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<<2010 Poomjai Nacaskul, Ph.D. | i | . !ก#$ .%. &''(>>
Financial Modelling with Copula Functions
Poomjai Nacaskul, Ph.D.
Bank of Thailand
Q1 2010
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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
< + =>?@ABAC 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;
& + =>?@DEFGHHIกกIBDC J (copula) KELMNกOPI?Q@NLMNกOPI? ก$RC >@ "#$ก&'()**() (functional) AS?PRCTGHUHF$@NDBBVMWGANก=NJ!XEB#Y N!HN C!IIZO DTก@@กกVE=[กEB#กDกDNT\IC AI]NS]H@HCT C=QH$^ก_^]Nก!IN= EGOD$ECNA`abS=B>]@N\HAS?Iก cJ\%!\OCR +,- .ก/& (Abe Sklar) ARdHB#ก=BJก=@dCH
( + =>?@=STHCJZS to learn how to (i) capture dependency structures
<<2010 Poomjai Nacaskul, Ph.D. | ii | . !ก#$ .%. &''(>>
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 particular copula against empirical data;
4th + to debunk the prevailing myth that the so-called Gaussian
copula was largely responsible for subprime CDO mispricing and ultimately the Global Financial Crisis of 2007+2009.
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 particular copula against empirical data;
l + =>?@$H$HN C=QHUJ to debunk the prevailing myth that the so-called Gaussian copula was largely responsible for subprime
CDO mispricing and ultimately the Global Financial Crisis of 2007+2009.
[PART 1]
Basics of Probabilistic (Financial) Modelling
Revisiting (the Notion of) Correlation
Introducing the Gaussian Copula
Defining Copulas Mathematically
Three Special Copulas
Sklarvs Theorem
[!RCAS? <]
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<<2010 Poomjai Nacaskul, Ph.D. | iii | . !ก#$ .%. &''(>>
Copula Density Function
Survival Copulas & Tail Dependence
Copula & Concordance (Measure)
[PART 2]
Copula Families
Archimedean Copulas
Multivariate Copulas
Elliptical Copulas
Modelling with Copula
Parametric Estimation Methodology for Copulas
Non-Parametric Copulas
Goodness-of-Fit Tests for Copulas
[PART 3]
Monte Carlo Simulation with Copulas
Financial Risk Modelling with Copulas
Credit Risk Modelling with Copulas
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<<2010 Poomjai Nacaskul, Ph.D. | iv | . !ก#$ .%. &''(>>
Detour in Credit Derivatives & Derivatives Pricing
Pricing Credit Derivatives with Copulas
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[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.
Wikipedia (2010), Copula (statistics), [http: //en.wikipedia.org/wiki/Copula_(statistics)].
Basics of Probabilistic (Financial) Modelling .%:)*?_0`$0a)*ก$%&'$()*+,,(-$*ก$%.*/0).8/*<3$@0F$&G.=b0
In probabilistic financial modelling, a quantity of interest (object under study) is generally represented as a random variable, generically X , whose likelihood of taking on particular (range of) values, i.e. expressed as a random variate, generically x , is summarised via the notion of probability distribution, specifically with a monotonically non-decreasing cumulative distribution
function (c.d.f.), generically:
กF$@NDBBANก=NJ=PJN CRE=Vw !J?NAS?=VwAS?! G>@CI\X#Q@Nก%กa IกXกD!NDAHCT!J?NAS?=STกCR5(67/.89: (random
variable) (AI?CdV=QST=Vw\IC@IกaGWR X ) _^?N CRE=VwAS?ES R=VwFC (J!IT) \RN >@Vกb@@กV97/.89: (random
variate) (AI?CdV=QST=Vw\IC@Iกa=$[ก x ) Iก!#VdCH\DC JQ@Nก/
7;ก7;$+'<$.=<5< (probability distribution) YT=E@TRNTJ?NHCT cumulative distribution function (c.d.f.) @I=VwLMNกOPIAN=STCD$EdR=Vw$B YTAI?CdV=QSTINS]:
(1) 44 344 21domain
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ariatevrandom
ariablevrandomfdc
ityimpossibilingxpresse
SupportxxXxF ℜ⊆∈≤≤≡≤ ,1)Pr()(0...
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<<2010 Poomjai Nacaskul, Ph.D. | v | . !ก#$ .%. &''(>>
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:
AI]NS] Gก/=>?@=VwdVdH EPH c.d.f. AS?=VwJO=PJNCJ= EGOAS?dHกกGVJIZO "#$ก&'()*7--?/+:5/<ก/ก@ก(-AB?/:<+5,/&
(parametric/parameterised function) AS? RdR=Vw$B \$@PRCNกGVJIZO
=>?@ ?<.(BC@;8* (support) Q@NกDกDN (Y=_^?N\ICDV!#RS:5/
6:*9;D+E* (probability measure) =VwBCก) =VwVE=|A+F5*(-
GAH (countable set) LMNกOPI?I]=STกCR probability mass function
(p.m.f.) _^?NC RกID$HCdHG?N GกJ!IT ]F#=Vw=_\VE=|A*(-G:9GAH ก[Iก=STกLMNกOPI?I]CR probability density function (p.d.f.) _^?NGVJIZOdH=ARกIBG?N
INI]=VVE!N OG$IกQ@NกF$@NDBB=PJN CRE=Vw @TRAS?กกFGCREPHLMNกOPI?\IC YT!J?NAS?\H@NAF\ _^?NdHDกRก/,*8:*
+'<$.=<5< (statistical inference) ก[ >@\H@NVEc R?/:<+5,/&ก@I*Aก/7;ก7;$ (distributional parameter)
AI]NGS] (\ICDV!#R H@AI]N c.d.f. D$EกกFG RJ=\@OGHI ^NI=Vwก89:ก/7;ก7;$ (family of distributions) YTAI?CdV=QSTINS]:
<<2010 Poomjai Nacaskul, Ph.D. | vi | . !ก#$ .%. &''(>>
(2) 321321
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It is from such integration that an expectation operator may be defined, generically:
D$EกกGVJIZO=PR=STCกI ก[!XJT5(6A@+*<*9
AI:B (expectation operator) INS]:
(3) [ ] ∫ℜ
=Ε duufugXg )|()()( θ [(]
And from there, the mth central moment is then defined, where w.l.o.g. letvs let ℜ= x , thus:
D$EกI]ก[dH=Vw mth central moment \RN _^?N@กFGGH ℜ= x _^?NAFGH=VwกcSAI?CdVdH ก[EdHJTINS]:
(4) ( ) K,4,3,2,)|(][)( =Ε−= ∫∞
∞−
mduufXuXM mm θ [l]
Hence our familiar notions of means, variance, and standard
deviation simply follow: INI]!J?NAS? #H= TdHDกR9+LM)B (means ) 6:7//6* (variance),
D$E 9+-M)B$+-*:5/N* (standard deviation) ก[\:
<<2010 Poomjai Nacaskul, Ph.D. | vii | . !ก#$ .%. &''(>>
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As a matter of fact, historically, our probabilistic grasps of nature have only relatively recently swung from expectation-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:
...
<<2010 Poomjai Nacaskul, Ph.D. | viii | . !ก#$ .%. &''(>>
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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 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.
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 rate are essentially theoretical and/or empirical in nature:
...
<<2010 Poomjai Nacaskul, Ph.D. | ix | . !ก#$ .%. &''(>>
(7) factorpriceprice
dividendpricepricereturn
dividend
price
price
todayyesterday
yesterdaytodaytodayyesterday
today
←+−
≡⇒
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,
[]
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.:
...
<<2010 Poomjai Nacaskul, Ph.D. | x | . !ก#$ .%. &''(>>
(9)
→)(
)(),(
yF
xFyxF
Y
X []
This simply corresponds to the notion of marginal distributions: ...
(10)
...
)()Pr()Pr(),()(
)()Pr()Pr(),()(
fdcariateuniv
onsdistributirginalma
Y
X
yFyYyYXyFyF
xFxXYxXxFxF
=≤=≤∩∞≤=∞=
=≤=∞≤∩≤=∞=
321 [<]
Working w.l.o.g. with continuous r.v.vs: ...
(11)
=⇒=
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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:
...
<<2010 Poomjai Nacaskul, Ph.D. | xi | . !ก#$ .%. &''(>>
(12) ),()(
)( ??? yxHyG
xF→
[<&]
QUIZ 1 What are the relationships amongst these items? Hint best to write each set of comparisons in terms of
equations.
