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<<2010 Poomjai Nacaskul, Ph.D. | i | ดร.พูมใจ นาคสกุล .. ๒๕๕๓>> Financial Modelling with Copula Functions Poomjai Nacaskul, Ph.D. Bank of Thailand and Mahanakorn University of Technology 17 December 2010 [LEARNING OBJECTIVES] 1 st – to review the significance of the analysis of co-movements amongst random variables as one of the cornerstones of modern finance, where traditionally Pearson product-moment correlation coefficient serves as sufficient statistics, and reveal the well-grounded notion of multivariate normal distribution essentially as a combined statement specifying both individual marginal distributions as well as the dependency structure; 2 nd – to introduce the concept of copula as a function of functions, i.e. a functional, that enables financial modellers to specify the dependency structure as a separate issue from the specification of individual distribution marginals, with insights provided through formal construction and basic theorems pioneered principally by the mathematician Abe Sklar; 3 rd – to learn how to (i) capture dependency structures in financial problems in terms of copulas, (ii) implement copula methodology in risk management and/or derivatives pricing applications, (iii) recognise the use of copulas in financial models adopted by global financial regulators as well as industry practitioners, and (iv) test the goodness of fit of a

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Page 1: Financial Modelling with Copula Functions Poomjai Nacaskul ... · Financial Modelling with Copula Functions Poomjai Nacaskul, Ph.D. Bank of Thailand and Mahanakorn University of Technology

<<2010 Poomjai Nacaskul, Ph.D. | i | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

Financial Modelling with Copula Functions

Poomjai Nacaskul, Ph.D.

Bank of Thailand

and

Mahanakorn University of Technology

17 December 2010

[LEARNING OBJECTIVES]

1st – to review the significance of the analysis of co-movements amongst random variables as one of the cornerstones of

modern finance, where traditionally Pearson product-moment correlation coefficient serves as sufficient statistics, and

reveal the well-grounded notion of multivariate normal distribution essentially as a combined statement specifying both

individual marginal distributions as well as the dependency structure;

2nd – to introduce the concept of copula as a function of functions, i.e. a functional, that enables financial modellers to

specify the dependency structure as a separate issue from the specification of individual distribution marginals, with

insights provided through formal construction and basic theorems pioneered principally by the mathematician Abe Sklar;

3rd – to learn how to (i) capture dependency structures in financial problems in terms of copulas, (ii) implement copula

methodology in risk management and/or derivatives pricing applications, (iii) recognise the use of copulas in financial

models adopted by global financial regulators as well as industry practitioners, and (iv) test the goodness of fit of a

Page 2: Financial Modelling with Copula Functions Poomjai Nacaskul ... · Financial Modelling with Copula Functions Poomjai Nacaskul, Ph.D. Bank of Thailand and Mahanakorn University of Technology

<<2010 Poomjai Nacaskul, Ph.D. | ii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

particular copula against empirical data.

[PART 1 - FOUNDATION]

Basics of Probabilistic (Financial) Modelling

Revisiting (the Notion of) Correlation

Introducing the Gaussian Copula

Defining Copulas Mathematically

Three Special Copulas

Sklar’s Theorem

Copula Density Function

Survival Copulas & Tail Dependence

Copula & Concordance (Measure)

[PART 2 - EXTENSION]

Copula Families

Archimedean Copulas

Multivariate Copulas

Page 3: Financial Modelling with Copula Functions Poomjai Nacaskul ... · Financial Modelling with Copula Functions Poomjai Nacaskul, Ph.D. Bank of Thailand and Mahanakorn University of Technology

<<2010 Poomjai Nacaskul, Ph.D. | iii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

Elliptical Copulas

Modelling with Copula

Parametric Estimation Methodology for Copulas

Non-Parametric Copulas

Goodness-of-Fit Tests for Copulas

[PART 3 - APPLICATION]

Monte Carlo Simulation with Copulas

Financial Risk Modelling with Copulas

Credit Risk Modelling with Copulas

Detour in Credit Derivatives & Derivatives Pricing

Pricing Credit Derivatives with Copulas

[REFERENCES]

[CLV] Cherubini, Luciano, Vecchiato (2004), Copula Methods in Finance, Chichester: John Wiley & Sons.

[JOE] Joe, Harry (1997), Multivariate Models and Dependence Concepts, Boca Raton: Chapman & Hall/CRC.

[NEL] Nelson, Roger B. (1999), An Introduction to Copulas, New York: Springer.

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<<2010 Poomjai Nacaskul, Ph.D. | iv | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

Wikipedia (2010), “Copula (statistics)”, [http: //en.wikipedia.org/wiki/Copula_(statistics)].

Basics of Probabilistic (Financial) Modelling

In probabilistic financial modelling, a quantity of interest (object under study) is generally represented as a random

variable, generically X , whose likelihood of taking on particular (range of) values, i.e. expressed as a random variate,

generically x , is summarised via the notion of probability distribution, specifically with a monotonically non-decreasing

cumulative distribution function (c.d.f.), generically:

(1) domain

X

ntycertaiingxpresse

ariatevrandom

ariablevrandomfdcityimpossibil

ingxpresse

SupportxxXxF ,1)Pr()(0...

Whenever/wherever possible, the c.d.f. used is one whose closed-formed, analytical expression is given by an integral of

a parametric/parameterised function, one that is non-negative over the range of integration.

When the support of the distribution (domain of the random variate with positive probability measure) is a countable set,

such a function is referred to as a probability mass function (p.m.f.), whence sums to one; when defined over an

uncountable support, it is referred to as a probability density function (p.d.f.), whence integrates to unity.

The primary task involved in probabilistic modelling is to specify the choice of function, whereupon the accompanying

chore of statistical inference is to estimate the value of the distributional parameter.

The resulting package, the random variable together with the c.d.f. and its parameterisation, signifies a family of

distributions, generically:

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<<2010 Poomjai Nacaskul, Ph.D. | v | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

(2)

parameter

onaldistributiupports

euncountabl

fdp

functionenonnegativ

u

functionparametric

ntegraliRiemann

xu

ondistributilExponentia

xexfduexFxXExpX 0,,)|(,)()Pr()(~

...

0

It is from such integration that an expectation (operator) may be defined, generically:

(3)

duufugXg )|()()(,

And from there, the mth central moment is then defined, where w.l.o.g. let’s let }{x , thus:

(4) ,4,3,2,)|(][)(

mduufXuXM mm

Hence our familiar notions of means, variance, and standard deviation simply follow:

(5)

duufuXordevstd

duufuXorariancev

duufuXormeans

formulaxpectationenotationnotion

XXX

XXX

X

)|(..

)|(

)|(][

22

2222

As a matter of fact, historically, our probabilistic grasps of nature have only relatively recently swung from “expectation-

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<<2010 Poomjai Nacaskul, Ph.D. | vi | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

based” view to “distribution-based” view.

It is no wonder that throughout the history of probabilistic modelling, one particular distribution stands out.

Not only is it phenomenologically one of the most prevalent in nature and theoretically one of the most relevant in

mathematics, the univariate Gaussian distribution, commonly known as the univariate normal distribution, depicted in

notations below, is notable for the very fact that it is parameterised by the very fundamental statistics of means and

variance themselves, thereby tying nicely and neatly together the “expectation-based” and “distribution-based”

perspectives:

(6)

viewbasedxpectatione

viewbasedondistributi

fdpionspecificat

parameterdisibutionnormal

ariateuniv

ariablevrandom

ddistributenormally

XX

xxfX

22

...

2

2

2

2

)(,][

21exp

21)(,,~

In financial modelling, a random variable usually represents one of four things: (i) a quoted price of a financial asset (i.e.

the ‘out-of-pocket’ expense of buying some financial security or the economic cost of bearing some financial contract) at

any given moment, (ii) an amount of net proceed (interest yielded on a coupon bond, dividend paid on an equity stock,

etc.), (iii) a rate of return from holding a financial asset over any given horizon, or (iv) value of a market-watched factor

that in turns (at least partially) determines market price/proceed/return variables.

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<<2010 Poomjai Nacaskul, Ph.D. | vii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

Whereas relations linking price, proceed, and return rate are quite definitional, i.e. proceeds together with changes in

price determine return rates, relationships between factor and price/proceed/return variables are essentially theoretical

and/or empirical in nature:

(7) factorpriceprice

dividendpricepricereturn

dividendprice

price

todayyesterday

yesterdaytodaytodayyesterday

today

???,,

,

In a financial economy, there are always more than one financial assets present, always more than one factors at work,

and probabilistic dependency relationships between returns of different financial assets, movements amongst a

multiplicity of factors that drive the market variables are likewise theoretical/empirical in nature.

As such, accurate, robust, and simple-to-interpret specification of dependency relationship as such will be of

fundamental advantage in probabilistic financial modelling.

In other words, over and above individual c.d.f., financial modelling requires the knowledge of the joint distribution

function (i.e. multivariate c.d.f.):

(8) 1),(0,Pr),( yxFyYxXyxF

Now, it is quite straight forward, given the joint distribution function, to recover the individual c.d.f.:

(9)

)()(

),(yFxF

yxFY

X

Page 8: Financial Modelling with Copula Functions Poomjai Nacaskul ... · Financial Modelling with Copula Functions Poomjai Nacaskul, Ph.D. Bank of Thailand and Mahanakorn University of Technology

<<2010 Poomjai Nacaskul, Ph.D. | viii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

This simply corresponds to the well known notion of marginal distributions:

(10)

...

)()Pr()Pr(),()()()Pr()Pr(),()(

fdcariateuniv

onsdistributirginalma

Y

X

yFyYyYXyFyFxFxXYxXxFxF

Working w.l.o.g. with continuous r.v.’s:

(11)

y

Y

fdcrginalma

Y

fdprginalma

Y

x

XXXy x

fdpntjoi

fdcntjoi dttfyFdsysfyf

dssfxFdttxfxf

dtdstsfyxF)()(),()(

)()(),()(

),(),(

......

......

However, in practice, such as in a ground-up model building exercise, individual c.d.f. specifications tend to become

available way ahead of that for the joint distribution function.

So it would be most useful if there exists a general, robust, and simple-to-implement method for defining joint probability

distribution in terms of individual c.d.f. specifications:

(12) ),()()( ??? yxH

yGxF

{QUIZ 1} What are the relationships amongst these items?

Hint – best to write each set of comparisons in terms of equations.

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<<2010 Poomjai Nacaskul, Ph.D. | ix | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

(a) )(xFX vs. ),( yxF vs. ),( yxf

(b) )(xFX vs. )Pr( YxX vs. )Pr( xX

(c) ),( yxf vs. )|( yxf vs. )|( yxF

Revisiting (the Notion of) Correlation

Upon encountering the very word correlation, for which many alternative definitions are gathered here [http:

//www.encyclo.co.uk/define/correlation], such as in a “correlation analysis” or a “correlation study of … and …”, no doubt

a great many will automatically think (a) the statistical relationship signifying how two random variables tend to “go

together”, worse (b) the average of product of relative deviations above/below respective means, even worse (c) the

expectation operator, or worst yet (d) the “rho” parameter.

In fact, all four concepts are quite correct, it’s just that as we go successively from (a) to (b) and to (c) and finally (d), the

definition becomes increasingly technical and mathematically specific, which (although normally a good thing) can prove

counterproductive, at times misleading, by inhibiting the generality by which we interpret, represent, capture, test, and

draw conclusion in our modelling methodology.

Our first task is to broaden, indeed question, our present understanding of what correlation entails, and the fundamental

role such an understanding plays in our conception of financial theories.

