<< стр. 7(всего 11)СОДЕРЖАНИЕ >>
might wonder why reduced-form models would be used in this context.
The answer partly resides in the fact that an important motivation for
using such models relates to an assumption proposed in the simple exercise
carried out in Chapter 16. There we made the assertion that default inten-
sities were independent of the riskless interest rate process. But one might
well suspect that the level of market interest rates and default probabilities
are correlated, in which case a credit risk modelвЂ”such as the reduced-form
and structural frameworks described in the current chapterвЂ”is needed.22
Other instances where a particular model for the forward default rate are
needed include some valuation problems for derivatives involving spread
optionality, some of which are discussed in Chapter 18.

17.2.6 Credit Default Swap Valuation
Models in the reduced-form tradition can be used to price both single-
and multi-issuer credit derivatives. As an illustration, we will use it to
value a vanilla credit default swap. As in Section 17.1.5, we will let S
denote the credit default swap premium for a contract that matures at
time T . For simplicity, we will continue to assume that the premium is
paid continuously and that the notional amount of the contract is \$1. In
addition, we will take both the riskless interest rate and default intensity
to be time invariant.
In the context of the basic reduced-form model, the present value of the
premium leg of the credit default swap can be written as:

T
S
Seв€’(r+О»)(sв€’t) ds = (1 в€’ eв€’(r+О»)(T в€’t) )
О·(t) = (17.55)
r+О»
t

22
We discussed some empirical results on the relationship between defaults and
interest rates in Chapter 15.
198 17. Main Credit Modeling Approaches

Notice above that, as in equation (17.48), the premium stream is discounted
at the risky rate r + О», reп¬‚ecting the uncertainty surrounding the default
event. (О» is the default intensity associated with the reference entity.)
To value the protection leg, we note that it is equivalent to a contingent
claim that pays (1 в€’ X) in the event of default before T , where X is the
recovery rate of the reference entityвЂ™s defaulted liabilities. The value of such
a claim is given by equation (17.49) with X replaced with 1 в€’ X. Letting
Оё(t) denote the value of the protection leg, we can write

О»(1 в€’ X)
(1 в€’ eв€’(r+О»)(T в€’t) )
Оё(t) = (17.56)
r+О»

A default swap typically has zero market value when it is set up, and
thus pricing such a contract is equivalent to п¬Ѓnding the value of S that
makes the two legs of the swap have equal value. This is given by

S = О»(1 в€’ X) (17.57)

which gives us the result that, for a given recovery rate and constant r and
О», the credit default swap premium tells us about the default intensity of
the reference entity. In other words, by assuming a value for the recovery
rate, we can use the above expression and observable default swap spreads
to infer the default intensity associated with the reference entity. Such a
CDS-implied default intensity can then be used in the valuation of other
credit derivatives instruments.
This simple CDS pricing exercise has made several restrictive assump-
tions, such as continuously paid premiums and constant interest rates and
default intensity. As discussed in Section 17.2.4, many of these assumptions
can be relaxed in order to bring the valuation exercise closer to reality.

17.3 Comparing the Two Main Approaches
There is no clear consensus in either the practitioner or academic litera-
tures about which of the two credit risk modeling approachesвЂ”structural
and reduced-formвЂ”is the most appropriate one. For instance, while the
structural form approach might be said to have the advantage of rely-
ing explicitly on the fundamentals underlying a given п¬ЃrmвЂ”as these are
reп¬‚ected in the п¬ЃrmвЂ™s balance sheetвЂ”such reliance could also be seen as
a drawback. Indeed, balance sheet information tends to become available
only on a quarterly basis, which could be a limitation if one is interested, for
instance, in accurately marking positions to market. (Moreover, balance-
sheet information can be quite noisy as the п¬Ѓnancial reporting scandals
of the early 2000s painfully reminded the markets.) At the same time,
17.3 Comparing the Two Main Approaches 199

reduced-form models might be of more limited value if oneвЂ™s interest is,
for example, in assessing how a change in the capital structure of a given
п¬Ѓrm may aп¬Ђect its п¬Ѓnancing costs, in which case careful use of a structural
model might be the most appropriate approach.
The two approaches can also be compared from a methodological per-
spective. For instance, those familiar with models of the term structure of
interest rates tend to feel more at ease, from a purely technical standpoint,
in the world of reduced-form models. Take, for instance, the relationship
between the survival probability, Q(t, T ), and the forward default rate,
h(t, s),

Q(t, T ) = eв€’
T
h(t,s)ds
t

As noted, this is similar to the mathematical relationship between zero-
coupon bond prices Z(t, T ) and time-t instantaneous forward interest rates.
Likewise, the mathematical treatment of default intensity, О», as well as its
relationship with h() and Q(), is analogous to the links among the spot
short-term interest rate, instantaneous forward interest rates, and zero-
coupon bond prices. Moreover, when О» is assumed to be stochastic, many
of the basic models for describing its evolution mirror common speciп¬Ѓca-
tions developed for interest rate models, such as the well-known yield curve
model of Cox, Ingersoll, and Ross (1985).
While п¬Ѓxed-income modelers might п¬Ѓnd substantial commonality
between techniques used in interest rate models and those used in the
reduced-form approach, equity-minded analysts will no doubt see familiar
ground in the structural approach. In addition to focusing on balance-sheet
information, to which many equity analysts are already used, a centerpiece
of the structural approach is the use of equity-based option theoretic results
to price defaultable debt instruments.
To sum up, from a methodological perspective, the pros and cons of
each approach have to be examined in the context in which the models
will be used, with certain models being more naturally suited for cer-
tain applications. In addition, some analysts might be more attracted to
one approach vs. the other based on how comfortable they feel with the
underlying methodological framework behind each class of models.
Empirically, while structural models are appealing in that they attempt
to link explicitly the likelihood of default of a given п¬Ѓrm to its economic and
п¬Ѓnancial condition, traditional forms of such models tend not to п¬Ѓt the data
as well as reduced-form models, especially, as noted in Section 17.1.4, in
relation to short-term credit spreads. Indeed, in the intensity-based frame-
work, defaults can happen suddenly and unexpectedly without having to be
presaged by observable phenomenon, such as the value of the п¬Ѓrm approach-
ing the default barrier in the typical BSM-style model. In this regard, what
may be characterized as another class of credit risk models has emerged.
200 17. Main Credit Modeling Approaches

These models attempt to combine the economic/intuitive appeal of struc-
tural models with the empirical plausibility of the intensity-based approach.
The model of Duп¬ѓe and Lando (2001) is one of the better-known works
in this strand of the literature.23
The Duп¬ѓe-Lando model can be thought of as a hybrid structural/
intensity-based model because it is essentially a п¬Ѓrst-passage model that,
contrary to standard structural models, also has an intensity-based inter-
pretation. In particular, rather than being given exogenously as in the pure
reduced-form approach, default intensity in the Duп¬ѓe-Lando model can be
calculated in terms of observable variables related to the balance-sheet
fundamentals of the п¬Ѓrm. Moreover, Duп¬ѓe and Lando argue that many
estimation methods used in the context of pure intensity-based models are
also applicable to the hybrid framework.
Essentially, a main thrust of this hybrid approach to credit risk modeling
is to assume that investors only have imperfect information about the true
п¬Ѓnancial condition of the п¬Ѓrm. For instance, investors may not know with
certainty just how far the value of the п¬Ѓrm is from its default boundary.
As a result, the possibility of a default in the very near term cannot be fully
these models tends to be more realistic than that implied by traditional
BSM-type models.24