(a) )(xFX vs. ),( yxF vs. ),( yxf (b) )(xFX vs. )Pr( ∞≤∩≤ YxX vs. )Pr( xX > (c + bonus) ),( yxf vs. )|( yxf vs. )|( yxF
...
Revisiting (the Notion of) Correlation -,-30(+03</1.%:)*)BEB9@?90D7
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 expectation operator, even worse (c) the expectation operator, or worst yet (d) the rho parameter.
In fact, all four concepts are quite correct, itvs just that as we go
...
<<2010 Poomjai Nacaskul, Ph.D. | xii | . !ก#$ .%. &''(>>
successively from (a) to (b) and to (c) and finally (d), the definition becomes increasingly technical and mathematically precise, 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, itvs 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 Galtonvs original introduction (albeit the notion correlation can be traced to as far back as Aristotle):
Aristotle (350 BC), Categories, translated by Edghill, Ella Mary (1928), available online via [http: //classics.mit.edu/Aristotle/categories.2.2.html].
Galton, Francis (1888), "Co-Relations and Their Measurement, Chiefly from Anthropometric Data", Proceedings of the Royal
<<2010 Poomjai Nacaskul, Ph.D. | xiii | . !ก#$ .%. &''(>>
Society, vol. 45, pp. 135-145, available online via [www.york.ac.uk/depts/maths/histstat/galton_corr.pdf].
Edgeworth, Francis Ysidro (1892), "Correlated Averages", Philosophical Magazine, series 5, vol. 34, no. 207 pp. 190-204, available online (w/ restricted access) via [http: //www.informaworld.com/smpp/content~db=all~content=a911033706?words=correlated,averages&waited=0].
Pearson, Karl (1920), Notes on the History of Correlation, Biometrika, vol. 13, no. 1, pp. 25-45, available online (w/ restricted access) via [http: //biomet.oxfordjournals.org/cgi/content/citation/13/1/25].
Rodgers, J. L. & Nicewander, W. A. (1988), Thirteen ways to look at the correlation coefficient, The American Statistician, vol. 42, no. 1, pp. 59-66, available online via [www.psych.umn.edu/faculty/waller/classes/.../RodgersNicewander.pdf].
Stigler, Stephen M. (1989), Francis Galtonvs Account of the Invention of Correlation, Statistical Science, vol. 4, no. 2 (May), pp. 73-79.
Rovine, M. J. & von Eye, A. (1997), A 14th Way to Look at a Correlation Coefficient: Correlation as the Proportion of Matches, The American Statistician, vol. 51, no. 1 (February), available online (w/ restricted access) via [http: //www.jstor.org/pss/2684692].
Dodge, Y. & Rousson, V. (2000), Direction dependence in a regression line, Communications in Statistics - Theory and
Methods, vol. 29, no. 9, pp. 1957-72, available online via [http: //www.informaworld.com/smpp/content~db=all~content=a780146860]
Piovani, Juan Ignacio (2008), The Historical Construction of Correlation as a Conceptual and Operative Instrument for Empirical Research, Quality and Quantity, vol. 42, no. 6 (December), pp. 757-777, available online (w/ restricted access) via
<<2010 Poomjai Nacaskul, Ph.D. | xiv | . !ก#$ .%. &''(>>
[http: //www.springerlink.com/content/jxq15v45405241t2].
Miller, Jeff (2009), Earliest Known Uses of Some of the Words of Mathematics (C), available online via [http: //jeff560.tripod.com/c.html].
Wikipedia (2010), Correlation and dependence, [http: //en.wikipedia.org/wiki/Correlation_and_dependence].
[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)
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dvvfvduufu
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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:
...
<<2010 Poomjai Nacaskul, Ph.D. | xv | . !ก#$ .%. &''(>>
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[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 isnvt so in the case of correlation:
...
<<2010 Poomjai Nacaskul, Ph.D. | xvi | . !ก#$ .%. &''(>>
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For instance, we might want to define expectation correlation measures with higher power moments and sample correlation statistics with higher power deviations:
...
<<2010 Poomjai Nacaskul, Ph.D. | xvii | . !ก#$ .%. &''(>>
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Or perhaps n even more general definition of powered correlation: ...
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[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 KendallUs rank
correlation coefficient, aka Kendall's tau.
[5] Indeed, the reason that (Pearsonvs) 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
...
<<2010 Poomjai Nacaskul, Ph.D. | xviii | . !ก#$ .%. &''(>>
distribution, only this time itvs 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 + bonus) in relation to Spearman's rank correlation coefficient, aka Spearman's rho
...
Introducing the Gaussian Copula a)+0G0'$ .ก$B7.oK:40;<=>($
Consider the rather well-known bivariate normal distribution, aka bivariate Gaussian distribution:
...
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YX
YX
onsdistributirginalma
normalariateuniv
YY
XX
disibutionnormal
ariatebiv
YYX
XYX
Y
X
vectorrandom
ddistributenormally
y
xyxyxf
Y
X
Y
X
−
−
−−⋅−
−=⇒
−∈⋅⋅==
ℜ∈
ℜ∈
Ν
Ν
Ν
−
−
+
Σ
µµ
σσσσ
µµρσπσ
ρσσρσσ
σσ
µµ
σµ
σµ
σσσσ
µµ
µ
[<]
<<2010 Poomjai Nacaskul, Ph.D. | xix | . !ก#$ .%. &''(>>
In Mathematica, for example, letvs define: ...
Fig.1 In[1]:= BivariateNormalPDF@x_, y_, µX_, µY_, σX_, σY_, ρ_D =
−
Hx−µXL2
σX2+Hy−µYL2
σY2−2 Hx−µXLHy−µYLρ
σXσY
2 I1−ρ2M
2 π 1 − ρ2 σX σY V-<
To view in 3D (w/ parameters subject to manipulation): ...
Fig.2
In[2]:= ManipulatePlot3DBivariateNormalPDF@D =
Manipulate@Plot3D@BivariateNormalPDF@x, y, µX, µY, σX, σY, ρD,
8x, µX −3 σX, µX+ 3 σX<,
8y, µY −3 σY, µY+ 3 σY<,
PlotRange → 80, 0.4<D,
"Bivariate Normal p.d.f.",
88µX, 0<, −5, 5, 0.5, Appearance → "Labeled"<,
88µY, 0<, −5, 5, 0.5, Appearance → "Labeled"<,
88σX, 1<, 0.5, 5, 0.5, Appearance → "Labeled"<,
88σY, 1<, 0.5, 5, 0.5, Appearance → "Labeled"<,
88ρ, 0.8<, −0.9, 0.9, 0.1, Appearance → "Labeled"<D
V-&
<<2010 Poomjai Nacaskul, Ph.D. | xx | . !ก#$ .%. &''(>>
Fig.3 Out[2]=
Biv ariate Normal p.d.f.
mX 0
mY 0
sX 1
sY 1
r 0.8
V-(
Or as a contour plot (w/ parameters subject to manipulation): ...
Fig.4
In[3]:= ManipulateContourPlotBivariateNormalPDF@D =
Manipulate@ContourPlot@BivariateNormalPDF@x, y, µX, µY, σX, σY, ρD,
8x, µX −3 σX, µX+ 3 σX<,
8y, µY −3 σY, µY+ 3 σY<,
PlotRange → 80, 0.4<D,
"Bivariate Normal p.d.f.",
88µX, 0<, −5, 5, 0.5, Appearance → "Labeled"<,
88µY, 0<, −5, 5, 0.5, Appearance → "Labeled"<,
88σX, 1<, 0.5, 5, 0.5, Appearance → "Labeled"<,
88σY, 1<, 0.5, 5, 0.5, Appearance → "Labeled"<,
88ρ, 0.8<, −0.9, 0.9, 0.1, Appearance → "Labeled"<D
V-l
<<2010 Poomjai Nacaskul, Ph.D. | xxi | . !ก#$ .%. &''(>>
Fig.5 Out[3]=
Biv ariate Normal p .d .f.
mX 0
mY 0
sX 1
sY 1
r 0.8
V-'
Working w.l.o.g. with standard normal marginals, i.e. with normalized r.v.vs ( ) XXXX σµ−a and ( ) YYYY σµ−a , effectively taking away the location-scale parameters, there is only one distributional parameter left, corresponding to the well-known (Pearson product-moment) correlation coefficient:
...
