Credit risk models : theory, applications and implementation

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Credit Risk Models:

Theory, Applications and Implementation

Master's Thesis in Financial Economics

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Theory, Appliations and Implementation

By:

Ulf Nore

Advisor:

Steinar Ekern

NHH, Spring 2010

Thisthesis waswritten asapartof the master'sprogram atNHH. Neithertheinstitution, the

advisor, nor the sensorsare- through theapprovalof this thesis- responsiblefor neitherthe

theoriesand methods used, nor results andonlusions drawn inthis work.

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List of Figures

1.1 Historialdefault rates. Soure: Moody's (2000). . . 12

1.2 One year transition matrix of Standard and Poor's redit ratings for the period1981-1996. Soure: CreditMetris. . . 13

1.3 PhysialumulativedefaultprobabilitiesforsomeratingslassesfromFig- ure 1.2.. . . 14

1.4 OutstandingCDSnotional. Soure: ISDA(http://www.isda.org/statistis/). 16 2.1 Binomialtree illustratingdisrete time default proess. . . 20

2.2 Priingin the binomialmodel. . . 20

2.3 HazardrateasOrnstein-Uhlenbekproessandorrespondingdefaulttime df. . . 25

2.4 The inversion method. . . 30

2.5 CID Simulation . . . 31

2.6 Defaulttimes simulation with two dierent opula funtions. . . 35

2.7 Bivariatenormal opulasimulation. . . 36

2.8 Bivariatet-opulasimulation with one degreeof freedom. . . 37

2.9 Estimatingdefault probabilityfrom a singlebond prie. . . 39

2.10 Estimatingdefault probabilityfrom a singlebond prie. . . 41

2.11 Cumulative probabilty funtionestimated usinga set of bond pries. . . 41

2.12 Floatinglegash ows. . . 42

2.13 Fixed Legash ows. . . 42

3.1 The density funtion for the number of asset defaults. . . 51

3.2 Samplepaths fromBlak-Cox simulation.. . . 55

4.1 The onvergene ofthe redued formsimulationmodelfor a simplesingle asset priingproblem. . . 61

4.2 Value of aone year CDS with ounterparty risk. . . 62

4.3 Code from CopulaExample. . . 62

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4.4 Pries of a rst of 5 and rst of 20 to default binary basket CDS as a

funtion of asset orrelation. . . 64

4.5 Basket values for dierent degrees of freedom ina opula model. . . 64

4.6 SeniorCDO tranhe values agains

ρ

^and

λ

^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ⁶⁵

4.7 Binary CDOpries for various hazardrates and orrelationoeients. . 66

4.8 Cash owstruture of a CDS. . . 66

4.9 A simpleCDO struture. . . 68

4.10 ExpetedlossesinCDOtranhesundervariousassetorrelations. N=20,000 simulations. . . 70

4.9 Expeted losses in aCDO asa funtion of asset orrelation. . . 72

A.1 Sampletrajetoryof a Brownian motion. . . 76

A.2 Geometri Brownian Motion Paths . . . 78

B.1 Priingerror(relativetoBlakSholesprie)inanaiveMonteCarlomethod. 84

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Introduction

Background

Reallingthe reditrisis asoneof the events deningof the lastdeade, itisnowonder

redit risk modeling has beome one of the entral researh areas in modern nane.

Leadingup toand even more aftertheaforementionedreditrisis, therehas been muh

debate on the need for regulating redit derivatives, an asset lass by many viewed as

important for understanding the bakground for the risis. A natural extension of this

debate is that of the value of the models used for valuingan risk managingsuh instru-

ments,in partiular withrespet tothe quality of redit ratings. The same models have

also seen appliations in banking apital regulation, another area where previously held

beliefs have been hallenged by these events.

Overview

The fous of this thesis is the two main lasses of redit risk models that appear in

the aademi literature and are used by pratitioners innanial institutions and redit

rating agenies. There is no "industry standard" priing model for redit derivatives or

risk management, in the manner of the Blak-Sholes Model for stok options. I will

therefore over qualitatively some of the variation in the eld. Beause of the limited

sopeofthis thesis, thefous ofthis presentation isonthebasi priniplesand methods,

whihare presented ina detailedand more formalway. I willalso outlinehow the basi

models an beextended.

Therst twohaptersprovideanintrodutiontothesetwomodelframeworks,known

asreduedformandstruturalmodels,respetively. Struturalmodelsbuildmoreorless

diretlyonoptionpriingtheory,andmakespeiassumptionsontheausalrelationship

between struturalvariablessuhasassetvalues,debt level,interestrateontheonehand

andrediteventsontheother,viewingarediteventmainlyasanendogenousevent -an

event that is explained inside the models by other variables. Redued form models, on

the other hand, see defaults as exogenous. No ausal relationships are assumed, we are

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only trying to obtain a probabilisti model based on available market data and ertain

assumptions about the data generating proesses. A brief disussion on some simple

tehniques for modelalibration isalso inluded.

Chapter 4 illustrates some of the theoretial onepts developed in the preeding

hapters by appliations to derivatives priing. Finally, the appendies ontain a brief

overview over some onepts in valuation theory and stohasti modeling that are used

throughout the thesis,as wellas the simulationmethodsused inmodelimplementation.

Methods

Inthe eld ofredit riskresearh,thereare numerous artilesand booksontainingana-

lytialresultsforhighlysophistiatedmodels. Whilereognizing thepratialusefulness

of suh ontributions, I believe there are ertain important advantages to fousing on a

numerialapproah.

The onstraints related to omputational osts that used to be the main problem

with numerialtehniques have beomeless important due tothe exponentialgrowth in

omputingpower. Seondly,itan oftenbeasimplermodellingtask toimplementanu-

merialapproahthantosearhforanalytialsolutionsformanyomplexproblems, and

it is often suient with a selet set of numerial methods for takling many problems.

Analytial approahes onthe other hand, oftenrequire onsiderablemathematial inge-

nuity and sophistiationthat may be beyond many pratitioners. Furthermore, asimple

numerial model an often easily be extend to more omplex ases without modifying

ore parts of the program.

Acknowledgements

I wish to thank my advisor, professor Steinar Ekern, not only for his guidane and

advie that has been invaluable formy work withthis thesis,but alsofor histeahing in

nanial theory and derivatives priing at NHH that stirred my interest in the elds of

mathematial and theoretial nane, hereunder the methods and problems I disuss in

this thesis.

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Chapter 1 Credit Risk - Empirical Data and Some Notes on Modeling

"Credit default swaps(CDSs) have

proved to beone ofthemost

suessfulnanial innovations of the

1990s."

Hull andWhite (2003)

1.1 Background

Finanial ativities reate wealth whenever they lead to a more produtive alloation of

apital and risk between the agents in the eonomy. For many agents, nanialinstitu-

tionsinpartiular,thehandlingofredit risk i.e. theriskofaborrowerbeing,totallyor

partially,unabletorepay aloan is anissue of utmostimportane. Untilquitereently,

managingreditrisk hasbeen diultdueto thelowliquidityof debt seurities, sothat

agents have been unable to redue their exposure to suh risk, either by selling debt

instruments or taking osetting positions in other instruments. While traditional debt

instruments,suh asorporatebonds, obviouslyare redit derivatives, they alsohavean

embedded interest risk element, whih make them less ideal for tradingand transferring

redit risk.