17.4 Ratings-based Models
Instead of allowing for the п¬Ѓrm to be only in one of two statesвЂ”default
and survivalвЂ”ratings-based models allow for a variety of states, where each
non-default state might correspond, for instance, to a given credit ratingвЂ”
such as AAA, A, BB+ , etc.вЂ”assigned to the п¬Ѓrm by a major credit rating
agency. Such models are not widely used for the direct pricing of vanilla
credit derivatives as most such instruments do not have payouts that are
ratings-dependent. Nonetheless, ratings-based models can be useful in the
context of credit derivatives that involve collateral requirements that are
linked to the credit rating of the counterparties.
We will not describe ratings-based models in detail in this book, but
simply highlight their main features and how they relate to the models
examined in previous sections. A well-known ratings-based model is that

23
Other related work includes Giesecke (2001) and Giesecke and Goldberg
(2004).
24
Within the structural framework, ZhouвЂ™s (1997) jump-diп¬Ђusion model, dis-
cussed in Section 17.2, constitutes an alternative approach to address the empirical
17.4 Ratings-based Models 201

of Jarrow, Lando, and Turnbull (1997). In what follows, we limit our-
selves mostly to providing a basic description of a discrete-time version
of the Jarrow-Lando-Turnbull (JLT) model. In so doing, we follow JLT
themselves, who also used a discrete-time setting to introduce their model.
Suppose that a given bond issuer can have one of J в€’ 1 credit ratings,
with 1 representing the highest credit quality and J в€’ 1 representing the
rating just prior to default. We will also allow for a Jth вЂњrating,вЂќ which
will correspond to default. Let П‰i,j represent the actual (not necessarily
risk-neutral) probability, based on all information available at time t, of
the п¬Ѓrm migrating from a rating of i at time t to one of j at time t + 1. For
simplicity we shall assume that these probabilities are time-invariant over
the horizon of interest, which we assume to span from time 0 to time U.
Let us deп¬Ѓne the J Г— J transition matrix в„¦ such that its (i, j)th element
is П‰i,j :

пЈ« пЈ¶
П‰1,1 П‰1,2 ... П‰1,J
пЈ¬ П‰2,1 П‰2,J пЈ·
П‰2,2 ...
пЈ¬ пЈ·
пЈ¬ пЈ·
в„¦ = пЈ¬ ... ... пЈ·
... ... (17.58)
пЈ¬ пЈ·
пЈ­П‰Jв€’1,1 . . . П‰Jв€’1,J пЈё
П‰Jв€’1,2
0 0 ... 1

We further assume that:

П‰i,j в‰Ґ 0 for all i, j, i = j (17.59)
J
П‰i,i в‰Ў 1 в€’ П‰i,j for all i (17.60)
j = 1, j = i

Equation (17.60) essentially acknowledges the fact that for any given п¬Ѓrm
rated i at time t, its time-t + 1 rating will have to be one of the J ratings.
In particular, the probability that the п¬Ѓrm will retain its i rating at t + 1
must be one minus the sum of the probabilities associated with migration
to any one of the remaining J в€’ 1 ratings.
In technical terms, the last row of в„¦ says that defaultвЂ”the Jth
вЂњratingвЂќвЂ”is an вЂњabsorbing state,вЂќ meaning that once the п¬Ѓrm enters into
a state of default, we assume that it will stay there with probability
П‰J,J = 1. In corporate п¬Ѓnance terms, the model assumes that there is
no reorganization after default.
Let П‰i,j (t, t + n) be the probability, conditional on information available
at time t, that the п¬ЃrmвЂ™s rating will change from i at time t to j at time
t+n. If в„¦(t, t+n) is the matrix such that its (i, j)th element is П‰i,j (t, t+n),
202 17. Main Credit Modeling Approaches

it can be shown that:

в„¦(t, t + n) = в„¦n (17.61)

i.e., ratings transitions are said to follow a Markovian process in that
the current ratings transition matrix is assumed to contain all currently
available relevant information regarding future ratings transitions.
Empirical estimates of в„¦ are published regularly by some of the major
credit-rating agencies based on actual rating changes in the universe of п¬Ѓrms
covered by these agencies. For pricing purposes, of course, what matter
are the risk-neutral transition probabilities П‰i,j , rather than the empirical
Лњ
probabilities in в„¦. JLT propose the following mapping between empirical
and risk-neutral probabilities:

П‰i,j (t, t + 1) = ПЂi (t)П‰i,j for all i, j, i = j
Лњ (17.62)

for ПЂi (t) в‰Ґ 0 for all i and t. Consistent with the discussion in Chapter 15,
ПЂi (t) can be thought of as a risk-premium-induced adjustment to the actual
transition probabilities.
The risk-neutral transition probabilities are assumed to satisfy condi-
tions analogous to those in equations (17.59) and (17.60). In addition, JLT
imposed the technical condition that П‰i,j (t, t+1) > 0 if and only if П‰i,j > 0,
Лњ
for 0 в‰¤ t в‰¤ U в€’ 1.
Лњ
Thus, we can also deп¬Ѓne the risk-neutral transition matrix в„¦(t, t + n),
and it should by now be clear that its (i, j)th entry is П‰i,j (t, t + n), which
Лњ
is the risk-neutral probability that the entity will migrate from a rating of
i at time t to one of j at time t + n. If we make the simplifying assumption
that both в„¦ and ПЂi are time invariant:

Лњ Лњ
в„¦(t, t + n) = в„¦(t, t + 1)n (17.63)

Suppose one is interested in the risk-neutral probability, conditional on
all information available at time t, that an i-rated п¬Ѓrm will survive through
some future date T . Given all the assumptions discussed thus far, this
probability is simply

Qi (t, T ) = 1 в€’ П‰i,J (t, T )
Лњ (17.64)

Thus, as discussed throughout this part of the book, and assuming that
riskless interest rates are independent of the stochastic process underlying
the ratings transitions of the п¬Ѓrm, the time-t price of a zero-recovery, zero-
coupon bond that will mature at time T with a face value of \$1 is:

Z0 (t, T ; i) = Z(t, T )[1 в€’ П‰i,J (t, T )]
d
Лњ (17.65)
17.4 Ratings-based Models 203

where we added the argument i to the zero-coupon bond price to indicate
that this bond was issued by a п¬Ѓrm that is currently rated i.
If Rd (t, T ; i) denotes the yield to maturity on this bond, we can use
results discussed in earlier sections in this chapter to derive the credit risk
spread associated with this п¬Ѓrm. In particular,

в€’ log(Z0 (t, T ; i)) + log(Z(t, T ))
d
R (t, T ; i) в€’ R(t, T ) =
d
T в€’t
в€’ log(1 в€’ П‰i,J (t, T ))
Лњ
= (17.66)
T в€’t

where we continue to assume that R(t, T ) в‰Ў в€’ log(Z(t,T )) is the yield to
T в€’t
maturity on a riskless zero-coupon bond with the same maturity date and
face value as the risky bond.
Thus far we have been using the model essentially to derive expressions
for prices and spreads that we were also able to examine with the modeling
approaches summarized in Sections 17.1 and 17.2. As their name suggests,
however, ratings-based models are particularly suitable for analyses involv-
ing yield spreads across diп¬Ђerent ratings. For instance, the model-implied
yield spread between two bonds rated i and j is

1 в€’ П‰j,J (t, T )
Лњ 1
Rd (t, T ; i) в€’ Rd (t, T ; j) = log (17.67)
1 в€’ П‰i,J (t, T ) T в€’ t
Лњ

