ńņš. 7 |

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 modelā”such as the reduced-form

and structural frameworks described in the current chapterā”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

Seā’(r+Ī»)(sā’t) ds = (1 ā’ eā’(r+Ī»)(T ā’t) )

Ī·(t) = (17.55)

r+Ī»

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 + Ī», reļ¬‚ecting the uncertainty surrounding the default

event. (Ī» 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 ā’ X) in the event of default before T , where X is the

recovery rate of the reference entityā™s defaulted liabilities. The value of such

a claim is given by equation (17.49) with X replaced with 1 ā’ X. Letting

Īø(t) denote the value of the protection leg, we can write

Ī»(1 ā’ X)

(1 ā’ eā’(r+Ī»)(T ā’t) )

Īø(t) = (17.56)

r+Ī»

A default swap typically has zero market value when it is set up, and

thus pricing such a contract is equivalent to ļ¬nding the value of S that

makes the two legs of the swap have equal value. This is given by

S = Ī»(1 ā’ X) (17.57)

which gives us the result that, for a given recovery rate and constant r and

Ī», 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 approachesā”structural

and reduced-formā”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 ļ¬rmā”as these are

reļ¬‚ected in the ļ¬rmā™s balance sheetā”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 ļ¬nancial 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 oneā™s interest is,

for example, in assessing how a change in the capital structure of a given

ļ¬rm may aļ¬ect its ļ¬nancing 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 ) = eā’

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, Ī», 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 Ī» is assumed to be stochastic, many

of the basic models for describing its evolution mirror common speciļ¬ca-

tions developed for interest rate models, such as the well-known yield curve

model of Cox, Ingersoll, and Ross (1985)[15].

While ļ¬xed-income modelers might ļ¬nd 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 ļ¬rm to its economic and

ļ¬nancial condition, traditional forms of such models tend not to ļ¬t 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 ļ¬rm 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 Duļ¬e and Lando (2001)[23] is one of the better-known works

in this strand of the literature.23

The Duļ¬e-Lando model can be thought of as a hybrid structural/

intensity-based model because it is essentially a ļ¬rst-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 Duļ¬e-Lando model can be

calculated in terms of observable variables related to the balance-sheet

fundamentals of the ļ¬rm. Moreover, Duļ¬e 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

ļ¬nancial condition of the ļ¬rm. For instance, investors may not know with

certainty just how far the value of the ļ¬rm is from its default boundary.

As a result, the possibility of a default in the very near term cannot be fully

discarded, and thus the pattern of short-term credit spreads generated by

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 ļ¬rm to be only in one of two statesā”default

and survivalā”ratings-based models allow for a variety of states, where each

non-default state might correspond, for instance, to a given credit ratingā”

such as AAA, A, BB+ , etc.ā”assigned to the ļ¬rm 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)[32] and Giesecke and Goldberg

(2004)[33].

24

Within the structural framework, Zhouā™s (1997)[76] jump-diļ¬usion model, dis-

cussed in Section 17.2, constitutes an alternative approach to address the empirical

implausibility of the short-term credit spreads implied by traditional structural models.

17.4 Ratings-based Models 201

of Jarrow, Lando, and Turnbull (1997)[45]. 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 ā’ 1 credit ratings,

with 1 representing the highest credit quality and J ā’ 1 representing the

rating just prior to default. We will also allow for a Jth ārating,ā which

will correspond to default. Let Ļi,j represent the actual (not necessarily

risk-neutral) probability, based on all information available at time t, of

the ļ¬rm 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 deļ¬ne the J Ć— J transition matrix ā„¦ such that its (i, j)th element

is Ļi,j :

ļ£« ļ£¶

Ļ1,1 Ļ1,2 ... Ļ1,J

ļ£¬ Ļ2,1 Ļ2,J ļ£·

Ļ2,2 ...

ļ£¬ ļ£·

ļ£¬ ļ£·

ā„¦ = ļ£¬ ... ... ļ£·

... ... (17.58)

ļ£¬ ļ£·

ļ£ĻJā’1,1 . . . ĻJā’1,J ļ£ø

ĻJā’1,2

0 0 ... 1

We further assume that:

Ļi,j ā„ 0 for all i, j, i = j (17.59)

J

Ļi,i ā” 1 ā’ Ļi,j for all i (17.60)

j = 1, j = i

Equation (17.60) essentially acknowledges the fact that for any given ļ¬rm

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 ļ¬rm 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 ā’ 1 ratings.

