
imp ortant applications when only one-sided credit risk is considered. For an
analysis of two sided default risk in an intensity-based framework, see Due
and Huang [6].
The pros and cons of mo deling default in an intensity-based framework as
opp osed to using the classical approach originating in Black and Scholes [2]
and Merton [20] are discussed elsewhere, see for example Due and Single-
ton [7],[8], and the review pap ers by Co op er and Martin [3] and Lando [17].
Here we simply note that this pap er and the work by Due and Singleton [8]
p ointtoone imp ortant advantage of the intensity-based framework, namely
the fact that the intensity based framework is capable of reducing the tech-
nical diculties of mo deling defaultable claims to the same one faced when
mo deling the term structure of non-defaultable b onds and related derivatives.
The structure of the pap er is as follows: In section 2 we outline the basic
construction of a Cox pro cess, concentrating on the rst jump time of sucha
pro cess. Knowing how to construct a pro cess means knowing how to simulate
it. Hence this section is imp ortant if one wants to implement the mo dels.
Furthermore, the explicit construction gives the key to proving the results
later on by - essentially - using prop erties of conditional exp ectations.
Section 3 then shows how to calculate prices of three 'building blo cks' from
which credit risky b onds and derivatives can be constructed. Several exam-
ples of claims are given which are constructed from these building blo cks.
Within this framework we have great exibility in how we sp ecify the be-
havior of the state variables which simultaneously determine the default free
term structure and the likeliho o d of default. When di usions are used as
state variables, the Feynman-Kac framework handles pricing of credit risky
derivative securities in virtually the same way as it handles derivatives in
which credit risk is not present. That particular setup is one of 'di usion
with killing' - a sp ecial case of the general framework. We also outline how
the notion of fractional recovery, as intro duced in Due and Singleton [8],
maybe viewed in terms of thinning of a p oint pro cess.
Section 4 is an applications of the metho dology. It is shown how a Marko-
vian mo del prop osed by Jarrow, Lando and Turnbull [13]may b e generalized
to include transition rates b etween credit ratings which dep end on the state 2
variables. One imp ortant asp ect of this generalization is that credit spreads
are allowed to uctuate randomly even between ratings transitions. Section
5 gives a concrete implementation of the generalized Markovian mo del by
considering an ane mo del for b oth the interest rate and the transition in-
tensities. It is shown how one can calibrate the mo del across all credit classes
simultaneously to t initial sp ot rate spreads and the sensitivities of these
spreads to changes in the short treasury rate. Section 6 concludes.
2 Construction of a Cox pro cess
Before setting up our mo del of credit risky b onds and derivatives, we give
b oth an intuitive and a formal description of how the default pro cess is
mo deled. The formal description is extremely useful when p erforming cal-
culations and simulations of the mo del, whereas the intuitive description is
helpful when the economic content of the mo dels are to b e understo o d.
Recall that an inhomogeneous Poisson pro cess N with non-negative
intensity function l satis es
k
R
t
Z
t
l u du
s
P N N = k = exp l u du k =0;1;::::
t s
k !
s
In particular, assuming N =0; we have
0
Z
t
P N = 0 = exp l u du :
t
0
Away of simulating the rst jump of N is to let E b e a unit exp onential
1
random variable and de ne
Z
t
= inf ft : l u du E g 2.1
1
0
A Cox pro cess is a generalization of the Poisson pro cess in which the
intensity is allowed to be random but in such a way that if we condition on
a particular realization l ;! of the intensity, the jump pro cess b ecomes an
inhomogeneous Poisson pro cess with intensity l s; ! :
In this pap er we will write the random intensity on the form
l s; ! = X
s 3
d d
where X is an R valued sto chastic pro cess and : R ! [0; 1 is a non-
negative, continuous function. The assumption that the intensity is a func-
tion of the current level of the state variables, and not the whole history, is
convenient in applications, but it is not necessary from a mathematical p oint
of view.
The state variables will include interest rates on riskless debt and may
include time, sto ck prices, credit ratings and other variables deemed relevant
for predicting the likeliho o d of default. Intuitively, given that a rm has
survived up to time t; and given the history of X up to time t; the probability
of defaulting within the next small time interval t is equal to X t +
t
o t :
Formally,we have a probability space ; F ;P large enough to supp ort
d
an R valued sto chastic pro cess X = fX : 0 t T g which is right-
t f
continuous with left limits and a unit exp onential random variable E which
1
d
is indep endent of X: Given also is a function : R ! R which we assume
is non-negative and continuous. From these two ingredients we de ne the
default time as follows:
Z
t
= inf ft : X ds E g: 2.2
s 1
0
This default time can be thought of as the rst jump time of a Cox pro cess
with intensity pro cess X : Note that this is an exact analogue to equation
s
2:1 with a random intensity replacing the deterministic intensity function.