Equations (17.66) and (17.67) can be used to identify potentially prof-
itable opportunities across diп¬Ђerent issuers with various credit ratings, by,
for instance, comparing model-implied spreads to the ones observed in the
market place. Alternatively, one may be interested in using equations like
(17.66) and (17.67) to calibrate the model to the data in order to use the
resulting risk-neutral transition probabilities to value п¬Ѓnancial instruments
and contracts with ratings-dependent payoп¬Ђs, such as bonds with ratings-
dependent coupons and credit derivatives contracts with ratings-linked
collateral requirements.
A full continuous-time version of the model described in this section
is provided in the original JLT paperвЂ”Jarrow, Lando, and Turnbull
(1997)вЂ”which also addresses calibration-related issues. Discussing the
technical details behind that version of the model, as well as model cali-
bration and other implementation topics, is outside the scope of this book.
Instead we limit ourselves to providing a very brief overview of some basic
concepts that are germane to the continuous-time speciп¬Ѓcation of the JLT
model. In particular, in the simplest case of time-invariant risk-neutral
transition probabilities, the J Г—J transition matrix for the continuous-time
204 17. Main Credit Modeling Approaches

version of the model can be written as:

в„¦(t, T ) = eО›(T в€’t) (17.68)

where the J Г— J matrix О› is typically called the generator matrix. The ith
diagonal element of О›, О»i,i , can be thought of as the exit rate from the
ith rating, and, for i = j, the (i, j)th element of О›, О»i,j , is the transition
rate between ratings i and j. The concepts of exit and transition rates are
analogous to that of default intensity, examined in Section 17.2. In this
sense, one can think of the JLT model as a generalized intensity-based
model, and, indeed, the JLT model is essentially an extension of the Jarrow
and Turnbull (1995) model.
The literature on ratings-based models is a vast one, and the uses and
implications of these models go well beyond the analysis and valuation of
credit derivatives. Other contributions to the literature include the work
of Kijima and Komoribayashi (1998), Lando and Skodeberg (2002),
Das and Tufano (1996), and Arvanitis, Gregory, and Laurent (1999).
Some structural credit risk models that are commercially available also
incorporate the analysis of ratings transitions. We summarize the main
features of a few well-known commercial models in Chapter 22.
18
Valuing Credit Options

Chapter 8 contained a basic discussion of the main features of spread and
bond options. In this chapter we describe a relatively simple framework
for valuing these instruments. We start Section 18.1 with a discussion of
forward-starting credit default swaps, introducing some concepts that will
come in handy in the valuation of credit default swaptions, the subject
of Section 18.2. Section 18.3 generalizes the valuation approach for credit
default swaptions so it can be used with other spread options. Extensions
and alternatives to the simple framework described in Sections 18.2 and
18.3 are brieп¬‚y discussed in Section 18.4. The valuation of bond options is
sketched out in Section 18.5.

18.1 Forward-starting Contracts
At the end of Chapter 7 we brieп¬‚y mentioned the forward-starting total
return swap, which is a contractual commitment to enter into a total return
swap at a п¬Ѓxed future date and at a predetermined spread. In this discus-
sion of valuation methods for credit default swaptions and other credit
options, we will meet two additional types of forward-starting contracts,
the forward-starting credit default swap and forward contracts involving
п¬‚oaters. The aims of such contracts are self-evident; they are agreements
to enter into a credit default swap and to buy and sell п¬‚oaters, respectively,
at future dates and at predetermined premiums (in the case of a CDS) and
206 18. Valuing Credit Options

As we shall see below, forward-starting credit default swaps can be
thought of as the underlying вЂњassetвЂќ in a credit default swaption, and
thus it will be instructive to have a basic understanding of how they are
valued before proceeding to examine the valuation of credit default swap-
tions. The same idea applies to forward contracts written on п¬‚oaters, and
so we shall examine them in some detail.

18.1.1 Valuing a Forward-starting CDS
Consider a forward-starting CDS agreement entered into at time t where
one party agrees to buy protection in a CDS that will start at the future
date T with a corresponding CDS premium of K, and premium payment
dates T1 , T2 , . . . , Tn . For simplicity, we assume that the notional amount of
the CDS is \$1.
From Section 16.2, we know that the time-t market value of such an
agreement to the protection buyer can be written as

n
W (t) = Z(t,Tj ){[Q(t,Tjв€’1 )в€’Q(t,Tj )](1в€’X)в€’Q(t,Tj )Оґj K} (18.1)
j =1

where Z(t, Tj ) corresponds to the proxy for a riskfree discount factorвЂ”
which, as discussed in Chapter 16, tends to be derived in practice from
the LIBOR/swap curve to reп¬‚ect the funding costs of the large banks that
tend to be most active in the CDS marketвЂ”and X and Q(t, Tj ) relate to,
respectively, the recovery rate of the reference entity (0 в‰¤ X < 1) and the
risk-neutral probability that the reference entity will survive through Tj ,
conditional on all information available at time t. Оґj is the accrual factor
for the jth premium payment (the number of days between the (j в€’ 1)th
and jth premium payment dates divided by the number of days in the year,
based on the appropriate day-count convention).
We can now introduce the notion of the forward CDS premium, which
can be thought of as the value of K in (18.1) such that the forward-starting
credit default swap has zero market value at time t. We shall let S F (t, T, Tn )
denote the forward CDS premium, as seen at time t, for a CDS contract
that will start at time T and have premium payment dates at T1 , T2 , . . . , Tn .
Solving (18.1) for K while requiring W (t) to be zero, we can write

n
Z(t, Tj )[Q(t, Tjв€’1 ) в€’ Q(t, Tj )](1 в€’ X)
j=1
F
S (t, T, Tn ) = (18.2)
n
Z(t, Tj )Q(t, Tj )Оґj
j=1

and substituting this last expression into (18.1) we arrive at a conve-
nient formula for the market value of a protection-buying position in a
18.1 Forward-starting Contracts 207

forward-starting credit default swap:

n
Z(t, Tj )Q(t, Tj )Оґj [S F (t, T, Tn ) в€’ K]
W (t) = (18.3)
j=1

which has the intuitive implication that the market value of a forward-
starting CDS depends crucially on the diп¬Ђerence between the corresponding
contract.1

18.1.2 Other Forward-starting Structures
The valuation of other credit-related forward-starting structures, such as
forward-starting asset swaps and forward contracts involving п¬‚oating-rate
notes, can be carried out using similar methods to the one just described
for forward-starting credit default swaps. Consider, for instance, a forward
contract to receive par for a п¬‚oating-rate note at a future date with a
prespeciп¬Ѓed spread over LIBOR. Assume, for simplicity, that the п¬‚oater
has a zero recovery rate. (This is a forward contract to sell a given п¬‚oater
for its par value at a future date at a predetermined spread.)
Recall, from Chapter 4, that the time-T market value of a just-issued
par п¬‚oater with a face value of \$1 and coupon payment dates T1 , T2 , . . . , Tn
can be written as
n
Z0 (T, Tj )Оґj [F в€— (T, Tjв€’1 , Tj ) + s(T, Tn )] + Z0 (T, Tn )
d d
1= (18.4)
j=1

where F в€— (T, Tjв€’1 , Tj ) is the point on the forward LIBOR curve, as seen
at time T , that corresponds to a loan lasting from the future date Tjв€’1
to Tj ; s(T, Tn ) is the par п¬‚oater spread, and, to simplify the notation,
Z0 (T, Tj ) в‰Ў Z(T, Tj )Q(t, Tj ).
d

Likewise, for a par п¬‚oater that pays LIBOR п¬‚at:

n
Z(T, Tj )Оґj F в€— (T, Tjв€’1 , Tj ) + Z(T, Tn )
1= (18.5)
j=1

which diп¬Ђers from the previous equation only because of the zero spread
and the choice of discount factors.