In technical terms, the last row of ā„¦ says that defaultā”the Jth

āratingāā”is an āabsorbing state,ā meaning that once the ļ¬rm enters into

a state of default, we assume that it will stay there with probability

ĻJ,J = 1. In corporate ļ¬nance terms, the model assumes that there is

no reorganization after default.

Let Ļi,j (t, t + n) be the probability, conditional on information available

at time t, that the ļ¬rmā™s rating will change from i at time t to j at time

t+n. If ā„¦(t, t+n) is the matrix such that its (i, j)th element is Ļi,j (t, t+n),

202 17. Main Credit Modeling Approaches

it can be shown that:

ā„¦(t, t + n) = ā„¦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 ā„¦ are published regularly by some of the major

credit-rating agencies based on actual rating changes in the universe of ļ¬rms

covered by these agencies. For pricing purposes, of course, what matter

are the risk-neutral transition probabilities Ļi,j , rather than the empirical

Ė

probabilities in ā„¦. JLT propose the following mapping between empirical

and risk-neutral probabilities:

Ļi,j (t, t + 1) = Ļi (t)Ļi,j for all i, j, i = j

Ė (17.62)

for Ļi (t) ā„ 0 for all i and t. Consistent with the discussion in Chapter 15,

Ļi (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 Ļi,j (t, t+1) > 0 if and only if Ļi,j > 0,

Ė

for 0 ā¤ t ā¤ U ā’ 1.

Ė

Thus, we can also deļ¬ne the risk-neutral transition matrix ā„¦(t, t + n),

and it should by now be clear that its (i, j)th entry is Ļi,j (t, t + n), which

Ė

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 ā„¦ and Ļi are time invariant:

Ė Ė

ā„¦(t, t + n) = ā„¦(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 ļ¬rm will survive through

some future date T . Given all the assumptions discussed thus far, this

probability is simply

Qi (t, T ) = 1 ā’ Ļi,J (t, T )

Ė (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 ļ¬rm, 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 ā’ Ļi,J (t, T )]

d

Ė (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 ļ¬rm 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 ļ¬rm. In particular,

ā’ log(Z0 (t, T ; i)) + log(Z(t, T ))

d

R (t, T ; i) ā’ R(t, T ) =

d

T ā’t

ā’ log(1 ā’ Ļi,J (t, T ))

Ė

= (17.66)

T ā’t

where we continue to assume that R(t, T ) ā” ā’ log(Z(t,T )) is the yield to

T ā’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 diļ¬erent ratings. For instance, the model-implied

yield spread between two bonds rated i and j is

1 ā’ Ļj,J (t, T )

Ė 1

Rd (t, T ; i) ā’ Rd (t, T ; j) = log (17.67)

1 ā’ Ļi,J (t, T ) T ā’ t

Ė

Equations (17.66) and (17.67) can be used to identify potentially prof-

itable opportunities across diļ¬erent 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 ļ¬nancial instruments

and contracts with ratings-dependent payoļ¬s, 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 paperā”Jarrow, Lando, and Turnbull

(1997)[45]ā”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 speciļ¬cation of the JLT

model. In particular, in the simplest case of time-invariant risk-neutral

transition probabilities, the J Ć—J transition matrix for the continuous-time

204 17. Main Credit Modeling Approaches

version of the model can be written as:

ā„¦(t, T ) = eĪ(T ā’t) (17.68)

where the J Ć— J matrix Ī is typically called the generator matrix. The ith

diagonal element of Ī, Ī»i,i , can be thought of as the exit rate from the

ith rating, and, for i = j, the (i, j)th element of Ī, Ī»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)[46] 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)[50], Lando and Skodeberg (2002)[52],

Das and Tufano (1996)[19], and Arvanitis, Gregory, and Laurent (1999)[3].

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 brieļ¬‚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 brieļ¬‚y mentioned the forward-starting total

return swap, which is a contractual commitment to enter into a total return

swap at a ļ¬xed 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

ļ¬‚oaters. The aims of such contracts are self-evident; they are agreements

to enter into a credit default swap and to buy and sell ļ¬‚oaters, respectively,

at future dates and at predetermined premiums (in the case of a CDS) and

spreads (the forward ļ¬‚oater contract).