[1] First of all, it’s perhaps useful to revisit how, i.e. historically, the very notion came to existence; here are some notable

papers on this topic, starting from Sir Francis Galton’s original introduction (albeit the notion correlation can be traced as

far back as to Aristotle):

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<<2010 Poomjai Nacaskul, Ph.D. | x | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

[2] Now, when we say “correlation” w/o further qualification, we generally mean (Pearson product-moment) correlation

coefficient, aka the Galton-Pearson r, which is defined between two random variables, generically YX , , in terms of

expectation:

(13)

dvvfvduufu

dvduvufvu

YX

YXorncorrelatio

dvduvufvuYXarianceovc

formulaxpectationenotaionnotion

YX

YX

YX

YXXY

YXYXXY

)|()|(

)|,(

)|,(

2222

In terms of sample statistics, i.e. a scalar quantities derived from actual paired observation data, the corresponding

notion is that of a sample correlation coefficient:

(14)

deviationandardstsample

n

iiny

meanssample

n

iin

n

iinx

n

iin

ncorrelatiosample

n

i yx

iixy

data

niii

yysyy

xxsxx

ssnyyxx

ryx

1

21

1

1

1

1

21

1

1

1

11

,

,

,)1(

,

[3] Yet note that while there are lower and higher-order measures of statistical deviations, corresponding to 1-norm, 2-

norm, … , all the way to -norm, the situation isn’t so in the case of correlation:

Page 11: Financial Modelling with Copula Functions Poomjai Nacaskul ... · Financial Modelling with Copula Functions Poomjai Nacaskul, Ph.D. Bank of Thailand and Mahanakorn University of Technology

<<2010 Poomjai Nacaskul, Ph.D. | xi | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

(15)

?max...max

?.

?4

)1(..2

?..1

,,1,,1

2

1

21

1

1

1

2

1

21

1

41

4

1

4

11

21

1

1

2

1

1

1

xxdevabsaxmxnorm

xx

xxmomentstdkxnormk

xx

xxkurtosisxnorm

ssnyyxx

rxxdevstdxnorm

xxdevabsmeanxnormncorrelatiodeviationnorm

iniini

kn

iin

n

i

kin

thkn

i

ki

n

iin

n

iinn

ii

n

i yx

iixy

n

iin

n

ii

n

iin

n

ii

For instance, we might want to define expectation correlation measures with higher power moments and sample

correlation statistics with higher power deviations:

(16)

,2,1,,

][][

][][

1

21

1

21

1

1

22

kyyxx

yyxx

YYXX

YYXXn

i

kin

n

i

kin

n

i

ki

kin

kk

kk

Page 12: Financial Modelling with Copula Functions Poomjai Nacaskul ... · Financial Modelling with Copula Functions Poomjai Nacaskul, Ph.D. Bank of Thailand and Mahanakorn University of Technology

<<2010 Poomjai Nacaskul, Ph.D. | xii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

Or perhaps an even more general definition of powered correlation:

(17)

lkjicbalkjicba

yyxx

yyxx

YYXX

YYXX

ln

i

kin

jn

i

iin

cn

i

bi

ain

lkji

cba

///)(,2,1,,,,,,

,,][][

][][1

1

1

1

1

1

1

1

1

11

1

[4] Instead, the only alternatives in currency (in use) are (non-parametric) rank correlation measures, in particular,

Spearman's rank correlation coefficient, aka Spearman's rho, and Kendall’s rank correlation coefficient, aka Kendall's tau.

[5] Indeed, the reason that (Pearson’s) correlation is foremost in our minds when it comes to our understanding of

multivariate random variables is probably the very same reason that means and variance are foremost in our grasp of

univariate random variables, namely the simultaneous appearance as key statistics and distributional parameter vis-à-vis

the normal/Gaussian distribution, only this time it’s the general multivariate, not univariate, version.

{QUIZ 2} Discuss the Pearson product-moment correlation:

(a) in relation to the 2-norm

(b) in relation to the property of symmetry

(c) in relation to Spearman's rank correlation coefficient, aka Spearman's rho

Introducing the Gaussian Copula

Consider the rather well-known bivariate normal/ Gaussian distribution, parameterised by the mean vector, , together

Page 13: Financial Modelling with Copula Functions Poomjai Nacaskul ... · Financial Modelling with Copula Functions Poomjai Nacaskul, Ph.D. Bank of Thailand and Mahanakorn University of Technology

<<2010 Poomjai Nacaskul, Ph.D. | xiii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

with the so-called variance-covariance matrix, :

(18)

...

1

2

2

2

5

2

2

2

2

21exp

12

1),(

]1,1[,,,

,,~

,~,,~

fdp

Y

X

YYX

XYXYX

YX

ionspecificatparameter

YXYXXY

YX

YX

onsdistributirginalma

normalariateuniv

YY

XX

disibutionnormal

ariatebiv

YYX

XYX

Y

X

vectorrandom

ddistributenormally

yx

yxyxf

Y

XYX

In Mathematica, for example, let’s define:

Fig.1 In[1]:= BivariateNormalPDFx, y, X, Y, X, Y,

xX2

X2 yY2

Y2 2 xXyY

XY

2 12

2 1 2 X Y

To view in 3D (w/ parameters subject to manipulation):

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<<2010 Poomjai Nacaskul, Ph.D. | xiv | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

Fig.2

In[2]:= ManipulatePlot3DBivariateNormalPDF ManipulatePlot3DBivariateNormalPDFx, y, X, Y, X, Y, ,x, X 3 X, X 3X,y, Y 3Y, Y 3Y,PlotRange 0, 0.4,

"Bivariate Normal p.d.f.",X, 0, 5, 5, 0.5, Appearance "Labeled",Y, 0, 5, 5, 0.5, Appearance "Labeled",X, 1, 0.5, 5, 0.5, Appearance "Labeled",Y, 1, 0.5, 5, 0.5, Appearance "Labeled",, 0.8, 0.9, 0.9, 0.1, Appearance "Labeled"

Fig.3 Out[2]=

Biv ariate Normal p .d .f.

mX 0

mY 0

sX 1

sY 1

r 0.8

Or as a contour plot (w/ parameters subject to manipulation):

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<<2010 Poomjai Nacaskul, Ph.D. | xv | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

Fig.4

In[3]:= ManipulateContourPlotBivariateNormalPDF ManipulateContourPlotBivariateNormalPDFx, y, X, Y, X, Y, ,x, X 3 X, X 3X,y, Y 3Y, Y 3Y,PlotRange 0, 0.4,

"Bivariate Normal p.d.f.",X, 0, 5, 5, 0.5, Appearance "Labeled",Y, 0, 5, 5, 0.5, Appearance "Labeled",X, 1, 0.5, 5, 0.5, Appearance "Labeled",Y, 1, 0.5, 5, 0.5, Appearance "Labeled",, 0.8, 0.9, 0.9, 0.1, Appearance "Labeled"

Fig.5 Out[3]=

Biv ariate Normal p.d .f.

mX 0

mY 0

sX 1

sY 1

r 0.8

Working w.l.o.g. with standard normal marginals, i.e. with normalised r.v.’s XXXX and YYYY ,

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<<2010 Poomjai Nacaskul, Ph.D. | xvi | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

effectively taking away the location-scale parameters, there is only one distributional parameter left, corresponding to the

well-known, aforementioned (Pearson product-moment) correlation coefficient:

(19)

yxyxyxf

YX

YX

parameterncorrelatiothe

ionspecificatparameter

onsdistributirginalma

normalariateuniv

andardst

disibutionnormal

ariatebivandardst

vectorrandom

ddistributenormally

andardst

2)1(2

1exp121),(

]1,1[,1,0~1,0~

,1

1,

00

~

2222

,1

{QUIZ 3} Suppose we are told that 2,0~ XX , 2,0~ YY , and ),( YXCorrel :

(a) Write out the variance-covariance matrix.

(b) Write out the covariance formula, i.e. XY , in the form of matrix multiplication.

(c) Simplify (b), i.e. multiply out all the terms, to obtain a scalar expression.

For convenience, let’s denote the bivariate standard normal c.d.f., the univariate standard normal c.d.f., and the inverse

(function) of the univariate standard normal c.d.f. thus:

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<<2010 Poomjai Nacaskul, Ph.D. | xvii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

(20)

uxxu

dssx

dtdststsyx

x

x

y x

)(sup)(,]1,0[:

2exp

21)(,]1,0[:

2)1(2

1exp121)|,(,]1,0[:

11

2

2222

2

Fig.6

In[4]:= StandardBivariateNormalPDFx, y, BivariateNormalPDFx, y, 0, 0, 1, 1,

Out[4]=

x2y22 x y 2 12

2 1 2

Fig.7 In[5]:= PhiInverseu 2 InverseErf1 2 u

Out[5]= 2 InverseErf1 2 u

Note that the joint distribution, i.e. joint c.d.f., can be re-written:

(21)

v u

y x

dtdsvuyxyvxu

dtdsyxyx

1 1

1 1

)(),()|,()()(

)(,)()|,(

11

)( )(11

Now note carefully how the above expressions make no use of the fact that the marginals are (standard) normal; in fact,

in their place any other univariate c.d.f. will do, hence the conclusion that X and Y need not be normally distributed at

all, for example, substituting )(xG for )(x and )(yH for )(y and the entire structure stands:

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<<2010 Poomjai Nacaskul, Ph.D. | xviii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

(22)

v u

yGG xHH

dtdsvuyxyHvxGu

dtdsyHxGyx

1 1

1 1

)(),()|,()()(

)(,)()|,(

11

)( )(11

This gives us a general methodology for using , the correlation parameter inherited from multivariate normal

distribution, to construct a different joint distribution function form any arbitrary marginals:

(23)

)()Pr(

)()(2

exp21

2)1(2

1exp121

,)()1(,)()(,)(

)Pr(Pr)Pr(

)(,)()Pr(

1)( 2

)(22

22

11111

11

1

1

yHyY

xGxGdss

dtdststs

xGxGHxG

YxXSupportYxXxX

yHxGyYxX

xG

xG

Y

Indeed this is the 1-parameter bivariate Gaussian copula function:

(24)

]1,1[,2)1(2

1exp121)(),(),(

,]1,0[]1,0[:1 1

2222

11

2

v u

dtdststsvuvuC

C

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[CLV: 112] Roncalli (2002) showed that this double integral expression (24) can be rewritten thus

(25)

u

v u

dssv

dtdststsvuvuC

C

02

11

2222

11

2

]1,1[,1

)()(

2)1(2

1exp121)(),(),(

,]1,0[]1,0[:1 1

Let’s retrace the steps, this time starting by defining 2 uniform random variables, 2 arbitrarily (unspecified) distributed

random variables, and 2 standard normally distributed random variables, the latter with correlation ]1,1[ :

(26)

21221121 ,Pr2,1,,Pr)1,0(~

,)()Pr(~,)()Pr(~

2,1,]1,0[,Pr)1,0(~

zzzZzZZZizzzZZ

SupportyyFyYDistrYSupportxxFxXDistrX

iuuuUUnifU

iiiii

YYY

XXX

iiii

Let the 1st uniform, 1st arbitrarily distributed, and 1st standard normally distributed random variates be tied together (map

1-to-1), and the 2nd uniform, 2nd arbitrarily distributed, and 2nd standard normally distributed random variates tied together

thus:

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(27)

....

2222

...

...

...

22

111111

,,,,

equivalentconsideredareeventsthesethatsense

thein

togethertiedare

ariablesvrandomThese

zZyYuUZYUzZxXuUZXU

Hence in terms of the corresponding random variates, the c.d.f., and the c.d.f. inverses:

(28)

21

21

21

21

2222

...

11

11

11

11...

1111

)()(,,

)()(,,

zyFu

zFyuFzyFuzyu

zxFu

zFxuFzxFuzxu

Y

YYY

inversefdc

X

XX

fdc

X

ariatevrandom

Moreover, these joint events are equivalent:

(29) 22112211 zZzZyYxXuUuU

And of course these joint probabilities are equal:

(30) 22112211 Pr)Pr(Pr zZzZyYxXuUuU

Finally, with some substitutions:

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(31)

...2

1112

11

21

)(

2211

]1,0[

21

11

21

211

1

21

)(

22112211

)(,)()()(Pr,PrPr

,Pr,PrPr

2

fdcoffunctionaasxpressede

YXYX

Gaussiannormalariatebivondistributiarbitraryntjoioverfunctionaasxpressede

Gaussiannormalariatebivondistributiuniformntjoi

yFxFyFZxFZzzzZzZyYxX

uuuZuZzzzZzZuUuU

So that there are two mutually consistent interpretations, namely (i) that a bivariate copula is a bivariate c.d.f. with uniform

marginals, and (ii) that it is a function that takes two univariate c.d.f.’s (each, in turn, just a function of one scalar random

variate) to produce a joint distribution (bivariate c.d.f.) thus:

(32)

)Pr()(,)()(,)(),(

]1,0[D:....'..:2

Pr,,

]1,0[]1,0[:..,...:1

11

2

221121

11

21

2

yYxXyFxFyFxFCyxC

Cfdcntjoiaproducetosfdcariateunivtwotakesthatfunctionalionnterpretati

uUuUuuuuC

Crginalsmauniformwitheiupportssquareunitoverfdcariatebivionnterpretati

YXYX

{QUIZ 4} Derive the bivariate Gaussian copula formula (at least describe the steps and rationales involved).