1
Note the similarity between(18.3) and the expression for marking to market a CDS
position in Chapter 16.
208 18. Valuing Credit Options

The time-t value of the latter par п¬‚oater, for t < T , can be shown to be
n
Z(t, Tj )Оґj F в€— (t, Tjв€’1 , Tj ) + Z(t, Tn )
Z(t, T ) =
j=1

which can be veriп¬Ѓed given the deп¬Ѓnition of forward LIBOR (see
Chapter 4):

Z(t, Tjв€’1 )
в€’1
F в€— (t, Tjв€’1 , Tj ) в‰Ў Оґj в€’1
Z(t, Tj )

As for the riskier par п¬‚oater, its time-t value becomes
n
Z d (t, Tj )[F в€— (t, Tjв€’1 , Tj ) + sF (t, T, Tn )]Оґj + Z0 (t, Tn )
d d
Z0 (t, T ) =
j=1

where sF (t, T, Tn ) is deп¬Ѓned as the forward par п¬‚oater spread associated
with this particular issuer, as seen at time t, for future borrowing between
times T and Tn .
Given the above, the task of valuing an arbitrary forward contract involv-
ing a п¬‚oater that will pay a spread of say K, which is not necessarily the
that, from the perspective of the party committed to selling the п¬‚oater, the
time-t value of such a contract can be written as
n
Z d (t, Tj )Оґj [sF (t, T, Tn ) в€’ K] + Z d (t, T )
FL
W (t) = (18.6)
j=1

which, again, has the simple intuition that a contract to sell a п¬‚oater at a
future date for parвЂ”in essence, a contract to pay a given spread starting
at some future dateвЂ”will have positive market value whenever the spread
K written into the contract is below the corresponding forward spread
associated with the issuer.
We carried out this discussion with a forward contract to sell a п¬‚oater.
The results would be entirely analogous for a forward-starting asset swap,
and we leave this exercise to the reader.

18.2 Valuing Credit Default Swaptions
Let W (t) be the time-t value, to a protection buyer, of a forward-starting
credit default swap. Continuing with the same setup introduced in the
18.2 Valuing Credit Default Swaptions 209

previous section, the CDS will start at a future time T , with payment
dates at T1 , T2 , . . . , Tn , and the premium is set at K. As a result:
n
Z(t, Tj )Q(t, Tj )Оґj [S F (t, T, Tn ) в€’ K]
W (t) =
j=1

which is simply (18.3).
Consider now a European option, written at time t, to buy protection in
the contract underlying the forward-starting CDS described in the previous
section. At time T , the exercise date of the option, S F (T, T, Tn ) = S(T, Tn ),
i.e., the forward premium converges to the spot premium, and the value of
the default swaption will be:
пЈ± пЈј
пЈІn пЈЅ
Z0 (T, Tj )Оґj [S(T, Tn ) в€’ K], 0
d
V (T ) = Max(W (T ), 0) = Max
пЈі пЈѕ
j=1
(18.7)

Equation (18.7) tells us that the holder of this credit default swaption
will exercise it only if the underlying CDS has positive market value at T ,
which is the case whenever the then-prevailing par CDS premium exceeds
the premium written into the option (otherwise the holder would be better
oп¬Ђ paying the prevailing CDS premium, S(T, Tn ), in a par CDS, which has
zero market value).
n d
One can think of the term j = 1 Z0 (T, Tj )Оґj in equation (18.7) as an
annuity factor that gives the time-T value of the entire stream of diп¬Ђerences
between the premium payments in a par CDS contract and the one speciп¬Ѓed
in the default swaption. If we let A(T, Tn ) denote this factor, we can write
the time-T value of the default swaption as

V (T ) = Max[A(T, Tn )(S F (T, T, Tn ) в€’ K), 0] (18.8)

To п¬Ѓnd the time-t value of the default swaption, it is convenient to rewrite
(18.8) as

V (T ) = A(T, Tn )Max[(S F (T, T, Tn ) в€’ K), 0] (18.9)

which tells us that the time-T value of the option is simply a function of
the present value of the diп¬Ђerence between the premium payments of the
two credit default swaps.
If we now recall that the time-t value of any п¬Ѓnancial asset is simply the
risk-adjusted expected present value of its cash п¬‚ow, we can write
в€—
V (t) = A(t, T )Et [Max(S F (T, T, Tn ) в€’ K, 0)] (18.10)
210 18. Valuing Credit Options

в€—
where Et [.] denotes the expected value of вЂњ.вЂќ conditional on information
available at time t, computed on the basis of probabilities that are appro-
priately adjusted for risk in a way that follows the spirit of the risk-neutral
probabilities discussed in Chapter 15.2
In order to derive a pricing formula for this default swaption, we need to
have an explicit assumption (a model) that describes the evolution of the
forward CDS premium over time. A common assumption is to assert that
S F (t, T, Tn ) is lognormally distributed, which allows one to use the option
pricing formula derived by Black (1976).3 If we let Пѓ(t, T, Tn ) denote
the volatility of percentage changes in S F (t, T, Tn ), we can write the Black
formula for a credit default swaption as:

V (t) = A(t, Tn )[S F (t, T, Tn )N (d1 ) в€’ KN (d2 )] (18.11)

with
в€љ
log(S F (t, T, Tn )/K)
d1 в‰Ў + .5Пѓ(T, Tn ) T в€’ t
Пѓ(T, Tn )2 (T в€’ t)

в€љ
log(S F (t, T, Tn )/K)
d2 в‰Ў в€’ .5Пѓ(T, Tn ) T в€’ t
Пѓ(T, Tn )2 (T в€’ t)

N (.) is the cumulative standard normal distribution, and we made the
simplifying assumption that Пѓ(t, T, Tn ) is time-invariant.
As with vanilla calls and puts, credit default swaption prices are strictly
increasing in the volatility of the relevant forward CDS premium. Other
basic features of call and put options, such as put-call parity, also hold.

18.3 Valuing Other Credit Options
The valuation of other credit options, such as an option to sell the п¬‚oater
underlying the forward contract discussed in Section 18.2, can be car-
ried out following essentially the same steps outlined for credit default
swaptions.

2
The reader with some familiarity with continuous-time п¬Ѓnance methods may recog-
в€—
nize the probability measure embedded in Et [.] as that corresponding to the so-called
вЂњannuity measureвЂќ (Hunt and Kennedy, 2000). Under this probability measure, both
V (t)
the relative price A(t,T ) and the forward par CDS premium S F (t, T, Tn ) follow a
n
random walk.
3
The Black pricing formula is a variant of the well-known Black-Scholes formula.
Black originally derived it for the pricing of options on futures contracts, but it can be
shown that it applies directly to the pricing of credit default swaptions and many other
related options. See Hull (2003) for a textbook discussion of the Black formula.
18.5 Valuing Bond Options 211

Consider, for instance, a put option on a п¬‚oater with a face value of \$1.
The option expires at date T and has a strike price of \$1. As discussed in
Chapter 8, this is basically a call option on the п¬‚oater spread.
At time T , the value of the put option will be

V F L (T ) = Max(1 в€’ W F L (T ), 0) (18.12)

i.e., the option holder will choose to sell the п¬‚oater for its face value only
if the market value of the п¬‚oater, W F L (T ), is less than \$1.
We can use the results of Section 18.1.2 to rewrite equation (18.12) as

V F L (T ) = A(T, Tn )Max[sF (T, T, Tn ) в€’ K, 0] (18.13)

and from here on one would proceed as in Section 18.2 to obtain the Black
pricing formula for this call spread option.