206 18. Valuing Credit Options

As we shall see below, forward-starting credit default swaps can be

thought of as the underlying āassetā 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 ļ¬‚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,Tjā’1 )ā’Q(t,Tj )](1ā’X)ā’Q(t,Tj )Ī“j K} (18.1)

j =1

where Z(t, Tj ) corresponds to the proxy for a riskfree discount factorā”

which, as discussed in Chapter 16, tends to be derived in practice from

the LIBOR/swap curve to reļ¬‚ect the funding costs of the large banks that

tend to be most active in the CDS marketā”and X and Q(t, Tj ) relate to,

respectively, the recovery rate of the reference entity (0 ā¤ X < 1) and the

risk-neutral probability that the reference entity will survive through Tj ,

conditional on all information available at time t. Ī“j is the accrual factor

for the jth premium payment (the number of days between the (j ā’ 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, Tjā’1 ) ā’ Q(t, Tj )](1 ā’ X)

j=1

F

S (t, T, Tn ) = (18.2)

n

Z(t, Tj )Q(t, Tj )Ī“j

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 )Ī“j [S F (t, T, Tn ) ā’ 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 diļ¬erence between the corresponding

forward CDS premium and the predetermined premium written into the

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 ļ¬‚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 ļ¬‚oating-rate note at a future date with a

prespeciļ¬ed spread over LIBOR. Assume, for simplicity, that the ļ¬‚oater

has a zero recovery rate. (This is a forward contract to sell a given ļ¬‚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 ļ¬‚oater with a face value of $1 and coupon payment dates T1 , T2 , . . . , Tn

can be written as

n

Z0 (T, Tj )Ī“j [F ā— (T, Tjā’1 , Tj ) + s(T, Tn )] + Z0 (T, Tn )

d d

1= (18.4)

j=1

where F ā— (T, Tjā’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 Tjā’1

to Tj ; s(T, Tn ) is the par ļ¬‚oater spread, and, to simplify the notation,

Z0 (T, Tj ) ā” Z(T, Tj )Q(t, Tj ).

d

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

n

Z(T, Tj )Ī“j F ā— (T, Tjā’1 , Tj ) + Z(T, Tn )

1= (18.5)

j=1

which diļ¬ers 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 ļ¬‚oater, for t < T , can be shown to be

n

Z(t, Tj )Ī“j F ā— (t, Tjā’1 , Tj ) + Z(t, Tn )

Z(t, T ) =

j=1

which can be veriļ¬ed given the deļ¬nition of forward LIBOR (see

Chapter 4):

Z(t, Tjā’1 )

ā’1

F ā— (t, Tjā’1 , Tj ) ā” Ī“j ā’1

Z(t, Tj )

As for the riskier par ļ¬‚oater, its time-t value becomes

n

Z d (t, Tj )[F ā— (t, Tjā’1 , Tj ) + sF (t, T, Tn )]Ī“j + Z0 (t, Tn )

d d

Z0 (t, T ) =

j=1

where sF (t, T, Tn ) is deļ¬ned as the forward par ļ¬‚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 ļ¬‚oater that will pay a spread of say K, which is not necessarily the

par spread, is relatively straightforward, and the reader can easily verify

that, from the perspective of the party committed to selling the ļ¬‚oater, the

time-t value of such a contract can be written as

n

Z d (t, Tj )Ī“j [sF (t, T, Tn ) ā’ K] + Z d (t, T )

FL

W (t) = (18.6)

j=1

which, again, has the simple intuition that a contract to sell a ļ¬‚oater at a

future date for parā”in essence, a contract to pay a given spread starting

at some future dateā”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 ļ¬‚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 )Ī“j [S F (t, T, Tn ) ā’ 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:

ļ£± ļ£¼

ļ£²n ļ£½

Z0 (T, Tj )Ī“j [S(T, Tn ) ā’ K], 0

d

V (T ) = Max(W (T ), 0) = Max

ļ£³ ļ£¾

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

oļ¬ 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 )Ī“j in equation (18.7) as an

annuity factor that gives the time-T value of the entire stream of diļ¬erences

between the premium payments in a par CDS contract and the one speciļ¬ed

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 ) ā’ K), 0] (18.8)

To ļ¬nd 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 ) ā’ 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 diļ¬erence between the premium payments of the

two credit default swaps.