<<2010 Poomjai Nacaskul, Ph.D. | xxii | . !ก#$ .%. &''(>>
(19)
( )( )
( )
⋅⋅−+⋅
−−
−=⇒
−∈
Ν
Ν
Ν
−
Σ
yxyxyxf
Y
X
Y
X
parameterncorrelatiothe
ionspecificatparameter
onsdistributirginalma
normalariateuniv
andardst
disibutionnormal
ariatebivandardst
vectorrandom
ddistributenormally
andardst
ρρρπ
ρρ
ρµ
2)1(2
1exp
12
1),(
]1,1[,1,0~
1,0~,
1
1,
0
0~
2222
,1
4342143421
44 344 21
48476v
[<]
QUIZ 3 Suppose we are told that ( )2,0~ XX σΝ , ( )2,0~ YY σΝ , and ρ=),( YXCorrel : (a) Write out the variance-covariance matrix. (b) Write out the covariance, XYσ (in quadratic form). (c + bonus) Simplify (b). Hint first multiply out all the terms.
...
For convenience, letvs denote especially the standard bivariate normal c.d.f., the univariate standard normal c.d.f., and the inverse of the standard univariate normal c.d.f. thus:
...
<<2010 Poomjai Nacaskul, Ph.D. | xxiii | . !ก#$ .%. &''(>>
(20)
( )
uxxu
dss
x
dtdststsyx
x
x
y x
=Φ=Φℜ→Φ
−=Φ→ℜΦ
⋅⋅−+⋅
−−
−=Φ→ℜΦ
ℜ∈
−−
∞−
∞− ∞−
∫
∫ ∫
)(sup)(,]1,0[:
2exp
2
1)(,]1,0[:
2)1(2
1exp
12
1)|,(,]1,0[:
11
2
2222
2
π
ρρρπ
ρ
[&]
Fig.6
In[4]:= StandardBivariateNormalPDF@x_, y_, ρ_D = BivariateNormalPDF@x, y, 0, 0, 1, 1, ρD
Out[4]=
−x2+y2−2 x y ρ
2 J1−ρ2N
2 π 1 −ρ2
V-1
Fig.7 In[5]:= PhiInverse@u_D = 2 InverseErf@−1 + 2 uD
Out[5]= 2 InverseErf@−1 + 2 uD V-2
Note that the joint c.d.f. can be re-written: ...
(21) ( ) ( )( )
( )( )
( )( )( )
∫ ∫
∫ ∫− −
− −
Φ
∞−
Φ
∞−
−−
ΦΦ
∞−
ΦΦ
∞−
−−
=ΦΦΦ=Φ⇒
Φ=
Φ=
=ΦΦΦΦΦ=Φ
v u
y x
dtdsvuyxyv
xu
dtdsyxyx
1 1
1 1
)(),()|,()(
)(
)(,)()|,(
11
)( )(11
K
K
ρρ
ρρ
[&<]
Now note carefully how the marginals need not even be univariate normal at all, other univariate c.d.f. will do, i.e. X and Y need not be normally distributed:
...
<<2010 Poomjai Nacaskul, Ph.D. | xxiv | . !ก#$ .%. &''(>>
(22) ( ) ( )( )
( )( )
( )( )( )
∫ ∫
∫ ∫− −
− −
Φ
∞−
Φ
∞−
−−
∞− ∞−
−−
=ΦΦΦ=Φ⇒
=
=
=ΦΦΦ=Φ
v u
yGG xHH
dtdsvuyxyHv
xGu
dtdsyHxGyx
1 1
1 1
)(),()|,()(
)(
)(,)()|,(
11
)( )(11
K
K
ρρ
ρρ
[&&]
This gives us a general method for using ρ , the correlation parameter inherited from standard bivariate normal distribution, to construct a different joint distribution function form any arbitrary marginals:
...
(23)
( ) ( )( )( )
( ) ( )( ) ( )( ) ( )( )
( )( )
( )
( )( )
)()Pr(
)()(2
exp2
1
2)1(2
1exp
12
1
,)()1(,)()(,)(
)Pr(Pr)Pr(
)(,)()Pr(
1)( 2
)(22
22
11111
11
1
1
yHyY
xGxGdss
dtdststs
xGxGHxG
YxXSupportYxXxX
yHxGyYxX
xG
xG
Y
==≤
=ΦΦ=
−=
⋅⋅−+⋅
−−
−=
∞ΦΦ=ΦΦΦ=∞ΦΦΦ=
∞≤∩≤=∈∩≤=≤
ΦΦΦ=≤∩≤
−Φ
∞−
∞
∞−
Φ
∞−
−−−−−
−−
∫
∫ ∫−
−
L
π
ρρρπ
ρρρ
ρ
[&(]
Indeed this is the 1-parameter bivariate Gaussian copula function: ...
<<2010 Poomjai Nacaskul, Ph.D. | xxv | . !ก#$ .%. &''(>>
(24) ( ) ( )( )( )
]1,1[,2)1(2
1exp
12
1)(),(),(
,]1,0[]1,0[:1 1
2222
11
2
−∈
⋅⋅−+⋅
−−
−=ΦΦΦ=
→
∫ ∫− −Φ
∞−
Φ
∞−
−− ρρρρπ
ρv u
dtdststsvuvuC
C
[&l]
[CLV: 112] Roncalli (2002) showed that this double integral expression (24) can be rewritten thus
... [&l]
(25) ( ) ( )
( )( )
∫
∫ ∫
−∈
−
Φ⋅−ΦΦ=
⋅⋅−+⋅
−−
−=ΦΦΦ=
→
−−
Φ
∞−
Φ
∞−
−−
− −
u
v u
dssv
dtdststsvuvuC
C
02
11
2222
11
2
]1,1[,1
)()(
2)1(2
1exp
12
1)(),(),(
,]1,0[]1,0[:1 1
ρρ
ρ
ρρρπ
ρ [&']
Letvs retrace the steps, this time starting by defining 2 uniform random variables, 2 arbitrarily distributed random variables, and 2 standard normally distributed random variables, the latter with correlation ]1,1[−∈ρ :
...
(26)
( )( )( )
( ) ( )[ ] ( ) ( )ρρ
θθ
21221121 ,Pr
2,1,,Pr)1,0(~
,)()Pr(~
,)()Pr(~
2,1,]1,0[,Pr)1,0(~
zzzZzZZZ
izzzZZ
SupportyyFyYDistrY
SupportxxFxXDistrX
iuuuUUnifU
iiiii
YYY
XXX
iiii
Φ=≤∩≤⇒=Ε
=ℜ∈Φ=≤⇒Ν
∈=≤⇒
∈=≤⇒
=∈=≤⇒
[&]
<<2010 Poomjai Nacaskul, Ph.D. | xxvi | . !ก#$ .%. &''(>>
Let the 1st uniform, arbitrarily distributed, and standard normally distributed random variates be tied together, and the 2nd uniform, arbitrarily distributed, and standard normally distributed random variates tied together thus:
...
(27)
444444 3444444 2143421....
2222
...
...
...
22
111111
,,
,,
equivalentconsideredareeventsthese
thatsense
thein
togethertiedare
ariablesvrandomThese
zZyYuUZYU
zZxXuUZXU
≤⇔≤⇔≤∋
≤⇔≤⇔≤∋
[&]
Hence in terms of the corresponding random variates, the c.d.f., and the c.d.f. inverses:
...
(28) ( ) ( ) ( )( )
( ) ( )
( ) ( ) ( )( )( ) ( )
=Φ=Φ
Φ==⇒Φ==∋
=Φ=Φ
Φ==⇒Φ==∋
−−
−−
−−
−−
21
21
21
21
2222
11
11
11
11
1111
)()(,,
)()(,,
zyFu
zFyuFzyFuzyu
zxFu
zFxuFzxFuzxu
Y
YYY
X
XXX
[&]
Moreover, these joint events are equivalent: ...
(29) 22112211 zZzZyYxXuUuU ≤∩≤⇔≤∩≤⇔≤∩≤ [&]
And of course these joint probabilities are equal: ...
<<2010 Poomjai Nacaskul, Ph.D. | xxvii | . !ก#$ .%. &''(>>
(30) ( ) ( )22112211 Pr)Pr(Pr zZzZyYxXuUuU ≤∩≤=≤∩≤=≤∩≤ [(]
Finally, with some substitutions: ...