When X is large, the integrated hazard grows faster and reaches the
s
level of the indep endent exp onential variable faster, and therefore the prob-
ability that is small b ecomes higher. Sample paths for which the integral
of the intensity is in nite over the interval [0;T ] are paths for which default
f
always o ccurs.
From the de nition we get the following key relationships:
Z
t
P >tj X = exp X ds t 2 [0;T ] 2.3
s s f
0st
0
Z
t
P >t = Eexp X ds t 2 [0;T ] 2.4
s f
0
What we have mo deled ab ove, is only the rst jump of a Cox pro cess.
When mo deling a Cox pro cess past the rst jump - something we will need in
Section 4 - one has to insure that the integrated intensity stays nite on nite
intervals if explosions are to b e avoided. One then pro ceeds as follows: Given 4
a probability space ; F ;P large enough to supp ort a standard unit rate
Poisson pro cess N with N = 0 and a non-negative sto chastic pro cess t
0
which is indep endentofN and assumed to b e right-continuous and integrable
on nite intervals, i.e.
Z
t
t := s ds < 1 t 2 [0;T]
0
Then 0 = 0 and has non-decreasing realizations. Now de ning
~
N := N t
t
wehaveaCox pro cess with intensity measure : For the technical conditions
we need to check to see that this de nition makes sense, see Grandell [12] pp.
9-16. This approach is also relevant when extending the mo dels presented in
this pap er to cases of rep eated defaults by the same rm.
3 Three basic building blo cks
The mo del for the default-free term structure of interest rates is given by
a sp ot rate pro cess, a money market account and an equivalent martingale
measure to which all exp ectations in this pap er refer:
Z
t
B t = exp r ds
s
0
!
B t
p t; T = E F
t
B T
The default time is mo deled using the framework describ ed in the pre-
vious section. The intensity pro cess is denoted : Hence we may write the
information at time t as
F = G _H
t t t
where pro cesses relevant in determining values of the sp ot rate and the hazard
rate of default are adapted to G and
H = f1 :0s tg
t
f sg 5
holds the information of whether there has b een a default at time t: Most
applications will be built around a state variable pro cess X , in which case
we write R X for the sp ot rate r at time t and the informational setup may
t t
be stated as
G = fX :0s tg
t s
H = f1 :0 s tg
t f sg
F = G _H
t t t
F then corresp onds to knowing the evolution of the state variables up to
t
time t and whether default has o ccurred or not.
Apart from the default characteristics and the default-free term structure
of interest rates what we need to know to price a defaultable contingent claim
are its promised payments and its payment in the event of default i.e. its
recovery . The basic building blo cks in constructing such claims are
1. X 1 : Apayment X 2G at a xed date which o ccurs if there has
T
f >T g
b een no default b efore time T . As an example think of a vulnerable
Europ ean option, i.e. an option whose issuer may default b efore the
option's expiration. A p ossible partial recovery can be captured in
two ways: Let be a G measurable random variable, and think of
T T
partial recovery either as a recovery at the time of maturity of the form
1 = 1 ; which consists of a non-defaultable claim and
T T T
f T g f >T g
our rst basic building blo ck, or a recovery at the time of default as in
the third building blo ck b elow.
2. Y 1 : A stream of payments at a rate sp eci ed by the G adapted
s f >sg t
pro cess Y which stops when default o ccurs. This may b e used to mo del
idealized swaps in whichwe think of payments taking place continually
in time. A p ossible recovery is captured by the next building blo ck.
3. A recovery payment at the time of default of the form Z where Z
is a G adapted sto chastic pro cess and where Z = Z ! : For a
t
!
contingent claim expiring at date T; think of Z as b eing 0 after time
T: This building blo ck mo dels a resettlement payment at the time of
default.
Note that the random variable X and the pro cess Y can be thought of
as promised payments, whereas the payment captured by Z is an actual
payment at the time of default. 6
The key advantage of the setup is that the calculation of prices b ecomes
similar to ordinary term structure mo deling:
Prop osition 3.1 Assume that the expectations
! !
Z
T
E exp r ds jX j
s
t
Z Z
T s
E jY j exp r du ds and
s u
t t
Z Z
T s
E jZ j exp r + du ds
s s u u
t t
are al l nite. The fol lowing identities hold for the three claims listed above:
! !
Z
T
E exp X 1 F 3.1 r ds
t s
f >T g
t
! !