Defining Copulas Mathematically

Here it’s perhaps useful to keep a neat distinction between the notion of a function from that of a functional and that of an

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operator.

We all know how function is essentially an unequivocal association i.e. a map from one set, the domain, to another, the

so-called co-domain, such that variables with same values get mapped exactly the same way (yields exactly the same

value, e.g. whenever the input is five, the output will be twelve).

On the other hand, technically a functional is a function whose domain is specifically a vector space and whose co-

domain is the field underlying said vector space, but for the present purpose think of a functional as a function of

functions (takes another function as an input to produce a value output).

Finally, an operator is essentially a function which acts on functions to produce yet another function:

(33)

t

xy

yx

dtfftffthgeffhoperator

dydxeyxfg

dydxeyxfggefgfunctional

xyxfyxyxfgeffunction

02121

2

1

0

1

02

1

0

1

01

212

)()()(*)(..:

),(

),(..,:

)(,),(..,:

So is copula a function, a functional, or an operator, i.e. letting }{F denote a “space of c.d.f.”, how do we mainly see the

“mapping action” of a copula?

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(34)

operatoranhenceFFC

functionalahenceFCfunctionahenceC

yxyFxFCyxF YX

,}{}{:,]1,0[}{:,]1,0[]1,0[:

,,)(),(),(2

2

2

???

Once again, referring to (12), the desire is to be able to construct a joint probability distribution from the marginals, i.e.

the 2k univariate c.d.f.’s, and that in essence is what a functional does.

Nonetheless for practical purposes, any interpretation will do, and in practice we usually see the term “copula function”,

or just “copula”, in use:

(35) )Pr()(),()()(

yYxXyFxFCFF

YXCopula

Y

X

Now let’s try constructing a bivariate copula from scratch, specifying the mathematical properties necessary to produce

a bivariate c.d.f., in particular:

(36)

)()(,1)(1),(

0)(,00),(,,)(),(),(

yFyFCxFxFC

yFCxFCyxyFxFCyxF

YY

XX

YX

YX

The first-lined property corresponds to saying that the copula function is “grounded”.

Moreover, corresponding to the monotonic, non-decreasing property of a univariate c.d.f. is the so-called “2-increasing”

property required of a bicariate copula function:

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(37) 0,,,,1010

1112212221

21

vuCvuCvuCvuCvvuu

Altogether:

(38)

.2,),1(,)1,(

,

]1,0[]1,0[: 2

ingincreasisCvvCuuC

groundedisC

C

{QUIZ 5} Describe the domain and co-domain (“range”) of the mapping of a trivariate copula )(),(),( yFxFwFC YXW .

Hint – in the form ??????: C , then explain.

Three Special Copulas

These 3 special copulas are considered the most fundamental.

[1] The independent copula, aka product copula, expresses the already familiar concept of statistical or probabilistic

independence:

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(39)

0,,,,

,1),1(,1)1,(

),0(000)0,(

),(

0

12

0

1211122122

11122122

vvuuvuvuvuvu

vuCvuCvuCvuCvvvCuuuC

vCvuuC

vuvuC

Whereas, [2] the minimum copula (Fréchet-Hoeffding lower bound) and [3] the maximum copula (Fréchet-Hoeffding

upper bound) bound all copulas, respectively, from below and from above, in the sense of expressing the Fréchet-

Hoeffding inequality over 2]1,0[ :

(40) },min{),(),(}1,0max{),(

,]1,0[,vuvuCvuCvuvuC

vu

This can be expressed rather elegantly in terms of the so-called concordance order relation, hence:

(41) ),(),(,]1,0[,),(),(,]1,0[,

2121

2121

vuCvuCvuCCvuCvuCvuCC

In particular, with minimum and maximum copulas at the ends, independent copula somewhere “in the middle”, all other

copulas fall somewhere in between:

(42) ),(),(),( vuCvuCvuC

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Note, however, such a ‘ ’ relation does not amount to there being a total order amongst all possible copulas. In other

words, for some pair of copulas (except the minimum/maximum), it’s possible that, away from the boundary of 2]1,0[ ,

one finds:

(43)

),(),(),(),(

)1,0(,,21

21

hgCfeCdcCbaC

ha

[CLV: 70] Random variables are said to be comonotone if C is their copula, and countermonotone if C is their copula,

both expressing the notion of perfect dependence (only one source of randomness, despite the designation of more than

one random variables) thus:

(44)

Vnamelyrandomnessofsourceoneonlyandone

Y

XYX

Unamelyrandomnessofsourceoneonlyandone

Y

XYX

UnifVyFVyYxFVxX

yFxFCyxF

UnifUyFUyYxFUxX

yFxFCyxF

,

,

)1,0(~,)(1Pr)Pr()(Pr)Pr(

)(),(),(

)1,0(~,)(Pr)Pr()(Pr)Pr(

)(),(),(

[NEL: 3] Parenthetically, copulas are related to the mathematical concept of triangle norms or t-norms, which arise within

the context of probabilistic measure spaces or PM spaces.

Some copulas are t-norms, and some t-norms are copulas.

Indeed, the minimum copula is identical to the formula for computing Lukasiewicz t-norm, the independent copula is

identical to the formula for computing product t-norm, and the maximum copula is identical to the formula for computing

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Gödel t-norm.

For reference, a t-norm is a function defined by 4 properties thus:

(45)

.

)1()1,()()),,(()),(,(

)(,,

)(),(),(

]1,0[]1,0[: 221121

212

elementidentityasactsaaTityassociativcbaTTcbTaT

tymonotonicibaTbaTbbaa

ityassociativabTbaT

T

{QUIZ 6} What are the 3 special copulas (write out the full mathematical expressions), and what’s so special about them?

Also, discuss in terms of concordance order the relation between each of the 3 special copulas and an arbitrary

”generic” copula C.

Sklar’s Theorem

[NEL: 3,14] Fundamental to copula mathematics is the Skalar’s Theorem, first published (in French) in 1959 by the

mathematician Abe Sklar, who, around that time, was working also on PM spaces.

This theorem states that, let F be a joint distribution function (bivariate c.d.f.) with margins G and H , then there exists

a copula ]1,0[]1,0[: 2 C such that:

(46) )(),(),(,, , yHxGCyxFSupportyx YX

Moreover, if G and H are both continuous, then C is unique.

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Conversely, if C is a copula, and G is a univariate c.d.f., as is H , then F , as defined by (46), becomes in and of itself

a joint distribution function (bivariate c.d.f.).

For instance, given )(~ ExpX , ),(~ baUnifY , and the independent copula:

(47) ],[),0[),(,1),(

]1,0[,,),(

],[,1)(

),0[,)(

bayxab

eyxF

vuuvvuC

bayab

yF

xexFx

Y

xX

In particular, suppose that )1,0(~ UnifX and )1,0(~ UnifY , then it is very natural to interpret any bivariate copula as

some bivariate joint distribution with uniform marginals (univariate c.d.f.’s):

(48)

]1,0[]1,0[),(,)Pr(

),()(),(

]1,0[,)()Pr()1,0(~]1,0[,)()Pr()1,0(~

vuvVuU

vuCvFuFC

vvvFvVUnifVuuuFuUUnifU VU

V

U

From all these then follow a number of identities:

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(49)

vvuCvuvVuU

vvuCvvVuU

vvuCuvVuU

vvuCvVuU

vuCvuvVuUvuCvvVuUvuCuvVuU

vuCvVuU

vVuUvVuUvVvvVuUvVuUuUu

vVuUvVuUvVvvVuUvVuUuUu

1),(1)|Pr(

),()|Pr(

1),()|Pr(

),()|Pr(

),(1)Pr(),()Pr(),()Pr(

),()Pr(

)Pr()Pr()Pr(1)Pr()Pr()Pr(1

)Pr()Pr()Pr()Pr()Pr()Pr(

{QUIZ 7} What is the Sklar’s theorem (write out the full mathematical expression)? Why is it so important? Also, discuss in

terms of “division of labour” in modelling with multivariate random variables. Hint – it’s sufficient to just state for the

bivariate case.

Copula Density Function

Just as we can ascribe a p.d.f. to a c.d.f., there is also a corresponding notion of copula density (function):

(50) vuvuCvuc

),(),(

2

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[CLV: 81] [NEL: 23] Indeed, this integrable (in calculus/measure-theoretic sense) aspect of the copula is referred to

formally as the absolutely continuous component ..caC of the copula C , whereupon C itself, in its most general form, is

said to be composed of this and/or the singular component ingularsC :

(51)

v uv u

ca

ingularsca

dtdststsCdtdstscvuC

vuCvuCvuC

0 0

2

0 0..

..

),(),(),(

),(),(),(

[CLV: 81] Note that the independent copula only has an absolutely continuous component; whereas, the minimum and

maximum copulas both contain only singular components:

(52)

),(),(00),(0},min{),(

),(),(00),(0}1,0max{),(

),(1),(1),(

0 0..

220 0

..

220 0

..

22

vuCvuCdtdsvuCvu

vuvu

vuC

vuCvuCdtdsvuCvu

vuvu

vuC

vuCuvdtdsvuCvu

uvvu

vuC

ingulars

v u

ca

ingulars

v u

ca

v u

ca

[CLV: 83] Finally, given a copula representation )(),(),( yFxFCyxF YX , the copula density function then yields the

following canonical representation of the joint (bivariate) p.d.f.:

(53)

tionrepresentacanonical

sfdprginalma

YX

densitycopula

YX

fdpntjoi

YXYX yfxfyFxFcyxf

yfxfyxfyFxFcvuc

.'.....

)()()(),(),()()(

),()(),(),(

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This can be seen to follow from the fact that )(1 UFX X and )(1 VFY Y

are individually normal, and hence

continuous, random variables, the transformation )(xFux X and )(yFvy Y are continuously differentiable

and strictly increasing, hence one-to-one, and so we have this multivariate transformation theorem to apply:

(54)

)()(

),()()()()(

),(

det

)(),()(),(),(

00

11

yfxfyxf

XYF

YXF

YYF

XXF

yxf

YV

XV

YU

XU

vFuFfyFxFcvucYXYXYX

tiontransformaariatebivthe

ofJacobian

YXYX

{QUIZ 8} Give (a) copula density and (b) canonical representation for the independent copula vuvuC ),( .

Survival Copulas & Tail Dependence

[NEL: 28] From actuarial science tradition, the focus is on survival time, a random variable T whose survival function, aka

survivor function or reliability function, i.e. the probability of surviving or “outlasting” beyond a certain point in time 0t

is denoted thus:

(55) )Pr()( tTtF

Not sure why one doesn’t often see the term “survival probability” used in this context!?

At any rate, we can also define joint survival function:

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(56) )(1,)(11)()(

)(),()(1)(11),()()(1)Pr(),(

yFxFCyFxFyFyFCyFxF

yxFyFxFyYxXyxF

YXYX

YXYX

YX

So for actuarial applications, it then makes sense to correspondingly define the survival copula:

(57)

)Pr(),(111)1,1(

)1,1(1),(

)Pr(

)Pr()Pr(

vVuUvuCvuvuC

vuCvuvuC

vVuU

vVuU

Then, using (57), we can rewrite (56) in a similarly compact expression:

(58) )(),(),(),(1)1,1(

)(1,)(11)()(),(yFxFCyxF

vuCvuvuCyFxFCyFxFyxF

YXYXYX

[NEL: 47] One particular example of this is the 2-parameter Marshall-Olkin copula:

(59) 1,0,,min,min),( 11 uvvuvuuvvuC OlkinMarshall

[CLV: 75] Note that for the minimum, independent, and maximum copulas, their survival copulas are the same as the

original copulas, for example, with the independent copula:

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(60) ),()1)(1(1),( vuCuvvuvuvuC

[CLV: 108] Related to the notion of survival copula, and quite relevant in financial risk applications, is the notion of tail

dependence or tail dependency which looks at the conditional probability of one random variable being extremely large,

given that the other one random variable is extremely large, or vice versa, hence symmetry in the definition.