(31)
( ) ( ) ( )( ) ( )( ) ( ) ( )( )
( ) ( ) ( )( ) ( )( ) ( ) ( )( )
44444 344444 21
444 8444 76444 8444 76
444 3444 21
444 8444 76444 8444 76
...2
1112
11
21
)(
2211
]1,0[
21
11
21
211
1
21
)(
22112211
)(,)()()(Pr
,PrPr
,Pr
,PrPr
2
fdcoffunctionaasxpressede
YXYX
Gaussiannormalariatebivondistributiarbitraryntjoioverfunctionaasxpressede
Gaussiannormalariatebivondistributiuniformntjoi
yFxFyFZxFZ
zzzZzZyYxX
uuuZuZ
zzzZzZuUuU
ρρ
ρρ
−−−−
−−−−
ΦΦΦ=Φ≤∩Φ≤=
Φ=≤∩≤=≤∩≤
ΦΦΦ=Φ≤∩Φ≤=
Φ=≤∩≤=≤∩≤
[(<]
So that there are two mutually consistent interpretations, namely that a copula is a bivariate c.d.f. over the unit-square support, i.e. with uniform marginals, and that it is a function that takes two univariate c.d.f.vs (each, in turn, just a function of one scalar random variate) to produce a joint distribution (bivariate c.d.f.) thus:
...
<<2010 Poomjai Nacaskul, Ph.D. | xxviii | . !ก#$ .%. &''(>>
(32) ( ) ( ) ( )( ) ( )
( ) ( )( ) ( ) ( )( ) )Pr()(,)()(,)(),(
]1,0[D:
....'..:2
Pr,,
]1,0[]1,0[:
..,...:1
11
2
221121
11
21
2
yYxXyFxFyFxFCyxC
C
fdcntjoiaproducetosfdcariateunivtwotakesthatfunctionalionnterpretati
uUuUuuuuC
C
rginalsmauniformwitheiupportssquareunitoverfdcariatebivionnterpretati
YXYX ≤∩≤=ΦΦΦ==
→ℜ⊆
≤∩≤=ΦΦΦ=
→
−
−−
−−
ρ
ρ [(&]
QUIZ 4 Derive the Gaussian copula (at least describe the steps and rationales involved).
...
Defining Copulas Mathematically ก'$E010/4$@-$*<p/JH$BJ%7AE2ก9,;<=>($
Here itvs perhaps useful to keep a neat distinction between the notion of a function, a functional, and an operator.
A 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).
Technically, a functional is a function whose domain is 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
...
<<2010 Poomjai Nacaskul, Ph.D. | xxix | . !ก#$ .%. &''(>>
function of functions.
Finally, an operator is essentially a function which acts on functions to produce yet another function:
(33)
( )
( )
( ) ∫
∫ ∫
∫ ∫
−==→
=
=
ℜ→
=+=ℜ→ℜ
+
+
t
xy
yx
dtfftffthgeffhoperator
dydxeyxfg
dydxeyxfg
gefgfunctional
xyxfyxyxfgeffunction
0
21212
1
0
1
0
2
1
0
1
0
1
212
)()()(*)(..:
),(
),(
..,:
)(,),(..,:
τττ
[((]
So is copula a function, a functional, or an operator, i.e. letting F denote a space of c.d.f., how do we see the mapping action of a copula?
...
(34) ( )
→
→
→
→ℜ∈=
operatoranhenceFFC
functionalahenceFC
functionahenceC
yxyFxFCyxF YX
,:
,]1,0[:
,]1,0[]1,0[:
,,)(),(),(2
2
2
??? [(l]
Once again, referring to (12), the desire is to be able to construct a joint probability distribution from the marginals, i.e. the 2≥k univariate c.d.f.vs, and that in essence is what a functional does.
... [<&]
<<2010 Poomjai Nacaskul, Ph.D. | xxx | . !ก#$ .%. &''(>>
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()(),()(
)(yYxXyFxFC
F
FYX
Copula
Y
X ≤∩≤= →
⋅
⋅ [(']
Now letvs construct a bivariate copula from scratch, specifying the mathematical properties necessary to produce a bivariate c.d.f., in particular:
...
(36) ( )( ) ( )( )( )
=
=
==
⇒ℜ∈=
)()(,1
)(1),(
0)(,00),(
,,)(),(),(
yFyFC
xFxFC
yFCxFC
yxyFxFCyxF
YY
XX
YX
YX [(]
The first property corresponds to saying that the 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 this copula function:
...
(37) ( ) ( ) ( ) ( ) 0,,,,10
1011122122
21
21 ≥+−−⇒
≤≤≤
≤≤≤vuCvuCvuCvuC
vv
uu [(]
<<2010 Poomjai Nacaskul, Ph.D. | xxxi | . !ก#$ .%. &''(>>
Altogether: ...
(38)
−
=
=∋→
.2
,),1(
,)1,(
,
]1,0[]1,0[: 2
ingincreasisC
vvC
uuC
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 ;<=>($?/.HqB$@J93
Letvs begin with 3 special copulas, considered the most basic.
[1] The independent copula, aka product copula, expresses the already familiar concept of statistical or probabilistic independence:
...
<<2010 Poomjai Nacaskul, Ph.D. | xxxii | . !ก#$ .%. &''(>>
(39) ( ) ( ) ( ) ( )
( )( )
≥−−=+−−=
+−−
=⋅=
=⋅=
=⋅==⋅=
∋⋅≡
≥≥
×
0
,,,,
,1),1(
,1)1,(
),0(000)0,(
),(
0
12
0
1211122122
11122122
4342143421vvuuvuvuvuvu
vuCvuCvuCvuC
vvvC
uuuC
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
≡≤≤−+≡
∈∀+−
[l]
This can be expressed rather elegantly in terms of the so-called concordance order relation:
...
(41) ),(),(,]1,0[,
),(),(,]1,0[,
2121
2121
vuCvuCvuCC
vuCvuCvuCC
≥∈∀⇔
≤∈∀⇔
f
p [l<]
In particular, with minimum and maximum copulas at the ends, independent copula in the middle, all other copulas fall somewhere
...
<<2010 Poomjai Nacaskul, Ph.D. | xxxiii | . !ก#$ .%. &''(>>
in between:
(42) ),(),(),( vuCvuCvuC +×− pLppLp [l&]
Note, however, such a relation does not amount to there being a total order amongst all possible copulas, i.e. for some pair of copulas (except the minimum/maximum), itvs possible that, away from the boundary of 2]1,0[ , one finds:
...
(43)
>
<∋∈∃
),(),(
),(),()1,0(,,
21
21
hgCfeC
dcCbaCha K [l(]
[CLV: 70] With that, 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, not two):
...
(44)
( ) ( )( )
( ) ( )( )
444444444 3444444444 21
44444444 344444444 21
Vnamelyrandomnessofsourceoneonlyandone
Y
XYX
Unamelyrandomnessofsourceoneonlyandone
Y
XYX
UnifVyFVyY
xFVxXyFxFCyxF
UnifUyFUyY
xFUxXyFxFCyxF
,
,
)1,0(~,)(1Pr)Pr(
)(Pr)Pr()(),(),(
)1,0(~,)(Pr)Pr(
)(Pr)Pr()(),(),(
≤−=≤
≤=≤⇒=
≤=≤
≤=≤⇒=
−
+
[ll]
<<2010 Poomjai Nacaskul, Ph.D. | xxxiv | . !ก#$ .%. &''(>>
[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 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
elementidentityasactsaaT
ityassociativcbaTTcbTaT
tymonotonicibaTbaTbb
aaityassociativabTbaT
T [l']
Wikipedia (2010), T-norm, [http: //en.wikipedia.org/wiki/T-norm].
Scholarpedia (2010), Triangular norms and conorms, [http: //www.scholarpedia.org/article/Triangular_norms_and_conorms].
QUIZ 6 What are the 3 special copulas (write out the full mathematical expressions), and whatvs so special about them?
...
<<2010 Poomjai Nacaskul, Ph.D. | xxxv | . !ก#$ .%. &''(>>
Also, discuss in terms of concordance order the relation between each of the 3 special copulas and an arbitrary generic copula C.
Sklarts Theorem -uqvK,-a)*Bก($%7
[NEL: 3,14] Fundamental to copula mathematics is the SkalarUs
Theorem, first published (in French) in 1959 by the mathematician Abe Sklar, who, around that time, was working also on PM spaces.
...
Sklar, Abe (1973), Random Variables, Joint Distributions, and Copulas, Kybernetica, vol. 9, pp. 449-460.
This theorem states that, let F be a joint distribution function (bivariate c.d.f.) with margins G and H , then there exists a copula ]1,0[]1,0[: 2 →C such that:
...
(46) ( ))(),(),(,, , yHxGCyxFSupportyx YX =∈∀ [l]
Moreover, if G and H are both continuous, then C is unique.