Z
T
= 1 E exp r + ds X G
s s t
f >tg
t
!
Z Z
T s
E Y 1 exp r du ds F 3.2
s u t
f >sg
t t
!
Z Z
T s
= 1 E Y exp r + du ds G
s u u t
f >tg
t t
Z
E exp r ds Z F 3.3
s t
t
!
Z Z
T s
= 1 E Z exp r + du ds G
s s u u t
f >tg
t t
Pro of. First we show that
!
Z
T
E 1 G _H =1 exp ds 3.4
f T g T t f >tg s
t
In the following computations we use the fact that the conditional exp ec-
tation is clearly 0 on the set f tg and that the set f >tg is an atom of 7
H : Therefore
t
E 1 jG _H
f T g T t
= 1 E 1 jG _H
f >tg f T g T t
P f T g\f >tgjG
T
= 1
f >tg
P >tjG
T
P T jG
T
= 1
f >tg
P >tjG
T
R
T
exp ds
s
0
= 1
f >tg R
t
exp ds
s
0
!
Z
T
= 1 exp ds
s
f >tg
t
which proves 3.4 .
Now consider 3.1 .
! !
Z
T
E exp r ds X 1 F
s t
f >T g
t
! ! !
Z
T
= E E exp r ds X 1 G _H jF
s f >T g T t t
t
! !
Z
T
= E exp r ds XE 1 G _H F
s f >T g T t t
t
! !
Z
T
= 1 E exp r + ds X F
s s t
f >tg
t
Now we want to replace the conditioning on F with conditioning on G .
t
Recall that E is a unit exp onential random variable which is indep endent
1
of the sigma eld G : In particular, E is indep endent of the sigma eld
T 1
R
T
exp r + ds X _G and therefore see Williams [21] 9.7 k
s s t
t
! !
Z
T
E exp r + ds G _ E X 3.5
s s t 1
t
! !
Z
T
= E exp r + ds X G :
s s t
t
But we also have the following inclusion among sigma elds:
G G _H G _ E : 3.6
t t t t 1 8
Combining 3.5 and 3.6 we conclude that
! !
Z
T
E exp r + ds X G _H
s s t t
t
! !
Z
T
: = E exp r + ds X G
s s t
t
which is what we wanted to show.
The pro of of 3.2 pro ceeds exactly as ab ove by rst conditioning on
G _H :
T t
For the pro of of 3.3 note that conditionally on G and for s > t the
T
density of the default time is given by
Z
s
@
P sj >t;G = exp du :
T s u
@s
t
Hence we nd
Z
E exp r ds Z F
s t
t
Z
= E E exp r ds Z G _H jF
s T t t
t
!
Z Z
T s
= 1 E Z exp r + du ds F
s s u u t
f >tg
t t
!
Z Z
T s
= 1 E Z exp r + du ds G
f >tg s s u u t
t t
where in the last line we have used the same argument as ab ove to replace
F by G
t t
We now turn to examples and illustrations of this framework.
Example 3.2 Option on credit spread.
In the examples considered ab ove the primary fo cus is securities with
credit risk. Note, however, that a mo del of the default intensity pro duces a
mo del for the yield spread for b onds of all maturities. Hence our framework
will easily pro duce mo dels for options on yield spreads as well.
Example 3.3 The independencecase.
A primary fo cus of this pap er is to allow for dep endence between default
intensities and state variables. Note that the indep endence case is easily 9
captured as well. In the state variable formulation wewould simply let X =
1 2 1 2
X ;X with X and X indep endent pro cesses and de ne the sp ot rate
1 2
as the pro cess r X and the default intensity pro cess as X : Of course,
either of these pro cesses maybe included as a state variable themselves.
Example 3.4 Feynman-Kac representation .
An imp ortant sp ecial case is when
G = fX :0stg 3.7
t s
d
where X is an R valued di usion pro cess with di usion matrix a x; t and
drift vector b x; t and in nitesimal generator
d d d
2
X XX
@F x 1 @ F x
2 d
+ b t; x F 2 C R A F x := a t; x
i t ik
2 @x @x @x
i k i
i=1 i=1
k =1
d d d
Let T = T and let f : R ! R; g :[0;T] R ! R and k : [0;T] R !
f
[0; 1 be continuous functions. Note that when 3.7 holds, the relations
1
3.1 , 3.2 and 3.3 are all of the form
!!
Z
T
t;x
f X exp k u; X du v t; x = E
T u
t
!