If we now recall that the time-t value of any ļ¬nancial asset is simply the

risk-adjusted expected present value of its cash ļ¬‚ow, we can write

ā—

V (t) = A(t, T )Et [Max(S F (T, T, Tn ) ā’ K, 0)] (18.10)

210 18. Valuing Credit Options

ā—

where Et [.] denotes the expected value of ā.ā 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)[8].3 If we let Ļ(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 ) ā’ KN (d2 )] (18.11)

with

ā

log(S F (t, T, Tn )/K)

d1 ā” + .5Ļ(T, Tn ) T ā’ t

Ļ(T, Tn )2 (T ā’ t)

ā

log(S F (t, T, Tn )/K)

d2 ā” ā’ .5Ļ(T, Tn ) T ā’ t

Ļ(T, Tn )2 (T ā’ t)

N (.) is the cumulative standard normal distribution, and we made the

simplifying assumption that Ļ(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 ļ¬‚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 ļ¬nance methods may recog-

ā—

nize the probability measure embedded in Et [.] as that corresponding to the so-called

āannuity measureā (Hunt and Kennedy, 2000[43]). 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)[41] for a textbook discussion of the Black formula.

18.5 Valuing Bond Options 211

Consider, for instance, a put option on a ļ¬‚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 ļ¬‚oater spread.

At time T , the value of the put option will be

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

i.e., the option holder will choose to sell the ļ¬‚oater for its face value only

if the market value of the ļ¬‚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 ) ā’ 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

credit spreads and, thus, CDS premiums are stochastic variables. Indeed,

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

stochastic forward CDS premiums is tantamount to admitting that default

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)[2] and Schonbucher (1999)[71], 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 ļ¬‚oating-rate bonds

and on credit default swaps. Similar methods apply to the valuation of

options written on ļ¬xed-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)[41] provides a useful

textbook discussion of the valuation of ļ¬xed-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 ļ¬rm will

default on its ļ¬nancial 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 ļ¬rms, 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 ļ¬rms 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)[57]. In addition, it seems plausible that default cor-

relation tends to be higher among ļ¬rms in the same industry than among

ļ¬rms in diļ¬erent lines of business altogether, although one can imagine

situations where a default by one ļ¬rm 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 coeļ¬cient, ĻY1 ,Y2 ,

involving two random variables Y1 and Y2 is deļ¬ned as

Cov[Y1 , Y2 ]

ĻY1 ,Y2 ā” (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 deļ¬ned below.

Mathematically, one can write an expression for the coeļ¬cient of default

correlation for any two entitiesā”called the pairwise default correlationā”

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 ĻA and ĻX 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 ĻA&X . These probabilities

are represented in the diagram in Figure 19.1. The area encompassed by the

rectangle in the ļ¬gure 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

ĻA&X that both AZZ Bank and XYZ Corp. will default.

1

In terms of the notation used in Part III, ĻA and ĻX can be written as 1 ā’ QA (t, T )

and 1 ā’ 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 deļ¬ne 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 deļ¬ned 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 ] = ĻA Ć— 1 + (1 ā’ ĻA ) Ć— 0 = ĻA

Ė (19.2)

Ė

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 ] ā” E (IA ā’ E[IA ])2 = ĻA (1 ā’ ĻA )

Ė Ė (19.3)

and the covariance can be written as

Cov[IA , IX ] ā” E (IA ā’ E[IA ])(IX ā’ E[IX ]) = E[IA IX ] ā’ E[IA ]E[IX ]

Ė Ė Ė Ė Ė Ė

But notice that IA IX is only nonzero when both IA and IX are 1, and that

happens with probability ĻA&X . Thus

Ė

E[IA IX ] = ĻA&X

218 19. The Basics of Portfolio Credit Risk

and we can write

Cov[IA , IX ] = ĻA&X ā’ ĻA ĻX (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:

ĻA&X ā’ ĻA ĻX

Cov[IA , IX ]

ĻA,X ā” = (19.5)

