Credit Models and the Crisis

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The Journal of Credit Risk (3981) Volume 6/Number 4, Winter 2010/11

Credit models and the crisis:

default cluster dynamics and thegeneralized Poisson loss model

Damiano Brigo

Department of Mathematics, Kings College London, Strand,

London WC2R 2LS, UK; email: [email protected]

Andrea Pallavicini

Banca Leonardo, Via Broletto 46, Milan 20121, Italy;

email: [email protected]

RobertoTorresettiQuaestio Capital Management, Via del Lauro 14, 20100 Milan, Italy;

email: [email protected]

We consider collateralized debt obligations (CDOs), analyzing their valuation

(both pre-crisis and in-crisis) with the generalized Poisson loss model. General-

izedPoisson lossis an arbitrage-free dynamic lossmodel capable of calibratingall

tranches for all maturities simultaneously. Alternative tranche analysis using the

implied copula framework or using historical estimation techniques highlights a

multimodal tail in the loss distribution underlying the CDO. An effective descrip-

tion of loss dynamics in terms of the generalized Poisson loss model explains

how the multimodal loss-probability distribution can be considered to arise fromexplicitly modeling the default of subsets of names within the CDOs pool of

names. Such default clusters may in turn be interpreted as sectors of the economy.

Our discussion is supported by abundant market examples through history, both

pre-crisis and in-crisis.

1 INTRODUCTION: PRE-CRISIS AND IN-CRISIS CREDIT MODELING

1.1 Bottom-up models

A common way of introducing dependence in credit derivatives modeling is by means

of copula functions. Typically a Gaussian copula is postulated on the exponentialrandom variables driving the default times of the names of the pool. In general, if we

The opinions expressed here are solely those of the authors and do not represent in any way those

of their employers.

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40 D. Brigo et al

try to model dependence by specifying dependence across single default times, we are

using the so called bottom-up framework, and the copula approach typically falls

within this framework. Such a general procedure cannot be extended in a simple way

to a fully dynamical model. We cannot do justice to the huge amount of literature on

copulas in credit derivatives here. We only mention that there have been attempts to

go beyond the Gaussian copula introduced to the CDO world by Li (2000) that have

led to the implied (base and compound) correlation framework, some important limits

of which have been pointed out in Torresetti et al (2006b). Li and Hong Liang (2005)

also proposed a mixture approach in connection with CDO-squareds. For results on

sensitivities computed with the Gaussian copula models, see, for example, Meng and

Sengupta (2008).

An alternative to copulas in the bottom-up context is to insert dependence among

the default intensities of single names (see, for example, Chapovsky et al (2007)).

Joshi and Stacey (2006) resort to modeling business time to create a default corre-

lation in otherwise independent single-name defaults using an intensity gamma

framework. Similarly, but in a firm-value-inspired context, Baxter (2007) introduces

Lvy firm-value processes in a bottom-up framework for CDO calibration. Lopatin

(2008) introduces a bottom-up framework effective in the CDO context as well, first

using single-name default intensities as deterministic functions of time and of the pool

default counting process, then focusing on hedge ratios and analyzing the framework

from a numerical performance point of view, showing that this model is interest-

ing despite the fact that it lacks an explicit modeling of single-name credit-spread

volatilities.

Albanese et al (2006) introduce a bottom-up approach based on structural model

ideas that can be made consistent with several inputs under both the historical and

the pricing measures that manages to calibrate CDO tranches.

At this point, it is useful to recall the concept of the implied copula (introduced

by Hull and White (2006) as a perfect copula): this is a non-parametric model

that can be used to deduce the shape of the risk-neutral pool loss distribution from

a set of market CDO spreads spanning the entire capital structure. The general use

of flexible systemic factors has been generalized and vastly improved by Rosen and

Saunders (2009), who also discuss the dynamic implications of the systemic factor

framework. In the context of one-factor models, though with a more econometric

flavor, Berd et al (2007) interpret the latent factor as the market return governed

by an asymmetric generalized autoregressive conditional heteroscedasticity model

and introduce the implied correlation surface as a mapping between the given loss-

generating model and a Gaussian benchmark.Factors and dynamics are also discussed

in Inglis et al (2008).

Our calibration results based on the implied copula already seen in Torresetti et al

(2006c) and published later in Torresetti et al (2009) and Brigo et al (2010) point out

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Credit models and the crisis: default cluster dynamics and the GPL model 41

that a consistent loss distribution across tranches for a single maturity features modes

in the tail of the loss distribution. These probability masses on the far-right tail imply

default possibilities for large clusters (possibly sectors of the economy) of names of

the economy. These results were originally published in 2006 on ssrn.com (see Brigo

etal (2006a,b)). We find the same features here using a completely different approach.

The implied copula can calibrate consistently across the capital structure but not

across maturities since it is a model that is inherently static. The next step is therefore

to introduce the dynamic loss model. This moves us into the so called top-down

framework (although dynamic approaches are also possible in the bottom-up context,

as we have seen in some of the above references).

1.2 Top-down framework

We could completely avoid single-name default modeling and instead focus on thepool loss and default counting processes, thereby considering a dynamical model at

the aggregate loss level, associated to the loss itself or to some suitably defined loss

rates. This is the top-down approach (see, for example, Bennani (2005); Giesecke

et al (2009); Sidenius et al (2008); Schnbucher (2005); Di Graziano and Rogers

(2005); Brigo et al (2006a,b); Errais et al (2009); Lopatin and Misirpashaev (2007)).

The first joint calibration results of a dynamic loss model across indexes, tranche

attachments and maturities (Brigo et al (2006a)) show that even a relatively simple

loss dynamics such as a capped generalized Poisson process suffices to account for

the loss-distribution dynamical features embedded in market quotes. This work also

confirms the implied copula findings of Torresetti et al (2006c), showing that the loss-

distribution tail exhibits a structured multimodal behavior implying non-negligibledefault probabilities for large fractions of the pool of credit references, which leaves

the potential for high losses implied by CDO quotes before the beginning of the

crisis. Cont and Minca (2008) use a non-parametric algorithm for the calibration

of top models, constructing a risk-neutral default intensity process for the portfolio

underlying the CDO to look for the risk-neutral loss process closest to a prior loss

process using relative entropy techniques (see also Cont and Savescu (2008)).

However, in general, to justify the down in top-down one needs to show that,

from the aggregate loss model, one can recover a posteriori consistency with single-

name default processes when they are not modeled explicitly. Errais et al (2009)

advocate the use of random thinning techniques for their approach (see also Halperin

and Tomecek (2008), who delve into more practical issues related to random thinning

of general loss models, and Bielecki et al (2008), who also build semi-static hedging

examples and consider cases where the portfolio loss process may not yield sufficient

statistics).

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42 D. Brigo et al

It is often notclear whether or not a fully consistent single-name default formulation

is possible given an aggregate model as the starting point.

There is a special bottom-up approach that can lead to a distinct and rich loss

dynamics. This approach is based on the common Poisson shock (CPS) framework

reviewed in Lindskog and McNeil (2003). This approach allows for more than one

defaulting name in small time intervals, which is in contrast with some of the above-

mentioned top-down approaches. In bottom-up language we see that this approach

leads to a MarshallOlkin copula linking the first jump (default) times of single names.

In top-down language, this model looks very similar to the generalized Poisson loss

(GPL) model of Brigo et al (2006a), where the number of defaults is not capped.

The problem with the CPS framework is that it allows for repeated defaults, which

is clearly wrong since one name could default more than once.

The CPS framework has been used in the credit derivatives literature by Elouer-

khaoui (2006), for example (see also the references therein). Balakrishna (2006)

introduces a semi-analytical approach, again allowing for more than one default in

small time intervals and hinting at its relationship with the CPS framework, that shows

some interesting calibration results. Balakrishna (2007) then generalizes this earlier

paper to include delayed default dependence and contagion.

1.3 Generalized Poisson loss and generalized Poisson

cluster loss models

Brigo et al (2007) address the repeated default issue in the CPS framework by con-

trolling the cluster default dynamics in order to avoid repetitions. They calibrate the

obtained model satisfactorily to CDO quotes across attachments and maturities, but

the combinatorial complexity of a non-homogeneous version of this model is forbid-

ding, and the resulting generalized Poisson cluster loss (GPCL) approach is hard to

use successfully in practice when taking single names into account.

In the context of the present paper, however, the GPL and GPCL models will still

be useful for showing how a loss-distribution dynamics consistent with CDO market

quotes should evolve.

As explained above, the GPL and GPCL models are dynamical models for loss that

have the ability to reprice all tranches and all maturities simultaneously. They mainly

differ in the way that they avoid repeated defaults, and by their stressing of the whole

pool or of individual clusters as fundamental objects. Here we employ a variant that

models theloss directly rather than by using thedefault counting process plus recovery.