A copula is said to be characterised by upper-tail dependence, or to exhibit upper-tail dependency, if the following limit of

a conditional probability term is non-zero:

(61)

]1,0[)()(Prlim)|Pr(lim)Pr(

)Pr(lim1

),(21lim1

)1,1(lim

)Pr()Pr(lim

)|Pr(lim)()(Prlim

11

11

1

11

1

1

11

1

wFYwFXwVwUwV

wVwUw

wwCww

wwCwU

wUwVwUwVwFXwFY

YXww

w

ww

w

wXY

wUpper

For instance, it’s very easy to verify the following:

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(62)

0

1121lim

1

)(),(21lim

1),(21lim

01

1221lim01

}1,0max{21lim1

),(21lim

01

)1(lim121lim

1),(21lim

111lim

1},min{21lim

1),(21lim

1

11

11

111

2

1

2

11

111

ww

w

www

wwwCw

www

wwww

wwwCw

ww

www

wwwCw

ww

wwww

wwwCw

ww

Gaussian

w

GaussianUpper

wwwUpper

wwwUpper

wwwUpper

Analogously, a copula is said to be characterised by lower-tail dependence, or to exhibit lower-tail dependency, if the

following limit of a conditional probability term is non-zero:

(63)

]1,0[)()(Prlim)|Pr(lim)Pr(

)Pr(lim1

),(lim

)Pr()Pr(lim

)|Pr(lim)()(Prlim

11

00

0

0

0

0

11

0

wFYwFXwVwUwV

wVwUwwwC

wUwUwV

wUwVwFXwFY

YXww

w

w

w

wXY

wLower

Similarly, it’s very easy to verify the following:

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(64)

00lim

)(),(lim),(lim

00lim}1,0max{lim),(lim

0limlim),(lim

1lim},min{lim),(lim

0

11

00

000

0

2

00

000

ww

ww

wwwC

wwww

wwwC

www

wwwC

ww

www

wwwC

ww

Gaussian

w

GaussianLower

wwwLower

wwwLower

wwwLower

{QUIZ 9} Verify for (a) the minimum copula }1,0max{),( vuvuC and (b) the maximum copula

},min{),( vuvuC , that their survival copulas are the same as the originals.

Copula & Concordance (Measure)

In general, joint probabilistic behaviour between two random variables X and Y will fall between two limiting cases: that

of complete independence (corresponding to C being their copula, whereupon X and Y are said to be independent),

and that of complete dependence (either positively, in which case C is their copula and X and Y are said to be

comonotone, or negatively, in which case C is their copula and X and Y are said to be countermonotone, either way

corresponding to the situation which reduces the number of random sources to just one).

Recall how these limiting cases correspond to the Pearson product-moment correlation value of 0 and 1, respectively.

This section generalises such a notion of measuring the degree of association between two random variables, whilst

keeping the desired fixtures that any such measure is bounded within 1 and equals 0 in the case of independence.

Whereas the Pearson product-moment correlation focuses on whether above-average values in one random variable

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tend to be associated with above-average values in the other random variable, let’s pursue here the idea of comparing

two pairs of realisations, ii yx , and jj yx , , to see whether whenever ji xx , this tends to be associated with

ji yy or instead with ji yy , and so on.

Given two realisations (random variates) each from two random variables, i.e. two joint events 11 yYxX and

22 yYxX , we say that the random variates are concordant or discordant, respectively according to the

following assignment rule, where let’s for now assume 21 xx and 21 yy :

(65)

)(0

)(0

21212121

2121

21212121

2121

pairdiscordantyyxxyyxxyyxx

pairconcordantyyxxyyxxyyxx

Given a sample consisting of 2n bivariate data niii yx 1, , one can compare, pair-wise, two data points

ijijii yxyx ,&, at a time (as such there will be a total of 2/)1( nn distinct comparisons), and add up the number

of instances of concordant pairs, c , versus the number of instances of discordant pairs, d , and define the Kendall’s tau

rank correlation coefficient or simply Kendall’s tau statistics for this sample set thus:

(66)

pairsdiscordantdpairsconcordantc

nndc

dcdc

##

,2/)1(

Then we can go back to the random variables (i.e. not random variates), define the probability of concordance,

econcordancPr , the probability of discordance, ediscordancPr , and their difference, which turns out to be just the

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probabilistic (population) counterpart to the Kendall’s tau “hat” (sample) statistics, so let’s denote it by “hatless” , thus:

(67)

ediscordanceconcordanc

YYXXYYXXYYXXediscordancYYXXeconcordanc

Pr

2121

Pr

2121

2121

2121

0Pr0Pr0PrPr0PrPr

Working with continuous r.v. leads to slightly simpler expression:

(68)

10Pr20Pr10Pr

0Pr10Pr

212121212121

21212121

YYXXYYXXYYXX

YYXXYYXX

Decomposing econcordancPr into probabilities of two joint events:

(69)

21211212

212121212121

PrPrPrPr0Pr

YYXXYYXXYYXXYYXXYYXX

Take the first term of the right:

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(70)

2

2

2

]1,0[

111112121212

),(),(

)(),()(),(

,PrPr

ialdiofferentcopula

copula

aldifferenticopula

YX

copula

XX

densitycopula

YX

vudCvuC

yFxFdCyFxFC

dydxyFxFcxYxXYYXX

[NEL: 127] Ultimately we have a very elegant theorem that tells us exactly how to arrive at this quantity.

In other words, noting that (70) is symmetric about whether 21 XX or 12 XX , so that (69) would have two identical

terms on the right, whence putting it them all back into (68) yields the Kendall’s tau-based measure of concordance for

the population as:

(71) 1),(),(42]1,0[

vudCvuC

[NEL: 129] Note that the double integral term can be interpreted as the expectation:

(72) )1,0(~,,),(),(),(2]1,0[

UnifVUVUCvudCvuC

[CLV: 98] For absolutely continuous copulas (w/ no singular component), we can substitute in for the copula differential

notation the more familiar double differentials:

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(73) dudvvuvuCdC

),(2

For when the copula has both absolutely continuous and singular components, or just the former, use the following

theorem instead:

(74)

22 ]1,0[]1,0[

),(),(411),(),(4 dudvv

vuCu

vuCvudCvuC

In other words, putting (71) and (74) together:

(75) 21),(),(),(),(

22 ]1,0[]1,0[

dudvv

vuCu

vuCvudCvuC

[CLV: 95] In general, other measures of concordance can be defined, each a function of how two random variables are

probabilistically joined up, hence equivalently a function of the two random variables as well as a function of the bivariate

copula, )(),( CYX , so long as they satisfy the following axiomatic properties:

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(76)

)()(

)(),(,lim),(),(lim)(),(),(),()(

0),(.,)()(),(),()()(]1,1[),()(

)(),()(,

2121 econcordancoforderCCCCviieconvergencuniformYXYXvuCvuCvi

YXYXYXvYXindepYXiv

symmetricXYYXiiinormalisedYXii

sscompleteneYXiYX

nnnnn

One nice thing about this (axiomatic definition) is that there is a theorem which guarantees that any (axiomatically

verified) measure of concordance will be invariant under increasing functions 2,1, igi :

(77) ),()(),(2,1, 21 YXYgXgingincreasigi

[CLV: 96] For example, consider a much simpler measure, called Blomqvist’s beta, which essentially looks at the value of

a bivariate copula at in the middle of the square:

(78)

11121

21,0max4

0121

214

1121,

21min4

121,

214'

C

C

C

CsBlomqvist

[CLV: 96] However, a more popular alternative to Kendall’s tau seems to be Spearman's rank correlation coefficient, or

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simply Spearman's rho, which can also be defined in terms of double integrals over the copula (written here w/o proof):

(79)

222 ]1,0[]1,0[]1,0[

'),(),(633),(123),(12 dudv

vvuCv

uvuCuvudCuvdudvvuCsSpearman

[CLV: 103] Although intuitive, it isn’t necessarily the case that the Pearson product-moment correlation would constitute a

measure of concordance proper, i.e. in the sense of satisfying (76), and in fact it doesn’t.

But that does not prevent us from writing its denominator, i.e. the covariance, in terms of copulas:

(80) XYXY Support

YXYXSupport

YX dxdyyFxFCyFxFCdxdyyFxFyxFYXCov )(),()(),()()(),(),(

Indeed, one major shortcoming of the standard correlation measure should be phrased in terms of the fact that because

it isn’t a measure of concordance proper, it isn’t invariant under nonlinear increasing functions in general, just linear

ones.

{QUIZ 10} Compare measure of concordance with the Pearson product-moment correlation. Hint – which one is more

general?

Copula Family

Recall how the definition of minimum, independent, and maximum copulas involve no parameter whatsoever, and how

for any other copula ]1,0[]1,0[: 2 C out there, it will always be bounded in the sense of concordance ordering CCC .

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Recall also how a copula is essentially a function (or a functional), and its main purpose, from a modelling perspective, is

to capture the joint distributional behaviours amongst random variables for whom we may have just the individual

univariate c.d.f.’s.

So it would make a lot of sense to develop, catalogue and extend toward a family of copulas defined as a collection of

parameterised functions, each a copula function proper, such that not only are different members of the family

distinguished by specific parametric values, but also let parametric inequality reflects the concordance order, which

preferably (at least for 1-parameter bivariate copula families) constitutes a total ordering within the family, thus:

(81)

...""..,|,|,

...""..,|,|,,,)|,(),(

2121

2121

21

trworderednegtivelyeivuCvuCor

trworderedpositivelyeivuCvuCvuCvuCC

A parametric copula family is said to be comprehensive if it includes (in the parametric limits) the minimum, independent,

and maximum copulas as its members:

(82)

),()|,(lim

),()|,(lim

),()|,(lim

,,

vuCvuC

vuCvuC

vuCvuC

One obvious method of constructing a family copula is to create a convex combination of minimum and maximum

copulas:

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(83)

""..,|,|,

]1,0[,),(),()1()|,(

2121 orderedpositivelyeivuCvuC

vuCvuCvuC

[CLV: 118] Along a similar distributional mixture approach, but with the bonus of also including the independent copula

as a member, hence making it comprehensive, is the 2-parameter Frechet family, whose member, the Frechet copulas

may have up to 2 terms for the component and 1 absolutely continuous component:

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(84)

}0,0{,}1,0{,}0,1{,

),()0,0|,(),()1,0|,(),()0,1|,(

...""..,,|,,|,...""..,,|,,|,

1]1,0[,

,),()1(),(),(),|,(),(

2121

2121

..

qpqpqp

vuCvuCvuCvuCvuCvuC

qtrworderednegativelyeiqpvuCqpvuCqqptrworderedpositivelyeiqpvuCqpvuCpp

qpqp

vuCqpvuCqvuCpqpvuCvuC

Frechet

Frechet

Frechet

componentssacomponentingulars

FrechetFrechet

{QUIZ 11} What makes a family of copulas comprehensive, and why would a comprehensive copula family be preferred

to one that isn’t?

Archimedean Copulas

[CLV: 120] [NEL: 89] One of the most, if not the most, general family of copula is the so-called Archimedean family, as

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appropriately named by Ling (1965).