In the last two deades, the way nanial institutions handle redit risk has been

alteredin afundamentalway by the introdutionof modern reditderivatives, the most

important being the redit default swap or CDS. The CDS is a simple instrument that

fora periodipayment guarantees protetionagainstthe redit riskof arefereneentity,

usually interms of some predened ash settlement between the issuer and the buyer of

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Figure1.1: Historial default rates. Soure: Moody's(2000).

redit protetion in event of default, wholly or partially overing the loss aused by the

redit event. An institutionhaving a largeredit exposure to somepartiular entity an

therefore use aCDS toneutralize this position. Furthermore, itis of ourse unneessary

toatually hold the underlyingbonds inorder toobtaina ertainrisk prole;tradingin

CDS's aloneis suient, astheseinstrumentsan beissued independently of whetheror

not the bonds are atually issued.

1.2 Data Sources and Some Empirical Facts About Credit Risk

From Figure 1.1, where the historial over-all US orporate default rates are plotted as

a time series together with the US Industrial Prodution Index (a measure of eonomi

growth),wegetafewimpressionsofsomepropertiesofdefaultrates. Thoughfairlyweak

(

− .14

^), ^there îs â ôrrelation ^between ^the ÎP îndex ând ^default ^rates. ^Strong êonomi

growthtendstogohandinhandwithlowdefaultrates,thoughtherehasbeen avariying

pattern with resepet to whether a weakening of the eonomy preeeds or follows an

inrease in default rates.

Another onept of key interest inreditrisk modeling inadditiontodefault rates is

defaultseverity,oftenreferredtoaslossgivendefault,usuallyaperentageofoutstanding

prinipal. Moody's (2000) have ompiled similar data for this quantity, and it exhibits

similar time series properties. On average, reovery rates are low near the bottom of

business yle ontrations and high after periodsof strong eonomi growth.

The ylial nature of reditrisk that isapparent fromFigure 1.1isalsoreminisent

of the problem of default orrelation or lustering, the fat that one default tends to

be followed by others. We an explain suh ausality by onsidering the dependene

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between rms in a supply hain; if a major buyer shuts down prodution, the suppliers

arealsomorelikelytodefault. Weanalsothinkofhowsimilarrmsdependonthesame

maroeonomi fators suh as fuel pries, and aggregate demand, et., and partiular

risk fators suhas trends orhypes.

1.2.1 Ratings Data and the Estimation of Default Probabilities

As defaults are infrequent low-probabilityevents,empirialdata ondefault probabilities

and interdependenes are hard to ompile. Of ourse, for a rm that has not defaulted,

we annot diretly observe its default probability as this is an event that only ours

one. Hene, we need to ome up with some estimates of these probabilities based on

data available for similar rms, or imply them from market pries using some priing

model.

AAA AA A BBB BB B CCC Default

AAA 90.81 8.33 0.68 0.06 0.12 0 0 0

AA 0.70 90.65 7.79 0.64 0.06 0.14 0.02 0

A 0.09 2.27 91.05 5.52 0.74 0.26 0.01 0.06

BBB 0.02 0.33 5.95 86.93 5.30 1.17 0.12 0.18

BB 0.03 0.14 0.67 7.73 80.53 8.84 1.00 1.06

B 0 0.11 0.24 0.43 6.48 83.46 4.07 5.20

CCC 0.22 0 0.22 1.30 2.38 11.24 64.86 19.79

Default 0 0 0 0 0 0 0 100.00

Figure 1.2: One year transition matrix of Standard and Poor's redit ratings for the

period 1981-1996. Soure: CreditMetris.

One ommon method for estimating suh probabilities is using data published by

ratingageniessuhasStandard&Poor. ConsiderFigure1.2whereaoneyeartransition

matrix of redit ratings is given. Entry

a ij

ⁱⁿ ^the ^table ^gives ^the probability of a rm going from rating

i

^to ^rating

j

ôver ^the ôurse ôf ône ^year. ^There âre ^several ^things

to note about suh data. We see that the default state is absorbing; one a rm has

defaulted, it willnever live again, and the probability of transitionfrom default to any

other rating is onsequently zero. Furthermore, the transitionprobabilities are physial

probabilities. This should be quite obvious as they are estimated from atual historial

data. They willtherefore generally dierfrom the risk neutraldefault probabilities that

an be impliedfrommarketpries. This methodis disussed inSetions 2.6.1 and 2.6.2.

Appendix A explainsthe distintionbetween risk neutraland physial probabilities.

Usingthismatrixitissimpletoomputethe n-yearprobabilitymatrix. If

T ₁

denotes the one year transition matrix, then the two year transition matrix is given by

T ₂ =

T ₁ · T ₁

. To see that this holds onsider the probability of starting in state AAA, and

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being in stateAAA aftertwo years. This isthe probability of stayingin AAA two years

in arowplus the probability of going from AAA to AA the rst year and bak toAAA

the seond, and soforth:

p ² (AAA | AAA) = p ² _AAA,AAA + p AAA,AA p AA,AAA + p AAA,A p A,AAA + ... + p AAA,CCC p CCC,AAA

In the same manner, we an nd the n-year transition probability matrix as

T _n = T ⁿ

1

^. Considering only the rightmost olumn of the matries

{ T ₁ , T ₂ , ..., T _n , }

^we ^have

estimates of the physial default probabilities for a rm of a given rating, for any time

horizon. Theumulativedensity funtionfollowingfromthismethodisplottedinFigure

1.3.

1 2 3 4 5 6 7 8 9

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Time in Years

Cumulative Default Probability

aaa aa a bbb bb

Figure1.3: PhysialumulativedefaultprobabilitiesforsomeratingslassesfromFigure

1.2.

Problems with Ratings Data

There are several reasons why probabilities implied from market data using models is

preferableto ratingsdata for the priing appliations:

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•

^Rating^agenies^reat^slower^than^the^marketⁱⁿantiipationoffuturereditquality.

Themost strikingexampleis thereent reditrisiswhere thesub-prime mortgage

baked seurities defaulted with atriple-A status.

•

^Firm^spei informationontained in marketpries is ignored;the default proba- bilitiesinferredfromratingsdataareaveragesoverapotentiallyveryheterogeneous

group of rms that are likelyexposed tovery dierent risk fators.

•

^The probabilities are physial, and an therefore not be used diretly as input to the valuation models asthey usually are stated.

1.2.2 Credit Derivatives Markets

Aswehaveseen,therearegoodargumentsforthatratingsdatamaynotbethebest data

soure for estimating default probabilities. Often, a better alternative is to use market

data. Therearethree importantmarkets fromwhihwe aninferredit riskinformation

usingthemodelingtoolsdisussedlater. Thesearetheequity,bondandreditderivatives

markets. This thesis explores some methods for implying redit risk information from

the seurities tradedin these markets.

Obviously,the quality of suh informationdepends ruiallyon the liquidityand the

transparenyofthenanialmarkets. Ifmarketpartiipantsareuninformedwithrespet

totheassetsthataretraded,themarketpriesdonotreetatualvaluesorprobabilities

andisthereforeworthless. Likewise, ifmarkets areilliquid,marketpriesmay notreet

atual asset values. The latter is often a problem with using bond pries whih is why

redit derivativepries are often preferredin estimating default probabilities.