ĻA (1 ā’ ĻA ) ĻX (1 ā’ ĻX )

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

horizonā”the region of overlap between areas A and X in Figure 19.1ā”the

higher their pairwise default correlation coeļ¬cient over that horizon.2

It can be shown that (19.5) is such that the pairwise default correla-

tion coeļ¬cient lies between 1 (perfect positive correlation) and ā’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, ĻA,X = 1, corresponds to

a scenario where the two entities have identical default probabilities,

ĻA = ĻX = Ļ , and where ĻA&X = Ļ . The case of perfect negative corre-

ĀÆ ĀÆ

lation corresponds to a scenario where ĻA&X is zero, its lower bound, and

ĻX = 1 ā’ ĻA , 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 ĻA&X = ĻA ĻX .

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 deļ¬ne 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)[55]

and Embrechts, McNeil, and Strautman (1999)[26].

3

One way to verify that 1 is the maximum value that ĻA&X can achieve is to convince

yourself that ĻA&X cannot be larger than Min[ĻA , ĻX ]. 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 Xā”or a default

correlation of nearly one according to the results discussed aboveā”but

where area A is larger than area Xā”AZZ bank is of an inferior credit

quality (greater default probability) than XYZ Corp. In the case depicted

in the ļ¬gure, 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 diļ¬culties

and challenges associated with modeling portfolio credit risk.4

The model builds on the seminal work of Black and Scholes (1973)[9]

and Merton (1974)[59], which we discussed in Part III. In particular, for a

hypothetical ļ¬rm i, we deļ¬ne 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)[37], Oā™Kane and Schlogl (2001)[64], Crosbie (2002)[16], and

many others.

220 19. The Basics of Portfolio Credit Risk

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

Default by ļ¬rm i at time T <=> Ri ā¤ Ci (19.6)

and the default probability associated with this ļ¬rm 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 ļ¬rms are included in the portfolio, we shall assume that the above

returns have already been standardized, i.e., for each ļ¬rm 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 ā’ Āµi

ĀÆ

Ri,t ā”

Ļi

ĀÆ

where Ri,t are the raw (ānon-standardizedā) returns associated with owning

ĀÆ

ļ¬rm i at time t, and Āµi and Ļi are the mean and standard deviation of Ri,t ,

respectively. Note that, by working with standardized returns, all the ļ¬rms

in the portfolio have the same expected returns and standard deviations of

returnsā”0 and 1, respectively.

Let us now consider a portfolio that contains debt instruments issued by

I diļ¬erent ļ¬rms, where default events for each ļ¬rm are deļ¬ned 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 ļ¬rms are correlated through dependence on one

common factor. A straightforward corollary of the ļ¬rst assumption is that,

for a given time horizon, the probability of default by any one single ļ¬rm

is given by

Ļi ā” Prob[default by ļ¬rm i] = Prob[Ri ā¤ 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 deļ¬ne it in discrete time as

Ri,t ā” log(Ai,t ) ā’ log(Ai,tā’1 )

where Ai,t is the value of ļ¬rm 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 ļ¬rm i.

Readers of earlier chapters will notice that we have shifted from an asset-value-based

deļ¬nition of defaultā”recall, for instance, that the Black-Scholes-Merton framework dis-

cussed in Chapter 17 modeled defaults as situations where the value of the ļ¬rm fell below

some critical levelā”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 ļ¬rm 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)ā”

and, indeed, throughout this part of the bookā”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 Ļi and then rely on (19.7) to back out a market-implied value

for Ci :

Ci = N ā’1 (Ļi ) (19.8)

where N ā’1 (.) is the inverse of N (Ā·), or the function that determines the

value of Ci in (19.7) that corresponds to a given value of Ļi .

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 ā’ Ī²i

2

Ri,t = Ī²i Ī±t + (19.9)

i,t

for all ļ¬rms 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 Ī±, which aļ¬ects all entities represented in the

portfolio, and on an entity-speciļ¬c factor i . For instance, the entities repre-

sented in the portfolio may all be sensitive to conditions in the overall stock

market, or, alternatively, Ī±t may stand for the current state of the economy.