The loss is modeled as the sum of independent Poisson processes, each associated

with the default of a different number of entities and capped at the pool size to avoid

infinite defaults. A possible interpretation of these driving Poisson processes is that

of defaults of sectors, although the sectors amplitudes vary in our formulation of the



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model pre-crisis and in-crisis. In the new in-crisis model implementation we fix the

amplitude of the loss triggered by each cluster of defaults a priori without calibrating

it, as we did in our earlier GPL and GPCL works. This makes the calibration more

transparent and the calibrated intensities of the sectors defaults easier to interpret.

We highlight how these models are able to reproduce the tail multimodal feature,

which the implied copula has proved is indispensable for accurately repricing market

spreads of CDO tranches on a single maturity.

We also refer to the related results of Longstaff and Rajan (2008), which point

in the same direction but add a principal component analysis on a panel of credit

default swap (CDS) spread changes and also give further comments on the economic

interpretation of the default clusters as sectors.An econometric investigation of cluster

defaults starting from the Poisson framework can be found in Duan (2009).

Incidentally, we draw the readers attention to the history of defaults, which points

to default clusters being concentrated in a relatively short time period (a few months):

the thrifts in the early 1990s at the height of the loan and deposit crisis, airlines after

2001 and, more recently, autos and financials give evidence of this. In particular,

from September 7, 2008 to October 8, 2008 a time window of one month we wit-

nessedseven credit events affectingmajor financial entities: Fannie Mae, Freddie Mac,

Lehman Brothers, Washington Mutual, Landsbanki, Glitnir and Kaupthing. Fannie

Mae and Freddie Mac conservatorships were announced on the same date (September

7, 2008) and the appointment of a receivership committee for the three Icelandic

banks (Landsbanki, Glitnir and Kaupthing) was announced between October 7 and 8.

Moreover, Standard & Poors (2009) issued a request for comments related to

changes in the rating criteria of corporate CDOs. Thus far, agencies had been adopting

a multifactor Gaussian copula approach to simulate the portfolio loss in the objective

measure. Standard & Poors proposed changing the criteria so that tranches rated

AAA would be able to withstand the default of the largest single industry in the

asset pool with zero recoveries. We believe that this is a move in the direction of

modeling the loss in the risk-neutral measure via GPL-like processes, given that the

proposed changes to Standard & Poors rating criteria imply the admittance of the

possibility that a cluster of defaults in the objective measure could exist as a stressed

but plausible scenario (see also Torresetti and Pallavicini (2007) for the specific case

of constant-proportion debt obligations).

Finally, we comment more generally on dynamical aggregate models and on the

difficulties encountered in deriving single-name hedge ratios when trying to avoid

complex combinatorial analysis. The framework therefore currently remains incom-

plete, because obtaining jointly tractable dynamics and consistent single-name hedges

that can be realistically applied on the trading floor remains a problem. We have pro-

vided some references for the latest research in this field above. We highlight, though,

that even simple dynamical models such as our GPL model or the single-maturity



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44 D. Brigo et al

implied copula model are enough to appreciate that the market quotes were imply-

ing the presence of large default clusters with non-negligible probabilities well in

advance of the credit crisis, as documented in 2006 and early 2007 (Brigo et al

(2006a,b, 2007)).

Finally, it is important to point out that most of the above discussion and references

(with very few exceptions) center on corporate CDOs mostly synthetic ones and

little literature is available for the valuation of CDOs on other asset classes, with

literature on possibly complex waterfalls and prepayment risk (including collateral-

ized loan obligations, residential mortgage-backed securities and CDOs of residential

mortgage-backed securities) generally relating to the asset class that triggered the cri-

sis. For manysuchdeals the main problem is oftenthe data. The scarce literature in this

area includes Jaeckel (2008), Papadopoulos and Tan (2007) and Fermanian (2009).

2 MARKET QUOTESFor single names, our reference products will be CDSs, but the most liquid multiname

credit instruments available in the market are credit indexes and CDO tranches (eg,

Dow JonesiTraxx (DJiTraxx) and CDX). We discuss these instruments below.

The procedure for selecting the standardized pool of names is the same for the

two indexes. Every six months a new series is rolled at the end of a polling process

managed by MarkIt where a selected list of dealers contributes the ranking of the

most liquid CDS.1

2.1 Credit indexes

The index is given by a pool of names 1 ; 2 ; : : : ; M , typically M D 125, each withnotional 1=M so that the total pool has unit notional. The index default leg consists

of protection payments corresponding to the defaulted names of the pool. Each time

one or more names default, the corresponding loss increment is paid to the protection

buyer until final maturity T D Tb arrives or until all the names in the pool have

defaulted.

In exchange for loss increase payments, a periodic premium with rate S is paid

from the protection buyer to the protection seller until final maturity Tb. This premium

is computed on a notional that decreases each time a name in the pool defaults,

decreasing by an amount corresponding to the notional of that name (without taking

out the recovery).

1 All credit references that are not investment grade are discarded. Each surviving credit reference

underlying the CDS is assigned to a sector. Each sector is contributing a predetermined number of

credit references to the final pool of names. The rankings of the various dealers for the investment

grade names are put together to rank the most liquid credit references within each sector.



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We denote by NLt the portfolio cumulated loss and by NCt the number of defaulted

names up to time t divided by M. Since, at each default, part of the defaulted notional

is recovered, we have 0 6 NLt 6 NCt 6 1. The discounted payout of the two legs of

the index is given as follows:

defleg.0/ WD

ZT0

D.0;t/ d NLt

prmleg.0/ WD S0 DV01.0/

DV01.0/ WD

bXiD1

iD.0; Ti /.1 NCTi /

where D.s;t/ is the discount factor (often assumed to be deterministic) between

times s and t and i D Ti Ti1 is the year fraction. In the second equation, DV01.0/

denotes the sum over all premium payment dates of the outstanding notional, weighted

by the relevant year fraction, and discounted back at time 0. The risk-neutral expec-

tation of such a quantity will provide the sensitivity of the premium leg to a unit

movement in the spread S0, all other things being equal (hence the name DV01,

although usually the spread is moved by 1 basis point (bp)). In the same equation,

the actual outstanding notional in each period would be an average over Ti1; Ti ,

but we have replaced it with 1 NCTi (the value of the outstanding notional at Ti ) for

simplicity, which is commonly done.

Note that, in contrast with the tranches (see Section 2.2), here the recovery is not

considered when computing the outstanding notional because it is only the number

of defaults that matters.

The market quotes the values of S0 that, for different maturities, balance the two

legs. If we have a model for the loss and the number of defaults, we may require that

the loss and the number of defaults in the model, when plugged into the two legs, lead

to the same risk-neutral expectation (and hence price):

S0 DE0

RT0 D.0;t/ d

NLt

E0Pb

iD1 iD.0; Ti /.1 NCTi /

(1)

where E denotes the expectation under the pricing risk-neutral measure.

Assuming deterministic default-free interest rates, we can rewrite:

S0 DRT0 D.0;t / dE0 NLt Pb

iD1 iD.0; Ti /.1 E0NCTi /

(2)

The assumption regarding deterministic default-free interest rates is a practical one

and allows us to obtain analytical or semi-analytical pricing formulas, as is usually



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46 D. Brigo et al

done in market practice. It can be relaxed to an assumption of independence between

defaults and default-free interest rates, with the consequence that the D.t;T/ terms of

the deterministic case become P . t ; T / D Et D.t; T /, the zero-coupon bond prices.

2.2 Collateralized debt obligation tranches

Synthetic CDOs with maturity T are contracts involving a protection buyer, a protec-

tion seller and an underlying pool of names. They are obtained by putting together a

collection of CDSs with the same maturity on different names, 1 ; 2 ; : : : ; M , typically

M D 125, each with notional 1=M, then tranching the loss of the resulting pool

between points A and B , with 0 6 A < B 6 1:

NLt WD1

B A. NLt A/1fA< NLt6Bg C .B A/1f NLt>Bg

A useful alternative expression is:

NLt WD1

B AB NL

0;Bt A NL

0;At (3)

Once enough names have defaulted and the loss has reached A, the count starts.

Each time the loss increases, the corresponding loss change rescaled by the tranche

thickness B A is paid to the protection buyer until maturity arrives or until the total

pool loss exceeds B , in which case the payments stop.

The discounted default leg payout can then be written as:

deflegA;B.0/ WD ZT

0D.0;t/ d NLt

As usual, in exchange for the protection payments, a premium rate SA;B0 , fixed at

time T0 D 0, is paid periodically at times T1; T2; : : : ; T b D T. Part of the premium

can be paid at time T0 D 0 as an upfront UA;B0 . The rate is paid on the surviving

average tranche notional. If we further assume that payments are made on the notional

remaining at each payment date Ti , rather than on the average in Ti1; Ti , the

premium leg can be written as:

prmlegA;B.0/ WD UA;B0 C S

A;B0 DV01A;B.0/

DV01A;B.0/ WD

bXiD1

iD.0; Ti /.1 NLTi /

When pricing CDO tranches, we are interested in the premium rate SA;B0 that sets

to zero the risk-neutral price of the tranche. The tranche value is computed taking the



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TABLE 1 Standardized attachment and detachment for the DJiTraxx Europe main and

CDX.NA.IG tranches.