First one needs to define a sort of generator:

(85)

""..,)0(0)1(

""..,))1(()()1()(]1,0[""..,)()(

]1,0[:

generatorstrictei

convexeibabaingdecreaseibaba

With the generator and its (pseudo) inverse, 1 , one can define an Archimedean copula quite simply:

(86) )()()|,(),( 1 vuvuCvuC nArchimedeanArchimedea

[CLV: 124] For instance, consider the 1-parameter Gumbel copula:

(87)

/1

1

1

/11

)ln()ln(exp

)ln()ln(

)()(

|,)|,(),(

exp)(

1,)ln()|()(

vu

vu

vu

vuCvuCvuC

ss

ttt

Gumbel

GumbelGumbelGumbel

GumbelnArchimedeaGumbelGumbel

Gumbel

GumbelGumbel

Consider also the 1-parameter Clayton copula:

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(88)

/1

111

1

/11

1

1,0max

11

)()(

|,)|,(),(

1,0max)(

}0{\),1[,1)|()(

vu

vu

vu

vuCvuCvuC

ss

ttt

Clayton

ClaytonClaytonClayton

ClaytonnArchimedeaClaytonClayton

Clayton

ClaytonClayton

Finally the 1-parameter Frank copula:

(89)

11

11ln

11ln

11ln

)()(

|,)|,(),(

11ln)(

}0{\,11ln

)|()(

1

1

1

11

eee

ee

ee

vu

vuCvuCvuC

ees

ee

tt

vu

vuFrank

FrankFrankFrank

FranknArchimedeaFrankFrank

sFrank

t

FrankFrank

[CLV: 127] Copulas from any of the three families (Gumbel, Clayton and Frank) are positively ordered w.r.t. the

respective “alpha” parameter, but in terms of other properties, these three popular Archimedean copula families do differ

quite a bit.

While two families (Clayton and Frank) are comprehensive, the other one (Gumbel) is not.

In fact, Gumbel copulas range from C to C , thereby ruling out altogether negative dependency.

In terms of tail dependence, Gumbel copulas have upper-tail dependency, i.e. 0GumbelUpper , Clayton copulas have lower-

tail dependency, i.e. 00 ClaytonLower , while Frank copulas have neither i.e. 0 Frank

LowerFrankUpper .

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{QUIZ 12} The Clayton copula family is comprehensive and incorporates/exhibits lower-tail dependency. Why would

such features be useful for financial modelling applications?

Multivariate Copulas

For the most cases, extensions from bivariate 2n to truly multivariate 2n copulas are quite obvious (although the

same perhaps cannot be said about the technical details needed in extending the 2-increasing condition to an n-

increasing version).

[CLV: 133] In particular, here are multivariate versions of the minimum, independent, maximum, Frechet, and

Archimedean copulas.

(90)

n

ii

nArchimedea

componentssacomponentingulars

FrechetFrechet

niii

n

ii

n

ii

uC

qpqp

CqpCqCpqpCC

uC

uC

uC

1

1

..1

1

1

)(

1]1,0[,

,)()1()()(),|()(

min)(

)(

1,0max)(

u

uuuuu

u

u

u

[CLV: 135] A multivariate version of the then bivariate Sklar’s theorem (46) is stated below.

Let F be a joint distribution function (multivariate c.d.f.) with margins nFF ,,1 , then there exists a copula

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]1,0[]1,0[: nC such that: :

(91)

n

nn

ii C

xF

xF

xF

CFSupport ]1,0[,)()(,

11

uuxx X

Moreover, if nFF ,,1 are all continuous, then C is unique.

Conversely, if C is a copula, and nFF ,,1 are univariate c.d.f.’s, then F , as defined by (91), becomes, in and of itself,

a joint distribution function (multivariate c.d.f.).

[CLV: 154] Moving right along, a multivariate version of the then bivariate copula density (function) (50) is given by:

(92) ni

n

uuuCc

1

)()( uu

From which a multivariate version of the then bivariate canonical representation (53) is given by:

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(93)

tionrepresentacanonical

sfdprginalma

n

iii

densitycopula

nn

ii

fdpntjoi

n

iii

nn

ii xf

xF

xF

xF

cfxf

f

xF

xF

xF

cc

.'..

1

11

...

1

11

)()()(

xxu

Other definitions follow in an equally straight forward manner.

For instance, a multivariate extension of a bivariate survival copula (57) is given by:

(94)

nn

uUuU

n

uU

n

uU

n

n

n

iin

uUuUuuCuuuuCC

CuuCuuuCC

nn

nn

11

Pr

1

PrPr

11

11

1

Pr,,1111,,1)(

)(11,,11,,)(

11

11

u1

u1u1u

Elliptical Copulas

Note in Fig.5 how the contour map of a bivariate normal/Gaussian distribution takes the elliptical shape, a feature that

motivates the general definition of a class of multivariate elliptical distributions, any whose member is parameterised by a

mean vector n and a positive definite (p.d.) or (at least) positive semi-definite (p.s.d.) matrix nn , i.e. one

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whose quadratic form with any non-zero vector is non-negative:

(95)

..0...0

,,dp

dsp

formquadratic

nnnn xx0xx

One of the ways an elliptical distribution is defined is via its p.d.f. which must be of the following form (recall that the

inverse of a p.s.d. matrix is also p.s.d.):

(96)

)()()(1 xxx gf

For example, the multivariate normal distribution is an elliptical distribution, with referred to as the variance-covariance

matrix:

(97)

,~)2(

)()(exp)(

)2()(

2/

121

2/

2/

Xxx

xnn

t

fetg

And so is the multivariate Student distribution:

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(98)

freedomofegreesddStudent

nf

tntg

nn

n

n

,,,~

)()(1)()2/()2/)(()(

)()2/(

1)2/)(()(

2/)(112/

2/

2/)(1

X

xxx

As is the multivariate logistic distribution.

(99)

,~)()(exp1

)()(exp)(

1)( 21

21

121

22/

2/

Logisticfe

etgt

t

Xxx

xxx

Once again, taking away the location-scale parameters, i.e. by setting/assuming O and setting/assuming has

been normalised into a correlation matrix R (all the diagonal elements are now 1’s), we can construct generically an

elliptical copula thus:

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(100)

1xxxxxx

uu

D)(..R

,RR)(,)()(

RR)(

,]1,0[]1,0[:

1

1

1

1

11

11

1 11

11

diagdspgfdfdxdxdxf

uF

uF

uF

F

u

u

u

CCC

C

nn iiuF

n

uF

i

uF

nn

ii

n

iellipticalelliptical

n

Thus Gaussian copula, Student’s t copula, and logistic copula, result, respectively, from when g represents, normal,

Student’s t, and logistic p.d.f.

While Gaussian and Student’s t copula are closely related, the key difference exploited in modelling is the fact that

Gaussian copula incorporates/exhibits no tail dependence; whereas, Student’s t copula does.

{QUIZ 13} Why do we name elliptical copulas “elliptical”?

Modelling with Copulas

Before the advent of copula, the tool kit for modelling the distribution of vector random variables was rather restricted to

just a few parametric families.

Beside multivariate normal (Gaussian), Student’s t, and beta (Dirichlet) distributions, there aren’t many multivariate

distribution families we can work with.

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At any rate, these specify that marginals come from the same family, i.e. “multivariate so-and-so” distribution is a

multivariate extension whereby individual marginals are by definition all from the “so-and-so” family (sometimes said that

univariate components appear as affine transformation of one another).

With copula, the scope for multivariate, probabilistic model building is broadened immensely, for now we are free to work

with marginals from different families, even using the copula to couple discrete marginals with continuous marginals

rather seamlessly.

One can have, for example, a bivariate distribution constructed from a bivariate Gaussian copula, one exponential

marginal and one Beta marginal.

Consider, for instance, the loan loss identity defined from a triplet of random variables: exposure at default, default event,

loss given default.

(101) figurepercentageobjectbooleanunitmonetary

efaultLossGivenDntDefaultEveDefaultExposureAtLoanLoss

Without copula, it’s often necessary to make simplifying assumptions, i.e. make exposure at default a deterministic

parameter , EAD , designate default event as a Bernoulli random variable D , parameterised by the single Probability of

Default or Default Probability (PD) parameter, and assume that this and the loss given default L , which may or may not

be Beta distributed, are in any case independent, no doubt such concessions are motivated not least by the

unavailability of bivariate Bernoulli-beta coupling:

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(102)

.).(

.).()(

)()(

vrbetapEAD

vrBernoulliLpEADceindependenLDEAD

EADrandomnonLDEADidentityLDEADLoanLoss

default

default

But copula mathematics can offer insights without making any kind of modelling assumption.

To demonstrate this point, let’s consider the definition of Value-at-Risk (VaR).

Suppose that over a given horizon asset '' A and '' B have a 99% VaR of ''AVaR and ''BVaR , respectively, then, without

any model assumption whatsoever, we can categorically place an upper bound on the probability of both assets falling

short of their respective VaR’s, simply by citing the Fréchet-Hoeffding inequality:

(103)

0}11.01.0,0max{}1,0max{),(1.0}1.0,1.0min{},min{),(

),()Pr(

Pr01.0)Pr(Pr01.0)Pr(Pr

''''''''

''''

''''

vuvuCvuvuC

vuCvVuU

VaRRVaRRvvVVaRRuuUVaRR

BBAA

BB

AA

Most appreciated is the fact that a copula-based methodology enables “decoupling” of the marginal model specification-

estimation-calibration stage from specifying-estimating-calibrating the joint probabilistic behaviour, thereby prescribing a

two-stage modelling process: [1] first model the individual distributions, then [2] proceed to model how their distributions

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join up.

Consider the problem of (parametric) estimation, i.e. given that the choice of marginals and the copula have been made,

determine the best values of the function parameters (both for copula function and marginal distributions) that best fits

the data.

Generically letting copulacopula and rginalsmarginalsma denote, respectively, the copula’ and marginals’ parameters,

the problem is to find:

(104) rginalsmacopularginalsma

copuladataFFitError

,)(minargˆ x

Rewriting the joint distribution in terms of copula and marginals, which in turn are rewritten explicitly with their respective

parameters reveals that the parametric estimation can indeed be performed in two stage, first over rginalsma and then

over copula , hence the Inference for the Margin (IFM) method [CLV: 156]:

(105)

dataxFxFCFitErrorii

dataxFxFFitErrori

xFxFCxFxFCFF

copularginalmannnrginalmacopula

rginalmannnrginalmarginalsma

copularginalmannnrginalmann

thst

copulacopula

thst

rginalsmarginalsma

thst

ˆ,,ˆminargˆ)(

,,minargˆ)(

,,,,|)(

111

111

11111xx

Parametric Estimation Methodology for Copulas

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What exactly do we mean by parametric estimation?

Given a data set in the form of d-point i.i.d. statistical sample dii 1x , and assume that a parametric multivariate

distribution |Xf has been chosen (model specification stage thus completed), and our task is to estimate the

parametric value based on data, in other words calculate dii 1

ˆˆ x .

For future references, we might as well also define statistical sampling, denoted djjX

1 for scalar/univariate case and

djj 1

X for vector/multivariate case, a set of i.i.d random variables whose realisations as random variates then comprise

the data sample, denoted djjx

1 for scalar/univariate case and d

jj 1x for vector/multivariate case:

(106)

n

ariatesvrandomdofsetsamplelstatistica

d

jjn

ji

j

d

jnj

ij

j

djj

nrealisatio

ariablesvrandom

dofsetsampling

lstatistica

djj

djj

nrealisatiodjj

x

x

x

xxX

x

x

x

x

xX ,

,

1,

,

,1

1

1

11

11

One of the most popular parametric estimation methodologies is based on the so-called Maximum Likelihood Method

(MLE) method.

Given one data point, the idea is to go with the distributional parameter which made the observed data point most likely.

Given 1d data points, the same thinking says one should go with the distributional parameter which made the

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observed data points most likely to have been generated in an i.i.d. process, hence the multiplication of individual

likelihoods.