Furthermore, redit default swap rates are usually quoted for a larger number of

maturities than bonds whihmeans a ner rediturve 1

. This approah isillustrated in

Chapter 2.

1.3 Modeling Credit Risk

Thetwolassesofmodelspresentedherean beseenasrepresenting twodierent"tradi-

tions". Struturalmodelsarestraightforwardextensionsoflassialoptionpriingtheory,

and was indeedone of the rst appliations of this theory outside ontingentlaims val-

uation (see for instane Merton (1974) and Blak and Cox (1976)). They rely expliitly

ona theory onthe ausalrelationship between asset pries 2

and bankrupty.

1

Theredit urve isommontermdesribingthetermstrutureof defaultprobabilities.

2

Or, in moreadvaned asessuh as Goldstein et al. (2001), the relationshipbetween ash ows,

interestrateset. andbankrupty.

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2001 0 2002 2003 2004 2005 2006 2007 2008 2009 2010 1

2 3 4 5 6 7 x 10 ⁴

Year

Outstanding CDS Notional, $10 9

Figure1.4: OutstandingCDSnotional. Soure: ISDA (http://www.isda.org/statistis/).

From a theoretial point of view, strutural models are for many reasons the "pre-

ferred"framework,asthey notonlyprovideaausalrelationshipbetween the strutural

variablesofthermandthedefaultprobabilities,butalsoaoherentframeworkforvalu-

ingany laimontherm'sassets. Furthermore,they anbeextendedinmanydiretions

inorporating, among other things, endogenous apital struture hanges, so that there

isaninterdependene between asset values andapitalstruture deisions. Suhmodels,

whihin the literature is referred to asdynami apital struture models, donot appear

oftenpratieinreditderivativesvaluationasthey arehardertoalibratethanthe sim-

pler stati apital struture modelsonsidered here. Therefore, itis assumed throughout

the disussion on strutural models in this thesis that the apital struture irrelevane

assumption 3

holds.

As pointed out by Vasiek (1984), modern strutural redit risk models are purely

quantitative, and is therefore radially dierent from "traditional" methods for asset

priing and reditvaluationthat relies onthe analyst's knowledge of arm's operations

to projet future ash ows under various senarios. However, the data used in the tra-

ditionalmethod,both abouttherm andthe markets inwhihitoperatesispresumably

publi information. Assuming aertain degreeof market eieny, this informationwill

alreadybereetedinthepriesoftherm'sassetsasreetedbydebtandequityvalues.

Redued formmodelsrepresentanapproahbasedonreliabilitytheory thatissimilar

3

Theassetvalueisindependentofthenanialstrutureofanentity.

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to modeling in insurane and operations management. They do not model ausal rela-

tionships between strutural variables, rather use default probabilities as inferred from

market pries. A default is onsidered "similar" to the ourrene of an event trigger-

ing a payment from an insurane ompany, or in the operations management ase, the

breakdown of a partiular mahine that is part of a prodution proess. There may be

several auses behind suh a breakdown; it may be due to human failing or have some

tehnial ause. In the model however, these are seen as random events ourring a-

ordingto some proess. From a modelingperspetive, we are interested indetermining

this proess than the atual ausality.

What separates the reditrisksettingfromthe operationsmanagementsettingisthe

role of interdependene. Default time interdependene is a major risk fator that must

be aounted for in redit portfolio valuation and risk assessment. While in a produ-

tion proess, simultaneous breakdowns may be preferable so that a total maintenane

an be performed, a large number of defaults ourring over a short period of time is

learlyproblematiforananialinstitutionwithalimitedashowandapitalreserve.

Another important problemis that inmany pratial problems the redit portfoliomay

ontain alarge numberof assets, sothat inorder to"sale down" the problemin suh a

way that we an make qualitativesense of the data, some redution of dimensionalityis

neessary. This topi is entralthroughoutthis thesis and aswe willsee, many dierent

methods are proposed in the literature. One standard method is to assume orrelation

arises through the individualassets' dependene ona set of systemi risk fators.

1.4 Evaluating Models

This thesis presents the twofundamental lasses of reditrisk models as wellas some of

theseveral extensionsofthesemodelsthathavebeenproposed. Fromapratialpointof

view, itisneessary tohavesomeriteriabywhihthesemodelsare evaluateddepending

ontheir appliation.

BasedonthenatureofdefaultssuggestedbyempirialstudiessuhasMoody's(2000),

we an speify requirements a model should be able to reprodue with respet to key

quantities like default rate orrelations and default probabilities. As demonstrated in

the CDO example in 4.4, multi-name redit derivative values are extremely sensitive to

default orrelations 4

. Furthermore, the analyst implementing the model is faed with

several important onstraints suh as:

•

^Sarity ôf ^data. ^Dataôn ^defaults îs^limited ⁱⁿ ^many ^respets. Ône ^may ^not ^have

suientlylongtimeseriesavailableortheremaybehangesinthedatagenerating

4

AsdisussedinHull(2007),thisisquitelearfrom theash owmehanisoftheseinstruments.

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proesses 5

sothat older observations are no longervalid. Hene, a modelwith few

parameters toestimateis tratable due to the unertainty in the estimates.

•

^Time ^onstraints ⁱⁿ implementing, testing and alibrating the models. A simple numerialmodelisoften simplertoverify against ananalyti base ase.

The lastpointshows that there isan important trade-obetween the rihness of the

modelandthetimespentonimplementingandmaintainingit. Thefoushereistherefore

the basi ases of the models that are treated thoroughly in a quantitative manner and

implemented numerially. Extending these is usually a quite straightforward issue of

adding more"bells and whistles" to the fundamentals.

5

Suhshiftsmaybeaused forinstane beausedbyregulatoryhanges.

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Chapter 2 Reduced Form Credit Risk Models

Thishapterprovidesanintrodutiontothetheorybehindoneofthetwostandardlasses

of redit risk models often referred to as redued form redit risk models. Aording to

Hulletal. (2006),this lassofmodelsislargelytheindustrystandardinreditderivative

modeling,primarily beause they are easyto tto observed marketpries.

This hapter also pays some attention to dierent methods for orrelation modeling

that are also used later onfor strutural models. Partiularattention is paid to the so-

alled opula approah that providesa tehnially eientmethodfor implementing the

multivariatedistributionof asetof assetsgiven themarginaldistributionsand estimates

of orrelations.

2.1 Single Credit Framework

Consider a single defaultable seurity and let

τ

^denote ^its ^survival ^time ^as ^measured

from

t = 0

^. ^On ^the ^lteredprobability spae

1

( P , F , Ω)

^. ^Here

P

denotes the riskneutral probabilitymeasure.

τ

^is^a^stopping^time^(a^random^variable)^with^respet^to^the^ltration

F t

^that ^represents ^the ^aumulated ^marketinformationavailable attime

t

^.