DJiTraxxEurope CDX.NA.IGmain (%) (%)

03 03

36 37

69 710

912 1015

1222 1530

22100 30100

(risk-neutral) expectation (in t D 0) of the discounted payout, which is the difference

between the default and premium legs above. We obtain:

SA;B0 D

E0RT0 D.0;t / d

NLt UA;B0

E0Pb

iD1 iD.0; Ti /.1 NLTi /

(4)

Assuming deterministic default-free interest rates, we can rewrite:

SA;B0 D

RT0 D.0;t/ dE0

NLt UA;B0Pb

iD1 iD.0; Ti /.1 E0NLTi /

(5)

The above expression can be easily recast in terms of the upfront premium UA;B0 for

tranches that are quoted in terms of upfront fees.

The tranches that are quoted on the market refer to standardized pools, standardizedattachmentdetachment points A and B and standardized maturities.

The standardized attachment and detachment differ slightly for the CDX.NA.IG

and the DJiTraxx Europe main tranches.2

For the DJiTraxx and CDX pools, the equity tranche .A D 0; B D 3%/ is quoted

by means of the fair UA;B0 , while assuming S

A;B0 D 500bps. Thereasonthat the equity

tranche is quoted upfront is to reduce the counterparty credit risk that the protection

seller is facing. All other tranches are generally quoted by means of the fair running

spread SA;B0 , assuming no upfront fee (U

A;B0 D 0). Following recent market turmoil,

the 36% and the 37% tranches have been quoted in terms of an upfront amount and

a running SA;B0 D 500bps given the exceptional risk premium priced by the market

for this tranche.

2 The attachment points of the CDX tranches are slightly higher, reflecting the average higher

perceived riskiness (as measured, for example, by the CDS spread or by balance-sheet ratios) of the

liquid investment grade North American names.



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3 A FULLY CONSISTENT DYNAMICAL MODEL:

THE GENERALIZED-POISSON LOSS MODEL

The GPL model can be formulated as follows. Consider a probability space supportinga number n of independent Poisson processes Z1; : : : ; Zn with time-varying and

possibly stochastic intensities 1; : : : ; n under the risk-neutral measure Q. The risk-

neutral expectation conditional on the market information up to time t , including the

pool loss evolution up to t , is denoted by Et . Intensities, if stochastic, are assumed to

be adapted to such information.

Define the stochastic process:

Zt WD

nXjD1

jZj.t / (6)

for positive integers 1; : : : ; n. In the following, we refer to the Zt process simply as

the GPL process. We will use this process as a driving process for the cumulated port-

folio loss NLt , which is the relevant quantity for our payouts. In Brigo et al (2006a,b),

the possible use of the GPL process as a driving tool for the default counting processNCt is illustrated instead.

The characteristic function of the Zt process is:

'Zt .u/ D E0eiuZt D E0E0e

iuZt j 1. t / ; : : : ; n.t/

where:

j.t / WD

Zt0

j.s/ ds; i D 1 ; : : : ; n

are the cumulated intensities of each Poisson process. Now, we substitute Zt,

obtaining:

'Zt .u/ D E0

nYjD1

E0eiu jZj .t/ j 1. t / ; : : : ; n.t/

D E0

nYjD1

'Zj .t/jj .t/.u j/

which can be directly calculated since the characteristic function 'Zj .t/jj .t/ of each

Poisson process, given its intensity, is known in closed form, leading to:

'Zt .u/ D E0 exp nX

jD1

j.t/.eiu j 1/ (7)

The marginal distribution pZt of the process Zt can be directly computed at any

time via an inverse Fourier transformation of the characteristic function of the process.



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The characteristic function 'Zt .u/ can be explicitly calculated for some relevant

choices of Poisson cumulated-intensity distributions (see, for example, Brigo et al

(2006a,b)).

3.1 Loss dynamics

The underlying GPL process Zt is non-decreasing and, given sufficiently large times,

it takes arbitrarily large values. The portfolio cumulated loss and the rescaled number

of default processes is non-decreasing, but limited to the interval 0; 1. Thus, we con-

sider the deterministic non-decreasing function W N[ f0g ! 0; 1 and we define

the cumulated portfolio loss process NLt as:

Lt WD L.Zt / WD min.Zt ; M0/ and NLt WD

Lt

M0(8)

where 1=M0

, M0

> M > 0, is the minimum jump for the loss process. M0

is clearlyrelated to the loss granularity.

Remark 3.1 Note that the loss is bounded within the interval 0; 1 by construction,

but that the possibility remains that the loss jumps more than M times, where M is

the number of names in the portfolio.If this is the case, we may checka posteriori that

the probability of such events is negligible. This is the case for all of our examples.

The marginal distribution of the cumulated portfolio loss process Lt can be easily

calculated. We obtain:

Lt D min.Zt ; M0/ D Zt1fZtM0g

Since Zt has a known distribution, the distribution of Lt can be easily derived as a

by-product. The related density (defined on integer values since the law is discrete)

is:

pLt .x/ D pZt .x/1fx M0g1fxDM0g

The density of NLt follows directly. Also, the intensity of Lt , ie, the density of the

absolutely continuous compensator of Lt (see, for example, Giesecke et al (2009)),

can be computed directly and is given by:

hL.t / D

nXjD1

min. j; .M0 Zt/

C/j.t / (9)

(seeBrigo etal (2006a,b) for details). The intensity h goes to zero when Z exceeds M0,

which corresponds to total loss, as expected. Furthermore, if all the possible integer

jump sizes between 1 and M0 are allowed, ie, if j D j and n D M0, the intensity

hL jumps whenever the cumulated portfolio loss process L jumps. The intensity



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50 D. Brigo et al

jumps downward, and this would seem to go in the opposite direction with respect

to self-excitedness, which is considered a desirable feature of loss models in general.

However, self-exciting features are embedded in our model, and embedded in the

possibility of having several defaults in small intervals, contrary to most approaches

to loss modeling. Consider, for example, just two names: instead of having the loss

of one name increase the likelihood of default (intensity) of a second name, we have

both names defaulting together immediately. This embeds self-excitedness, although

in an extreme way. Finally, to view the effect of single names on each other, we need

a more sophisticated formulation based on the common Poisson shocks framework,

leading to the GPCL model, which is analyzed in Brigo et al (2007) and is presented

below with a few updates.

Example 3.2 It is worth presenting an example to further clarify the GPL for-

mulation. Let us take an example considering a GPL model where there are three

independent Poisson processes Z1, Z2 and Z3 correspondingly multiplied by sizes1, 10 and 40, leading to:

Zt D 1Z1.t / C 10Z2.t/ C 40Z3.t/

with constant deterministic intensities 1 D 4%, 2 D 1:5% and 3 D 0:75%.

Assume a 40% deterministic recovery, and take one scenario where the three processes

have eachjumped once before the current time. Theloss of the standardized DJiTraxx

pool of 125 names associated to this scenario with one jump of Z1, Z2 and Z3,

respectively, would be 0.48%, 4.8% and 19.2%, respectively. The expected loss in

one year for the uncapped loss process is:

0:2352% D .0:0048 0:04/ C .0:048 0:015/ C .0:192 0:0075/

In Figure 1 on the facing page we plot the resulting default counting distribution on

four increasing maturities: 3, 5, 7 and 10 years. We note that the modes on the right

tail of the distribution become more evident as the maturity increases. These bumps

are the corresponding probabilities of a jump of a higher amplitude occurring.

From the distribution we also note that the probability is not simply assigned to

the higher-amplitude jumps (10 and 40 defaults in our example) but also to a number

of defaults immediately above: this would be the probability of having jumps of both

a high amplitude (either 10 or 40) and of a lower amplitude (1 in our case), where

jumps with lower amplitudes have higher probability.

Furthermore, we note that some second-order modes start to appear as maturity

increases. These are the bumps corresponding to multiple jumps in the higher-size

(10 and 40), smaller probability and GPL components.

Let us now assume we are to price two tranches: 24.8% and 1019.2%. The

expected tranche loss of these tranches, and, ultimately, their fair spread, will depend

primarily on the probability mass lying above the tranche detachment.



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FIGURE 1 Default counting M NCT probability distribution associated to a GPL model

featuring three independent Poisson processes with jump amplitudes 1 D 1, 2 D 10 and

3 D 40, with constant deterministic intensities 1 D 4%, 2 D 1:5% and 3 D 0:75%.

10

8

6

4

2

00 10 20 30 40 50 60

3 year

%

(a)

10

8

6

4

2

00 10 20 30 40 50 60

%

5 year

(b)

10

8

6

4

2

00 10 20 30 40 50 60

%

7 year

(c)



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52 D. Brigo et al

FIGURE 1 Continued.