The likelihood function and its generally more practical derivation, the log-likelihood function, are first defined below:

(107)

d

jjd

d

jjdd

d

jj

ffldatalllllikelihoodoglii

fdatalllikelihoodi

1

1

1

11

1

lnln))(ln()|()()(

)|()()(

xx

x

The maximum likelihood estimator is then found by way of optimisation:

(108)

d

jjdMLE fdatall

1

1 lnmaxarg)|(maxargˆ

x

Of course, the nicest thing about copula is that, by way of the canonical representation (53) (93), the log-likelihood also

separates nicely:

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(109)

d

j

n

irginalmaijiid

d

jrginalsmacopulajd

d

j

n

irginalmaijiicopularginalsmajd

d

j

n

irginalmaijiicopularginalmanjnnrginalmajd

sfdprginalmaparametric

n

irginalmaii

densitycopulaparametric

copularginalmannnrginalma

fdpntjoiparametric

rginalsmacopula

th

th

ththst

thst

xfc

xfc

xfxFxFcll

xfxFxFcf

1 1,

1

1

1

1 1,

1

1 1,,1,11

1

.'..

1111

...

ln,ln

lnln

,,ln)(

,,,

u

u

x

The fact that the log-likelihood separates into two parts, the first depending on both the copula’s parameter copula and

the marginals’ parameter rginalsma , the second only on the marginals’ once again suggests a two-stage parametric

estimation, hence the Canonical Maximum Likelihood (CML) method [CLV: 160], which in a sense represents a MLE

specialisation of IFM (105):

(110)

d

jcopularginalmanjnnrginalmajdcopula

d

j

n

irginalmaijiidrginalsma

thst

copulacopula

th

rginalsmarginalsma

xFxFcii

xfi

1,1,11

1

1 1,

1

ˆ,,ˆlnminargˆ)(

lnminargˆ)(

But converting the marginally distributed statistical sample data djj 1

x into uniformly distributed points djj 1

u can be

achieved, as per non-parametric method, without relying on any modelling assumption whatsoever, i.e. by using the so-

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called empirical distribution function (empirical c.d.f.), defined via the indicator function }1,0{'','': FALSETRUE1 :

(111)

functionindicator

boolean

d

jnjndnnEmpirical

d

jijidiiEmpirical

d

jjdEmpirical

djj FALSETRUEb

otwTRUEb

b

axaF

axaF

axaF

'','',.0

''1}{,

1,

1

1,

1

11,1

111

1

1

1

1

1

x

Using the univariate empirical distribution functions, then not only is it possible to decompose the parametric estimation

problem into 2 stages, it’s also possible to perform the 2nd phase in parallel, hence a 1-stage problem:

(112)

d

jcopulajnnEmpiricaljEmpiricaldcopula xFxFcii

copulacopula 1,,11

1|||| ,,lnminargˆ)(

Of course, given any specific data set, we don’t expect the two estimates to be the same, but they ought to be fairly

close:

(113) copulacopula ˆˆ||

Non-Parametric Copulas

Taking the idea of non-parametric statistics even further, let’s pursue the idea of a non-parametric copula.

Just as the simplest of non-parametric univariate distribution, i.e. empirical c.d.f., is obtained by using the data points

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themselves, so too is the simplest of non-parametric copulas obtained in a similar manner, as follows.

For a given univariate statistical sample djjx

1, let’s define (univariate) order statistics d

jjx1)( and (univariate) rank

statistics djjr

1 thus:

(114)

},,1{,,

},,1{

)(

1

)()()1(1)(

11

dtjtrxx

statisticsrankdr

xxrrr

statisticsorderxxxxxX

jtj

j

lklkdjj

djdjj

djj

nrealisatiodjj

For multivariate case, the situation, and hence notation, is a little bit more complicated, as the ranks, and hence orders,

can be different for each dimension },,1{ ni .

For a given multivariate statistical sample djj 1

x , define order statistics and rank statistics that achieve ordering/ranking

within each of the dimension },,1{ ni .

Still, we can define (multivariate) order statistics djj 1)(

x and (multivariate) multivariate rank djj 1

r statistics thus:

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(115)

},,1{,,

},,1{

,,1,

,,1,

)(,

1

)()()1(1)(

1,

,

,1

11

dtjtx

d

ni

ni

x

x

x

ijitji

nj

ilikilikdjj

idiji

djj

d

jjn

ji

j

djj

nrealisatiodjj

rx

r

xxrrr

xxxx

xX

[CLV: 161] Then define Deheuvels’ empirical copula thus:

(116)

1,,,,0,

,,,,

1

1 1

1

1 1

1

1 1,

1

1 1,

11

dt

dt

dttt

xxdt

dt

dtC

nid

j

n

iiijd

d

j

n

iiijd

d

j

n

iitjid

d

j

n

iitjid

niEmpirical ii

r1rI

x1xI

Goodness-of-Fit Tests for Copulas

Not only is it possible to specify the copula and estimate its parameters separately from specifying and estimating the

parameters for the marginals, it is also possible to perform a Goodness-of-Fit tests (GoF).

Malevergne & Sornette (2003) adapted the Kolmogorov as well as Anderson-Darling distances as their distributional test

metrics.

Meanwhile, Mashal & Zeevi(2002) and Chen, Fan, Patton (2004) exploited the fact that the Student’s t distribution is a

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heavy-tailed generalisation of (and therefore embeds as a special case) the normal distribution.

Perhaps one of the simplest methods, first proposed in a bivariate context by Nacaskul & Sabborriboon (2009), is to

transform the data into the unit hyper-cube (a square if we’re talking just bivariate copulas) using the empirical marginals.

This unit hyper-cube is then chopped up into mini hyper-cubes (mini squares or rectangles if we’re talking just bivariate

copulas) which are then treated as data bins, then test each proposed copula (function as well as parameterization) by

comparing expected frequencies (under the hypothesis of the proposed copula being the right one) verses observed

frequencies, a la the well-known Chi-square GoF test for category data.

Along the same line, Arnold, Helen (2006) had earlier noted how the Chi-square GoF statistics could be used to test a

proposed copula against the null hypothesis of independence (independent copula being the right one).

Fig.8

-30%

-25%

-20%

-15%

-10%

-5%

0%

5%

10%

15%

-25% -20% -15% -10% -5% 0% 5% 10% 15%

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Scatter Plots of "NYSE Index" vs. "Coca Cola" returns – ‘actual’ (left) & ‘[0,1]’ (right)

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<<2010 Poomjai Nacaskul, Ph.D. | lxiii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>

Fig.9

-25%

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

25%

-30% -25% -20% -15% -10% -5% 0% 5% 10% 15%

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Scatter Plots of "SET Index" vs. "Siam Cement" returns – ‘actual’ (left) & ‘[0,1]’ (right)

{QUIZ 14} Does the Goodness-of-Fit test methodology described by Nacaskul & Sabborriboon (2009) and/or Arnold

(2006) require additional assumptions regarding the marginal distributions? Why/why not?

Monte Carlo Simulation with Copulas

Recall how probability is concerned with the distributional and expectation properties of random variables, such as those

comprising our statistical sampling djj 1

X , and statistics is concerned with how to infer the distributional and

expectation properties of the random variables given the empirical data observed in the form of our statistical sample

djj 1

x , Monte Carlo simulation describes a methodology by which we a computer algorithm is used to generate a

sequence of hypothetical events and artificial data, hence our randomly-generated random variates djj 1

x in manner

consistent with the specified distributional and expectation properties.

In other words, probability tells us how a so-and-so distribution would appear, statistics tells us which so-and-so

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distribution best accounts for the appearance of observed data, while Monte Carlo simulation generates numerical

examples consistent with whatever so-and-so distribution was specified.

In univariate setting, the task involves two steps: (1) generating a generic sequence of pseudo-random numbers djju

1

(random in the sense that one cannot predict the next number) and (2) transforming these pseudo-random numbers into

random variates djjx

1 with the specified distributional and expectation properties.

The first step involves the pseudo-random number generator (PRNG), whose generated 1d i.i.d. pseudo-random

numbers lie uniformly distributed between zero and one:

(117)

djUnifUdii

djUnifUdiiuUuUjuPRNGi

jpseudo

jjjpseudojpseudo

,,1,)1,0(~...

,,1,)1,0(~...,PrPr,1,)(

The second step simply involves the c.d.f. inverse, of whatever distribution desired, noting how:

(118)

xXuU

uFUFuUxFXFxX )()(Pr)Pr()()(Pr)Pr( 11

So that a sequence of non-uniform pseudo-random variates can be generated and stored as artificial data:

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(119)

,1,)(~...

,1,)(~...,PrPr,1,)(

jDistrXdii

jDistrXdiixXxXjxuii

jpseudo

jjjpseudojpseudojpseudo

Altogether:

(120) ,1,1

)()( juFxuPRNG jpseudojpseudoiijpseudoi

As with multivariate analysis where multivariate distribution can be simplified by decomposing it into (1) the marginals

and (2) the copula, so too within the context of Monte Carlo simulation is multivariate pseudo-random number generation

considerably simplified if the task can be broken down into ensuring that (1) individually each of the 1n components of

a generated pseudo-random vector obeys the marginal distribution while (2) together as a whole vector they obey the

copula function.

Overall the process still looks the same, except a series of pseudo-random vectors, as opposed to mere scalars, are

generated:

(121)

,1,

,1

,1

,11

1

)(

,

,

,11

)(

j

uF

uF

uF

u

u

u

u

u

u

PRNG

jnpseudon

jipseudoi

jpseudo

jpseudoii

jnpseudo

jipseudo

jpseudo

jnpseudo

ipseudo

pseudo

jpseudoixu

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The key feature here is that kpseudoU and klpseudoU are allowed to not be independent of one another.

(122) lknlkuUuUuUuU llpseudokkpseudollpseudokkpseudo ,},,1{,,PrPrPr

This is done inside the first step, essentially separating it into two sub-routine steps:

(123) ,1,)()2()1(

jPRNG jpseudoiijpseudoijpseudoixuv

This additional step jpseudojpseudo uv is necessary to allow the introduction of dependence structure via copula.

For elliptical copulas, notably the Gaussian copula, this step is greatly simplified by way of Cholesky decomposition of

the p.d. (positive definite) correlation matrix R into a product of a lower triangular matrix L and its transpose L .

Given a p.d. matrix nnM in general, Cholesky decomposition is accomplished as follows:

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(124)

niill

llmllllllllm

lmlllllllm

nil

llmllllllm

lmlllllm

nilm

llllm

mlllm

LL

l

ll

lll

lll

ll

l

mmm

mmm

mmm

M

ii

i

k kilkliliiliiliililli

i

k ikiiiiiiiiiiiiii

iiiiiii

iiiii

nn

inii

ni

nnnin

iii

nnnin

iniii

ni

,,1),(0

,,3),2(0

,,2),1(,0

00

00

00

1

11,2211

1

12

2211

22

1212232221212

22122222222122122

11

1121111

1111111111

1111

1

1

11

1

1

1111

Summarised thus:

(125)

ii

i

k kilklili

i

k ikiiii

lllm

lnil

lml

ni 1

1

1

12

,},,1{

,

,},,1{

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Or in the form of a double loop algorithm:

(126)

ii

i

k kilklili

i

k ikiiii lllm

lnilforlmlnifor1

11

12 ,,,1;,,,1

(127)

,1,L

0L,,LLR

11

T

1

1

1

11

11

j

x

x

x

v

v

v

x

x

x

v

v

v

z

z

z

u

u

u

PRNG

jiji

uniformdependent

npseudo

ipseudo

pseudo

jnpseudo

ipseudo

pseudo

jpseudo

normalandardstcorrelated

jpseudo

jnpseudo

ipseudo

pseudo

jpseudo

normalandardsttindependen

jnpseudo

ipseudo

pseudo

jnpseudo

ipseudo

pseudo

jpseudo

uniformtindependen

jnpseudo

ipseudo

pseudo

jpseudo

ij

vzx

zu

One of the computationally more efficient alternative methods of generating standard normal random variates from

uniform ones is the Box-Muller transform, which utilises two independent uniform random variates to generate two

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independent standard normal random variates at a time (efficiency comes from the simple analytical expression, not from

the fact that two random variates are generated together):

(128)

1001

,00

~...

2sinln2

2cosln2

21

21

2

1

2

1

Z

z

pseudo

normalandardsttindependen

transformMullerBox

pseudo

pseudopseudo

uniformtindependen

pseudo

pseudo

dii

uu

uuzz

uu

PRNG

{QUIZ 15} What are Excel commands for (a) generating a uniform random variate and (b) transforming a uniform

random variate into standard normal random variate?