Conversely, if C is a copula, and G and H are univariate c.d.f.vs, then F , as defined by (46), becomes, in and of itself, a joint distribution function (bivariate c.d.f.).
For instance, given )(~ λExpX , ),(~ baUnifY , and the
... [l]
<<2010 Poomjai Nacaskul, Ph.D. | xxxvi | . !ก#$ .%. &''(>>
independent copula:
(47) ],[),0[),(,1
),(
]1,0[,,),(
],[,1
)(
),0[,)(
bayxab
eyxF
vuuvvuC
bayab
yF
xexFx
Y
xX
×∞∈−
−=⇔
∈=
∈−
=
∞∈⋅=⋅−
⋅−
λ
λλ
[l]
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.vs):
...
(48) ( )
×∈
≤∩≤=
=
⇒
∈==≤⇒
∈==≤⇒
]1,0[]1,0[),(
,)Pr(
),()(),(
]1,0[,)()Pr()1,0(~
]1,0[,)()Pr()1,0(~
vu
vVuU
vuCvFuFC
vvvFvVUnifV
uuuFuUUnifU VU
V
U [l]
From this then follow a number of identities: ...
<<2010 Poomjai Nacaskul, Ph.D. | xxxvii | . !ก#$ .%. &''(>>
(49) 4444444 84444444 76
4444444 34444444 21
v
vuCvuvVuU
v
vuCvvVuU
v
vuCuvVuU
v
vuCvVuU
vuCvuvVuU
vuCvvVuU
vuCuvVuU
vuCvVuU
vVuUvVuUvVv
vVuUvVuUuUu
vVuUvVuUvVv
vVuUvVuUuUu
−+−−
=>>
−=≤>
−−
=>≤
=≤≤
⇓
+−−=>∩>
−=≤∩>
−=>∩≤
=≤∩≤
⇒
>∩>+>∩≤=>=−
>∩>+≤∩>=>=−
≤∩>+≤∩≤=≤=
>∩≤+≤∩≤=≤=
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(
[l]
QUIZ 7 What is the Sklarvs theorem (write out the full mathematical expression), and whatvs so special about it? 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 56*ก789:0<3$@E0$+0F0;<=>($
Just as we can ascribe a p.d.f. to a c.d.f., there is also a ...
<<2010 Poomjai Nacaskul, Ph.D. | xxxviii | . !ก#$ .%. &''(>>
corresponding notion of copula density (function):
(50) vu
vuCvuc
∂∂∂
≡),(
),(2
[']
[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
dtdsts
tsCdtdstscvuC
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:
...
<<2010 Poomjai Nacaskul, Ph.D. | xxxix | . !ก#$ .%. &''(>>
(52)
),(),(00),(0,min),(
),(),(00),(01,0max),(
),(1),(1),(
0 0
..
220 0
..
220 0
..
22
vuCvuCdtdsvuCvu
vu
vu
vuC
vuCvuCdtdsvuCvu
vu
vu
vuC
vuCuvdtdsvuCvu
uv
vu
vuC
ingulars
v u
ca
ingulars
v u
ca
v u
ca
++++
−−−−
×××
=≠=⋅=⇒∂∂
∂=
∂∂∂
=≠=⋅=⇒=∂∂
−+∂=
∂∂
∂
==⋅=⇒=∂∂
∂=
∂∂∂
∫ ∫
∫ ∫
∫ ∫
['&]
[CLV: 83] Finally, given a copula representation ( ))(),(),( yFxFCyxF YX= , the copula density then yields the
following canonical representation of the joint (bivariate) p.d.f.:
...
(53) ( ) ( )44444 344444 21
448447644 844 76876
tionrepresentacanonical
sfdprginalma
YX
densitycopula
YX
fdpntjoi
YXYX yfxfyFxFcyxf
yfxf
yxfyFxFcvuc
.'.....
)()()(),(),()()(
),()(),(),( ⋅⋅=⇒== ['(]
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=a and )(yFvy Y=a are continuously differentiable and strictly
increasing, hence one-to-one, and so the joint (bivariate) p.d.f.vs are related by way of the multivariate transformation theorem:
...
<<2010 Poomjai Nacaskul, Ph.D. | xl | . !ก#$ .%. &''(>>
(54)
( ) ( ))()(
),()()()()(
),(
det
)(),()(),(),(
00
11
yfxf
yxf
X
YF
Y
XF
Y
YF
X
XFyxf
Y
V
X
VY
U
X
U
vFuFfyFxFcvuc
YXYXYX
tiontransformaariatebivthe
ofJacobian
YXYX =
∂
∂⋅
∂
∂−
∂
∂⋅
∂
∂=
∂∂
∂∂
∂∂
∂∂
==−−
4342143421
43421
['l]
Makarov, Roman (2009), Transformation Theorem. Bivariate (Multivariate) Normal Distribution, Lecture 18, available online via www.wlu.ca/documents/38249/lecture-18-print.pdf].
QUIZ 8 Give (a) copula density and (b) canonical representation for the independent copula vuvuC ⋅=× ),( .
...
Survival Copulas & Tail Dependence ;<=>($+EF*ก$%)4>F%)1ก9,<3$@B9@?90D7A0BF30E$*
[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 0>t is denoted thus:
...
(55) )Pr()( tTtF >≡ ['']
Not sure why one doesnvt often see the term survival probability ...
<<2010 Poomjai Nacaskul, Ph.D. | xli | . !ก#$ .%. &''(>>
used in this context!?
At any rate, we can also define joint survival function:
(56) ( ) ( ) ( )( ) ( )( ))(1,)(11)()(
)(),()(1)(11
),()()(1)Pr(),(
yFxFCyFxF
yFyFCyFxF
yxFyFxFyYxXyxF
YXYX
YXYX
YX
−−+−+=
+−−−−=
+−−=>∩>≡
[']
So for actuarial applications, it then makes sense to correspondingly define the survival copula:
...
(57)
444444444444 8444444444444 76
43421
444444 3444444 21
)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
vuCvuvuC
yFxFCyFxFyxFYX
YXYX =⇒
+−−=−−
−−+−+= [']
[NEL: 47] One particular example of this is the 2-parameter ...
<<2010 Poomjai Nacaskul, Ph.D. | xlii | . !ก#$ .%. &''(>>
Marshall-Olkin copula:
(59) 1,0,,min,min),( 11 <<== −−−−− βαβαβα 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:
...
(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:
...
<<2010 Poomjai Nacaskul, Ph.D. | xliii | . !ก#$ .%. &''(>>
(61)
( )
( ) ]1,0[)()(Prlim)|Pr(lim)Pr(
)Pr(lim
1
),(21lim
1
)1,1(lim
)Pr(
)Pr(lim
)|Pr(lim)()(Prlim
11
11
1
11
1
1
11
1
∈>>=>>=>
>∩>=
−+−
=−
−−=
>>∩>
=
>>=>>≡
−−
→→
→
→→
→
→
−−
→
−−
−
−−
−
−−
wFYwFXwVwUwV
wVwUw
wwCw
w
wwCwU
wUwV
wUwVwFXwFY
YXww
w
ww
w
wXY
wUpperλ
[<]
For instance, itvs very easy to verify the following: ...
(62)
( )0
1
121lim
1
)(),(21lim
1
),(21lim
01
1221lim0
1
1,0max21lim
1
),(21lim
01
)1(lim
1
21lim
1
),(21lim
11
1lim
1
,min21lim
1
),(21lim
1
11
11
111
2
1
2
11
111
=−
+−=
−
ΦΦΦ+−=
−+−
=−
−+−==
−−++−
=−+−
=−−
=−+−
=−
+−
=−−
=−
+−=
−+−
=
−−−
−−−
−−−
−−−
→
−−
→→
→→
−
→
−
→→
×
→
×
→→
+
→
+
w
w
w
www
w
wwCw
w
ww
w
www
w
wwCww
w
w
ww
w
wwCww
w
w
www
w
wwCw
ww
Gaussian
w
GaussianUpper
wwwUpper
wwwUpper
wwwUpper
ρλ
λ
λ
λ
[&]
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:
...
<<2010 Poomjai Nacaskul, Ph.D. | xliv | . !ก#$ .%. &''(>>
(63)
( )
( ) ]1,0[)()(Prlim)|Pr(lim)Pr(
)Pr(lim
1
),(lim
)Pr(
)Pr(lim
)|Pr(lim)()(Prlim
11
00
0
0
0
0
11
0
∈<<=<<=>
<∩<=
−=
<<∩<
=
<<=<<≡
−−
→→
→
→
→
→
−−
→
++
+
+
+
++
wFYwFXwVwUwV
wVwUw
wwCwU
wUwV
wUwVwFXwFY
YXww
w
w
w
wXY
wLowerλ
[(]
Similarly, itvs very easy to verify the following: ...
