financial modelling with copula functions poomjai nacaskul ... · financial modelling with copula...
TRANSCRIPT
<<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
<<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
<<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.
<<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:
<<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-
<<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.
<<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
<<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.
<<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):
<<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:
<<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
<<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
<<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):
<<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):
<<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 ,
<<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:
<<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:
<<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
<<2010 Poomjai Nacaskul, Ph.D. | xix | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
[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:
<<2010 Poomjai Nacaskul, Ph.D. | xx | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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:
<<2010 Poomjai Nacaskul, Ph.D. | xxi | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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
<<2010 Poomjai Nacaskul, Ph.D. | xxii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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?
<<2010 Poomjai Nacaskul, Ph.D. | xxiii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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:
<<2010 Poomjai Nacaskul, Ph.D. | xxiv | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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:
<<2010 Poomjai Nacaskul, Ph.D. | xxv | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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
<<2010 Poomjai Nacaskul, Ph.D. | xxvi | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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
<<2010 Poomjai Nacaskul, Ph.D. | xxvii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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.
<<2010 Poomjai Nacaskul, Ph.D. | xxviii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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:
<<2010 Poomjai Nacaskul, Ph.D. | xxix | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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
<<2010 Poomjai Nacaskul, Ph.D. | xxx | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
[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
.'.....
)()()(),(),()()(
),()(),(),(
<<2010 Poomjai Nacaskul, Ph.D. | xxxi | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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:
<<2010 Poomjai Nacaskul, Ph.D. | xxxii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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:
<<2010 Poomjai Nacaskul, Ph.D. | xxxiii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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:
<<2010 Poomjai Nacaskul, Ph.D. | xxxiv | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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:
<<2010 Poomjai Nacaskul, Ph.D. | xxxv | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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
<<2010 Poomjai Nacaskul, Ph.D. | xxxvi | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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
<<2010 Poomjai Nacaskul, Ph.D. | xxxvii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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:
<<2010 Poomjai Nacaskul, Ph.D. | xxxviii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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:
<<2010 Poomjai Nacaskul, Ph.D. | xxxix | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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:
<<2010 Poomjai Nacaskul, Ph.D. | xl | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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
<<2010 Poomjai Nacaskul, Ph.D. | xli | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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 .
<<2010 Poomjai Nacaskul, Ph.D. | xlii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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:
<<2010 Poomjai Nacaskul, Ph.D. | xliii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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:
<<2010 Poomjai Nacaskul, Ph.D. | xliv | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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
<<2010 Poomjai Nacaskul, Ph.D. | xlv | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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:
<<2010 Poomjai Nacaskul, Ph.D. | xlvi | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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 .
<<2010 Poomjai Nacaskul, Ph.D. | xlvii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
{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
<<2010 Poomjai Nacaskul, Ph.D. | xlviii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
]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:
<<2010 Poomjai Nacaskul, Ph.D. | xlix | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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
<<2010 Poomjai Nacaskul, Ph.D. | l | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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:
<<2010 Poomjai Nacaskul, Ph.D. | li | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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:
<<2010 Poomjai Nacaskul, Ph.D. | lii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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.
<<2010 Poomjai Nacaskul, Ph.D. | liii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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:
<<2010 Poomjai Nacaskul, Ph.D. | liv | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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
<<2010 Poomjai Nacaskul, Ph.D. | lv | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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
<<2010 Poomjai Nacaskul, Ph.D. | lvi | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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
<<2010 Poomjai Nacaskul, Ph.D. | lvii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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:
<<2010 Poomjai Nacaskul, Ph.D. | lviii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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-
<<2010 Poomjai Nacaskul, Ph.D. | lix | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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
<<2010 Poomjai Nacaskul, Ph.D. | lx | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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:
<<2010 Poomjai Nacaskul, Ph.D. | lxi | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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
<<2010 Poomjai Nacaskul, Ph.D. | lxii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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)
<<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
<<2010 Poomjai Nacaskul, Ph.D. | lxiv | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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:
<<2010 Poomjai Nacaskul, Ph.D. | lxv | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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
<<2010 Poomjai Nacaskul, Ph.D. | lxvi | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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:
<<2010 Poomjai Nacaskul, Ph.D. | lxvii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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{
<<2010 Poomjai Nacaskul, Ph.D. | lxviii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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
<<2010 Poomjai Nacaskul, Ph.D. | lxix | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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
<<2010 Poomjai Nacaskul, Ph.D. | lxx | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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
<<2010 Poomjai Nacaskul, Ph.D. | lxxi | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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
<<2010 Poomjai Nacaskul, Ph.D. | lxxii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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.
<<2010 Poomjai Nacaskul, Ph.D. | lxxiii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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
<<2010 Poomjai Nacaskul, Ph.D. | lxxiv | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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
<<2010 Poomjai Nacaskul, Ph.D. | lxxv | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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)(
<<2010 Poomjai Nacaskul, Ph.D. | lxxvi | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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:
<<2010 Poomjai Nacaskul, Ph.D. | lxxvii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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
<<2010 Poomjai Nacaskul, Ph.D. | lxxviii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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.,
<<2010 Poomjai Nacaskul, Ph.D. | lxxix | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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
<<2010 Poomjai Nacaskul, Ph.D. | lxxx | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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
<<2010 Poomjai Nacaskul, Ph.D. | lxxxi | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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
<<2010 Poomjai Nacaskul, Ph.D. | lxxxii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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
<<2010 Poomjai Nacaskul, Ph.D. | lxxxiii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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:
<<2010 Poomjai Nacaskul, Ph.D. | lxxxiv | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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
<<2010 Poomjai Nacaskul, Ph.D. | lxxxv | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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.
<<2010 Poomjai Nacaskul, Ph.D. | lxxxvi | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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
<<2010 Poomjai Nacaskul, Ph.D. | lxxxvii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
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.
<<2010 Poomjai Nacaskul, Ph.D. | lxxxviii | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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):
<<2010 Poomjai Nacaskul, Ph.D. | lxxxix | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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):
<<2010 Poomjai Nacaskul, Ph.D. | xc | ดร.พมใจ นาคสกล พ.ศ. ๒๕๕๓>>
(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?