18.4 Alternative Valuation Approaches
An implicit (but fundamental) assumption made in this chapter is that
it would make no sense to write an option on something that behaves
deterministically. Going back to results derived in previous chapters,
however, which suggested a close relationship between CDS premiums
and the default probabilities of the reference entity, the assumption of
probabilities are themselves stochastic.
Indeed, an alternative approach to valuing spread and bond options is
to model the stochastic behavior of default intensity directly, often jointly
with the behavior of short-term interest rates, and use the resulting frame-
work to derive the option prices of interest. As discussed by Arvanitis and
Gregory (2001) and Schonbucher (1999), stochastic default proba-
bility models can also be used to value more complex credit options than
the ones examined in this chapter, such as Bermudan options and other
structures with more than one exercise date.

18.5 Valuing Bond Options
We discussed above the valuation of options written on п¬‚oating-rate bonds
and on credit default swaps. Similar methods apply to the valuation of
options written on п¬Ѓxed-rate bonds. For instance, one may assume that
the forward price of the bond is lognormally distributed and then use the
Black formula as in Sections 18.2 and 18.3. Hull (2003) provides a useful
textbook discussion of the valuation of п¬Ѓxed-rate bond options.
Part IV

Introduction to Credit
Modeling II: Portfolio
Credit Risk

213
19
The Basics of Portfolio Credit Risk

The credit risk models we have examined thus far in this book have all
focused on single default events, or on the likelihood that a given п¬Ѓrm will
default on its п¬Ѓnancial obligations within a given period of time. We shall
now shift gears, so to speak, and take a quick tour of approaches and tech-
niques that are useful for modeling credit risk in a portfolio setting. As we
saw in Chapters 9, 10, and 14, two key concepts in the modeling of portfolio
credit risk are default correlation and the loss distribution function. The
basic model discussed in this chapter allows us to take a closer quantitative
look at these concepts.
Before we proceed, however, one caveat is in order. The discussion in
this part of the book only scratches the surface of what has become a large
and growing technical literature on the modeling of portfolio credit risk.
Our goal is only to introduce the reader to some of the main issues and
challenges facing both academics and practitioners in the real world when it
comes to modeling the default risk embedded in a portfolio of credit-related
instruments.

19.1 Default Correlation
Intuitively, default correlation is a measure related to the likelihood that
two or more reference entities will default together within a given hori-
zon. The higher the default correlation between two п¬Ѓrms, the higher the
chances that default by one of them may be accompanied by a default
by the other. Given that corporate defaults are relatively rare events,
216 19. The Basics of Portfolio Credit Risk

empirical measures of default correlation are not easy to come by. The
available evidence suggests that default correlation tends to be higher for
lower quality credits for higher borrowers, presumably because less-credit
worthy п¬Ѓrms are more sensitive to the ups and downs of the economy,
and that the extent of default correlation depends on the time horizon in
question (Lucas, 1995). In addition, it seems plausible that default cor-
relation tends to be higher among п¬Ѓrms in the same industry than among
п¬Ѓrms in diп¬Ђerent lines of business altogether, although one can imagine
situations where a default by one п¬Ѓrm strengthens the position of its main
competitor, making it less likely to default.

19.1.1 Pairwise Default Correlation
From elementary statistics, we know that the correlation coeп¬ѓcient, ПЃY1 ,Y2 ,
involving two random variables Y1 and Y2 is deп¬Ѓned as

Cov[Y1 , Y2 ]
ПЃY1 ,Y2 в‰Ў (19.1)
Var[Y1 ] Var[Y2 ]

where Cov[Y1 , Y2 ] denotes the covariance between Y1 and Y2 , and Var[Y1 ]
and Var[Y2 ] stand for the variances of Y1 and Y2 , respectively, all of which
are deп¬Ѓned below.
Mathematically, one can write an expression for the coeп¬ѓcient of default
correlation for any two entitiesвЂ”called the pairwise default correlationвЂ”
in terms of their respective default probabilities. To see this, we shall
take another look at the two hypothetical entities we have been analyzing
throughout this book: XYZ Corp. and AZZ Bank.
Using notation introduced earlier, we shall let П‰A and П‰X denote the
(risk-neutral) default probabilities of AZZ and XYZ, respectively, over a
given time horizon.1 As for the probability that both AZZ Bank and XYZ
Corp. will default together, we shall denote it as П‰A&X . These probabilities
are represented in the diagram in Figure 19.1. The area encompassed by the
rectangle in the п¬Ѓgure represents all possible survival and default outcomes
associated with AZZ and XYZ over a given time horizon. Those outcomes
that involve a default by AZZ are shown within area A and those involving
defaults by XYZ are represented by area X. For simplicity, we shall assume
that the area of the rectangle is equal to 1 and thus we can think of the
areas A and X as the probabilities that AZZ Bank and XYZ Corp. will
default, respectively, within the prescribed time horizon. Furthermore, the
region of overlap between areas A and X corresponds to the probability
П‰A&X that both AZZ Bank and XYZ Corp. will default.

1
In terms of the notation used in Part III, П‰A and П‰X can be written as 1 в€’ QA (t, T )
and 1 в€’ QX (t, T ), respectively, where, for instance, QA (t, T ) is the risk-neutral proba-
bility, conditional on all information available at time t, that AZZ will survive through
some future time T . As in previous chapters, we continue to denote the current time as t.
19.1 Default Correlation 217

FIGURE 19.1. Diagrammatic Representation of Default Probabilities

Let us now deп¬Ѓne IX as a random variable that takes the value of 1
in the event of default by XYZ Corp. over a given horizon and 0 other-
wise. IA is likewise deп¬Ѓned for AZZ Bank. In technical terms, IX and IA
are called indicator functions regarding default events by XYZ and AZZ.
It is straightforward to see that the expected values of IX and IA are
simply the associated default probabilities over the horizon of interest. For
instance, given that IA can only take two values, and relying on risk-neutral
probabilities, its expected value is:

E[IA ] = П‰A Г— 1 + (1 в€’ П‰A ) Г— 0 = П‰A
Лњ (19.2)

Лњ
where E[IA ] is the expectation of IA , computed on the basis of risk-neutral
probabilities.
Again using basic results from elementary statistics, we can also compute
the variance of IA and the covariance between IA and IX . The variance is

Var[IA ] в‰Ў E (IA в€’ E[IA ])2 = П‰A (1 в€’ П‰A )
Лњ Лњ (19.3)

and the covariance can be written as

Cov[IA , IX ] в‰Ў E (IA в€’ E[IA ])(IX в€’ E[IX ]) = E[IA IX ] в€’ E[IA ]E[IX ]
Лњ Лњ Лњ Лњ Лњ Лњ

But notice that IA IX is only nonzero when both IA and IX are 1, and that
happens with probability П‰A&X . Thus

Лњ
E[IA IX ] = П‰A&X
218 19. The Basics of Portfolio Credit Risk

and we can write

Cov[IA , IX ] = П‰A&X в€’ П‰A П‰X (19.4)

We can now derive an expression for the pairwise default correlation for
AZZ Bank and XYZ Corp. entirely in terms of their respective default
probabilities:

П‰A&X в€’ П‰A П‰X
Cov[IA , IX ]
ПЃA,X в‰Ў = (19.5)
П‰A (1 в€’ П‰A ) П‰X (1 в€’ П‰X )
Var[IA ] Var[IX ]