To be consistent with the deļ¬nition of Ri,t as a random variable with

zero mean and unit variance, we further assume that Ī±t and, for all i, i,t

have also been standardizedā”they have zero mean and unit varianceā”and

that they are independently and normally distributed.6 We also assume

that, for any two ļ¬rms i and j, j,t and i,t are independently distributed

and that Ī±t and i,t , and thus Ri,t are serially uncorrelated, i.e.,

Ė Ė

E[Ī±t Ī±tā’s ] = E[ i,t i,tā’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

Ė

unit variance of returns. E[Ri,t ] = 0 follows directly from the zero-mean assumption on

Ī± and i . To see that the variance of Ri,t remains unit, simply note that we can write

it as

Var[Ri,t ] = Ī²i E[Ī±2 ] + (1 ā’ Ī²i )E[ 1 ā’ Ī²i E[Ī±t

2Ė 2Ė 2Ė

2

i,t ] + 2Ī²i i,t ]

t

which is equal to one given the variance and independence assumptions regarding Ī±t

and i,t .

7

The concept of serial independence is discussed brieļ¬‚y in Appendix B.

222 19. The Basics of Portfolio Credit Risk

In light of equation (19.9), we can show that Ī²i 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 ) ā” E[(Ri,t ā’ E[Ri,t ])(Rj,t ā’ E[Rj,t ])] = Ī²i Ī²j

Ė Ė Ė (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, Ī±, returns involving any two

ļ¬rms 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 |Ī±t ] = Ī²i Ī±t

Ė (19.11)

Var[Ri,t |Ī±t ] = 1 ā’ Ī²i

2

(19.12)

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

conditionally uncorrelated:

Cov[Ri,t , Rj,t |Ī±t ] = 1 ā’ Ī²i 1 ā’ Ī²j E[

2Ė

2

i,t j,t ] =0 (19.13)

where E[Ri,t |Ī±t ] is the expected value of Ri,t conditioned on the time-t

Ė

value of Ī±, and Var[.|Ī±t ] and Cov[.|Ī±t ] are analogously deļ¬ned. Note that

(19.13) follows from the fact that i,t and j,t are mutually independently

Ė

distributed, i.e., E[ i,t j,t ] = 0.

We can also compute conditional default probabilities over given hori-

zons for any ļ¬rm in the portfolio. For instance, for Ī±T = Ī±t , the time-t

probability that the ļ¬rm will default at time T can be written as

ā¤ Ci ā’Ī²i Ī±t

ā

Ļi (Ī±t ) ā” Prob[default by ļ¬rm i at T |Ī±t ] = Prob i,T 2

1ā’Ī²i

(19.14)

which, unlike the expression for unconditional default probabilities, equa-

tion (19.7), depends importantly on Ī²i .

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 āĪ²ā

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 ) = Ī²i Ī²j (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

Ļi&j ā’ Ļi Ļj

Ļi,j = (19.16)

Ļi (1 ā’ Ļi ) Ļj (1 ā’ Ļj )

where we have already seen that the model implies that the probabilities

Ļi and Ļj can be written as N (Ci ) and N (Cj ), respectively. Now, Ļi&j

denotes the probability that both i and j will default over the time horizon

of interest. In the context of the model,

Ļi&j ā” Prob[Ri ā¤ Ci and Rj ā¤ Cj ] (19.17)

but, given that Ri and Rj are individually normally distributed with corre-

lation coeļ¬cient Ī²i Ī²j , they are jointly distributed according to the bivariate

normal distribution. Thus,

Ļi&j = N2 (Ci , Cj , Ī²i Ī²j ) (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 , Ī²i Ī²j ) ā’ N (Ci )N (Cj )

Ļi,j = (19.19)

N (Ci )(1 ā’ N (Ci )) N (Cj )(1 ā’ N (Cj ))

Equation (19.19) shows the explicit link between the return correlation

parameters, Ī²i and Ī²j , 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 probabilitiesā”Ļi = Ļj = .05ā”and

return correlationsā”Ī²i = Ī²j = Ī², and Ī² varies from 0 to 1. The ļ¬gure

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 (Ī²) and Pairwise Default Correlation

between Ī² and Ļi,j , we shall couch the discussion of default correlation in

this part of the book mostly in terms of Ī². We do this simply for analytical

convenience.