Z Z
T s
t;x
+E g s; X exp k u; X du ds 3.8
s u
t t
2
Hence, under suitable regularity conditions on the di usion co ecients
and on the functions f; g; k; the exp ectation in 3.8 is also a solution satis-
fying an exp onential growth condition to the Cauchy problem
@v
d
+ kv = A v + g [0;T] R
t
@t
d
v T; x = f x x 2 R
and we see that the credit risky securities can be mo deled using exactly the
same techniques as default-free securities.
1
Note that in this section we will explicitly write time dep endence of payments, sp ot
rates and hazard rates. In previous sections time dep endence is p ossible simply by let-
ting one of the state variables b e time - here we prefer to separate it from the di usion
comp onents.
2
See Karatzas and Shreve [15] and Due [5] for results and references to some of the
literature on which combinations of conditions one can imp ose to obtain this corresp on-
dence. 10
Example 3.5 Fractional recovery and thinning.
In Due and Singleton [8], and Due, Schro der and Skiadas [9] a notion
of fractional recovery is considered which allows not only the intensity but the
recovery rate to enter into the expression for the default adjusted sp ot rate
pro cess. Here wesketchhow this maybegiven a simple interpretation in the
Cox pro cess framework through the notion of thinning. Receiving a fraction
of pre-default value in the event of default of a contract is equivalent, from
t
a pricing p ersp ective, to receiving the outcome of a lottery in which the full
pre-default value is received with probability under the martingale measure
and 0 is received with probability1 ; i.e the event of default has b een
t t
retained with probability 1 . This in turn may be viewed as a default
t
pro cess in which there is 0 recovery but where the default intensity has b een
thinned using the pro cess ; pro ducing a new default intensity of 1 :
In the case of a claim with promised payment X and fractional recovery at
T
the default date of one then obtains that with Z = c ; the price of
this claim at time t is given as
! !
Z Z
T
c = E exp r ds X 1 + exp r ds Z 1 F
t s f >T g s ft< T g t
t t
! !
Z
T
= 1 E exp r + 1 ds X G :
s s t
f >tg
s
t
4 A generalized Markovian mo del
In this section, as a concrete application of the framework presented ab ove.
we show how to generalize the mo del describ ed in Jarrow, Lando and Turn-
bull [13] for pricing defaultable b onds. It is shown how the mo del can be
generalized to incorp orate state dep endence in transition rates and risk pre-
mia allowing for sto chastic changes in credit spreads even between ratings
transitions.
There, the time to default is mo deled as the absorption time in state K
of a Markov chain with generator matrix
0 1
1 12 13 1K
B C
B C
21 2 23 2K
B C
B C
. . .
.
. . . .
A = B C 4.1
.
. . .
B C
B C
@ A
K 1;1 K 1;2 K 1 K 1;K
0 0 0 11
0 for i 6= j
ij
and
K
X
= ; i =1;:::;K 1:
i ij
j =1;j 6=i
In the study of Fons and Kimball [10] it is do cumented that default rates of
lower rated rms show signi cant time variation and even for the top rated
rms for which default is extremely rare the standard deviations of default
rates may still be signi cant in relative terms. Also, Du ee [4] shows that
noncallable b ond yield spreads fall when the Treasury term structure yield
increases. Whether this variation can be captured through variation in the
state variables or whether a parametrization in terms of constant intensities
as in Jarrow, Lando and Turnbull [13] p erforms equally well is an interesting
empirical issue which we are currently checking. However, it is necessary
with an extension if one is to take into account the random uctuations on
credit spreads for constant time to maturity b onds.
Here, we will show how the Markovian mo del can be generalized to in-
corp orate random transition rates and random default intensities. We will
start with a fairly general mo del and then imp ose sp ecial structure to gain
computational advantages.
First, de ne the matrix
1 0
X X X X
1 s 12 s 13 s 1K s
C B
C B
X X X X
21 s 2 s 23 s 2K s
C B
C B
. . .
.
. . . .
C A s = B
.
X
. . .
C B
C B
X X X X
A @
K 1;1 s K 1;2 s K 1 s K 1;K s
0 0 0
and assume that
K
X
X = X ; i =1;:::;K 1:
i s ij s
j=1;j 6=i
It is not essential that the random entries A s are written as functions
X
of state variables, but wecho ose this formulation since this we b elieve will b e
the interesting one to work with in practice. Also, we could of course include
time as one of the state variables. Heuristically,we think of, say, X t as
1 s
the probability that a rm in rating class 1 will jump to a di erent class or 12
default within the small time interval t: We now give the precise mathe-
matical construction and note again that this is essential if one wants to run
simulations with the mo del:
Let E ;:::;E ;E ;:::;E ;::: b e a sequence of indep endent unit ex-
11 1K 21 2K
3
p onential random variables.