10

8

6

4

2

00 10 20 30 40 50 60

%

10 year

(d)

3.2 Model limits

The GPL model that we have introduced can currently be viewed as a particularly

simple parametrization of the market-implied loss-distribution dynamics. A positive

feature is that the loss changes only by positive jumps, which should be the case

in any sensible loss model. Furthermore, this choice allows us to achieve a good

calibration to market data, as we will see in the following section. However, we

are not making explicit assumptions on two important issues: firstly, we have not

addressed possible ways of making our model consistent with single-name dynamics,

and secondly, we have not explained how to choose a full-featured pool spread and

recovery dynamics. One possibility would be to make the intensities in the Poissonprocesses driving Z stochastic and to consider more general transformations of Z to

obtain the loss process.

Since we are focusing only on the calibration of CDO tranches, which depend

only on the loss marginal distribution, we will avoid discussing such problems here

(we direct the interested reader to Brigo et al (2006a,b) for an extensive analysis

of candidate spread and recovery dynamics, and to Brigo et al (2007) for further

discussion, including consistency with single-name data).We pointout that significant

progress and testing in loss modeling will only be possible when more liquid market

quotes for tranche options and forward-start tranches are available.

3.3 Model calibration

We work with the basic GPL model specification as given by the driving GPL process

Z in (6), which we use to model the pool loss through (8). In this basic formulation,

each Poisson mode Zj has a deterministic piecewise-constant intensity j.t/.



15/44


Given that we have modeled the pool loss NLt directly, we do not completely charac-

terize the rescaled default counting process NCt , but instead give only its expectations.

Indeed, such expectationsare the only information on default counting that are implicit

in index market quotes (Equation (1)), whereas tranche quotes (Equation (4)) depend

only on the loss and not on default counting explicitly. We therefore assume:

E0 NCt WD1

1 RE0 NLt ; 0 6 R < 1 E0 NLTb (10)

where the range of definition of the constantR is taken in order to ensure that, at each

time t , the expected value of the rescaled number of defaults is greater than or equal

to the cumulated portfolio loss, and that both are smaller than or equal to one. Note

that we avoid introducing an explicit dynamics for the recovery rate (see Brigo et al

(2006a,b, 2007) for an initial discussion on recovery dynamics). Our R here can be

interpreted as a kind of average recovery rate.

3.4 Detailed calibration procedure

The model parameters found using the calibration procedure are the amplitudes j 2

fm 2 N W m 6 M0g, j D 1 ; : : : ; n, and the cumulated intensities j.T /, which are

real non-decreasing piecewise linear functions in the tranche maturity.

The optimal values for the amplitudes are selected in the following way.

1) Fix the minimum jump size to 1=M0 by choosing the integer M0 > M > 0.

2) Find the integer value for 1 by calibrating the cumulated intensity 1 for each

value of 1 in the range 1;M0, all other modes being set to zero. Calibration isobtained by minimizing the objective function (11) below. Finally, choose the 1for which the calibration error is at its minimum when using the corresponding

1. We call this chosen 1 the best integer value for 1.

3) Add the amplitude 2 and find its best integer value by calibrating the cumulated

intensities 1 and 2, starting from the previous value for 1 as a guess, for

each value of2 in the range 1;M0.

4) Repeat the previous step for i with i D 3 and so on by calibrating the cumu-

lated intensities 1; : : : ; i , starting from the previously found 1; : : : ; i1

as an initial guess, until the calibration error is under a given threshold or untilthe intensity i can be considered negligible.

5) Checka posteriori that the probability of having more than M jumps is negli-

gible and that the value ofR is within the arbitrage-free range given in (10).



16/44

54 D. Brigo et al

TABLE 2 DJiTraxx index and tranche quotes in basis points on May 13, 2005, with

bidask spreads in parentheses.

Maturities Attachment 3 5 7 10detachment years years years years

Index 38 54 65 77(4) (1) (3) (2)

Tranche

03 2,060 4,262 5,421 6,489(100) (118) (384) (124)

36 72 173 398 590(10) (68) (40) (20)

69 28 57 141 188(6) (6) (17) (15)

912 13 31 72 87(2) (5) (20) (15)

1222 3 21 42 60(1) (3) (13) (10)

Index and tranches are quoted through the periodic premium, while the equity tranche is quoted as an upfront

premium (see Section 2).

The objective function f to be minimized in the calibration is the squared sum of

the errors shown by the model to recover the tranche and the index market quotes

weighted by market bidask spreads:

f.;/ DXi

"2i ; "i Dxi . ; / xmidi

xbidi xaski

(11)

where the xi (with i running over the market quote set) are the index values S0for DJiTraxx index quotes and either the periodic premiums S

A;B0 or the upfront

premium rates UA;B for the DJiTraxx tranche quotes.

3.5 Calibration results

The GPL model is calibrated to the market quotes observed weekly from May 6,

2005 to October 18, 2005. Following Albanese et al (2006), we take R D 30% as

our reference value for the recovery rate in the DJiTraxx Europe market for spot

and forward contracts. The quality of our calibration below is not altered if we select

a value R D 40% resembling the recovery typically used in simplified quoting

mechanisms in the market (see Brigo et al (2006a,b, 2007) for examples of this). We

start with M0 D 200, corresponding to a minimum loss jump size of 50bps.



17/44


TABLE 3 (a) Calibration error "i in (11) (calculated with respect to the bidask spread)

for tranches quoted on May 13, 2005; (b) cumulated intensities (integrated up to tranche

maturities) of the GPL model with M0 D 200.

(a)


Index 0.0 0.1 0.3 0.0

Tranche

03 0.0 0.1 0.2 0.2

36 0.0 0.0 0.2 0.0

69 0.0 0.0 0.3 0.1912 0.1 0.1 0.1 0.4

1222 0.0 0.0 0.2 0.3

(b)

.T / 3 5 7 10

years years years years

1 1.955 3.726 4.464 7.694

3 0.000 0.062 0.305 0.305

8 0.016 0.033 0.109 0.10912 0.004 0.013 0.026 0.026

19 0.006 0.006 0.017 0.017

72 0.000 0.009 0.026 0.049

185 0.000 0.002 0.002 0.008

Each row corresponds to a different Poisson component with jump amplitude . Recovery rate is 30%.

As a first example, consider the calibration date May 13, 2005 (see Table 2 on the

facing page). In Table 3 we list the calibration result and the values of the calibrated

parameters. The calibration errors " are very low for all maturities. Note that a cal-

ibration error smaller than one means that the difference between the market quote

and the model price is smaller than the bidask spread.

Consider, as a second example, the calibration date October 11, 2005 (see Table 4

on the next page). In Table 5 on page 57, we list the calibration results and the values of

the calibrated parameters. The calibration errors show that the 10-year equity tranche



18/44

56 D. Brigo et al

TABLE 4 DJiTraxx index and tranche quotes in basis points on October 11, 2005, with

bidask spreads in parentheses.


Index 23 38 47 58(2) (1) (1) (1)

Tranche

03 762 3,137 4,862 5,862(26) (26) (76) (74)

36 20 95 200 515(10) (1) (3) (10)

69 7 28 43 100(6) (1) (2) (4)

912 12 27 54(2) (4) (5)

1222 7 13 23(1) (2) (3)

is not correctly priced. We find such mispricing in many calibration examples, in

particular after October 2005.

For the minimum loss jump size 1=M0, besides 50bps, we try the values 2bps and

10bps, corresponding, respectively, to M0 equal to 5,000 and 1,000. As can be seen

from Table 6 on the facing page, the 10-year maturity tranches (in our experience themost difficult to calibrate) are stable through the three different loss sizes, suggesting

that going below 50bps does not add much flexibility to the model. This is confirmed

by further tests. In particular, the difference between the M0 D 1;000 calibration

and the M0 D 5;000 calibration is always small. Furthermore, the behavior of the

mean calibration error, ie, of the mean of the absolute values of the "i across time and

quotes, for the three different choices of M0 is quite similar and within one bidask

spread.

We also note that as the minimum jump size decreases (granularity increases),

the loss distribution becomes noisier, due to the presence of small amplitudes. Fur-

thermore, very small modes, appearing when the minimum jump size is as small as

a few basis points, may violate the requirement that the loss process jumps fewer

than M times (see Remark 3.1). We also try calibrations with M0 less than 200, ie,

with a minimum loss jump greater than 50bps. In this case the calibration error grows

quickly. Indeed, the minimum jump size, in this case, becomes greater than the typical



19/44


TABLE 5 (a) Calibration error "i in (11) (calculated with respect to the bidask spread) for

tranches quoted on October 11, 2005; (b) cumulated intensities (integrated up to tranche

maturities) of the GPL model with M0 D 200.

(a)


Index 0.0 0.0 0.1 0.1

Tranche

03 0.1 0.1 1.2 2.1

36 0.1 0.1 0.3 1.0

69 0.0 0.1 0.3 0.9

912 0.4 0.8 0.81222 0.0 0.0 0.0

(b)

.T / 3 5 7 10

years years years years

1 0.441 2.498 4.466 7.555

2 0.435 0.435 0.435 0.671

11 0.004 0.023 0.023 0.023

22 0.000 0.001 0.006 0.030

29 0.000 0.000 0.001 0.00132 0.001 0.004 0.004 0.004

192 0.000 0.001 0.005 0.011

The three-year maturity quotes lack two tranches.