Equation (19.5) formalizes an intuitive and key result regarding pairwise
default correlations. In particular, other things being equal, the higher the
probability that any two entities will default together over the prescribed
horizonвЂ”the region of overlap between areas A and X in Figure 19.1вЂ”the
higher their pairwise default correlation coeп¬ѓcient over that horizon.2
It can be shown that (19.5) is such that the pairwise default correla-
tion coeп¬ѓcient lies between 1 (perfect positive correlation) and в€’1 (perfect
negative correlation). In the former case, the two entities either default or
survive together; in the latter case, only one of the entities will survive
through the end of the relevant horizon and a position in a bond issued
by one entity can be used as a hedge against default-related losses in an
otherwise comparable bond issued by the other.
The case of perfect positive correlation, ПЃA,X = 1, corresponds to
a scenario where the two entities have identical default probabilities,
П‰A = П‰X = П‰ , and where П‰A&X = П‰ . The case of perfect negative corre-
ВЇ ВЇ
lation corresponds to a scenario where П‰A&X is zero, its lower bound, and
П‰X = 1 в€’ П‰A , i.e., there is no region of overlap between areas A and X in
Figure 19.1 and area X takes up the entire portion of the rectangle that is
not encompassed by area A.3
In addition to the two polar cases of perfect positive and negative correla-
tion, another special case is the situation of zero default correlation between
any two entities. As can be seen in equation (19.5), this corresponds to the
case where П‰A&X = П‰A П‰X .
Before we proceed, we can use a variant of Figure 19.1 to verify a result
advanced in Chapter 9. We claimed in that chapter that as the default

2
Equation (19.5) is only one of several possible ways to deп¬Ѓne default correlation. For
instance, an alternative approach would be to focus on the correlation of default times.
Technically oriented readers interested in this topic may wish to consult Li (2000)
and Embrechts, McNeil, and Strautman (1999).
3
One way to verify that 1 is the maximum value that ПЃA&X can achieve is to convince
yourself that П‰A&X cannot be larger than Min[П‰A , П‰X ]. Hint: Look at the diagram in
Figure 19.1.
19.1 Default Correlation 219

FIGURE 19.2. Diagrammatic Representation of a Case of High Default Correla-
tion

correlation between two entities approaches 1, the probability that both
entities will default over a given period approaches the default probability
of the entity with the higher credit quality. This is illustrated in Figure 19.2,
which shows a near complete overlap between areas A and XвЂ”or a default
correlation of nearly one according to the results discussed aboveвЂ”but
where area A is larger than area XвЂ”AZZ bank is of an inferior credit
quality (greater default probability) than XYZ Corp. In the case depicted
in the п¬Ѓgure, all scenarios where XYZ Corp. defaults are also scenarios
where AZZ Bank defaults. Hence the region of overlap between areas A
and X is the same as area X.

19.1.2 Modeling Default Correlation
Thus far we have essentially limited ourselves to a portfolio with debt
instruments issued by only two entities. How do we examine default cor-
relation in more realistic settings, which often involve a large number of
entities? We discuss below a very simple modeling framework that allows us
to start tackling this issue and that should expose many of the diп¬ѓculties
and challenges associated with modeling portfolio credit risk.4
The model builds on the seminal work of Black and Scholes (1973)
and Merton (1974), which we discussed in Part III. In particular, for a
hypothetical п¬Ѓrm i, we deп¬Ѓne default as a situation where the return Ri on

4
Versions of the simple model discussed in this section can be found in Gupton,
Finger, and Bhatia (1997), OвЂ™Kane and Schlogl (2001), Crosbie (2002), and
many others.
220 19. The Basics of Portfolio Credit Risk

owning the п¬Ѓrm falls below a given threshold Ci at a given date T .5 Thus,

Default by п¬Ѓrm i at time T <=> Ri в‰¤ Ci (19.6)

and the default probability associated with this п¬Ѓrm is simply the
probability that its return will fall short of Ci over the relevant time horizon.
To simplify the analysis, especially when debt instruments issued by
several п¬Ѓrms are included in the portfolio, we shall assume that the above
returns have already been standardized, i.e., for each п¬Ѓrm i in the portfolio,
we express returns in terms of deviations from the sample mean and divide
the result by the standard deviation of returns:
Ri,t в€’ Вµi
ВЇ
Ri,t в‰Ў
Пѓi
ВЇ
where Ri,t are the raw (вЂњnon-standardizedвЂќ) returns associated with owning
ВЇ
п¬Ѓrm i at time t, and Вµi and Пѓi are the mean and standard deviation of Ri,t ,
respectively. Note that, by working with standardized returns, all the п¬Ѓrms
in the portfolio have the same expected returns and standard deviations of
returnsвЂ”0 and 1, respectively.
Let us now consider a portfolio that contains debt instruments issued by
I diп¬Ђerent п¬Ѓrms, where default events for each п¬Ѓrm are deп¬Ѓned as in equa-
tion (19.6). Two key assumptions of the model we are about to introduce
are that (i) individual returns Ri are normally distributed, and (ii) returns
(and thus defaults) across п¬Ѓrms are correlated through dependence on one
common factor. A straightforward corollary of the п¬Ѓrst assumption is that,
for a given time horizon, the probability of default by any one single п¬Ѓrm
is given by

П‰i в‰Ў Prob[default by п¬Ѓrm i] = Prob[Ri в‰¤ Ci ] = N (Ci ) (19.7)

where N (.) is the cumulative distribution function of the standard normal
distribution.

5
Strictly speaking, Ri could be more properly called a log-return given that we
implicitly deп¬Ѓne it in discrete time as

Ri,t в‰Ў log(Ai,t ) в€’ log(Ai,tв€’1 )

where Ai,t is the value of п¬Ѓrm i at time t. In the remainder of this book, however, we
will economize on terminology and simply call Ri,t the return associated with п¬Ѓrm i.
Readers of earlier chapters will notice that we have shifted from an asset-value-based
deп¬Ѓnition of defaultвЂ”recall, for instance, that the Black-Scholes-Merton framework dis-
cussed in Chapter 17 modeled defaults as situations where the value of the п¬Ѓrm fell below
some critical levelвЂ”to one that is based on (log) returns. These two approaches to spec-
ifying defaults are perfectly consistent with one another: In the Black-Scholes-Merton
framework, the value, Ai , of the п¬Ѓrm was assumed to be lognormally distributed, which
has the implication that log-returns are normally distributed, as assumed here.
19.1 Default Correlation 221

Here we pause to note that the probabilities involved in equation (19.7)вЂ”
and, indeed, throughout this part of the bookвЂ”correspond to the concept
of risk-neutral probabilities, which we discussed in Part III. Thus, if they
are available, one can use market prices of liquid assets, such as credit
default swap premiums for certain reference names, as a means of arriving
at a value for П‰i and then rely on (19.7) to back out a market-implied value
for Ci :

Ci = N в€’1 (П‰i ) (19.8)

where N в€’1 (.) is the inverse of N (В·), or the function that determines the
value of Ci in (19.7) that corresponds to a given value of П‰i .
Going back to the second assumption (correlation through dependence
on a common factor), we complete the model by making the following
assumption regarding the evolution of returns:

1 в€’ ОІi
2
Ri,t = ОІi О±t + (19.9)
i,t

for all п¬Ѓrms in the portfolio, i.e., for i = 1, 2, . . . , I.
Equation (19.9) says that individual returns, and thus the likelihood of
default, depend on a factor О±, which aп¬Ђects all entities represented in the
portfolio, and on an entity-speciп¬Ѓc factor i . For instance, the entities repre-
sented in the portfolio may all be sensitive to conditions in the overall stock
market, or, alternatively, О±t may stand for the current state of the economy.
To be consistent with the deп¬Ѓnition of Ri,t as a random variable with
zero mean and unit variance, we further assume that О±t and, for all i, i,t
have also been standardizedвЂ”they have zero mean and unit varianceвЂ”and
that they are independently and normally distributed.6 We also assume
that, for any two п¬Ѓrms i and j, j,t and i,t are independently distributed
and that О±t and i,t , and thus Ri,t are serially uncorrelated, i.e.,