Assume that a rm starts out in rating class : De ne
0
Z
t
= inf ft : X ds E g i =1;:::;K
;i ;i s 1i
0 0
0
and let
= min :
;i
0 0
i6=
0
mo dels the time the rating changes from the initial level and Here,
0
0
this o ccurs the rst time a set of 'comp eting risks' which tries to pull the
rating away from exp eriences a jump. The new rating level may then be
0
written as
= arg min :
;i
1
0
i6=
0
and if the value of is K this corresp onds to default. Now let
1
Z
t
= inf ft : X ds E g i =1;:::;K
;i ;i s 2i
1 1
0
and
= min :
;i
1 1
i6=
1
Hence the next jump o ccurs as time , the new rating level is
1
= arg min
;i
2
1
i6=
1
and so forth. We use to denote the time of default, i.e.
= inf ft : = K g
t
With this construction we obtain a continuous time pro cess which condi-
tionally on the evolution of the state variables is a non-homogeneous Markov
3
Strictly sp eaking, we will only need K-1 exp onentials for each jump but the notation
will b e simpler by ignoring this for now. 13
chain. Conditionally on the evolution of the state variables, the transition
probabilities of this Markov chain satisfy
@P s; t
X
= A s P s; t :
X X
@s
It is worth noting that the solution of this equation in the time-inhomogeneous
case is typically not equal to
Z
t
P s; t = exp A u du
X X
s
which is most easily seen by recalling that when A and B are square matrices,
we only have
exp A + B = exp A exp B
when A and B commute.
Foraway of writing the solution using pro duct integral notation, see for
example of Gill and Johansen [11].
5 Pricing b onds and derivatives in the gen-
eralized Markov mo del
A general b ond pricing formula for defaultable b onds with zero recovery is
then given in the following
Prop osition 5.1 Consider a zero coupon bond maturing at time T issued
by a rm whose rating at time t is i and whose credit rating and default
intensity evolves as described by the process above. Then given zerorecovery
in default, the price of the bond is given as
! !
Z
T
i
v t; T = E exp R X ds 1 P t; T G
s X i;K t
t
where P t; T is the i; K 'th element of the matrix P t; T :
X i;K X
Pro of.
!
1
f >T g
i i
v t; T = B t E F
t
B T 14
! !
Z
T
i
= E 1 exp R X ds F
f >T g s t
t
! !
Z
T
i
F = E exp R X ds E 1 G _F
t s T t
f >T g
t
! !
Z
T
= E exp R X ds 1 P t; T G
s X i;K t
t
where we use the same argumentas earlier to replace F by G :
t t
While this approach pro duces mo dels that can easily b e simulated using
the construction of the pro cess given ab ove, it is of interest to derive mo dels
which, although more restrictive in their assumptions, o er computational
tractability and p ossibly even analytical solutions. One way to do this is
to imp ose more structure on the conditional intensity matrix such that the
expression for P s; t simpli es:
X
d
Lemma 5.2 For i = 1;:::;K 1; let : R ! [0; 1 be a non-negative
i
function de ned on the state space of X and assume that for almost every
sample path of X; we have
Z
T
f
X ds < 1:
s
i
0
Let X denote the K K diagonal matrix diag X ;:::; X ;0 : For
s s s
1 K 1
each path of X assume that the time dependent generator matrix has the rep-
resentation
1
A s =B X B ;
X s
where B is a K K matrix whose columns consist of K eigenvectors of
A s : Also, de ne the diagonal matrix
X
0 1
R
t
exp X du 0 0
u
1
s
B C
B C
.
.
B C
.
0 0
B C
E s; t =
X
B C
R
.
t
B C
.
. exp X du 0
u
@ A
K 1
s
0 0 1
1
Then with P s; t = BE s; t B ; P s; t satis es Kolmogorov's back-
X X X
ward equation
@P s; t
X
= A s P s; t
X X
@s 15
and P s; t is the transition probability of an inhomogeneous Markov chain
X
on f1;:::;Kg:
Pro of.
Since
A s B = B s
X
X
we see that
@P s; t
X
1
= B s E s; t B
X
X
@s
1
= A s BE s; t B
X X
= A s P s; t
X X
which shows that P s; t is indeed a solution to the backward equation.
X
From section 4.4 in Gill and Johansen [11] we know that under our assump-
tions on A s this equation has a unique solution, and the solution is a
X
transition matrix for an inhomogeneous Markov chain with the density of
the 'intensity measure' given by A s : Hence P s; t de nes a Markov
X X
chain, as was to b e shown.