Financial Risk Modelling with Copulas

First up, let’s go over some backgrounds before we bring in copulas.

Risk is defined by a triplet of possibility, probability; and utility.

By possibility, we mean there must be more than one possible outcomes involved.

Mathematically, this corresponds to the notion of a measurable set , , which itself comprises of the set, i.e. the

sample space, , representing the (infinite/uncountable) universe of possible outcomes in all its infinite details, and the

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sigma-algebra, i.e. the event set , , representing the set of events, themselves referred to mathematically as

measurable sets, whereby (a) an empty set, corresponding to non-event, is included in , (b) if an event is defined

(“something happening”), so is it’s complement (“that something not happening”), and (c) for any (possibly infinite)

collection of events defined, their intersection is also a defined event.

The term uncertainty may also be used, whence risk becomes a triplet of uncertainty, probability, and utility, but because

there are many concepts of uncertainties, depending on interpretations, let’s not use this term here.

By probability, we mean there is to be a function, called probability measure, , which assigns to each measurable set

or event in a number (a) between zero and one (b) such that these values assigned to disjoint events simply add up.

It is then up to us (not mathematics) to interpret what probability measure means to us: frequency of occurrences, as per

classical statistics, or a degree of belief that an event will take place, as per Bayesian statistics, and so on.

Any such triplet ,, is referred to in mathematics as a probability space.

By utility, we mean there exists a kind of preference structure, }{ , that essentially allows us not only to rank whether a

given outcome, once realised, is desired when compared to another, but also even when one alternatives (generally

both) is yet uncertain to occur, hence probabilistic in nature, i.e. with associated probability assigned to it by .

In short, with risk, (future) reality must contains (possibly infinite, possibly uncountable) alternatives, whose probabilities

add up to one, and whose realisations or likelihood of being realised are subject to preference, hence the preference of

“upside risks” over “downside ones”, and so on.

Financial Risk means that (a) preference structured is defined with reference to financial outcomes, (b) randomness arise

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from/are rooted in financial market/institution variables/factors, (c) the situation can be managed/mitigated by means of

financial techniques/tools, and/or (d) the problem is seen as/deemed to be intrinsic/integral to financial

markets/institutions.

Market Risk is defined as the opportunity/possibility & probability of financially relevant gains/losses due to movements in

the financial-market and monetary-economic variables, namely interest/exchange rates, equity/commodity prices, etc.

Credit Risk is defined as the opportunity/possibility & probability of financially relevant losses (but occasionally gains)

due to credit events: (w.r.t. bank portfolio) defaults on loans as well as counterparty/settlement failures, (w.r.t. bond

portfolio) defaults on interest/principal payments as well as credit-rating downgrades, (w.r.t. derivatives portfolio) single-

obligor as well as multi-obligor events, and so on.

Operational Risk is defined as the opportunity/possibility & probability of financially relevant losses due to failures, frauds,

and/or errors as well as random accidents, natural catastrophes, and/or manmade disasters, whence leading to

damages, disruptions, and/or incursions, thereby negatively impacting financial conditions, business conduct, and/or

institutional integrity overall.

Risk Management (Process) comprises 4 steps: identify/define/indicate, measure/assess/monitor,

mitigate/control/manage, and review/analyse/report.

Risk Modelling, the act/process/activity of building/testing/implementing a Risk Model, indeed pertains to all 4, although

in terms of model development, it’s usually centred on/associated with the 2nd phase.

For some, it might be useful to pursue further distinction, i.e. between modelling risk dynamics/factors, the part of risk

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modelling discipline concerned with the nature of risk factors themselves (which theoretically appears much the same to

everyone regardless), and modelling risk exposures/positions, the part of risk modelling discipline that has to do with how

the (nature of) risky environment transpires to become financial-economic costs or benefits to us, given the structure of

our financial positions, which is what expose us to the risk dynamics/factors to begin with (hence fundamentally effects

each financial portfolio uniquely).

Sometime the source of randomness, that which constitutes our risk factor, or the nature of our pay-off as a function of

that randomness, that which constitutes our risk exposure, is collectively/generically referred to as our risk drivers.

(129)

)()(

/

/

//)(,//)(

,//)(,//)(

)(

,,,

}{,,

sderivativefinancialviacontingentsinstrumentltraditionaviadirect

positionsxposureserisk

latentobservable

factorsdynamicsriskdriversriskModelsRisk

reportanalysereviewivmanagecontrolmitigateiii

monitorassessmeasureiiindicatedefineidentifyi

rocessPManagementRisk

RiskslOperationaRisksCreditRisksMarketRiskFinancial

Risk

StructurereferenceP

Utility

SpacerobabilityP

robabilityPyPossibilit

With that, now let’s bring in copulas.

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Here the first distinction to make is between cases of there being one or multiple risk driver(s).

The second distinction to make is between cases involving multiple risk drivers which are all independent or otherwise.

Let’s also make the third distinction between when dependent risk drivers are essentially “multiplied” together, such as

random defaults and random losses given defaults, and when dependent risk drivers are essentially “added” together,

such as a portfolio of return-correlated assets.

And perhaps it’s useful to make the forth and final distinction between when portfolio risk drivers are simply multivariate

normal random variables, or otherwise.

(130)

copulaotwdriversriskadditive

copuladriversrisktivemultiplicadriversriskdependent

driversrisktindependen

driversriskaggregatemultiple

driverriskindividualinglesModelsRisk

.,~

/

/

X

{QUIZ 16} Think of an instance where copula enables us to capture market and other risk (credit/counterparty,

operational, reputational, etc.) w/o having to assume independence?

Credit Risk Modelling with Copulas

First up, let’s go over some backgrounds before we bring in copulas.

Recall that with single-borrower loans, primary focus in given to assessing the probability that that particular loan will

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default, hence the PD parameter.

With a portfolio of single-borrower loans, each of which may have different maturities, primary focus then turns to

assessing the probability that over a given (investment) horizon, there will have been more than one occurrences of loan

defaults, hence default correlation, although a more precise/technically accurate (albeit rather clumsy) term should be

something like multi-default dependency structure.

In the narrowest sense, a default correlation is defined as per the Pearson’s product-moment definition.

Given a basket of just 2 loans, each with respective PD, i.e. each a Bernoulli random variables, then default correlation is

by definition:

(131)

ncorrelatiodefault

Defaultii DDppppp

pppipBernoulliD 11Pr,11

2,1,~ 21122211

2112

In the broadest sense, default correlation refers conceptually to the way occurrences of individual defaults are not wholly

independent events, hence the application of copula is motivated by the practical needs for more general multi-default

dependency structures.

Again, given a basket of 2 loans:

(132) 212112 1Pr ppDDpncorrelatiodefault

For basket of 2n loans, this amounts to saying that even if individual PD parameters are equal, the total number of

defaults (each a Bernoulli random variable) will not add up into a Binomial random variable (sum of i.i.d. Bernoulli random

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variables):

(133)

),(~,,1,)(~1

pnBinXDnipBernoulliDncorrelatiodefaultn

iii

Because defaults are not normal random variables, and we ought to be free as to how to arrive at the quantity ijp , it

would be difficult to get anywhere with default correlation without copulas.

In fact, in order to induce some kind of dependency structure between defaults, it’s better to think of a default process in

general before recapitulating back into simple ‘yes’/’no’ default event (a Boolean random variable).

Think of default process as the process of dying in the biological world.

Everybody dies, the question is when.

Then instead of working with a Bernoulli random variable, let’s talk in term of a positive continuous default time or time-to-

default or time-until-default random variable, 0T , whose c.d.f. is then called default-time c.d.f.

The time-to-default concepts then recapitulates back to default event once we specified a time interval, our

(investment/loan) horizon, usually a year.

(134) yearTDp 1Pr1Pr

The flipped side to the default-time c.d.f. is called the survival function:

(135) tTtFtTtS 1)(1Pr)(

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From which it follows that the default time p.d.f. can be written in terms of either:

(136) )(')(')( tStFtf

One popular (and intuitive) way of modelling a default process is to consider the asset value of a going concern as a

stochastic process, i.e. a family of random variables indexed by time, 0, tX t , whence defining time-to-default

random variable as a stopping time reached when the asset value dips below a certain default threshold (which one may

think of as total liability of the firm), hence the framework goes by the name of Asset Value Model (AVM) methodology.

(137) 0,inf0,00

thresholddefaultXTtimestoppingtX ttt

In contrast, a reduced-form methodology does not delve into how asset value evolves as a process, instead approaches

the default time random variable summarily by way of so-called hazard/failure rate, which is referred to in this context

(credit risk modelling as opposed to reliability theory/modelling, from which the term hazard/failure rate originates) as

default intensity., where we begin with a constant, i.e. “time-homogeneous”, default intensity parameter, i.e.

0)( t .

(138) )()(')exp()(')exp()(

tStSttSttS

From which it then follows that default arrival is a homogeneous Poisson process, in turn, implying that default time is

exponentially distributed, thus:

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(139) )(~)exp(1)()exp()(1)Pr(1)Pr()( ExpTttFttFtTtTtS

Then use this “lambda” as the definition of default intensity, allowing it to be, not just constant, but a function of time,

hence not merely a hazard rate, but a hazard rate function, from which it then follows that default arrival follows a non-

homogeneous Poisson process thus:

(140)

t

dsstStStSt

0

)(exp)()()(')(

Note how this default intensity or hazard rate function can then be interpreted as an instantaneous default rate,

conditional on having survived up to time 0t .

To see that, first rewrite )(t as follows:

(141) )(1

)()(1

)(')()(')(

tFtf

tFtF

tStSt

Recalling that, by definition:

(142) t

tTttTt

tFttFtFtftt

PrPrlim)()(lim)(')(00

Use Bayes theorem to arrive at the instantaneous default probability, conditional on having survived up to time 0t :

(143)

)(1

)()(limPr

PrlimPr

PrlimPrlim0000 tF

tFttFtT

ttTttT

tTttTtTttTtttt

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To convert from default probability to default rate (default probability per unit time), one simply divides (143) through by

the time increment, which in our case is t (and appears inside the limit), eventually, with (142) recovering the

expression for )(t :

(144)

)()(1

)()(1

)()(limPr

lim00

ttF

tftFt

tFttFt

tTttTtt

In any event, the similarity between the default intensity or hazard rate in credit risk modelling and the short rate in

interest rate risk modelling is striking, and indeed Duffie & Singleton (1999) went on to prove that defaultable bonds can

be valued, within the short-rate framework, as if it were default-free, but with the hazard rate, (presumed independent of

the short rate) added to the short rate in the time value discounting (each under risk neutral expectation):

(145)

eBondDefaultFredsssreBondDefaultabldssrt rate

discountodifiedm

ratehazard

t

rateshort 00

)()(exp)(exp

Finally, it’s possible to generalise this “lambda” into being some non-negative stochastic process, from which it then

follows that default arrivals becomes a doubly stochastic Poisson process, perhaps better known as the Cox process.

With that, now let’s bring in copulas.

Given the individual default processes in terms of individual default-time c.d.f.’s, alternatively in terms of survival

functions, we can then employ a copula, alternatively a survival copula (57)(94), to construct a joint default-time c.d.f.,

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alternatively a joint survival function:

(146) nnn

nnn

tTtTttSfunctionsurvivalntjoitTtTttFfdcdefaultntjoi

111

111

Pr,,Pr,,...

In other words, credit correlation modelling then becomes a matter of specifying and parameterising the appropriate

copula/survival copula used to couple together individual default-time c.d.f.’s/survival functions:

(147)

ban

n

iii

nnb

n

iii

nnb

n

iii

nnn

nnan

CCiffttFtS

tFtTtTtStFtFCtS

CCtStSCtS

copulasurvivalaingustStSCttScopulaaingustFtFCttF

,,1

)()Pr()Pr(1)(1,,1

)(1)(1,,11

)(,,,,)(,,,,

11

111

111

111

111

u1u1u

In particular, Li (2000) proposed using a Gaussian copula construction of joint default-time c.d.f. where, in a bivariate

case:

(148) 11,,,

...