We are now interested in aframework inwhih probabilististatements about

τ

^an

be made. Therefore let

F (T ) = P (τ ≤ T )

^be ^the ^umulative distribution funtion (df) of thedefault time,ie. theprobability thatthe time ofdefault oursbeforeapartiular

time

T

^. Ân êquivalent ^statement îs ^the ^survival ^funtion

S(T ) = 1 − F (T )

^whih ^is

the probability that a seurity does not default prior time

T

^. ^Closely ^related ^to

F (T )

^is

the probability density funtion (pdf)

f (t) = ^dF _dt ^(t)

^that ^an ^beinterpreted as the default probability onaninnitesimallysmalltime interval aroundsome point intime

t

^.

1

SeeAppendixAforsomebakgroundandreferenesonthis terminology.

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2.1.1 A Binomial Model of Credit Risk

As an illustration, onsider a bond with fae value 100, and 6% annual oupon rate

paid annually until maturity in year 3. Let

q = λ∆t = .08

^be ^the ^onditional ^risk

neutral default probabilityonanintervaloflength

∆t = 1

^,^i.e. ^the probabilityof default ourring during the interval, onditioned on survival up to the start of the interval.

Furtherassumethat if defaultours, thevaluereovered isonstant

R = 40

^paid ^at^the

end of the year asillustrated in Figure2.1.

40 1.05

6 1.05

40 1.05 ²

6 1.05 ²

106 1.05 ³

40 1.05 ³

q 1 − q

(1 − q ) ² q (1 − q )

(1 − q ) ³ q (1 − q ) ²

Figure2.1: Binomialtree illustratingdisretetime default proess.

AsdisussedinAppendixA,weanvaluethis riskybondbydisountingtheexpeted

ashowsbytheriskfreeinterestrate,hereassumedtobe5%withdisreteompounding.

The example is summarized in Figure 2.2. The probabilities in row 2 and 4 are the

umulativesurvivalandannualdefaultprobabilities,respetively. Toarriveattheresults

here, the onditional probability of default in a partiular year is the probability of

surviving up tothat year times the probability of defaultingin that partiular year.

Year 1 2 3

Cash ow, survival 6 6 106

CumulativeProbability 0.9200 0.8464 0.7787

Cash ow, default 40 40 40

AnnualProbability 0.0800 0.0736 0.0677

Expeted Cash Flow 8.7200 8.0224 85.2494

Disounted Cash Flow 8.3048 7.2766 73.6416

Expeted NPV 88.2230

Figure2.2: Priing inthe binomialmodel.

In omparison,thepresentvalueofariskfreebond withthesameash owstruture

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is

6 · 1.05 ⁻ ¹ + 6 · 1.05 ⁻ ² + 106 · 1.05 ⁻ ³ = 102.7232

^,^so^the ^risk ^premium^on^the ^risky^bond

is

13.5003

^. Ît îs âssumed ^throughout ^this ^thesis ^that ^redit ^risk îs ^the ônly ^risk ^fator.

In reality, suh a prie dierene is usually explained in terms of other, additional risk

fators, liquidity riskbeing the most important.

Mixed Probability Binomial Models

In many valuation problems, the binomial model is an exellent tool; its primary ad-

vantage being its tehnial simpliity and intuitive nature. It is the among the simplest

derivative priing models to understand, explain and implement numerially, yet pow-

erful enough to to repliate the results from simulation models in many ases given a

suiently smallstep size.

The key problemwiththismodelasitisformulatedaboveisthat itdoesnot aount

for dependene between default times whih is, as mentioned in the introdution, one

of the most important risk fators that any redit risk model must handle well if it is

to be applied to portfolio modeling. One ommon extension of the binomial model is

to randomize the default probability

q

^to ^mimi^dependene ^between ^the ^binomial ^trees

representing the variousrms in the portfolio.

While suh binomial models are used in pratie, the next setions, take a dierent

approah tomodeling orrelationthat uses aontinuous time framework.

2.1.2 The Hazard Rate Function

Akeyquantityofinterest 2

istheinstantaneousdefaultprobabilityonditionalonsurvival

up to a ertain point in time

t

^. ^This probability is often referred to as the hazard rate funtion. It isdened asthe limitofthe probabilityof survivalonaninterval

(t, t + ∆t)

^,

given

τ > t

^, ^as

∆t

^approahes ^zero:

Denition 2.1.1. Hazard Rate Funtion

Let

F (t)

^be ^the ^umulative distribution funtion of the default time

t

^and

f(t)

^its

derivative, then the hazard rate funtion

λ(τ)

^is ^dened ^as:

λ(τ) = lim

∆t → 0

P [t < τ < t + ∆t | τ > t] = f (τ)

1 − F (τ ) = f(τ )

S(τ)

^(2.1.1)

Thelastequalityanbeseenbywritingouttheprobabilitiesasintegralsandapplying

the fundamental theorem of alulus tothe numerator and reognizing the denominator

as

1 − F (t)

^.

2

Thisisbeauseitspeiesthedefaultgeneratingproessin thismodelframework.

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Note that we are yet to speify the funtional form of

F

^,

f

^and

λ

^as ^we ^have ^so

far only dealt with them abstratly. In the the example in Setion 2.1.1,

λ

^is ^assumed

onstant and

F (t)

^is^on ^the ^form:

F (n∆t) = λ∆t + (1 − λ∆t)λ∆t + (1 − λ∆t) ² λ∆t + ... + (1 − λ∆t) ⁿ λ∆t

Here

λ∆t

^is ^the probability of defaulting on an interval of length

∆t

^. ^In ^the ^next

setion we onsider a modelwhere

λ

âts âs ^the ^parameter ⁱⁿ â ôntinuous ^default ^time

distribution.

2.1.3 The Poisson/Cox Process

As initially noted, we want to provide some model of defaults as the ourrene of a

disrete and rare event without, as in the strutural models onsidering the underlying

eonomi proesses driving these events. A simple example of a proess satisfying these

requirements is the Poisson proess

N (t)

^whih îs â ôntinuous ^time, ^disrete ^spae

ounting proess. Wewant todenethe defaultof asset

i

^as^the^rst ^jump^of^the ^proess

N i (t)

^. ^Theinterdependene between the rms inthe portfoliois given by the orrelation struture of a set of Poisson proesses.

Walpoleet al. (2007)denes the Poisson proess interms of three key properties:

Denition 2.1.2. Poisson Proess

Let

I

^be ^the ^indiator ^funtion ^assoiated ^with ^the ^stopping ^time

τ

^. ^The ^Poisson

proessisa funtion

F : Ω → N ⁺

mapping thesample spae tothe set ofpositiveintegers suh that:

N(t) = X n

i=1

I τ i ≤ t

^(2.1.2)

satisfying the following properties:

1. The Markov property or "memorylessness": the number of events ourring on a

time interval

[t 0 , t 1 ]

^is independent of the number of events ourring on any other disjoint time interval

[T 1 , T 2 ]

^.³

2. The probability of an event ourring on a partiular time interval is proportional

to the length of the interval.

3. The probability of more than one event ourring an an innitesimal time interval

is negligible.

3

Inpartiular,anyeventourringonatimeintervalstartingat

t

^isindependentof

F ^t

^(here: ^the^set

ofinformationrevealedtothemarket(historialdefault data)).