The assumption that the eigenvectors of A s exist and are not time
X
varying is a strong assumption. The simpli cation is, however, profound as
well as we see in the next result which delivers a b ond pricing mo del which
mo dels b onds of all credit categories simultaneously:
Prop osition 5.3 Assume that transitions between rating classes condition-
al ly on X are governed by the generator matrix
1
A s = B X B
X s
where B and X are de ned as in Lemma 5.2. Assume in addition, that
s
0
the K th row of A s is 0 corresponding to the state of default being an
X
absorbing state . The spot rate process for default free debt is given by R X :
The price at time t of a zero coupon bond with maturity T , issued by a rm
in rating class i and for simplicity with zero recovery in default is given by
! !
Z
K 1
T
X
i
v t; T = E exp X R X du G
u u t
ij j
t
j=1
where the coecients are de ned below.
ij 16
Pro of. Conditionally on X the probability of defaulting b efore T given a
credit rating of i at time t and hence no previous default since default in
0
this mo del is an absorbing state is equal to the i; K th entry of the matrix
P t; T : This entry has the form
X
Z
K
T
X
1
P t; T = b exp X du b ;
X i;K ij u
j
jK
t
j =1
1
0 1
where b denotes the j; K th entry of B and using the fact that the rows
jK
1
of A sum to 0 and that the last rowofA is 0; one maycheck that b b =
X X iK
KK
1
1: De ning = b b wemay therefore write the probability of no default
ij
ij
jK
as
Z
K 1
T
X
1 P t; T = exp X du 5.1
X i;K u
ij j
t
j =1
Let sup erscript i on the exp ectation op erator signify a time t rating of the
issuer of i: The sup erscript is removed when the exp ectation do es not dep end
on this information. Now combine Prop osition 5.1 and 5.1 to obtain
! !
Z
K 1
T
X
i
v t; T = E exp X R X du G
u u t
ij j
t
j=1
We have expressed the price of b ond in credit class i as a linear combi-
nation of functionals of the form
Z
t
E exp X R X du :
u u
j
0
Note that by using the results on ane term structure mo dels letting and
R be ane functions of di usions with ane drift and volatility we obtain
here a class of mo dels whose b ond prices are expressed as sums of ane
mo dels.
Tokeep the exp osition simple wehavelooked at zero coup on b onds with
zero recovery. The mo dels ab ove, however, also allowus to price more com-
plicated contracts whose contractual payo are linked to the rating of the
issuer. Examples include credit options and swaps with credit triggers. This
exercise can be carried out for all the three building blo cks that we priced
in Section 3. Consider an example linked to credit insurance in which the
holder of the contingent claim receives a payment which dep ends on state
variables and on the rating of some company at a xed time T: Here, the 17
contract itself need not have any credit risk in that it may easily pay out
an amount in the event of default of the rm whose rating determines the
payout.
Prop osition 5.4 Let the model be as in the previous proposition. Consider
a contingent claim whose payo at time T is given as
C = C X ; :
T T
T
Assume that the rating at time t of the rm under consideration is i: The
price of this claim at time t is given by
! !
Z
K K
T
X X
i 1
: C = E b b C X ;m exp X R X du G
ik T u u t
k
t km
t
m=1
k =1
Pro of. First, note that
Z
K
T
X
1
P t; T = b b exp X du ;
X i;m ik u
k
km
t
k =1
and therefore
! !
Z
K
T
X
i i
C = E C X ;m P t; T exp R X du G
T X i;m u t
t
t
m=1
! !
Z
K
T
X
= E C X ;m P t; T exp R X du G
T X i;m u t
t
m=1
! !
Z Z
K K
T T
X X
1
= E C X ;m b b exp X du exp R X du G
T ik u u t
k
km
t t
m=1
k=1
! !
Z
K K
T
X X
1
= E b b C X ;m exp X R X du G
ik T u u t
k
km
t
m=1
k =1
which is the desired result.
Note that this pricing formula allows payments to be directly linked to
the rating status of a certain risky rm.
Clearly, the parametric assumptions we imp ose to get the very tractable
mo del ab ove are strong. By sto chastic scaling of the eigenvectors we allow
2
sto chastic changes the K 1 transition and default intensities to vary in
a K 1 dimensional subspace and this will allow sto chastic uctuations to 18
a ect higher rated and lower rated rms di erently. Empirical analysis of
this form is currently b eing carried out in separate work.