TABLE 6 GPL calibration error for different minimum loss sizes 1=M0 with respect to the

bidask spread for 10-year tranches on October 11, 2005.

Attachmentdetachment 50bps 10bps 2bps

03 2.1 1.8 1.8

36 1.0 1.0 1.0

69 0.9 0.9 0.9

912 0.8 0.9 0.8

1222 0.0 0.2 0.0

Recovery rate is 30%.



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58 D. Brigo et al

TABLE 7 Values of the Poisson amplitudes =M0 for different values of the minimum loss

jump 1=M0.

(a) 50bps

Poisson amplitudes 1 2 3 4 5 6 7

Date (%) (%) (%) (%) (%) (%) (%)

May 6, 2005 0.50 1.50 4.00 6.00 9.50 39.50 92.50

September 2, 2005 0.50 1.00 4.00 5.50 12.50 39.00 100.00

October 11, 2005 0.50 1.00 5.50 11.00 14.50 16.00 96.00

(b) 10bps

Poisson amplitudes 1 2 3 4 5 6 7

Date (%) (%) (%) (%) (%) (%) (%)

May 6, 2005 0.10 1.50 4.60 5.90 9.60 39.60 53.00

August 5, 2005 0.20 1.10 1.40 8.10 11.30 49.00 62.40

October 11, 2005 0.10 0.70 1.00 6.30 11.50 14.50 93.70

(c) 2bps

Poisson amplitudes

1 2 3 4 5 6 7Date (%) (%) (%) (%) (%) (%) (%)

May 6, 2005 0.02 1.50 5.26 9.64 17.58 39.64 99.78

August 12, 2005 0.38 1.06 1.14 7.38 12.24 41.34 99.80

October 3, 2005 0.02 0.98 1.16 7.52 9.74 43.34 65.16

October 11, 2005 0.16 0.68 1.00 6.30 10.98 14.46 94.90

Only the calibration dates between May 6, 2005 and October 18, 2005 where the =M0 values change are listed.

portfolio loss given when one name defaults. 50bps seems, then, to be a reasonable

reference value.

Also, the values of the Poisson amplitudes are quite stable across the calibration

dates. Indeed, in six months we observe at most four changes in their values, as shown

in Table 7.

The loss distribution implied by the GPL model is multimodal and the probabil-

ity mass moves toward larger loss values as the maturity increases. These features



21/44


FIGURE 2 Loss-distribution evolution of the GPL model with a minimum jump size of

50bps at all the quoted maturities up to 10 years, drawn as a continuous line.

0 0.02 0.04 0.06 0.08 0.100

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Loss

0.10 0.15 0.20 0.25 0.300

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0x 10

3

Loss

(a)

3 years5 years7 years10 years

(b)

3 years5 years7 years10 years

are shared by different approaches. For instance, static models, such as the implied

default-rate distribution in Torresetti et al (2006c), suggest multimodal loss distribu-

tions, as mentioned in the introduction regarding the implied copula. The evolution

of the implied loss distribution is shown in Figure 2.

The dynamic credit correlation model of Albanese et al (2006) shows implied

loss distributions whose modes tend to group as the maturity increases, leading to a



22/44

60 D. Brigo et al

distribution approaching normality. The GPL model reproduces this behavior (see,

for example, Brigo et al (2006a,b)).

4 APPLICATIONS TO MORE RECENT DATA ANDTHE CRISIS

In this section we check whether the critical features that we discussed regarding

implied correlation and the subsequent elements coming from more advanced models

are still present in-crisis after mid 2007. We will observe that the features are still

present and are often amplified in the market after the beginning of the crisis.

4.1 Customizing the generalized Poisson loss model

to deal with sectors

We now consider the GPL model to assess how well it performs in-crisis. We change

the model formulation slightly in order to have a model that is more in line with the

current market, while maintaining all the essential features of the modeling approach.

Compared with the model described in Section 3, we introduce the following modifi-

cations. We fix the jump amplitudes a priori rather than calibrating them through the

detailed calibration procedure seen in Section 3.4, and we associate different recov-

eries to different . Fixing the before calibration will make the model less flexible

but quicker to calibrate and possibly more stable. To fix the a priori, we reason as

follows. Fix the independent Poisson jump amplitudes to the levels just above each

tranche detachment, when considering a 40% recovery.

For the DJiTraxx index, for example, this would be realized through jump ampli-

tudes ai D i=125, where:

5 D roundup125 0:03

1 R

; 6 D roundup

125 0:061 R

7 D roundup

125 0:09

1 R

; 8 D roundup

125 0:12

1 R

9 D roundup

125 0:22

1 R

; 10 D 125

and, in order to have more granularity, we add the sizes 1, 2, 3 and 4:

1 D 1; 2 D 2; 3 D 3; 4 D 4

In total we have n D 10 jump amplitudes. We then modify the obtained sizes slightlyin order to account for CDX attachments that are slightly different. Eventually, we

obtain the set of amplitudes:

125 ai i 2 f1;2;3;4;7;13;19;25;46;125g



23/44


We want to associate a recovery of 40% to all of these amplitudes , except 125, to

which we want to associate a recovery of 0% in order to size the premium quoted

by the market for super-senior tranches. We start by considering the default counting

process which, by introducing the non-Armageddon indicator It , can be cast in the

following form:

NCt D 1 It .1 Nct/ (12)

with:

Nct WD min

n1XiD1

aiZi .t/;1

; It WD 1fZn.t/D0g

We refer to the component associated with n D 10 D 125 as the Armageddon

mode, since whenever Zn D Z10 jumps, the whole pool defaults. We refer to the

jump Zn D Z10 as an Armageddon event. Indeed, whenever theArmageddon com-

ponent Zn jumps for the first time, the default counting processN

Ct jumps to the entirepool size and every name in the pool has defaulted, with no more defaults being

possible.

We then introduce the stopping time O as the minimum time between the Armaged-

don jump event and the time when the reduced pool without the Armageddon compo-

nent has completely defaulted. This is also the time when the full pool has defaulted:

O D inf

t W

nXiD1

aiZi .t/ > 1

We note that I is 0 after the Armageddon event, and otherwise it is equal to 1, while

Nc is equal to 1 after the reduced pool is over, so, after some algebra, we obtain:

NCt D 1fO6tg C Nct1fO>t g (13)

Note that, as expected, NC depends on Nc only at terminal time t .

To derive the loss process with a zero recovery for the Armageddon event, we first

calculate the evolution of the counting process:

d NCt D It d Nct dIt.1 Nct/

Here, notation such as d NCt denotes the left increment d NCt D NCt NCt, since we are

considering right-continuous jump processes. Intuitively, this quantity is zero if NC

does not jump at t , whereas it is one if NC jumps at t .We then apply the recovery only to the terms coming from jumps that are not due

to the Armageddon event:

d NLt D .1 R/It d Nct dIt.1 Nct/



24/44

62 D. Brigo et al

We cannow integrate the previous equation to obtain theloss process. To manipulate

the above equation we observe three things. Firstly, I is zero after the Armageddon

event, otherwise it is equal to one. Secondly, dI is non-zero only at the Armageddon

event. Finally, Nc is equal to one and d Nc is zero after the reduced pool is over:

NLt D .1 R NcO/1fO6tg C .1 R/ Nct1fO>t g (14)

Note that NL now depends on Nc both at full-pool exhaustion time O and at terminal

time t .

Remark 4.1 (Interpretation of the model) Whenever the Armageddon component

Zn jumps the first time, we will assume that the recovery rate associated to the

remaining names defaulting in that instant will be zero. The pool loss, however, will

not always jump to one, since there is the possibility that one or more names already

defaulted before the Armageddon component Zn jumped, and that they defaulted

with recovery rate R. If, at a given instant t , the whole pool defaults, ie, NCt D 1, this

may happen in one of two ways.

1) Zn jumped by t . In this case, the portfolio has been wiped out with the help of

an Armageddon event. Note that, in this case, d NCt D d NLt ifZn jumps at t . In

fact, the recovery associated to the pool fraction defaulting in that instant will

be equal to zero.

2) Zn has not jumped by t . In this case the portfolio has been wiped out, not with

the help of anArmageddon event, but because of defaults of more small or large

sectors that do not comprise the whole pool. Note that, in this case, the loss is

less than the whole notional, as all these defaults had recovery R > 0.

In this way, whenever Zn jumps at a time when the pool has not yet been wiped

out, we can rest assured that the pool loss will be above 1 R. We do this because

the market in 2008 quoted CDOs with prices assuming that the super-senior tranche

would be impacted to a level that was impossible to reach with recoveries fixed at

40%. For example, there was a market for the DJiTraxx five-year 60100% tranche

on March 25, 2008 quoting a running (bid) spread of 24bps bid.