...'

1

...'

1

...

fdcnormalariatebiv

ariatevrandomuniform

fdctimedefaultsB

BB

ariatevrandomuniform

fdctimedefaultsA

AA

fdctimedefaultntjoi

BA tFtFttF

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Then by invoking the 1-year horizon, is interpreted as the default-time correlation (actually the original paper uses the

term “survival time correlation”), i.e. in the sense of:

(149) BA

BABA TVarTVar

TTCovFFF

,

,)1(,)1()1,1( 11

Li (2000) went on to remark that in reality this parameter is generally “much smaller” than the more ubiquitous asset

correlation, which, provided some additional information regarding the individual capital structures, can in turn be

derived from equity (return) correlation, which is readily available on a historical/implied basis.

In any event, with Gaussian copula, a Monte Carlo simulation approach to simulating default times expediently begins

with generating correlated multivariate standard normal random variates, as per (127), from which dependent default

times can then be imputed.

(150)

,1,

1

1

11

111

j

xF

xF

xF

t

t

t

x

x

x

timesdefaultdependent

jnsimulatedn

isimulatedi

simulated

jnsimulated

isimulated

simulated

jsimulated

normalandardstcorrelated

jnpseudo

ipseudo

pseudo

jpseudo tx

{QUIZ 17} What is the difference between default correlation 2211

2112

11 ppppppp

Default

and the expression

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21

21 ,TVarTVar

TTCov

(where 1T and 2T represent time-to-defaults)?

Detour in Credit Derivatives & Derivatives Pricing

First up, let’s go over some backgrounds before we bring in copulas.

Recall that (financial) derivatives are financial instruments (securities, contracts, bilateral exposures) with no intrinsic

claim values, whose prices in theory derive deterministically (by way of mathematical formulas) from other underlying

stochastic processes (financial assets, capital/commodities market indices, monetary/economic numbers, and so on,

also referred to generically as underlying assets), although in practice may be subject to non-deterministic market

dynamics and/or liquidity adjustment factors of their own.

Hence the term applies to financial options, swaps, and contingent claims in general.

Early on, derivatives were generally underlined by stochastic processes derived from equity stock prices, foreign

exchange rates, and various interest rates, hence clearly driven by market risks; whereas, later on, newer classes of

credit derivatives, so called because they are not so much underlined as defined vis-à-vis credit events, emerged and

gained popularity.

Early on, derivatives were generally underlined by single stochastic processes; whereas, later on, newer classes of

basket derivatives, so called because they are not underlined by an individual asset but in terms of basket reference,

emerged and gained popularity.

Two types of basket credit derivatives are most prominent, namely Basket Default Swaps (BDS), which is a multi-asset

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generalisation of the single asset Credit Default Swaps (CDS), and Collateralized Debt Obligations (CDO).

One of the most familiar forms of BDS is the so-called 1st-to-default CDS (1tD-CDS), where a credit event is defined by

the first default (if any) amongst a basket of referenced names, and the rather obvious generalisation into the 2nd-to-

default CDS (2tD-CDS) and eventually nth-to-default CDS (NtD-CDS) versions.

In any event, the basic set up is that of contingency claim analysis, diagrammatically depicted thus:

(151)

problem

t

assumption

marketfreearbitrage

conditionarbitrageno

upset

TTTT

T

t

t

C

sSnrealisatioonlconditionaknownSCCunknownSknownS

processstochastictS

?

0,

/

Whereas the prices of underlying assets are subject principally by financial markets’ demand/supply pressures, the

prices of derivatives are determined by a more exact mechanism.

In essence, because derivatives exist alongside underlying factor, but without, as it were, introducing additional source

of randomness, it is possible to apply the so-called arbitrage/replication/hedging argument to argue that its present price

can be determined exactly from the present realisation of all relevant random variables (i.e. the random variates)

because an arbitrage-free risk-less (i.e. all risks perfectly hedged) strategy can be devised to replicate exactly the future

pay-outs, hence exact valuation, of said derivatives.

Let’s hide a lot of possibly very dense, very technical, and very complicated details, and summarise by saying that with

an Equivalent Martingale Method (EMM) of options pricing, any derivatives can be priced as an expectation of contingent

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pay-offs discounted at the risk-free rate of return, taken against a so-called risk-neutral (probability) measure:

(152)

)(exp),(..

)(..,)(),(

,),(

/

Q}{

tTrTtBei

billtreasurycouponzeroviagediscountfreeriskcreditTtB

SCC

CTtBC

freerisk

indexassetunderlying

T

functioninisticetermd

offpayinalterm

T

Tt

One of the most basic, widely variable, and familiar of all credit derivatives is the ubiquitous (single-asset) CDS where the

protection buyer pays premiums to the protection seller until such time as the contract expires or the credit event

triggered (generally corresponding to whenever referenced credit/name defaults on any of its liabilities), whichever

comes first, and the protection seller stands ready to compensate the protection buyer for such loss (generally net of

recovery) should the credit event be so triggered.

Starting from 00 t , let mtttt m,,,0 121 be the payment dates for the premium leg of the deal, whose

present value at the start of the contract is then given, in terms of risk-neutral survival function, by:

(153) tTtSremiumPCDSNotionaltStDiscountremiumsPPVm

i

paymentpremium

fractionyear

annuallyxpressedeprotectionof

valueface

functionsurvival

neutralrisk

ii

Q}{Q}{

11

Q}{1 Pr)(,'

Conversely, the present value of the protection leg is given, in terms of risk-neutral default c.d.f., by:

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(154) tTtFoveryRateecRNotionaltFTDiscountrotectionPPVpaymentprotection

defaultgivenlossprotectionofvalueface

fdcdefault

neutralrisk

m

Q}{Q}{

%

...

Q}{ Pr)(,1

Equating (153) with (154) yields the CDS premium the start of the contract:

(155)

m

iii

m

tStDiscount

overyRateecRtFTDiscountremiumPCDS

11

Q}{1

Q}{ 1'

Alternatively, the present value of the protection leg could be broken down according to the same time bucketing as with

the premium leg:

(156)

m

i

paymentprotection

defaultgivenlossprotectionofvalueface

yprobabilitdefault

neutralrisk

iii overyRateecRNotionaltStStDiscountrotectionPPV1 %

Q}{1

Q}{ 1

Equating (153) with (156) in lieu of (154) then yields the CDS premium the start of the contract:

(157)

m

iii

m

iiii

tStDiscount

overyRateecRtStStDiscountremiumPCDS

11

Q}{1

1

Q}{1

Q}{ 1'

This CDS premium is fixed at the start of the contract, and subsequently the marked-to-market (M2M) value of the

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contract is the difference between the present value of the premium leg and the present value of the protection leg.

Pricing Credit Derivatives with Copulas

With 1tD-CDS, the pricing methodology essentially retains the same structure as when pricing (single-asset) CDS, with 2

key differences.

The major difference is that survival time is redefined to be the minimum of the survival times (or second smallest for 2tD-

CDS and so on).

The minor difference is that recovery rates may differ for each asset in the referenced basket.

Let’s deal with the major issue of redefining survival time, which is where copula comes in.

Just as with single-underlying derivatives, where pricing formulas depend wholly on the return volatility parameter, i.e.

regardless of mean return, so do formulas for pricing basket derivatives depend most critically on the structure of

dependency amongst the underlying factors, hence the importance of copula specification for pricing basket derivatives,

and especially so in the case of basket credit derivatives, where risk drivers are certainly not multivariate normal random

variables.

In essence, we need to carry Sklar’s theorem (46)(91) through to a risk-neutral setting, replacing real-world probability,

retrogressively referred to as physical probability, with risk-neutral definition.

So instead of “physical” copula, we shall work with risk-neutral copula, and instead of “physical” copula density, we shall

work with risk-neutral copula density, and so on.

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But what is the copula applied to?

Once again, recall how instead of working with Bernoulli random variables representing defaults and default correlation

in that sense (131), we shall deal with default times, which, for the simplest case of constant hazard rates, are

exponential random variables.

In essence, we need a copula representation of a joint c.d.f. with exponential marginals.

Let the referenced basket consist of 1k assets, whose default times are thus designated by the following random

vector.

(158) kiExpT

T

T

T

iik

k

i ,,1,~

1

T

First, let’s start with another random vector, one whose components are independent and exponentially distributed:

(159)

kji

jisSsS

sSsSExpS

S

S

S

jjii

jjii

iik

k

i ,,1,,

,PrPr

Pr~

)(

1

S

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Now use individual exponential c.d.f.’s to transform the components into independent uniform random variables:

(160)

kiUnifUdii

S

S

S

SF

SF

SF

U

U

U

i

kk

ii

kk

ii

k

i ,,1,)1,0(~...

exp1

exp1

exp1 11111

U

Using inverse normal c.d.f. then begets a vector of i.i.d. standard normal random variables:

(161)

kiZdii

S

S

S

SF

SF

SF

U

U

U

Z

Z

Z

i

kk

ii

kk

jj

k

i

k

i ,,1,)1,0(~...

exp1

exp1

exp1

1

1

111

1

1

111

1

1

11

1

Z

Now multiply this vector by the Cholesky decomposition L of the correlation matrix TLLR :

(162) k[0,1]R,)R,0(~LT NXZX

Then reverse the mappings, first to get back to uniform random vector.

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(163)

kiUnifV

X

X

X

V

V

VcopulaGaussianaei

structuredependenceawithendowednow

i

k

i

k

i ,,1,)1,0(~)..(

11

V

Then to exponential random vector whose components are now endowed with the dependence structure of a Gaussian

copula:

(164)

kiExpT

X

X

X

V

V

V

VF

VF

VF

T

T

TcopulaGaussianaei

structuredependenceawithendowednow

ii

k

k

i

i

k

k

i

i

kk

ii

k

i ,,1,~

1ln

1ln

1ln

1ln

1ln

1ln

)..(

1

1

1

1

1

1

11

11

T

When pricing via EMM, our task then involves Monte Carlo simulation to generate default-time random variates,

essentially starting from (160):

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(165)

sim

rrpseudorpseudo

sim

rrkpseudo

ipseudo

pseudo

rkpseudo

ipseudo

pseudo

rpseudo

sim

rrkpseudo

ipseudo

pseudo

u

u

u

z

z

z

u

u

u

#

1T

#

1

1

1

11

1

#

1

1

L

zxz

For each simulation run simr #,,1 , use the following joint default times.

(166)

sim

rrk

kpseudo

i

ipseudo

pseudo

rk

kpseudo

i

ipseudo

pseudo

rkpseudok

ipseudoi

pseudo

rkpseudo

ipseudo

pseudo

rpseudo

x

x

x

v

v

v

vF

vF

vF

t

t

t

#

1

1

1

1

1

1

1

11

11

1ln

1ln

1ln

1ln

1ln

1ln

t

For instance, for pricing 1tD-CDS, register, for each simulation run, the minimum default time.

(167) sim

rkrpseudoirpseudorpseudoirpseudo t#

11min ,,,,min

ttt

We can now use (153) to calculate the present value of the premium leg (in an expression which involves an indicator

function):

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(168)

sim

r

m

i

paymentpremium

fractionyear

annuallyxpressedeprotectionof

valuefacettirsim remiumPCDStDNotionaltDiscountremiumsPPVirpseudo

#

1

11 '1

1min

1

Then use (156), which for simulation purpose is somewhat preferable to (154), to calculate the present value of the

protection leg thus:

(169)

sim

r

m

i

paymentprotection

defaultgivenlossprotectionofvalueface

tttirsim overyRateecRNotionaltDiscountrotectionPPVirpseudoi

#

11 %

11

min

1

The (M2M) present value of the 1tD-CDS, as a function of premium is then simply:

(170)

sim

rrsimrsimsimsim rotectionPPVremiumsPPVCDStDPV

#

1#

11

Conversely, the initial premium is set such that the above expression is exactly zero at the start of the contract.

{QUIZ 18} Describes the procedure for pricing a 2tD-CDS (second-to-default basket default swap) assuming that you

already have 10,000 simulated default times for each of the 5 assets in the reference basket?