(24)

Some Properties of the Poisson Distribution

Two importantonsequenes of this denition are:

•

^The probability distributionof N(t) isthe Poisson distribution,that is, the probability of exatly

k

êvents ôurring ôn â ^time întervâl ôf ^length

τ

^is ^then ^given ^by

the probability mass funtion of the Poisson distribution:

F (T, k) = P [N (t + T ) − N (t) = k] = e ⁻ ^λT (λT ) ^k

k!

^(2.1.3)

•

^In ^partiular ^we ^see ^that ^the probability that no defaults our on a given time intervalis given by:

F (T, 0) = P [N (t + T ) − N (t) = 0] = e ⁻ ^λT

^(2.1.4)

That is, the probability distribution of the waiting time until the rst ourrene

is anexponential distribution with parameter

λ

⁴^.

The lastpointaboveisimportantasweinterpretthetime

τ ₁

^of ^rst^jump^as^the ^time

of default. The time to default (or survival time) is therefore exponentially distributed

with a mean

1 λ

^and ^variane

1 λ ²

^. ^Note ^that ^weôuld âlso ^start ^with ^the âssumption ^that

timetodefaultisexponentiallydistributed,andthenarriveattheabovedenitionofthe

Poisson proess.

We an show the latter by onsidering a disrete setting where

λ(t)h

^denotes ^the

probability of surviving on an interval

[t, t + h]

^onditional ^on ^no ^previous ^default. ^The

umulativeprobability of surviving up to time

t

^is

p s (t)

^. ^It ^follows ^that:

p _s (t + h) − p _s (t) = − λ(t)p _s (t)h

Taking the limitas

h → 0

^:

dV

dt = − λ(t)p s (t)

whihhas the solution:

p s (t) = e ⁻ ^R ⁰ ^t ^λ(s)ds

We say that

N (t)

îs â ôunting ôr "jump" proess. We interpret the time

τ

^of ^the

ourrene of the rst "jump" of the proess

N (τ)

^as ^default.

4

Fornotationalsimpliity,

λ

îsâssumedônstant^here.

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The Poisson proess is entirely speied by a single parameter

λ

^, ^the ^hazard ^rate,

often referred to as the proess' intensity, whih is as the name indiates, a measure of

the frequeny of events ourring. The Poisson proess, or asit is sometimes alled, the

(time)homogeneousPoissonproessisapartiularase ofthe moregeneralCox proess,

where

λ(t) = λ

îsâônstant. ^Laterôn,

λ(t)

îs^dened ⁱⁿ^termsôfâ^stohasti^dierential

equation soas toallow forrandom variationsin default intensities.

ThedefaultofasinglereditisinthisframeworkgivenastherstjumpofthePoisson

proesswhihisthe rst passagetimeto

N (t) = 1

^,

τ

^dened^similarly^to^a^default ⁱⁿ^the

Blak-Cox model:

τ = inf { t ∈ R ⁺ | N (t) = 1 }

^(2.1.5)

The Credit Curve

The notion of a term struture of default intensities or, more olloquially, redit urve

is ourring frequently in the literature on redit risk. Similarly to the yield urve in

interest ratemodeling, expressing theyield onashortinterval

[t, t + dt]

^, ^the ^redit^urve

is the instantaneous default probability or hazard rate on a short interval. The redit

urve does of ourse ontain preisely the same information as the survival or default

time distributions.

The Cox Process

The above Poisson model an be generalized to allowing for a time varying and even

stohasti default intensity. This type of proess is referred to as a Cox proess or a

non-homogeneous Poisson proess. For instane, we ould allow

λ = λ(t)

^to^be^given ^by

the followingstohasti dierentialequation (SDE):

dλ(t, λ(t)) = µ(t, λ(t))dt + σ(t, λ(t))dW (t)

^(2.1.6)

where

W (t)

îs^the^standardûnivariate^Wiener^proess^dened ⁱⁿÂppendixÂ. ^Wêân

thinkoftheproessdrivingthisasthe"stateoftheeonomy",where

λ(t)

^will^be^inversely

relatedtostatevariablessuhasGDPgrowth,reditspreadsandsoforth. Oneapproah

tomimi the yliality apparent in atualdefault data is touse a mean-reverting SDE,

suh asthe Ornstein-Uhlenbek proess dened inAppendix A.

Fromtheinstantaneousdefaultprobabilityitisasimplemattertoderiveanexpression

forthe probabilityof aseurity surviving onatime interval

[t, T ]

^onditional^on^no^prior

default as the "sum"of allthe instantaneousdefault probabilities:

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p s (t, T ) = P [τ > T | τ > t] = E

exp

− Z T

t

λ(s)ds F (t)

(2.1.7)

The probabilityof default ourring onthe same interval isdenoted

p _d (t, T )

^:

p d (t, T ) = 1 − p s (t, T )

^(2.1.8)

These integrals are not neessarily simple or even possible to evaluate analytially.

This depends on the funtional form of

λ

^. ^However, ^simple ^numerial ^methods ^often^do

a good jobapproximatingthem.

In the homogeneous ase (onstant

λ

^),^the ^survivalprobability an be simplied:

p s (t, T ) = e ⁻ ^λ(T ⁻ ^t)

^(2.1.9)

and likewise the umulativedefault probability:

p d (t, T ) = 1 − e ⁻ ^λ(T ⁻ ^t)

^(2.1.10)

1 1.5 2 2.5 3 3.5 4 4.5 5

0 0.05 0.1 0.15 0.2 0.25

Time in years

λ − instantaneous default probability

1 1.5 2 2.5 3 3.5 4 4.5 5

0 0.1 0.2 0.3 0.4

Time in years p d − cumulative default probability

Figure 2.3: Hazard rate as Ornstein-Uhlenbek proess and orresponding default time

df.

Figure 2.3 illustrates the relationship between hazard rates and the umulative de-

fault probability. Here the hazard rate funtion is given by the stohasti dierential

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equation

5

dλ t = α(λ 0 − λ t )dt + σdW t

^. ^As ^before, ^the ^survival probability is

p s (0, t) = exp[ − R t

0 λ(s)ds]

^. Conversely, the umulative default probability

p d (0, t) = 1 − p s (0, t)

^.

The top gure shows a partiular trajetory for the mean-reverting default intensity

proess. To ompute the integral behind the seond gure, the midpoint method for

numerial integration 6

is used. Note how the df below is at in the times where the

default intensity is low and steep later on when

λ

îs ^high. ^For â ^simulation^model, ît îs

neessary tosimulate a large number of trajetories for

λ

^.

Summary

Toonlude the disussion here,werestate somekeypointsthatare entraltothesimu-

lationalgorithmslateron. Withaonstanthazardrate

λ

^,^time^to^default^isharaterized by an exponential distribution. The propertiesof this distribution is summarizedbelow.

•

^Cumulativeprobability distributionof defaulting prior to

t

^:

F (t) = 1 − e ⁻ ^λt

^.

•

Correspondingprobabilitydensity funtion

f(t) = λe ⁻ ^λt

^.

•

^Mean ^survival ^time:

1/λ

^and ^variane:

1/λ ²

^.

2.2 Cash Flow Pricing in a Reduced Form Model

From the above framework it is possible to work out formulas priing risky ash ows

using its default probability and an interest rate model. Consider rst the simple ase

of nding the time

t

^value ^of ^a defaultablezero ouponbond

G(t, T )

^paying âûnit âsh

ow attime

T

ôntingent ôn^survivalând ^nothing ôtherwise⁷^.