19.2 The Loss Distribution Function

We ļ¬rst met the concept of the loss distribution function in this book

in Chapter 10, where we discussed portfolio default swaps. To recap, an

informal deļ¬nition 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,

Ī²i = Ī²

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 ļ¬rst 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 Ī±, which corre-

sponds to the concept of the conditional loss distribution function. Given

that, conditional on Ī±, 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 Ī±.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!

|Ī± ā” Prob [k defaults|Ī±] = Ļ(Ī±)k [1 ā’ Ļ(Ī±)]Iā’k

Prob L =

k!(I ā’ k)!

I

(19.20)

where

C ā’ Ī²Ī± C ā’ Ī²Ī±

Ļ(Ī±) ā” Prob ā¤ =N (19.21)

i,t

1 ā’ Ī²2 1 ā’ Ī²2

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, Ī±, 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 Ī² of 0.5. The ļ¬gure depicts three cases, one with Ī± set at its average

value of zero, and the others corresponding to Ī± set at plus and minus one

standard deviation, ā’1 and 1, respectively.12

As can be seen in Figure 19.4, the conditional loss distribution func-

tion ļ¬‚attens out as Ī± declines. This is consistent with the intuition that,

with positive dependence of returns on Ī±, conditional default probabilities

increase for all entities in the portfolio as Ī± decreases. Lower values of Ī±

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 āfatter tailsā 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, Ī±, 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 diļ¬erent 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 Ī± 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 hereā”equation (19.24) belowā”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 ālaw,ā 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 Ī±, and the weights

are given by the probability density function of Ī±. Thus,

ā

Prob [k defaults |Ī± = y] n(y)dy

Prob [k defaults ] = (19.22)

ā’ā

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:

ā

I!

Ļ(y)k (1 ā’ Ļ(y))Iā’k n(y)dy

Prob [k defaults ] = (19.23)

k!(I ā’ k)!

ā’ā

ā

where Ļ(y) ā” N Cā’Ī²y

, as we saw earlier.

2

1ā’Ī²

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:

ā

K

I!

Prob [l ā¤ K] = Ļ(y)k (1 ā’ Ļ(y))Iā’k n(y)dy (19.24)

k!(I ā’ k)!

ā’ā

k=0

where Prob[l ā¤ 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

ļ¬rm-speciļ¬c factors to be fully stochastic makes the mathematical analysis

of credit risk in portfolios substantially more involving than in the case of

ļ¬xed values for the common factor, especially for portfolios with a large

number of assets. One technique that greatly simpliļ¬es the analysis of

relatively homogeneous portfolios with many assets is the so-called large-

portfolio approximation methodā”see, for instance, Vasicek (1987)[73].14

As the name suggests, the main thrust of this approach is to assume that

the portfolio has a suļ¬ciently 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

Īø ā” E[L|Ī±] ā Ļ(Ī±)

Ė (19.25)

where Īø and L are the expected and actual percentage loss in the portfolio,

respectively.

Note that for any given value of Īø, one can back out the implied value

of Ī± upon which the conditional expectation in (19.25) is based:

Ī± = Ļ ā’1 (Īø) (19.26)

where Ļ ā’1 (.) is the inverse function of Ļ(Ī±).16

14

Our presentation of the large-portfolio approximation method partly follows Oā™Kane

and Schlogl (2001)[64].

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 signiļ¬cant, 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 Ī± 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, Īø, because, for positive Ī², individual

returns will be farther away from the default boundary C than implicit in

(19.25). Mathematically, this can be summed up as:

Ī± ā„ Ļ ā’1 (Īø) <=> L ā¤ Īø (19.27)

Thus, we can make the following probabilistic statement:

Prob[Ī± ā„ Ļ ā’1 (Īø)] = Prob[L ā¤ Īø] (19.28)

Given that Ī± was assumed to be normally distributed, and relying on

the symmetric nature of the normal probability density function,17

Prob[Ī± ā„ Ļ ā’1 (Īø)] = Prob[Ī± ā¤ ā’Ļ ā’1 (Īø)] = N (ā’Ļ ā’1 (Īø))

Thus we arrive at the result:

Prob[L ā¤ Īø] = N (ā’Ļ ā’1 (Īø)) (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 ā’ Ī²Ī±

Īø=N (19.30)

1 ā’ Ī²2

and thus we can write

ńņš. 7 |