(64)
( )0
0lim
)(),(lim
),(lim
00
lim1,0max
lim),(
lim
0limlim),(
lim
1lim,min
lim),(
lim
0
11
00
000
0
2
00
000
==ΦΦΦ
=
==−+
=
===
====
+++
+++
+++
+++
→
−−
→→
→→
−
→
−
→→
×
→
×
→→
+
→
+
ww
ww
w
wwC
ww
ww
w
wwC
ww
w
w
wwCw
w
w
ww
w
wwC
ww
Gaussian
w
GaussianLower
wwwLower
wwwLower
wwwLower
ρλ
λ
λ
λ
[l]
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.
...
<<2010 Poomjai Nacaskul, Ph.D. | xlv | . !ก#$ .%. &''(>>
Copula & Concordance (Measure) ;<=>($ก9,(@$J%391)<)0<)%7+10o7
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 tend to be
...
<<2010 Poomjai Nacaskul, Ph.D. | xlvi | . !ก#$ .%. &''(>>
associated with above-average values in the other random variable, letvs 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 letvs for now assume 21 xx ≠ and 21 yy ≠ :
(65)
( )( )
( )( ) )(0
)(0
21212121
2121
21212121
2121
pairdiscordantyyxxyyxx
yyxx
pairconcordantyyxxyyxx
yyxx
⇒<−−⇒
<∩>
>∩<
⇒>−−⇒
<∩<
>∩>
[']
Given a sample consisting of 2≥n bivariate data ( ) n
iii 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 ,
...
<<2010 Poomjai Nacaskul, Ph.D. | xlvii | . !ก#$ .%. &''(>>
and define the KendallUs tau rank correlation coefficient or simply KendallUs tau statistics for this sample set thus:
(66)
=
=
−−
=+−
=pairsdiscordantd
pairsconcordantc
nn
dc
dc
dc
#
#,
2/)1(τ []
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 probabilistic (population) counterpart to the Kendallvs tau hat (sample) statistics, so letvs denote it by hatless τ , thus:
...
(67)
( ) ( )( )( )( ) ( )( )( )
( )( )( )( )
( )( )( )( )
4444 34444 214444 34444 21ediscordanceconcordanc
YYXXYYXX
YYXXediscordanc
YYXXeconcordanc
Pr
2121
Pr
2121
2121
2121
0Pr0Pr
0PrPr
0PrPr
<−−−>−−≡
<−−≡
>−−≡
τ []
Working with continuous r.v. leads to slightly simpler expression: ...
(68) ( )( )( ) ( )( )( )
( )( )( ) ( )( )( )( ) ( )( )( ) 10Pr20Pr10Pr
0Pr10Pr
212121212121
21212121
−>−−=>−−−−>−−=
⇓
>−−−=<−−
YYXXYYXXYYXX
YYXXYYXX
τ []
<<2010 Poomjai Nacaskul, Ph.D. | xlviii | . !ก#$ .%. &''(>>
Decomposing ( )econcordancPr into probabilities of two joint events:
...
(69) ( )( )( ) ( ) ( )( ) ( )21211212
212121212121
PrPr
PrPr0Pr
YYXXYYXX
YYXXYYXXYYXX
<∩<+<∩<=
<∩<+>∩>=>−− []
Take the first term of the right: ...
(70)
( ) ( ) ( ) ( )( )
( ) ( )
∫∫
∫∫
∫∫
⋅=
⋅=
⋅⋅<∩<=<∩<
ℜ
ℜ
2
2
2
]1,0[
111112121212
),(),(
)(),()(),(
,PrPr
48476876
444 8444 7644 844 76
44 844 76
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 Kendallvs tau-based measure of concordance for the population as:
... [][][]
<<2010 Poomjai Nacaskul, Ph.D. | xlix | . !ก#$ .%. &''(>>
(71) 1),(),(42]1,0[
−≡ ∫∫ vudCvuCτ [<]
[NEL: 129] Note that the double integral term can be interpreted as the expectation:
...
(72) [ ] )1,0(~,,),(),(),(2]1,0[
UnifVUVUCvudCvuC Ε=∫∫ [&]
[CLV: 98] For absolutely continuous copulas (w/ no singular component), we can substitute in for the copula differential notation the more familiar double differentials:
...
(73) dudvvu
vuCdC ⋅
∂∂∂
=),(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 dudv
v
vuC
u
vuCvudCvuCτ [l]
In other words, putting (71) and (74) together: ... [<][l]
<<2010 Poomjai Nacaskul, Ph.D. | l | . !ก#$ .%. &''(>>
(75) 2
1),(),(),(),(
22 ]1,0[]1,0[
=⋅∂
∂⋅
∂∂
+ ∫∫∫∫ dudvv
vuC
u
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 Μ=Μ=Μ , so long as they satisfy the following axiomatic properties:
...
(76)
( )( ) ( ) )()(
)(),(,lim),(),(lim)(
),(),(),()(
0),(.,)(
)(),(),()(
)(]1,1[),()(
)(),()(
,
2121 econcordancoforderCCCCvii
econvergencuniformYXYXvuCvuCvi
YXYXYXv
YXindepYXiv
symmetricXYYXiii
normalisedYXii
sscompleteneYXi
YX
nnn
nn
Μ≤Μ⇒
Μ=Μ⇒=
Μ−=−Μ=−Μ
=Μ⇒
Μ=Μ
−∈Μ
Μ∃
∀
∞→∞→
p
[]
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, =ig i :
...
<<2010 Poomjai Nacaskul, Ph.D. | li | . !ก#$ .%. &''(>>
(77) ( ) ),()(),(2,1, 21 YXYgXgingincreasig i Μ=Μ⇒= []
[CLV: 96] For example, consider a much simpler measure, called BlomqvistUs beta, which essentially looks at the value of a bivariate copula at in the middle of the square:
...
(78)
( )
( )
( )
−=−
−+⋅=
=−⋅⋅=
+=−
⋅=
⇒−
≡
−
×
+
1112
1
2
1,0max4
012
1
2
14
112
1,
2
1min4
12
1,
2
14'
C
C
C
CsBlomqvist
β
β
β
β []
[CLV: 96] However, a more popular alternative to Kendallvs tau seems to be Spearman's rank correlation coefficient, or 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
v
vuCv
u
vuCuvudCuvdudvvuCsSpearmanρ []
Fredricks, G.A. & Nelsen, R.B. (2007), On the Relationship between Spearmanvs rho and Kendallvs tau for pairs of continuous random variables, Journal of Statistical Planning and Inference, vol. 137, no. 7, pp. 2143-50.
<<2010 Poomjai Nacaskul, Ph.D. | lii | . !ก#$ .%. &''(>>
[CLV: 103] Although intuitive, it isnvt 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 doesnvt.
But that does not prevent us from writing its denominator, i.e. the covariance, in terms of copulas:
... []
(80) ( ) ( ) ( )( )∫∫∫∫ ×−=−=XYXY Support
YXYX
Support
YX dxdyyFxFCyFxFCdxdyyFxFyxFYXCov )(),()(),()()(),(),( []
Indeed, one major shortcoming of the standard correlation measure should be phrased in terms of the fact that because it isnvt a measure of concordance proper, it isnvt 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 <%),<%93;<=>($
Recall how the definition of minimum, independent, and maximum copulas involve no parameter whatsoever, and how for any other
...
<<2010 Poomjai Nacaskul, Ph.D. | liii | . !ก#$ .%. &''(>>
copula ]1,0[]1,0[: 2 →C out there, it will always be bounded in the sense of concordance ordering +− CCC pp .
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.vs.
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) ( ) ( )
( ) ( )
>⇒<
<⇒<
∀∋==
ρρρρρ
ρρρρρ
ρρρ...""..,|,|,
...""..,|,|,
,,)|,(),(
2121
2121
21
trworderednegtivelyeivuCvuC
or
trworderedpositivelyeivuCvuC
vuCvuCC [<]
A parametric copula family is said to be comprehensive if it includes (in the parametric limits) the minimum, independent, and
...