Letting

P (t, T )

^denote^the^risk-free ^disount^funtion^we^have^the^following^whih^is^a

diretappliationoftheriskneutralpriingframeworkdesribed earlier 8

foradefaultable

zero ouponbond:

G(t, T ) = E [P (t, T ) |F (t)] = P (t, T )p s (t, T ) = e ⁻ ^R ^t ^T (r(s)+λ(s))ds

(2.2.1)

When both the hazard and interest rates are stohasti proesses, there is a resem-

blane between the above priing equation and the bond priing expressions found in

5

Thereisaveryimportantproblemtonoteaboutusingthispartiularproessasamodelfordefault

intensities; namelythatitis not stritly non-negative,learlyatoddswiththedenition ofthehazard

rateasaprobability.

6

SeeCheneyandKinaid(2007).

7

This assumptionwill berelaxed lateron. Inthe mostgeneralasethefration lostto bankrupty

ost

α(t)

îs^speiedâsâ^stohasti ^proess.

8

Under the standard assumptions of arbitrage free markets, the same results hold for almost any

proessforassetvalues.

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multi-fator interest rate models 9

. In the ase of onstant default intensity and interest

rates we get a very simple priingequation:

G(t, T ) = e ⁻ ^(r+λ)(T ⁻ ^t)

From these equations, it is reasonable to interpret

λ

âs â ^risk ^premium. Ûsing ^these

equations,anyotherdefaultableseurityanbepriedsimilarlytotheabovezerooupon

bond.

2.2.1 Recovery Rates

The above example is learly stylized as it assumes that reovery rates are zero; either

thereisaunitashowattimeTorthereisnoashow. Thisisofourseunrealisti,and

asinthestruturalmodelsofChapter3,weanintrodueareoveryvalueproportionate

tothe faevalue of the bond.

This approah isknown asreovery of faevalue (RFV), and is perhapsthe simplest

possible approah,in partiular whenthe fration reovered isonstant. More advaned

models mayapply reovery ofmarketvalueormodelthe frationreovered asastohas-

ti proess. Hull (2006) disusses a number of dierent models of reovery rates with

referenes tothe literature.

Let

α

^denote^the ^fration ^reovered,

τ

^the ^stopping^time^indiating^default, ^the^value

of a defaultablezero oupon bond with unit fae value is now given as:

G(t, T ) = E [P (t, T ) + αP (t, τ ) |F (t)]

^(2.2.2)

While a losed form expression an be derived for the above expetation, I will only

onsider an intuitive numerialmethod of evaluating the integrals using a midpoint ap-

proximationand ompute the expetations by Monte Carlosimulation.

2.3 Default Correlation and Model Implementation

Now that a redued model of default probability and single entity or asset priing has

been established, the key problem still remains, namely speifying dependene or asso-

iation struture between default times. While the primary question of interest is the

orrelations between default times, it is important to stress that it is not the only. In

more advaned models we are also interested in the relationship between variables suh

9

Even thoughthere isawell-establishedtheoryonmulti-fator interestratemodels,workingoutan

analytiexpressioninthemostgeneralasewithorrelatedratesisnon-trivial.

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as default, reovery, interest rates, et. In this thesis, the main onern is default time

dependene.

Before we an start implementing a model, an appropriate measure of interdepen-

dene must be hosen. Whereas this is a relatively simple matter in terms of strutural

models, whereitisoneusually an settlewiththe orrelation

< dA 1 , dA 2 >

^between ^two

It proesses (see Shreve (2004) for rigorous denition), there are several approahes to

modelingasset prieinterdependene inredued formmodels. As disussed in Li(2000)

andElizalde(2005a),oneouldhoose thestandardPearsonorrelationoeientgiven,

in the bivariatease, as:

ρ _XY = cov[X, Y ] σ X σ Y

Translating this into our framework of defaultable seurities, we an let

1 A (t)

^and

1 B (t)

^denote ^two îndiator ^random ^variables ^taking ôn ^the ^value ône îf êntity Â ôr ^B,

respetively, have defaulted by time t. Letting

p _A (t)

^be ^the probability that A defaults prior totime

t

^:

var(1 i ) = p A (t)(1 − p A (t))

and:

cov[1 i , 1 j ] = p ij − p i p j

we get the following:

ρ XY = p AB − p A p B

p p A p B (1 − p A )(1 − p B )

(2.3.1)

For a partiular lass of multivariatedistributions, known as elliptial distributions,

whih inludesthe importantGaussian distribution, the orrelation oeient (or more

generally, the orrelation matrix) fully determines the dependene struture. However,

it an be problemati due to its linearity whih means that we an have a fully deter-

ministi relationship between two variables yet zero orrelation. A simple illustration

is if

X ∼ Φ(0, 1)

^and

Y

îs ân êven ^funtion ôf

X

^, ^for ^instane,

Y = X ²

^. ^Obviously^,

this is problemati, as we want a zero orrelation oeient to signify that there is no

assoiation between the variables. This is a key problem that is disussed later in the

setiononopulas. SueittosayfornowthatthePearsonorrelationmeasureremains

importantinthis analysis,in partiular as aninput tothe opula models.

2.3.1 Simulating Defaults – The Inversion Method

Wehave nowovered suient detailto develop asimple simulationalgorithmwhen we

know the funtional form and parameters of

λ(t)

âs ^well âs ^the ôrrelation ^matrix

Σ

.

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Let

X

^be ^a^random ^variable ^and

F

^be ^some ^assoiated ^umulativedistribution funtion funtion (df). Sine

F

^is^a non-dereasing funtion, ithas aninverse

F ⁻ ¹

^:

F ⁻ ¹ (q) = inf { x : F _X (x) ≥ q }

^(2.3.2)

From the denition of the df and the properties of the uniform distribution, the

following important relationship that is entral to the simulation algorithms applied to

redued formmodels follows. Let

U

^be âûniform ^random^variable ôn ^the întervâl

[0, 1]

^.

Then we have the following relationship:

P [X ≤ x] = P [F ⁻ ¹ ( U ) ≤ x]

^(2.3.3)

= P [F (F ⁻ ¹ ( U )) ≤ F (x)]

^(2.3.4)

= P [ U ≤ F (x)]

^(2.3.5)

= F (x)

^(2.3.6)

The rst equalityuses the fatthat

X = F ⁻ ¹ ( U )

^. ^T^o^see ^this, ^onsider ^the ^partiular

ase ofdefaulttimes. Nowthe domainof

F

^is

R ⁺

anditsrangeis

[0, 1]

^(by^the ^denition

of a probability). The inverse

F ⁻ ¹

^, ^therefore, ^must ^transform ^elements ⁱⁿ

[0, 1]

^onto

R ⁺

aording to the df. The last equality above follows from the property of the uniform

distribution on

[0, 1]

^, ^that

P ( U < u) = u

^.

So we have that

X

^and

F _X ⁻ ¹ ( U )

^have ^the ^same ^df. ^Thus ^random^variables^with ^any

givendfanbesimulatedbydrawinguniformrandomvariablesandapplyingtheinverse

df. This algorithmisknown asthe inversion method 10

.