In the dynamic loss model, recovery can be made a function of the default rate NC

(or other solutions are possible; see Brigo et al (2006b) for more discussion). Here

we use the above simple methodology to allow losses of the pool to penetrate beyond

1 R and hence severely affect even the most senior tranches, in line with market

quotations.



25/44


4.2 Numerical calculation of loss distribution

We know how to calculate the distribution of both NCt and NLt . The distribution of the

counting process may be directly calculated from Equation (12) as the distribution ofa reduced GPL, ie, a GPL where the jump Zn is excluded, whose counting process

will be Nct .

The distribution of the loss process can be calculated starting from Equation (14).

A simple approach involves explicitly calculating the forward Kolmogorov equation

satisfied by the probability distribution:

p NLt .x/ D QfNLt D xg;

d

dtp NLt .x/ D

Xy

At.x;y/p NLt .y/ (15)

where x and y are generic states of the loss process and where the generator matrix

At is given by:

At.x;y/ WD limt!0

Qf NLtCt D x j NLt D yg 1fxDyg

t

This procedure is allowed since, at each point in time, starting only from the value

of the loss process, it is possible to say whether the Armageddon event happened, so

we are able to reduce the loss process, which depends on the stopping time O, to a

Markov process. In order to do so, we consider the following states for the pool loss

process divided by M:NLt 2 A[B

whereA is the set of states where the Armageddon event has not happened, andB is

the set of states where the Armageddon event has happened. The two sets are defined

as follows:

A WD

0; .1 R/

1

M; .1 R/

2

M; : : : ; . 1 R/

M 1

M; 1 R

B WD

.1 R/

M 1

MC

1

M; : : : ; . 1 R/

1

MC

M 1

M; 1

Note that the two sets are disjoint if R > 0, so if we know the value of NL only at time

t , we can state whether the Armageddon event has happened.

Example 4.2 Consider a pool of six names with the following GPL modes with

constant deterministic default intensities:

Zt D Z1.t/ C 3Z2.t / C 6Z.t /

1 D 0:020; 3 D 0:010; D 0:003



26/44

64 D. Brigo et al

TABLE 8 Generator matrix for a simple GPL process of three amplitudes for a pool of six

names with 40% recovery for each mode, and the Armageddon mode, whose recovery is

zero.

y x 0 10 20 30 40 50 60 100

0 3.30 0.00 0.00 0.00 0.00 0.00 0.00 0.00

10 1.00 3.30 0.00 0.00 0.00 0.00 0.00 0.00

20 0.00 1.00 3.30 0.00 0.00 0.00 0.00 0.00

30 2.00 0.00 1.00 3.30 0.00 0.00 0.00 0.00

40 0.00 2.00 0.00 1.00 3.30 0.00 0.00 0.00

50 0.00 0.00 2.00 0.00 1.00 3.30 0.00 0.00

60 0.00 0.00 0.00 2.00 2.00 3.00 0.00 0.00

67 0.00 0.00 0.00 0.00 0.00 0.30 0.00 0.00

73 0.00 0.00 0.00 0.00 0.30 0.00 0.00 0.0080 0.00 0.00 0.00 0.30 0.00 0.00 0.00 0.00

87 0.00 0.00 0.30 0.00 0.00 0.00 0.00 0.00

93 0.00 0.30 0.00 0.00 0.00 0.00 0.00 0.00

100 0.30 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Starting states (y) on columns, arrival states (x) on rows (see Equation (15)). See text for intensity and amplitude

data. All values in the table are in percent.

where we have called Z.t / the mode corresponding to the Armageddon event. We

consider a recovery of 40% for the first two modes and a zero recovery for the last

mode. The loss states are NLt 2 A[B with:

AWD f0%; 10%; 20%; 30%; 40%; 50%; 60%g

B WD f67%; 73%; 80%; 87%; 93%;100%g

We can directly calculate the generator matrix since the pool loss may only jump

via the three GPL modes, and each mode only corresponds to one loss transition. We

display the matrix entries in Table 8, with starting states on columns and arrival states

on rows. Note that the main diagonal is calculated to ensure probability conservation

through time.

If no default has happened at time t D 0, the boundary condition for our Kol-

mogorov equation is p NL0.x/ D 1fxD0g, so we can integrate Equation (15) by means

of matrix exponentiation:

p NLt .x/ D exp Zt

0

Xy

Au.x;y/ du

p NL0.y/ D exp

tXy

A.x;y/

p NL0.y/

where the last step is due to the fact that, in this case, the generator matrix does

not depend on time. Matrix exponentiation can be quickly computed with the Pad



27/44


FIGURE 3 Loss distribution evolution up to 10 years predicted by the GPL model for a

pool of six names.

0

2040

6080

100

2

4

6

8

10

0

Years to maturity

Attachment points (%)

Loss

(%)

10

5

0

40% recovery for each mode except the Armageddon mode, whose recovery is zero. See text for intensity and

amplitude data.

approximation (Golub andVan Loan (1983)), leading to a closed-form solution for the

probability distribution p NLt .x/. This distribution can then be used in the calibration

procedure. In Table 9 on the next page and in Figure 3, we show the loss distribution

for our example.

4.3 Calibration results

Bearing this interpretation of the modes in mind, we decided to select the GPL ampli-

tudes by choosing the independent Poisson jump amplitudes to be at the level just

above each tranche detachment when using a 40% recovery. This led to the specific

values of5; : : : ; 10 that we introduced earlier in the paper. With an eye on the vari-

ety of shapes of the implied copula historical calibrated distributions, as summarized

in Brigo et al (2010) and in Torresetti et al (2006c), we added four amplitudes (from

one to four), corresponding to a small number of defaults.



28/44

66 D. Brigo et al

TABLE 9 Loss distribution numerical values for 5-year and 10-year maturity dates, pre-

dicted by the GPL model for a pool of six names.

0 5 10years years years(%) (%) (%)

0% 100.00 84.79 71.89

10% 0.00 4.24 7.19

20% 0.00 0.11 0.36

30% 0.00 8.48 14.39

40% 0.00 0.42 1.44

50% 0.00 0.01 0.07

60% 0.00 0.46 1.72

67% 0.00 0.00 0.00

73% 0.00 0.00 0.02

80% 0.00 0.07 0.2587% 0.00 0.00 0.00

93% 0.00 0.04 0.12

100% 0.00 1.38 2.55

40% recovery for each mode, except the Armageddon mode whose recovery is zero.

We now present the goodness of fit of this GPL through history. We measure the

goodness of fit by calculating, for each date, the relative mispricing:

MisprRel

A;B

D

8

SA;B;ask0

0 otherwise

where SA;B;theor0 is the tranche theoretical spread as in Equation (4), where, for the

calculation of the expectations of both numerator and denominator, we take the loss

distribution resulting from the calibrated GPL.

In Figure 4 on the facing page and Figure 5 on page 68 we present the relative

mispricing for all tranches, for all maturities and for both indexes throughout the

sample (March 2005 to June 2009).

We note that, while the 5-year tranches could be repriced fairly well in the cur-

rent credit crisis for both DJiTraxx and CDX, the 10-year tranche calibrations for

both indexes have sensibly been made less precise following the collapse of Lehman

Brothers. Recently, with the stabilization of credit markets we again see fairly precise

calibration results (ie, a relative mispricing in the 24% range) for all tranches and

for both indexes and both maturities.



29/44


FIGURE 4 Relative mispricing resulting from the GPL calibration: DJiTraxx.

20

15

10

5

0

5

10

15

20

03/2005 01/2006 11/2006 09/2007 08/2008 06/2009

03%36%

69%912%

1222%22100%

Index

%

(a)

20

15

10

5

0

5

10

15

20

03/2005 01/2006 11/2006 09/2007 08/2008 06/2009

%

03%36%

69%912%

1222%22100%

Index

(b)

To highlight where the problems in calibration come from, we have grouped mis-

pricings across three different categories.

Instrument: we have grouped all tranches, independent of maturity and seniority, into

one group and we have put the remaining calibrated instruments, ie, the 5-year and10-year indexes, in the other group.

Maturity: we have grouped all tranches and indexes in two groups according to their

maturity (5 and 10 years).



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68 D. Brigo et al

FIGURE 5 Relative mispricing resulting from the GPL calibration: CDX.

20

15

10

5

0

5

10

15

20

%

03/2005 01/2006 11/2006 09/2007 08/2008 06/2009

03%36%69%912%1222%22100%Index

(a)

20

15

10

0

5

10

15

20

%

03%36%

69%

912%1222%

22100%

Index

5

01/2006 11/2006 09/2007 08/2008 06/200903/2005

(b)

Seniority: we have grouped all tranches (leaving out the indexes) into three categories

depending on the seniority of the tranche in the capital structure.

1) Equity: equity tranche for both the DJiTraxx and CDX indexes.

2) Mezzanine: comprising the two most junior tranches after the equity tranche.