Лњ Лњ
E[О±t О±tв€’s ] = E[ i,t i,tв€’s ] =0

for all nonzero values of s.7

6
It is easy to verify that (19.9) is consistent with the assumption of zero mean and
Лњ
unit variance of returns. E[Ri,t ] = 0 follows directly from the zero-mean assumption on
О± and i . To see that the variance of Ri,t remains unit, simply note that we can write
it as

Var[Ri,t ] = ОІi E[О±2 ] + (1 в€’ ОІi )E[ 1 в€’ ОІi E[О±t
2Лњ 2Лњ 2Лњ
2
i,t ] + 2ОІi i,t ]
t

which is equal to one given the variance and independence assumptions regarding О±t
and i,t .
7
The concept of serial independence is discussed brieп¬‚y in Appendix B.
222 19. The Basics of Portfolio Credit Risk

In light of equation (19.9), we can show that ОІi plays a crucial role
in capturing the extent of return correlation among any two entities in
the portfolio. In particular, the covariance between Ri,t and Rj,t can be
written as

Cov(Ri,t , Rj,t ) в‰Ў E[(Ri,t в€’ E[Ri,t ])(Rj,t в€’ E[Rj,t ])] = ОІi ОІj
Лњ Лњ Лњ (19.10)

which, given the assumption of unit variances, also corresponds to the
correlation between Ri,t and Rj,t .
From a modeling standpoint, one advantage of the above framework is
that, for a given value of the common factor, О±, returns involving any two
п¬Ѓrms in the portfolio, and thus their respective default events, are uncor-
related.8 To verify the conditional independence of defaults, we start by
computing the conditional mean and variance of returns under the model:

E[Ri,t |О±t ] = ОІi О±t
Лњ (19.11)

Var[Ri,t |О±t ] = 1 в€’ ОІi
2
(19.12)

and it is then straightforward to see that returns are indeed pairwise
conditionally uncorrelated:

Cov[Ri,t , Rj,t |О±t ] = 1 в€’ ОІi 1 в€’ ОІj E[
2Лњ
2
i,t j,t ] =0 (19.13)

where E[Ri,t |О±t ] is the expected value of Ri,t conditioned on the time-t
Лњ
value of О±, and Var[.|О±t ] and Cov[.|О±t ] are analogously deп¬Ѓned. Note that
(19.13) follows from the fact that i,t and j,t are mutually independently
Лњ
distributed, i.e., E[ i,t j,t ] = 0.
We can also compute conditional default probabilities over given hori-
zons for any п¬Ѓrm in the portfolio. For instance, for О±T = О±t , the time-t
probability that the п¬Ѓrm will default at time T can be written as

в‰¤ Ci в€’ОІi О±t
в€љ
П‰i (О±t ) в‰Ў Prob[default by п¬Ѓrm i at T |О±t ] = Prob i,T 2
1в€’ОІi
(19.14)

which, unlike the expression for unconditional default probabilities, equa-
tion (19.7), depends importantly on ОІi .

8
Throughout this chapter we assume that Ci is a deterministic variable for all i.
19.1 Default Correlation 223

19.1.3 Pairwise Default Correlation and вЂњОІвЂќ
We have shown that the model-implied correlation between returns on any
two assets in the portfolio is:

Cor(Ri,t , Rj,t ) = Cov(Ri,t , Rj,t ) = ОІi ОІj (19.15)

How does that relate to the concept of pairwise default correlation dis-
cussed in Section 19.1.1? There we saw that we can write the default
correlation between any two entities i and j in the portfolio as

П‰i&j в€’ П‰i П‰j
ПЃi,j = (19.16)
П‰i (1 в€’ П‰i ) П‰j (1 в€’ П‰j )

where we have already seen that the model implies that the probabilities
П‰i and П‰j can be written as N (Ci ) and N (Cj ), respectively. Now, П‰i&j
denotes the probability that both i and j will default over the time horizon
of interest. In the context of the model,

П‰i&j в‰Ў Prob[Ri в‰¤ Ci and Rj в‰¤ Cj ] (19.17)

but, given that Ri and Rj are individually normally distributed with corre-
lation coeп¬ѓcient ОІi ОІj , they are jointly distributed according to the bivariate
normal distribution. Thus,

П‰i&j = N2 (Ci , Cj , ОІi ОІj ) (19.18)

where N2 () is the cumulative distribution function of the bivariate normal
distribution.9
We can now write out the default correlation between any two entities
in the portfolio entirely in terms of model parameters:

N2 (Ci , Cj , ОІi ОІj ) в€’ N (Ci )N (Cj )
ПЃi,j = (19.19)
N (Ci )(1 в€’ N (Ci )) N (Cj )(1 в€’ N (Cj ))

Equation (19.19) shows the explicit link between the return correlation
parameters, ОІi and ОІj , and the degree of default correlation between ref-
erence entities i and j. Figure 19.3 illustrates the nature of this link for a
particular parameterization of default probabilitiesвЂ”П‰i = П‰j = .05вЂ”and
return correlationsвЂ”ОІi = ОІj = ОІ, and ОІ varies from 0 to 1. The п¬Ѓgure
shows that default correlation increases monotonically with return correla-
tion, but the relationship is very nonlinear. Given this one-to-one mapping

9
See Appendix B.
224 19. The Basics of Portfolio Credit Risk

FIGURE 19.3. Return Correlation (ОІ) and Pairwise Default Correlation

between ОІ and ПЃi,j , we shall couch the discussion of default correlation in
this part of the book mostly in terms of ОІ. We do this simply for analytical
convenience.

19.2 The Loss Distribution Function
We п¬Ѓrst met the concept of the loss distribution function in this book
in Chapter 10, where we discussed portfolio default swaps. To recap, an
informal deп¬Ѓnition of the loss distribution function would say that, for a
given portfolio, it is the function that assigns probabilities to default-related
losses of various magnitudes over a given time horizon.
We will continue to build on the simple modeling framework introduced
in the previous section to take a closer look at the loss distribution function
and its relation to default correlation. For convenience, however, we will
make a few additional simplifying assumptions. First, we assume that the
portfolio is composed of a set of homogeneous debt instruments, i.e., for
all i,

ОІi = ОІ

Ci = C
19.2 The Loss Distribution Function 225

Second, we assume that each entity represented in the portfolio corresponds
to an equal share of the portfolio. Henceforth, we will refer to such a
portfolio as an equally weighted homogeneous portfolio.
The п¬Ѓrst assumption ensures that all entities represented in the port-
folio have the same default probability over the time period of interest;
the second guarantees that there is a one-to-one correspondence between
the number of defaults in the portfolio and the size of the percentage
default-related loss in the portfolio. In particular, in a portfolio with, say,
I reference entities, the probability of k defaults among the entities in the
portfolio is equivalent to the probability of a 100 k percent default-related
I
loss in the portfolio.10

19.2.1 Conditional Loss Distribution Function
Armed with the tools developed thus far we can compute the probability
distribution of default-related losses for a given value of О±, which corre-
sponds to the concept of the conditional loss distribution function. Given
that, conditional on О±, individual returns are independently and normally
distributed, a basic result from statistics says that the number of defaults
in the portfolio is binomially distributed for a given value of О±.11 Thus, if
we let L denote the percentage (default-related) loss in the portfolio over,
say, the next year, we can write the conditional probability of a given
loss as

k I!
|О± в‰Ў Prob [k defaults|О±] = П‰(О±)k [1 в€’ П‰(О±)]Iв€’k
Prob L =
k!(I в€’ k)!
I
(19.20)

where

C в€’ ОІО± C в€’ ОІО±
П‰(О±) в‰Ў Prob в‰¤ =N (19.21)
i,t
1 в€’ ОІ2 1 в€’ ОІ2

is the conditional probability of default of each reference entity represented
in the equally weighted homogeneous portfolio.
Equation (19.20) is essentially the expression for the conditional loss
distribution function implied by (19.9), applied to the case of an equally-
weighted homogeneous portfolio. To highlight its dependence on the
common factor, О±, Figure 19.4 shows conditional loss distribution func-
tions corresponding to a period of, say, one year for a portfolio with 20