For example: we an generate two orrelated uniform random vetors

[ U 1 , U 2 ]

^. ^As-

suming asset 1 has a

t 5

distributed returns while asset 2 is normally distributed, we set

X 1 = t ⁻ ₅ ¹ ( U ¹ )

^and

X 2 = Φ ⁻ ¹ ( U ¹ )

^. ^Using ^this ^we ^let ^the ^above ^df

F (t) = e ⁻ ^λt

^be ^the

survivalfuntion,ie. probabilityofnodefaultpriortotime

x

^. ^The^inverse^of^this^funtion

is:

T = − ln(p s ) λ

Sine

p s

^is ^a probability we an generate default times by simulating a set of

[0, 1]

uniform random variates

{ u 1 , u 2 , ..., u n }

^and transforming them by the formula:

T i =

− ^ln(u _λ ⁱ ⁾

^. ^This ^method ^is^disussed ^further ⁱⁿ^Setion ^4.1.

10

As an aside, theinversionmethod an beveryuseful when simulating aportfolio of assets where

theindividualassetshavedierent(marginal)probabilitydistributions. Forexample,ifweassumetwo

assetsAand Bhavenormallyand

t 5

distributedreturns,weangeneratetwouniformrandomvetors

{ u 1 , u 2 }

^and^let^the^return^vetors^be

R A = Φ ⁻ ¹ (u ¹ )

^and

R B = t ⁻ 5 ¹ (u ² )

^.

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−3 −2 −1 0 1 2 3 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(a)Transforming

Φ(0, 1)

^random^variates^to^uniform

(0,1)variates.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 10 20 30 40 50 60 70

u~U(0,1)

τ ~Exp( λ )

(b) Transforming uniform variates to exponential

variateswithparameter

λ

^.

Figure2.4: The inversion method.

2.4 Conditionally Independent Defaults

Wenow turn to the rst tehnique for dealingwith orrelation modeling. The ore idea

behindonditionallyindependentdefaults-CID-models,isthatdefaultsareindependent

onditionedonthe realizationofaset ofsystemifatorsthatdetermine thehazardrate.

Suhfators may be GDP, the short interest rate 11

, reditspreads, et. To illustratethe

tehnique we let

λ(t)

^be ^a ^stohasti ^proess. ^Firm

i

îs âssumed ^to ^default ât ^time

τ

given by:

τ = inf

t : Z t

0 λ(t)dt ≥ E i

(2.4.1)

Where

E i

îs ân ûnitary exponentially distributed random variable (

E i ∼ e ^Z ^0,1

^), ^and

E i

^and

E j

^are independent for

i 6 = j

^.

Illustration

Mostauthors,suhasDuee(1999)useratherompliatedmodelstodetermine

λ

^relying

on multi-fator tehniques from term struture modeling. To illustrate, we onsider a

simpliedmodel, where the hazardrate is a zero drift geometri Brownian motion with

onstant volatility:

dλ(t, λ(t)) = λ(t)σdW (t)

11

Duee(1999)proposesamodelontheform

λ i (t) = λ ^∗ i (t) + αs 1 (t) + βs 2 (t)

^where^the

s i

^are^fators

inferredfrom atwo-fatormodel oftheshort rate.

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0 0.1 0.2 0.3 0.4 0.5

0 20 40 60 80 100

(a)Twosamplepathsofhazardrateproess.

0 100 200 300 400 500

0 20 40 60 80 100

(b)Defaulttimesimulationhistograms.

Figure2.5: CID Simulation

The default riterionfor rm

i

îs ^stillâs^given ⁱⁿÊquation ^2.4.1. ^Wê ^then ^have ^that:

P [τ i > t] = exp

− Z t

0 λ(t)dt

(2.4.2)

sothat

1 t>τ i

îsâ^Cox, ôr^doubly^stohasti^Poisson^proess. ^The^simulationâlgorithm

is summarizedbelow:

1. Generate one path of

λ

^and approximate the integral inEquation 2.4.1.

2. Generate

N

exponentialrandomvariatesanddeterminethe timeofdefaultaord- ingto Equation2.4.1.

3. Repeat step 1 and 2 above.

Two sample pats and default time histograms are plotted inFigure 2.5.

2.5 Copula Functions

2.5.1 Definition and Some Central Properties

A popular methodfor orrelationmodeling inthe redued form framework isthe opula

method,a methodthat uses atransformationof aset of marginaldistributionstoreate

a joint distribution. This setion will present the fundamentals of opula theory and

some partiular opula funtions illustrating the basi onept as well as the breadth of

modelsavailable. Thenextsetionshows howitanbeappliedtopriingproblemsusing

simulation ina redued formmodel.

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Several good referenes on opula theory and its appliations in nanial modeling

are available, hereunder Nelsen (1999) and Li (2000). A omprehensive artile on the

measuring and modeling of orrelated risks is Wang (1998). Elizalde 2005a ontains a

omprehensive list of referenes to further artiles on this eld. Finally, many software

pakages andnanialalgorithmslibrariessuh asMATLABand QuantLibontain rou-

tines for opulamodels that are omprehensively doumented.

We start by a denition:

Denition 2.5.1. Copula

An-dimensionalopulaisdenedasthejointumulativedensityfuntion

C : [0, 1] ⁿ → [0, 1]

ôf â ûniformly distributed random vetor

U ∈ R ⁿ

:

C(u 1 , u 2 , ..., u n , Σ) = P {U ¹ ≤ u 1 , ..., U ^N ≤ u n }

^(2.5.1)

A opula is therefore amultivariatedistribution funtion with uniformlydistributed

marginals. An important result in the theory of opulas states that the marginaldistri-

butions and the dependene between the set of variables an be separated. Firstly, we

an use opulas to linka set of marginaldistributions toa joint distribution:

C(F 1 (x 1 ), ..F n (x n ) = P [ U 1 ≤ F 1 (x 1 ), ..., U n ≤ F n (x n )]

^(2.5.2)

= P [F ₁ ⁻ ¹ ( U ¹ ) ≤ x 1 , ..., F _n ⁻ ¹ ( U ⁿ ) ≤ x n ]

^(2.5.3)

= P [X ₁ ≤ x ₁ , ..., X _n ≤ x _n ]

^(2.5.4)

= F (x 1 , ..., x n , Σ )

^(2.5.5)

Forinstane, inthe bivariatease withX andY randomvariableswithmarginaldfs

F _X

^and

F _Y

^:

C(x, 1) = P [ U ≤ x, U ≤ 1] = x

^.

The following theorem, rst proven by Sklar, shows the the onverse alsoholds; any

multivariatedistributionfuntionan, underertaintehnialassumptionsbewrittenas

a opula.

Theorem 2.5.2. (Sklar) Let

G

^be ^an n-dimensional distribution funtion with ontin- uous marginals

F 1 , ..., F n

^. ^Then ^there ^exists ^an n-dimensional opula

C

^suh ^that:

G(x 1 , ..., x n ) = C(F 1 (x 1 ), ..., F n (x n ))

^(2.5.6)

If we onsider two bivariate uniform random variables on

[0, 1]

^,

X

^and

Y

^, ^with ^the

opula funtion

C(x, y, ρ) = P (X < x, Y < y | ρ)

^,^we ^observe ^that:

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• C(x, 1, ρ) = P (X < x, Y < 1 | ρ) = P (X < x) = x

^, îe. ^we ân ôbtain ^the ôf â

variable

X

^by êvaluating^the ôpula^when âllôther ^parameters âre ^1.