<<2010 Poomjai Nacaskul, Ph.D. | liv | . !ก#$ .%. &''(>>
maximum copulas as its members:
(82)
=
=
=
∋∃
−
→
×
→
+
→
+×−
−
×
+
),()|,(lim
),()|,(lim
),()|,(lim
,,
vuCvuC
vuCvuC
vuCvuC
ρ
ρ
ρ
ρρρ
ρρ
ρρ
ρρ
[&]
One obvious method of constructing a family copula is to create a convex combination of minimum and maximum copulas:
...
(83)
( ) ( )444444444444 8444444444444 76
4444444444 34444444444 21
""..,|,|,
]1,0[,),(),()1()|,(
2121 orderedpositivelyeivuCvuC
vuCvuCvuC
ρρρρ
ρρρρ
<⇒<
⇓
∈⋅+⋅−= +−
[(]
[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. | lv | . !ก#$ .%. &''(>>
(84)
( ) ( )( ) ( )
44 844 76
44444 344444 21
44444 844444 76
44444444444444 344444444444444 21
44444444444444 844444444444444 76
4444444444444444444 34444444444444444444 21
444 8444 764444 84444 76
0,0,
1,0,
0,1,
),()0,0|,(
),()1,0|,(
),()0,1|,(
...""..,,|,,|,
...""..,,|,,|,
1
]1,0[,,),()1(),(),(),|,(),(
2121
2121
..
=
=
=
⇓
=
=
=
⇓
>⇒<
<⇒<
⇓
≤+
∈⋅−−+⋅+⋅==
××
−−
++
×
−
+
×−+
qp
qp
qp
vuCvuC
vuCvuC
vuCvuC
qtrworderednegativelyeiqpvuCqpvuCqq
ptrworderedpositivelyeiqpvuCqpvuCpp
qp
qpvuCqpvuCqvuCpqpvuCvuC
Frechet
Frechet
Frechet
componentssacomponentingulars
FrechetFrechet
[l]
QUIZ 11 What makes a family of copulas comprehensive, and why would a comprehensive copula family be preferred to one that isnvt?
...
Archimedean Copulas ;<=>($+,,)$%7<K@K1KB
<<2010 Poomjai Nacaskul, Ph.D. | lvi | . !ก#$ .%. &''(>>
[CLV: 120] [NEL: 89] One of the most, if not the most, general family of copula is the so-called Archimedean family, as appropriately named by Ling (1965).
...
Ling, C.H. (1965), Representation of Associative Functions, Publicationes Mathematicae Debrecen, vol. 12, pp. 189-212.
First one needs to define a sort of generator: ...
(85)
+∞=
=
−+≥−+⇒∈
>⇒<
∋ℜ→ +
""..,)0(
0)1(
""..,))1(()()1()(]1,0[
""..,)()(
]1,0[:
generatorstrictei
convexeibaba
ingdecreaseibaba
φφ
λλφφλλφλφφ
φ [']
With the generator and its (pseudo) inverse, ( ) 1−φ , one can define an Archimedean copula quite simply:
...
(86) ( ) ( ))()()|,(),( 1 vuvuCvuC nArchimedeanArchimedea φφφφ +== − []
[CLV: 124] For instance, consider the 1-parameter Gumbel copula: ...
<<2010 Poomjai Nacaskul, Ph.D. | lvii | . !ก#$ .%. &''(>>
(87) ( ) ( )
( )( ) ( )( ) ( )
( )( )
−+−−=
−+−=
+=
==
⇒
−=
≥−=
=
−
−
−
ααα
ααα
α
φ
φφφ
φα
φ
α
αφφ
/1
1
1
/11
)ln()ln(exp
)ln()ln(
)()(
|,)|,(),(
exp)(
1,)ln(
)|()(
vu
vu
vu
vuCvuCvuC
ss
t
tt
Gumbel
GumbelGumbelGumbel
GumbelnArchimedeaGumbelGumbel
Gumbel
GumbelGumbel
[]
Consider also the 1-parameter Clayton copula: ...
(88) ( )( ) ( )
( )( ) ( )( ) ( ) ( )( )
( )
−+=
−+−=
+=
==
⇒
+⋅=
+∞−∈−=
=
−−−
−−−
−
−−
−
ααα
αα
ααα
αα
φ
φφφ
φα
αφ
α
αφφ
/1
111
1
/11
1
1,0max
11
)()(
|,)|,(),(
1,0max)(
0\),1[,1
)|()(
vu
vu
vu
vuCvuCvuC
ss
t
tt
Clayton
ClaytonClaytonClayton
ClaytonnArchimedeaClaytonClayton
Clayton
ClaytonClayton
[]
Finally the 1-parameter Frank copula: ...
(89)
( ) ( )( )
( )( ) ( )
( )
( )( )
+
−
−−−=
−
−−
−
−−=
+=
==
⇒
+−−=
ℜ∈
−
−−=
=
−
⋅−⋅−
−
⋅−
−
⋅−−
−
−−−
−
⋅−
11
11ln
1
1ln
1
1ln
)()(
|,)|,(),(
11ln)(
0\,1
1ln
)|()(
1
1
1
11
α
αα
α
α
α
α
α
αα
α
α
φ
φφφ
φα
φ
α
αφφ
e
ee
e
e
e
e
vu
vuCvuCvuC
ees
e
e
tt
vu
vuFrank
FrankFrankFrank
FranknArchimedeaFrankFrank
sFrank
t
FrankFrank
[]
[CLV: 127] Copulas from any of the three families (Gumbel, ...
<<2010 Poomjai Nacaskul, Ph.D. | lviii | . !ก#$ .%. &''(>>
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. 0>Gumbel
Upperλ , Clayton copulas have lower-tail dependency, i.e. 00 >⇒> Clayton
Lowerλα , while Frank copulas have neither i.e. 0== Frank
LowerFrankUpper λλ .
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 ;<=>($A0E($4 x J93+=%
For the most cases, extensions from bivariate 2=n to truly multivariate 2>n copulas are quite obvious (although the same
...
<<2010 Poomjai Nacaskul, Ph.D. | lix | . !ก#$ .%. &''(>>
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
n
iii
n
ii
n
ii
uC
qp
qpCqpCqCpqpCC
uC
uC
uC
1
1
..
1
1
1
)(
1
]1,0[,,)()1()()(),|()(
min)(
)(
1,0max)(
φφu
uuuuu
u
u
u
444 8444 76444 8444 76
[]
[CLV: 135] A multivariate version of the then bivariate Sklarvs theorem (46) is stated below.
Let F be a joint distribution function (multivariate c.d.f.) with margins nFF ,,1 K , then there exists a copula ]1,0[]1,0[: →nC such that::
... [l]
<<2010 Poomjai Nacaskul, Ph.D. | lx | . !ก#$ .%. &''(>>
(91)
( )
( )
( )
n
nn
ii C
xF
xF
xF
CFSupport ]1,0[,)()(,
11
∈=
=∈∀ uuxx X
M
M
[<]
Moreover, if nFF ,,1 K are all continuous, then C is unique.
Conversely, if C is a copula, and nFF ,,1 K are univariate c.d.f.vs, 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
uuu
Cc
∂∂∂∂
≡LL1
)()(
uu [&]
From which a multivariate version of the then bivariate canonical representation (53) is given by:
... ['(]
<<2010 Poomjai Nacaskul, Ph.D. | lxi | . !ก#$ .%. &''(>>
(93)
( )
( )
( )( )
( )
( )
( )
( )
4444 34444 21
48476
4484476
M
M
M
M
tionrepresentacanonical
sfdprginalma
n
iii
densitycopula
nn
ii
fdpntjoi
n
iii
nn
ii xf
xF
xF
xF
cf
xf
f
xF
xF
xF
cc
.'..
1
11
...
1
11
)()(
)( ∏∏ =
=
⋅
=⇒=
= xx
u [(]
Other definitions follow in an equally straight forward manner.
For instance, a multivariate extension of a bivariate survival copula (57) is given by:
... [']
(94)
( ) ( )
( )( )
( )
( )( )
( )
4444444444444444444 84444444444444444444 76
L44 344 21
K876
LK
4444444444444 34444444444444 21
KK
L
nn
uUuU
n
uU
n
uU
n
n
n
iin
uUuUuuCuuuuCC
CuuCuuuCC
nn
nn
>∩∩>=+−−++−=−−=−
⇓
−+−=−−+−≡=
>∪∪>−
>>
Τ
=∑
11
Pr
1
PrPr
11
11
1
Pr,,1111,,1)(
)(11,,11,,)(
11
11
u1
u1u1u
[l]
Elliptical Copulas ;<=>($+,,)K(/=J/<)(
Note in Fig.5 how the contour map of a bivariate normal/Gaussian distribution takes the elliptical shape, a feature that motivates the
...