Allowing the generator to be sp eci ed more freely will complicate com-
putation but simulation is by no means the only way out: Consider again
the equation
@P s; t
X
= A s P s; t :
X X
@s
To compute the matrix P s; t ; use the approximation based on a partition
X
s = s
0 1 n
P t; t = I and
X
P s ;t = P s ;t A s P s ;t s s i =0;:::;n 1:
X i X i+1 X i+1 X i+1 i i+1
The entries in this approximation will all b e functions of values of the pro cess
X at n + 1 distinct values, which again brings us back to a standard numerical
problem involving only the di usion X:
6 Fitting spreads and spread sensitivities
We consider in this section an illustrative implementation of the generalized
Markovian mo del presented ab ove. A full implementation and testing of the
mo dels will b e the topic of subsequentwork and therefore the chosen param-
eters and prices should be seen primarily as illustrative and as providing a
way for the reader to check the calibration pro cedure.
The goal is to show how the generalized mo del building on a Vasicek
mo del of the riskless term structure can be adapted in the approach using
mo di ed eigenvalues of the generator matrix to pro duce a mo del which ts
short credit spreads and changes in short spreads as a function of changes
in the short rate for all credit classes simultaneously. The generalization to
multifactor mo dels with ane structure will b e apparent.
Assume that the sp ot rate pro cess for riskless debt is given as
dr = b a r dt + dB :
s t t
Now calibrate the mo del to data by p erforming the following steps:
1. Estimate an empirical generator matrix A and assume that there exists
a matrix B whose columns are the eigenvectors of A: 19
2. Let
r = + r ; j =1;:::;K 1
s j s
j j
where ; are constants.
j
j
3. Let r denote a K K diagonal matrix whose rst K 1 diagonal
s
elements are given as ab ove and with the last diagonal element equal
to zero. De ne
1
A s = B r B :
X s
4. Cho ose the parameters ; such that 'twocharacteristics'
j j=1;:::;K 1
j
of each yield curve are satis ed simultaneously across credit classes.
An example follows shortly.
By this sp eci cation of the generator we may encounter negative tran-
sition intensities but just as we may cho ose to work with Gaussian interest
rates as an approximation, despite the fact that this means negativeinterest
rates with p ositive probability, we will allow the approximation of the de-
fault intensity under the risk neutral measure to be negative with p ositive
probability. The constants will b e chosen in practice such that the intensities
are negative with very small probability.
In our example we work with sp ot spreads and sensitivities in the sp ot
spreads to changes in r as the characteristics we wish to match. This choice
has the advantage of pro ducing a simple linear equation to determine and
: To see this, consider the expression for the prices at time 0 of defaultable
b onds with zero recovery in this sp ecial setup:
! !
Z
K 1
T
X
i
v t; T = E exp + r r ds G
j s s t
ij j
t
j =1
! !
Z
K 1
T
X
= E exp 1 r ds G :
j s t
ij j
t
j =1
De ne the sp ot spread for class i as
!
@
i i
s r = lim log v t; T r :
t t
T !t
@T
Now we nd 20
Prop osition 6.1
K 1
X
i
s r = r
t t
ij j
j =1
K 1
X
@
i
s r = :
t j
ij
@r
t
j=1
Pro of. Note that
@
i
log v t; T
@T
@ 1
i
= v t; T
i
v t; T @T
! !
Z
K 1
T
X
1
+ 1 r exp G = E + r r ds
j T t j s s
j ij j
i
v t; T
t
j =1
K 1
X
+ 1 r !
j t
ij j
T !t
j =1
P
K 1
and using the fact that =1 we nd after subtracting the sp ot rate
ij
j =1
that the spread is
K 1
X
i
s r = + r
t j t
ij j
j=1
where in the last limit argument we use dominated convergence for condi-
tional exp ectations. Now di erentiate this expression to obtain
K 1
X
@
i
s r =
t j
ij
@r
t
j=1
so determines the spread sensitivity to changes in the sp ot rate.