•

^If

X

^and

Y

^are independent, then

C(x, y, ρ) = P (X < x) P (Y < y) = xy

^.

•

^With^perfet orrelation,

C(x, y, ρ) = P (X < x) P (Y < y) = min(x, y)

Why Use Copula Models?

While the theory of opulas may perhaps seem unneessarily omplex at rst sight, the

key point to the above disussion about what a opula atually does, namely reating a

multivariatejointdistributionthat isonsistent withthe speiedmarginaldistributions

of the systemi and idiosynrati fators. While we have ertain "simple" multivariate

distributions that an be used togenerate multivariate data suh as adefault times of a

portfolio, this set is limited. Furthermore,most simple methodsimpose restritions that

are important in pratie, the most important being that the marginals must have the

same univariate distribution. For example, the multivariate Gaussian distribution has

univariateGaussian marginals.

Forinstane, onsider the orrelationstruturethat willbeusedmuhlater oninthe

disussion onstruturalmodels. Let

X i

^be^the^random^variable^that^determines^the ^time

of default forrm

i

^. Îtîs â^funtion ôf â^systemi ^risk^fator

Y

^and ^anidiosynrati risk fator

ǫ i

^where

Y

^and

ǫ i

^are independent:

X i = ρ i Y i + q

1 − ρ ² _i ǫ i

Now, the hoie of marginal distribution for

Y

^and

ǫ i

^will ^determine ^the ^opula

uniquely. If for instane both

Y

^and

ǫ i

^are ^standard ^normally distributed, a Gaussian opula willresult. For any other hoie of distributions, adierentopula is the result.

To summarize,whatistratable about theopula approah isthatitprovidessimple

method tospeify a multivariate joint distributionfor any set of marginaldistributions.

2.5.2 Some Classes of Copula Functions

For the purpose of this thesis we onsider three opula funtionsthat appear frequently

in the nanial literature in general, and partiularlyin that on redued form models -

normal, t- and mixed normal opulas. These are under no irumstanes the only ones

available, but they are omparatively simple to estimate and implement with standard

software. Furthermore, the basi properties of these distributions are well known from

fundamentalprobabilitytheory. Forfurtherdisussion onopulamodels see forinstane

Li (2000)and Elizalde(2005a)and souresited therein.

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Denition 2.5.3. Normal Copula

Let

Φ ^N

^denote ^the N-dimensional normal umulative distribution funtion, the N- dimensional normal or Gaussianopula

C ^N

^is ^given^by:

C ^N (u 1 , u 2 , ..., u N ) = Φ ^N (Φ ⁻ ¹ (u 1 ), Φ ⁻ ¹ (u 2 ), ..., Φ ⁻ ¹ (u N ), Σ)

^(2.5.7)

As a partiular example wenote the bivariatenormal opula given by:

C ² (u 1 , u 2 ) = Φ ² (Φ ⁻ ¹ (u 1 ), Φ ⁻ ¹ (u 2 ), ρ)

Inasimilarfashion tothatabove,weandenethe NdimensionalStudenttopulawith

v

^degrees ^of^freedom.

Denition 2.5.4. Student t Copula

Let

t ^N _v

^denote^the^student^t ^umulativedistributionfuntionwith

v

^degrees^of ^freedom.

Then the N dimensional t-opula

C ^t

^is^dened ^by:

C ^t (u ₁ , u ₂ , ..., u _N ) = t ^N _v (t ⁻ _v ¹ (u ₁ ), t ⁻ _v ¹ (u ₂ ), ..., t ⁻ _v ¹ (u _N ))

^(2.5.8)

Typially, for nanial appliations,

v

îs ^hosen ^to â ^low ^number ^suh âs ⁵ ôr ³

produing a fat tailed distribution (higher risk of extreme losses and gains). As the

number of degrees of freedom gets very high, the distribution onverges to a normal

distribution.

Finally, we onsider two opulas that are somewhat dierent from the two previous.

The rst approah follows fromthe two last properties of opulas at the end of Setion

2.5.1, that

C(x, y, 1) = min(x, y)

^and

C(x, y, 0) = xy

^. ^Consider ^next ^a ^weighted ^om-

bination of these two funtions

ρ

^be ^the ^weight ^assigned ^to ^the ^rst. ^We ^onsider ^the

bivariate normalase:

Denition 2.5.5. Mixed Bivariate Copula

Let

(x, y)

^be â ^set ôf ^random ^variables ^that âre independent. A opula is then given by

C 1 = xy

^. ^Let

(v, w)

^be ^two ^perfetly ôrrelated ^random ^variables. Ânother ôpula îs

then given by

C 2 = min(x, y)

^. ^If

0 < ρ ≤ 1

^,

C(u, v) = (1 − ρ)uv + ρ min(u, v) = (1 − ρ)C ₁ + ρC ₂

^(2.5.9)

denes a mixed bivariate opula.

Finally, asan illustrationof the breadth of opula funtions available as alternatives

to the more ommon normal and t-opulas, we onsider a type of opula that is not

determined by the standard orrelationoeient.

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Denition 2.5.6. Clayton Copula

Let

u

^and

v

^be ^uniform^random ^variables ^on

[0, 1]

^and

0 < θ < ∞

^be ^a ^onstant. ^The

funtion

C(u, v)

^denes â ^bivariate ^Clayton ôpula îf:

C(u, v) = (u ⁻ ^θ + v ⁻ ^θ − 1) ⁻ ¹ ^θ

^(2.5.10)

The parameter

θ

^is ^here ^a ^parameter determining the dependene between the two variables,where

θ = 0

^meansindependentmarginals. Contrarytothe opulasabove,the Claytonopula doesnot allowfor negativeorrelation. However, as Trivedi andZimmer

(2005)states,itexhibits stronglefttaildependene whihmakesitanappropriatemodel

for redit risk. This type of dependene is important in redit risk modeling; one one

rm defaults, ithas onsequenes for otherrms it is doingbusiness with.

0 10 20 30 40 50

0 5 10 15 20 25 30 35 40 45 50

Default time Asset 1

Default time Asset 2

(a)Normalopula.

0 10 20 30 40 50

0 5 10 15 20 25 30 35 40 45 50

Default time Asset 1

Default time Asset 2

(b)

t 1

^-opula.

Figure 2.6: Default timessimulation with two dierent opula funtions.

Figure2.6illustratedefaulttimesgeneratedusingtheinversionmethodfromabivari-

ate normalopula versus default times froma

t 1

^-opula. Ît îs âpparent ^that ^the ^normal

opula yield muh more sattered default times than the t-opula that exhibits more of

a default lustering.

Figures 2.7-2.8 are plots of the random variates from bivariate opulas themselves.

Notie the dierene between the Gaussian and the t-opula; while the rst tend to

satter the observations more, the t-opula gives a "learer" pattern. For a orrelation

oeient of .8, the band formed in the t-opula example is muh slimmer than in the

Gaussianase.

Credit risk models : theory, applications and implementation