10
For added convenience, we are assuming a zero recovery rate for all entities in the
portfolio.
11
See Appendix B for an overview of the binomial distribution.
226 19. The Basics of Portfolio Credit Risk

FIGURE 19.4. Conditional Loss Distribution Functions for an Equally Weighted
Homogeneous Portfolio

reference entities, each with an individual default probability of 5 percent
and a ОІ of 0.5. The п¬Ѓgure depicts three cases, one with О± set at its average
value of zero, and the others corresponding to О± set at plus and minus one
standard deviation, в€’1 and 1, respectively.12
As can be seen in Figure 19.4, the conditional loss distribution func-
tion п¬‚attens out as О± declines. This is consistent with the intuition that,
with positive dependence of returns on О±, conditional default probabilities
increase for all entities in the portfolio as О± decreases. Lower values of О±
in this case pull individual returns closer to their default thresholds, C,
increasing the likelihood of larger losses in the portfolio and thus allowing
for вЂњfatter tailsвЂќ in the conditional loss distribution function.

19.2.2 Unconditional Loss Distribution Function
When pricing multi-name credit derivatives such as basket swaps and port-
folio default swaps it is often the unconditional loss distribution function
that will matter most. As its name suggests, this concept is not predicated
on any one particular value of the common factor, О±, and thus it fully

12
Figure 19.4 illustrates an additional use of the modeling framework examined in
this chapter, which is to assess likely losses in the portfolio under diп¬Ђerent scenarios
involving the common factor, a practice commonly called scenario stress testing.
19.2 The Loss Distribution Function 227

takes on board the reality that future values of О± are themselves generally
subject to substantial uncertainty.
In what follows we shall use the model to derive an analytical solution
for the unconditional loss distribution function of a homogeneous portfolio.
The technical requirements for this derivation are perhaps a bit beyond
the scope of this book, but we should note that readers less interested
in statistical and mathematical details may skip this subsection entirely
without any fear of missing out on what is to come in the remainder of
Part IV. Indeed, the main reason for obtaining an expression for the loss
distribution hereвЂ”equation (19.24) belowвЂ”is so we can use it later on to
check the accuracy of the large-portfolio approximation and simulation-
based methods described in sections 19.2.3 and 19.4, respectively. It is the
latter method, in particular, that will be our tool of choice throughout most
of this part of the book, not just for deriving loss distribution functions,
but also for valuing multi-name credit derivatives.
To derive an expression for the unconditional loss distribution, we appeal
to a basic result from statistics, known as the law of iterated expectations.
According to this вЂњlaw,вЂќ the unconditional expectation of having, say, k
defaults in the portfolio is given by the probability-weighted average of
conditional probabilities of having k defaults. These latter probabilities are
computed over all possible values of the common factor О±, and the weights
are given by the probability density function of О±. Thus,

в€ћ
Prob [k defaults |О± = y] n(y)dy
Prob [k defaults ] = (19.22)
в€’в€ћ

where n(.) is the probability density function of the standard normal
distribution.13
Substituting equations (19.20) and (19.21) into (19.22) we obtain the
following expression for the unconditional probability of k defaults in the
portfolio:

в€ћ
I!
П‰(y)k (1 в€’ П‰(y))Iв€’k n(y)dy
Prob [k defaults ] = (19.23)
k!(I в€’ k)!
в€’в€ћ

в€љ
where П‰(y) в‰Ў N Cв€’ОІy
, as we saw earlier.
2
1в€’ОІ

13
See Appendix B for a quick review of key results from statistics.
228 19. The Basics of Portfolio Credit Risk

Given equation (19.23), the analytical solution for the loss distribution
function of a homogeneous portfolio is:

в€ћ
K
I!
Prob [l в‰¤ K] = П‰(y)k (1 в€’ П‰(y))Iв€’k n(y)dy (19.24)
k!(I в€’ k)!
в€’в€ћ
k=0

where Prob[l в‰¤ K] denotes the probability that the number of defaults
in the portfolio will be equal to or less than K. Equation (19.24) is a
somewhat cumbersome expression that can be solved, for instance, via
numerical integration, provided I is not too large.

19.2.3 Large-Portfolio Approximation
As we saw in the previous section, allowing for both the common and the
п¬Ѓrm-speciп¬Ѓc factors to be fully stochastic makes the mathematical analysis
of credit risk in portfolios substantially more involving than in the case of
п¬Ѓxed values for the common factor, especially for portfolios with a large
number of assets. One technique that greatly simpliп¬Ѓes the analysis of
relatively homogeneous portfolios with many assets is the so-called large-
portfolio approximation methodвЂ”see, for instance, Vasicek (1987).14
As the name suggests, the main thrust of this approach is to assume that
the portfolio has a suп¬ѓciently large number of reference entities so that
the expected fraction of entities defaulting over a given time horizon can
be approximated by the corresponding individual default probabilities of
the entities.15 In terms of the conditional default probabilities derived in
Section 19.2.1, this implies

Оё в‰Ў E[L|О±] в‰€ П‰(О±)
Лњ (19.25)

where Оё and L are the expected and actual percentage loss in the portfolio,
respectively.
Note that for any given value of Оё, one can back out the implied value
of О± upon which the conditional expectation in (19.25) is based:

О± = П‰ в€’1 (Оё) (19.26)

where П‰ в€’1 (.) is the inverse function of П‰(О±).16

14
Our presentation of the large-portfolio approximation method partly follows OвЂ™Kane
and Schlogl (2001).
15
Readers with some familiarity with statistics will recognize the (conditional) law of
large numbers at work here.
16
We will ignore the approximation error embedded in (19.25) from now on. Obviously,
that error can be signiп¬Ѓcant, especially for small portfolios, which, as we will see in
Section 19.4, can be examined with alternative methods.
19.2 The Loss Distribution Function 229

Suppose now that the actual value of О± turns out to be larger than the
one used in (19.25). Other things being equal, the actual percentage loss, L,
will be smaller than the expected loss, Оё, because, for positive ОІ, individual
returns will be farther away from the default boundary C than implicit in
(19.25). Mathematically, this can be summed up as:

О± в‰Ґ П‰ в€’1 (Оё) <=> L в‰¤ Оё (19.27)

Thus, we can make the following probabilistic statement:

Prob[О± в‰Ґ П‰ в€’1 (Оё)] = Prob[L в‰¤ Оё] (19.28)

Given that О± was assumed to be normally distributed, and relying on
the symmetric nature of the normal probability density function,17

Prob[О± в‰Ґ П‰ в€’1 (Оё)] = Prob[О± в‰¤ в€’П‰ в€’1 (Оё)] = N (в€’П‰ в€’1 (Оё))

Thus we arrive at the result:

Prob[L в‰¤ Оё] = N (в€’П‰ в€’1 (Оё)) (19.29)

which is an approximate expression for the unconditional loss distribution
of the large homogeneous portfolio.
To write out (19.29) explicitly in terms of the parameters of the
model, note that, ignoring any errors introduced by the large portfolio
approximation, equations (19.21) and (19.25) imply

C в€’ ОІО±
Оё=N (19.30)
1 в€’ ОІ2

and thus we can write
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