Assume that we observe at time 0 when the level of interest rates is r ; a
0
vector of sp ot yield spreads given by the vector
1 K 1
b b b
s = s r ;:::;s r
0 0 0
and that we have estimated a vector of sensitivities given by
1 K 1
c c c
ds = ds r ;:::;ds r :
0 0 0 21
As a consequence of the prop osition, if we can calibrate our mo del to having
the correct spread initially and the correct sensitivities by solving the systems
b
+ r = s
0 0
c
= ds
0
where
=
ij
i;j =1;:::;K 1
= ;:::; and
1 K 1
= ;:::; :
1 K 1
We give an illustrative example:
In Jarrow, Lando and Turnbull [13] the generator matrix is given by
1 0
0:1153 0:1019 0:0083 0:0020 0:0031 0:0000 0:0000 0:0000
C B
0:0091 0:1043 0:0787 0:0105 0:0030 0:0030 0:0000 0:0000
C B
C B
C B
0:0010 0:0309 0:1172 0:0688 0:0107 0:0048 0:0000 0:0010
C B
C B
0:0007 0:0047 0:0713 0:1711 0:0701 0:0174 0:0020 0:0049
C B
C A = B
C B
0:0005 0:0025 0:0089 0:0813 0:2530 0:1181 0:0144 0:0273
C B
C B
0:0000 0:0021 0:0034 0:0073 0:0568 0:1928 0:0479 0:0753
C B
C B
0:0000 0:0000 0:0142 0:0142 0:0250 0:0928 0:4318 0:2856
A @
0:0000 0:0000 0:0000 0:0000 0:0000 0:0000 0:0000 0:0000
Assuming a Vasicek mo del under the risk neutral measure with parame-
ters
a = 0:05
b = 0:01
= 0:015
r = 0:05
0
Let the spreads in basis p oints for classes AAA down to CCC be given
4
as
b
s = 16; 20; 27; 44; 89; 150; 255
0
4
These spreads are consistent with JP Morgan's CreditMetrics estimates for spreads of
Industrial b onds with maturity one year as of July 11, 1997. The sp ot rates will of course
slightly di erent 22
5
and for the purp ose of illustration let the spread sensitivities b e given as
c
ds = 0:2; 0:3; 0:4; 0:5; 0:6; 1:0; 2:0
0
With these we can calibrate the mo del and get estimates of ; :
= 0:1745; 0:1687; 0:1175; 0:0934; 0:0831; 0:0721; 0:0325
= 2:8004; 2:7181; 1:8026; 1:4139; 1:2640; 1:1200; 0:5348 :
The resulting term structures of yield spreads are shown in Figure 1 and
Figure 2.
It is imp ortant to stress that this metho d of adjusting the eigenvalues,
while convenient b ecause of preserving closed-form solutions, is problematic
in the sense that one cannot be sure that desirable sto chastic monotonicity
prop erties of the rating and default pro cess are preserved. Therefore, when
inserting the estimated parameters into the pricing relation, the yield curves
of di erent rating categories may cross. If one allows all entries of the ran-
dom generator to vary freely and not only in a space of dimension equal
to set number of rating classes this can be controlled. It is also p ossible
to randomize only the default intensities asso ciated with each class in a way
which leaves the transition intensities b etween non-default states xed. This
metho d will give a natural way of incorp orating fractional recovery rates
into the intensities. In b oth cases several numerical issues arise which will
be treated in separate work.
7 Conclusion
Wehave laid out a framework which is convenient for analyzing nancial in-
struments sub ject to credit risk through counterparty default and derivatives
whose underlying is a credit risk variable such as a credit spread. We take
into account the p ossible correlation b etween default free b onds and default
probabilities by letting interest rates and hazard rates be governed by com-
mon state variables. The main feature of the framework is that it reduces
the technical issues of mo deling credit risk to the same issues we face when
mo deling the ordinary term structure of interest rates. As a main application
5
Both Longsta and Schwartz [18] and Du ee [4] nd that spreads are negatively
correlated with the level of interest rates and that the sensitivity is stronger for lower
rated b onds. 23
we showhow to generalize a mo del of Jarrow, Lando and Turnbull [13] to al-
low for sto chastic transition intensities between rating categories. A main
application of this mo del is pricing of claims in which the credit rating of the
defaultable party enters explicitly. An implementation is given in a simple
one factor mo del in which the ane structure gives closed form solutions.
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1991. 25 Yield spreads for investment grade debt after calibration
BBB
A
AA Yield spread in percent
AAA 0.2 0.3 0.4 0.5 0.6 0.7
0246810
Time to maturity
Figure 1: Yield spreads on defaultable zero coup on b onds as a function
of time to maturity for investment grade ratings. The yields have b een
calibrated using an empirically observed generator and a risk-adjustment to
match b oth observed sp ot spreads and changes in sp ot spreads due to changes
in short treasury rates. 26 Yield spreads for speculative grade debt after calibration
CCC
B Yield spread in percent
BB 1.0 1.5 2.0 2.5
0246810
Time to maturity
Figure 2: Yield spreads on defaultable zero coup on b onds as a function
of time to maturity for sp eculative grade ratings. The yields have b een
calibrated using an empirically observed generator and a risk-adjustment to
match b oth observed sp ot spreads and changes in sp ot spreads due to changes
in short treasury rates. 27