For the DJiTraxx index this means the 36% and 69% tranches, whereas

for the CDX this means the 37% and 710% tranches.

3) Senior: comprising the remaining most senior tranches. For the DJiTraxx

this means the 912%, 1222% and 22100% tranches, whereas for the CDX

this means the 1015%, 1530% and 30100% tranches.



31/44


FromFigure 6 onthe nextpage and Figure 7 onpage 71we notethat the GPL model

produced calibrations that resulted in a relative mispricing that was larger (though of

relatively small magnitude) in the period from June 2005 to October 2006. For both

the DJiTraxx and the CDX pools the mispricing could be ascribed to the 10-year

tranches, in particular to the equity and mezzanine tranches.

We also note that from October 2008 to June 2009 (ie, after the default of Lehman

Brothers), the GPL calibration again resulted in a non-zero but still fairly contained

relative mispricing that, in this case, could be ascribed to both the 5-year and 10-year

tranches independent of their positions in the capital structure (equity, mezzanine or

senior).

5 ADDING SINGLE-NAME AND CLUSTER DYNAMICS:

THE GENERALIZED-POISSON CLUSTER LOSS MODEL

The GPL model introduces a simple mechanism to avoid repeated defaults by cap-

ping the default counting process to the number of names in the pool. However, this

approach prevents us from extending the model to incorporate single names, since

we are limiting the number of defaults irrespective of which name is defaulting.

In Brigo et al (2007), to overcome this issue we introduced the GPCL model,

starting from the standard CPS framework with repeated defaults and using it as an

engine to build a new model for (correlated) single-name defaults, clusters defaults

as well as for the default counting process and the portfolio loss process. Two possible

strategies (aside from the GPL capping function) were introduced to deal with single

names.

We now review the GPCL strategy based on an adjustment at the single-name levelto avoid repeated defaults, leading to (correlated) single-name default processes. This

approach leads to a less clear cluster dynamics in terms of the original cluster repeated

default processes, but it shows interesting properties at the single-name level. We refer

the reader to Brigo et al (2007) for the strategy based directly on cluster dynamics.

5.1 Common Poisson shock basic framework

In the CPS framework we consider the observation that a single names default can

be originated by different events or factors. Occurrence of the event/factor number

e is modeled as a jump of independent Poisson processes Ne, e D 1 ; : : : ; m. Each

event can be triggered many times r D 1 ; 2 ; : : : as jumps go on. The r th jump ofNe

triggers a default event for name k with probability per;k

. Note that a defaulted name

k may default again. We address this limitation later.

The CPS framework may accommodate systemic and idiosyncratic factors in a

natural way. Consider, for instance, the following two cases:



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70 D. Brigo et al

FIGURE 6 DJiTraxx relative mispricing resulting from the GPL calibration grouped by

(a) maturity (5 and 10 years), (b) instrument type (index and tranches) and (c) seniority

(equity, mezzanine and senior).

12

10

8

6

4

2

003/2005 10/2005 06/2006 02/2007 10/2007 06/2008 02/2009

10 year all5 year all

(a)

12

10

8

6

4

2

003/2005 10/2005 06/2006 02/2007 10/2007 06/2008 02/2009

TrancheIndex

%

%

(b)

12

10

8

6

4

2

003/2005 10/2005 06/2006 02/2007 10/2007 06/2008 02/2009

MezzanineEquity

%

14

(c)

Senior



33/44


FIGURE 7 CDX relative mispricing resulting from the GPL calibration grouped by (a)

maturity (5 and 10 years), (b) instrument type (index and tranches)and (c)seniority (equity,

mezzanine and senior)

12

10

8

6

4

2

0

03/2005 10/2005 06/2006 02/2007 10/2007 06/2008 02/2009

10 year all5 year all

(a)

12

10

8

6

4

2

003/2005 10/2005 06/2006 02/2007 10/2007 06/2008 02/2009

TrancheIndex

%

%

(b)

12

10

8

6

4

2

003/2005 10/2005 06/2006 02/2007 10/2007 06/2008 02/2009

MezzanineEquity

%

14

(c)

Senior



34/44

72 D. Brigo et al

1) e is a totally systemic factor ifper;1 D per;2 D D p

er;n D 1;

2) e is a totally idiosyncratic factor if there exists a k such that, for all j k, we

obtain per;k D 1 and p

er;j D 0.

We define the dynamics for the single-name default process Nk, jumping each time

name k defaults, as:

Nk.t/ WD

mXeD1

Ne.t/XrD1

Ier;k

where Ier;k

is a Bernoulli variable with probabilityQfIer;k

D 1g D per;k

. Note that the

process Nk itself turns out to be Poisson. Furthermore, by differentiating the above

relationship, we obtain:

dNk.t/ D

m

XeD1 Ie1CNe.t/;k dN

e.t /

so that, if we consider two names k and h, we have that the respective default counting

processes Nk and Nh are not independent, since their dynamics is explained by the

same driving events Ne.t /.

Thecore of the CPS framework involves mapping thesingle-name default dynamics

(which consist of the dependent Poisson processes Nk) into a multiname dynamics

explained in terms of independent Poisson processes QNs , where s is a subset (or

cluster) of names of the pool, defined as follows:

QNs.t / D

m

XeD1Ne.t/

XrD1 Xs0s.1/js

0jjsj

Yk02s0Ier;k0

where jsj is the number of names in the cluster s. In a summation, s 3 k means

that we are adding up across all clusters s containing k, k 2 s means that we are

adding across all elements k of cluster s, jsj D j means that we are adding across all

clusters of size j , and, finally, s0 s means that we are adding up across all clusters

s0 containing cluster s as a subset.

Theproof of the independence of QNs for different subsets s canbe found in Lindskog

and McNeil (2003). Note that a jump in an QNs process means that all the names in

the subset s, and only those names, have defaulted at the jump time. We denote by Qsthe intensity of the Poisson process QNs.t /, and we assume it to be deterministic for

the time being. We refer to Brigo et al (2007) for model extensions.

5.2 Cluster processes as common Poisson shock building blocks

We do not need to remember the above construction. All that matters for the following

developments are the independent clusters default Poisson processes QNs.t /. These



35/44


can be taken as fundamental variables from which (correlated) single-name defaults

and default counting processes follow. The single-name dynamics can be derived

based on these independent QNs processes in the so-called fatal shock representation

of the CPS framework:

Nk.t/ DXs3k

QNs.t / or dNk.t / DXs3k

d QNs.t / (16)

where the second equation is the same as the first one but in instantaneous jump

form. We now introduce the process Zj.t/, which describes the occurrence of the

simultaneous default of any j names whenever it jumps (with jump size 1):

Zj.t/ WDXjsjDj

QNs.t/ (17)

Note that each Zj.t /, being the sum of independent Poisson processes, is itself Pois-

son. Furthermore, since the clusters corresponding to the different Z1; Z2; : : : ; ZMdo not intersect, the Zj.t / are independent Poisson processes.

The multiname dynamics, that is, the default counting process Zt for the whole

pool, can be easily derived by carefully adding up all the single-name contributions:

Zt WD

MXkD1

Nk.t / D

MXkD1

Xs3k

QNs.t/ D

MXkD1

MXjD1

Xs3k;jsjDj

QNs.t / D

MXjD1

jXjsjDj

QNs.t/

leadingto the relationship that links the setof dependentsingle-name default processes

Nk with the set of independent and Poisson distributed counting processes Zj:

MXkD1

Nk.t / DMX

jD1

jZj.t/ DW Zt (18)

Hence, the CPS framework offers us a way of consistently modeling the single-

name processes along with the pool-counting process, taking into account the cor-

relation structure of the pool, which remains specified within the definition of each

cluster process QNs . Note, however, that the Zt=M process is not, properly speaking,

the rescaled number of defaults NCt , since the former can increase without limit, while

the latter is bounded in the 0; 1 interval. We address this issue in the next section,

along with the issue of avoiding repeated single-name defaults.

5.3 The single-name adjusted approach

In order to avoid repeated defaults in single-name dynamics, we can introduce con-

straints on the single-name dynamics ensuring that each single name makes only one

default. Such constraints can be implemented by modifying Equation (16) in order to



36/44

74 D. Brigo et al

allow for only one default. Given the same repeated cluster processes QNs as before,

we define the new single-name default processes N1k

replacing Nk as solutions of the

following modification of Equation (16) for the original Nk :

dN1k .t / WD .1 N1k .t

//Xs3k

d QNs.t/

DXs3k

d QNs.t /Ys3k

1f QNs.t/D0g (19)

Remark 5.1 (Interpretation) This equation amounts to saying that name k jumps

at a given time if some cluster s containing k jumps (ie, QNs jumps) and if no cluster

containing name k has ever jumped in the past.

We can compute the new cluster defaults QN1s consistent with the single names

N1k

as:

d QN1s .t/ D Y

j2sdN

1j .t/ Y

j2sc.1 dN

1j .t// (20)

where sc is the set of all names that do not belong in s.

Now we can use Equation (18), with the N1k

replacing the Nk , to calculate how the

new counting processes Z1j are to be defined in terms of the new single-name default

dynamics:

MXkD1

dN1k .t/ D

MXkD1

.1 N1k .t//

Xs3k

d QNs.t /

D

M

XkD1

.1 N1k .t//

M

XjD1

Xs3k;jsjDj

d QNs.t /

D

MXjD1

XjsjDj

d QNs.t /Xk2s

.1 N1k .t//

D

MXjD1

XjsjDj

d QNs.t /Xk2s

Ys03k

1f QNs0 .t/D0g

This expression should match:

dZ1.t / WD

Xjj dZ1j.t /

so that the counting processes are to be defined as:

dZ1j.t / WD1

j

XjsjDj

d QNs.t/Xk2s

Ys03k

1f QNs0 .t/D0g(21)



37/44


The intensities of the above processes can be directly calculated in terms of the

density of the process compensator. By direct calculation, we obtain:

hN1k

.t/ D Ys3k

1f QNs.t/D0gXs3k

Qs.t/

hZ1j

.t/ D1

j

XjsjDj

Qs.t /Xk2s

Ys03k

1f QNs0 .t/D0g

9>>>>=>>>>;

(22)

where, in general, we denote by hX.t/ the compensator density of process Xat time t ,

referred to as intensity ofX, and where Qs is the intensity of the Poisson process QNs .

If we consider the repeated Poisson cluster default building blocks QNs to be

exogenously given, the model N1k

; QN1s ; Z1j is a consistent way of simulating the

single-name processes, the cluster processes and the pool-counting process from the

point of view of avoiding repeated defaults. In particular, we obtain:

NCt WD1

M

Xk

N1k .t / D1

MZ1t 6 1

Note, however, that the definition ofN1k

in (19), even if it avoids repeated defaults of

single names, is not consistent with the spirit of the original repeated cluster dynamics.

Consider the following example.

Example 5.2 (Single names versus clusters) Consider two clusters s D f1;2;3g

and z D f3;4;5;6g. Assume that no names defaulted up to time t except for cluster z,

in that in a single past instant preceding t names 3, 4, 5 and 6 (and only these names)

defaulted together (ie, QNz jumped at some past instant). Now suppose that, at time t ,

cluster s jumps, ie, names 1, 2 and 3 (and only these names) default, so QNs jumps for

the first time. Does name 2 default at t ?

According to our definition of N12 , the answer is yes, since no cluster containing

name 2 has ever defaulted in the past. However, we have to be careful in interpreting

what is happening at cluster level. Indeed, clusters z and s cannot both default since,

in this way, name 3 (in both clusters) would default twice. So we see that the actual

clusters default dynamics of this approach, implicit in Equation (20), does not have

a clear intuitive link with processes QNs . This is why, in Brigo et al (2007), we present

a second strategy to avoid repeated defaults that is also consistent at the cluster level.

5.4 Homogeneous pool limit

To simplify the parameters and combinatorial complexity, we may assume that the

cluster intensities Qs depend only on the cluster size jsj D j . Then it is possible to



38/44

76 D. Brigo et al

directly calculate the intensity of the pool-counting process C D Z1 as:

hZ1.t/ D 1 Z

1t

M Xj j M

j ! Qjwhere Qj is the common intensity of clusters of size j .

We see that the pool-counting process intensity hZ1 is a linear function of the

counting process C D Z1 itself, as would be expected from general arguments for

a pool of independent names (again with homogeneous intensities). In such a pool,

a default of one name does not affect the intensity of default of other names, and

the pool intensity is the common homogeneous intensity multiplied by the number

of outstanding names. Each new default simply diminishes the pool intensity of one

common intensity value and the pool intensity is always proportional to the number

(fraction) of outstanding names .1 NC /.

5.5 Credit default swap calibration

In this section we tackle the single-name CDS calibration using the above strategy for

avoiding repeated defaults. In such a strategy, the compensator for the single default

event of name k is given by Equation (22). This leads to the following expression for

the default leg of a CDS on name k with deterministic recovery R paying protection

up to maturity T, under deterministic risk-free interest rates, and assuming that the

cluster intensities Qs are deterministic:

E0defleg.0/ D LGD

ZT0

P.0;u/E0dN1k .u/

D LGDZT0

P.0;u/E0hN1k

.u/ du

D LGD

ZT0

P.0;u/Ys3k

E01f QNs.u/D0gXs3k

Qs.u/ du

D LGD

ZT0

P.0;u/Ys3k

eQs.u/

Xs3k

Qs.u/ du

D LGD

ZT0

P.0;u/ exp

Xs3k

Qs.u/

Xs3k

Qs.u/ du

which is the same as the default leg in a standard deterministic intensity single-name

model for the CDS when the intensity of name k at time t is given by:

hCDSk .t / WDXs3k

Qs.t /



39/44


In particular, if cluster intensities Qs are also constant in time, besides being deter-

ministic, and if we consider CDSs with continuous payments in the default leg, we

can see that single-name consistency with CDSs on name k having fair running spread

SkCDS.0;T/ for maturity T is achieved with:

Xs3k

Qs DSkCDS.0;T/

LGD

Imposing this (linear) constraint on the Q0s for all k values in the CDO calibration

ensures single-name consistency with CDSs. Of course, there is no need to assume

constant cluster intensities, so the method also works when one tries to fit the term

structure of both CDSs and CDOs. It is sufficient to go through the obvious general-

ization for piecewise constant cluster intensities.

This highlights a great advantage of this capping strategy for avoiding repeated

defaults: it makes single-name CDS calibration very easy to achieve.

5.6 Customizing the generalized Poisson cluster loss model to

deal with sectors

As was previously done for the GPL model, we can split the default counting process

and the loss process to factor out the Armageddon event in order to assign a zero

recovery to such a mode to allow for a better calibration of super-senior tranches.

We consider the rescaled pool-counting process NCt and the corresponding reduced

process Nc1

t , which can be defined as:

NCt WD1

M

MXjD1

jZ1j.t /

Nc1t WD1

M 1

M1XjD1

j dZ1j.t /

Starting from the counting process, we obtain:

d NCt D

1

M

MXjD1

j dZ1j.t /

D1

M

MXjD1

XjsjDj

d QNs.t/Xk2s

Ys03k

1f QNs0 .t/D0g



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78 D. Brigo et al

D 1f QN .t/D0g1

M

M1XjD1

XjsjDj

d QNs.t /Xk2s

Ys03k;s0

1f QNs0 .t/D0g

C 1f QN .t/D0g dQN .t/ 1

M

Xk2

Ys03k;s0

1f QNs0 .t/D0g

D 1f QN .t/D0g d Nc1t C 1f QN .t/D0g d

QN .t/.1 Nc1t/

and, by introducing the non-Armageddon indicator I1t , we obtain:

d NCt D I1t d Nc

1t C .1 Nc

1t/I

1t dI

1t ; I

1t WD 1fZ1

M.t/D0g D 1f QN .t/D0g

where is the set of all possible names.

For the GPL model we then introduce the stopping time O as the minimum time

between the Armageddon event time and the time when the reduced pool is over:

namely, the time when the full pool is over. Now we can integrate in order to obtain thecounting process by observing the following: firstly, I1 is zero after the Armageddon

event, otherwise it is equal to one; secondly, d I1 is non-zero only at the Armageddon

event; and finally, Nc is equal to one and d Nc is zero after the reduced pool is over.After

some algebra, we obtain:

NCt D 1fO6tg C Nc1t 1fO>t g

Note that NCt depends on the value of Nc1 only at time t and that the equation relating

the two counting processes has the same form as the corresponding Equation (13) of

the GPL model.

The same line of reasoning used for the GPL model may be repeated for the GPCLloss process as well by considering a zero recovery for the Armageddon event. We

obtain:

d NLt D .1 R/I1t d Nc

1t C .1 Nc

1t/I

1t dI

1t

and, by integrating:

NLt D .1 R Nc1O/1fO6tg C .1 R/ Nc

1t 1fO>t g

we obtain an equation relating the loss of the pool to the counting process of a reduced

GPCL model, which has the same form as the corresponding Equation (14) of the GPL

model, allowing for the same numerical techniques to calculate the loss distribution.

6 CONCLUSIONS AND FUTURE DEVELOPMENTS

In this paper we have considered CDOs. We analyzed their valuation (both pre-crisis

and in-crisis) using our pre-crisis GPL model, an arbitrage-free dynamic loss model



41/44


capable of calibrating all the tranches for all the maturities simultaneously. We have

also confirmed the CDO model-implied loss-distribution features that we had already

highlighted in Brigo et al (2006a): we found a multimodal loss probability distribution

that can also be obtained independently within the implied copula framework. This

behavior can be associated to the default of clusters of names within the CDO pool,

which, in turn, may be interpreted as default of sectors of the economy. We repe

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Credit Models and the Crisis