<<2010 Poomjai Nacaskul, Ph.D. | lxii | . !ก#$ .%. &''(>>
general definition of a class of multivariate elliptical distributions, any whose member is parameterised by a mean vector nℜ∈µv and a positive definite (p.d.) or (at least) positive semi-definite
(p.s.d.) matrix nn×ℜ∈Σ , i.e. one whose quadratic form with any non-zero vector is non-negative:
(95)
Σ⇒>
Σ⇒≥Σ≠ℜ∈∀∋ℜ∈Σ Τ××
..0
...0,,
dp
dsp
formquadratic
nnnn321 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) ( )
Σ
−Σ−=
−Τ )()()(
1 µµ vvxx
xg
f []
Owen, J. & Rabinovitch, R. (1983), On the Class of Elliptical Distributions and their Applications to the Theory of Portfolio Choice, The Journal of Finance, vol. 38, no. 3 (Jun), pp. 745-52.
Valdez, Emiliano A. (2005), Tail Conditional Variance for Elliptically Contoured Distributions, Belgian Actuarial Bulletin, vol. 5, no. 1, pp. 26-36.
For example, the multivariate normal distribution is an elliptical ...
<<2010 Poomjai Nacaskul, Ph.D. | lxiii | . !ก#$ .%. &''(>>
distribution, with Σ referred to as the variance-covariance matrix:
(97) ( ) ( )ΣΝ⇒
Σ
−Σ−−=⇒=
−Τ−
,~)2(
)()(exp)(
)2()(
2/
121
2/
2/
µπ
µµπ
vvv
Xxx
xnn
t
fe
tg []
And so is the multivariate Student distribution: ...
(98)
( )
( )
( ) freedomofegreesddStudent
nf
tntg
nn
n
n
=Σ
⇓
−Σ−+Σ⋅⋅Γ
+Γ=
⇓
Σ⋅⋅Γ
++Γ=
+−Τ
+−
υµ
µµυπυ
υ
υπυ
υ
υ
υ
υυ
,,,~
)()(1)()2/(
)2/)(()(
)()2/(
1)2/)(()(
2/)(112/
2/
2/)(1
v
vv
X
xxx
[]
As is the multivariate logistic distribution. ...
(99) ( )( )
( )( )( )Σ⇒
−Σ−−+Σ
−Σ−−∝⇒
+∝
−Τ
−Τ
−
−
,~)()(exp1
)()(exp)(
1)(
2121
121
22/
2/
µµµ
µµ vvv
vv
Logisticfe
etg
t
t
Xxx
xxx []
Owen, J. & Rabinovitch, R. (1983), On the Class of Elliptical Distributions and their Applications to the Theory of Portfolio Choice, The Journal of Finance, vol. 38, no. 3 (Jun), pp. 745-52.
Valdez, Emiliano A. (2005), Tail Conditional Variance for Elliptically Contoured Distributions, Belgian Actuarial Bulletin, vol. 5,
<<2010 Poomjai Nacaskul, Ph.D. | lxiv | . !ก#$ .%. &''(>>
no. 1, pp. 26-36.
Once again, taking away the location-scale parameters, i.e. by setting/assuming O=µv and setting/assuming Σ has been normalised into a correlation matrix R (all the diagonal elements are now 1vs), we can construct generically an elliptical copula thus:
...
(100) ( )
( )
( )
( )( )( )( ) ( )
====
=
==
→
−Τ
∞− ∞− ∞−
−
−
−
∫ ∫∫ ∫− − −
1xx
xxxx
uu
D)(
..R,
R
R)(,)()(
RR)(
,]1,0[]1,0[:
1
1
1
1
11
11
1 11
11
diag
dspgfdfdxdxdxf
uF
uF
uF
F
u
u
u
CCC
C
nn iiuF
n
uF
i
uF
nn
ii
n
iellipticalelliptical
n
LLLL
M
M
M
M
[<]
Thus Gaussian copula, StudentUs t copula, and logistic copula, result, respectively, from when g represents, normal, Studentvs t, and logistic p.d.f.
While Gaussian and Studentvs t copula are closely related, the key difference exploited in modelling is the fact that Gaussian copula incorporates/exhibits no tail dependence; whereas, Studentvs t
...
<<2010 Poomjai Nacaskul, Ph.D. | lxv | . !ก#$ .%. &''(>>
copula does.
QUIZ 13 Why do we name elliptical copulas elliptical? ...
Modelling with Copulas ก$%&'$()*+,,1234;<=>($
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), Studentvs t, and beta (Dirichlet) distributions, there arenvt many multivariate distribution families we can work with.
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).
...
Wikipedia (2010), Dirichlet distribution, [http: //en.wikipedia.org/wiki/Dirichlet_distribution].
Wikipedia (2010), Joint probability distribution, [http: //en.wikipedia.org/wiki/Joint_probability_distribution].
<<2010 Poomjai Nacaskul, Ph.D. | lxvi | . !ก#$ .%. &''(>>
Wikipedia (2010), Multivariate normal distribution, [http: //en.wikipedia.org/wiki/Multivariate_normal_distribution].
Wikipedia (2010), Multivariate Student distribution, [http: //en.wikipedia.org/wiki/Multivariate_Student_distribution].
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) 444 3444 214434421444 3444 21figurepercentageobjectbooleanunitmonetary
efaultLossGivenDntDefaultEveDefaultExposureAtLoanLoss ××≡ [<<]
Without copula, itvs 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
...
<<2010 Poomjai Nacaskul, Ph.D. | lxvii | . !ก#$ .%. &''(>>
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:
(102)
[ ] [ ][ ][ ] [ ]
[ ].).(
.).(
)(
)(
)(
vrbetapEAD
vrBernoulliLpEAD
ceindependenLDEAD
EADrandomnonLDEAD
identityLDEADLoanLoss
default
default
βαα+
⋅⋅=
Ε⋅⋅=
Ε⋅Ε⋅=
−⋅Ε⋅=
⋅⋅Ε=Ε
[<&]
But copula mathematics can offer insights without making any kind of modelling assumption.
To demonstrate this point, letvs 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
...
<<2010 Poomjai Nacaskul, Ph.D. | lxviii | . !ก#$ .%. &''(>>
respective VaRvs, simply by citing the Fréchet-Hoeffding inequality:
(103)
( )( )( )
=−+=−+=≥
===≤=≤∩≤=
≤∩≤
==≤=≤
==≤=≤
−
+
011.01.0,0max1,0max),(
1.01.0,1.0min,min),(),()Pr(
Pr
01.0)Pr(Pr
01.0)Pr(Pr
''''''''
''''
''''
vuvuC
vuvuCvuCvVuU
VaRRVaRR
vvVVaRR
uuUVaRR
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 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 Θ∈θ and rginalsmarginalsma Θ∈θ denote, respectively, the copulav and marginalsv parameters, the problem is to find:
...
<<2010 Poomjai Nacaskul, Ph.D. | lxix | . !ก#$ .%. &''(>>
(104) ( ) rginalsmacopularginalsma
copuladataFFitError Θ×Θ=Θ∈
==
Θ∈ θ
θθθ
θ,)(minargˆ x [<l]
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 rginalsmaΘ and then over copulaΘ , hence the Inference for the Margin (IFM) method [CLV: 156]:
...
(105)
( ) ( ) ( )( ) ( ) ( )( )
( ) ( )( ) ( ) ( )( )
44444444444444444 844444444444444444 76
K
K
44444444444444444 344444444444444444 21KK
=
=
⇓
===
Θ∈
Θ∈
dataxFxFCFitErrorii
dataxFxFFitErrori
xFxFCxFxFCFF
copularginalmannnrginalmacopula
rginalmannnrginalmarginalsma
copularginalmannnrginalmann
thst
copulacopula
thst
rginalsmarginalsma
thst
θθθθ
θθθ
θθθθθ
θ
θ
ˆ,,ˆminargˆ)(
,,minargˆ)(
,,,,|)(
111
111
11111xx
[<']
Parametric Estimation Methodology for Copulas %G.,K4,3/DKA0ก$%=%G.@/0<F$?$%$@/.J)%7AE2ก9,;<=>($
What exactly do we mean by parametric estimation?
Given a data set in the form of d-point i.i.d. statistical sample d
ii 1=x , and assume that a parametric multivariate distribution ( )θ|Xf has been chosen (model specification stage thus
...
<<2010 Poomjai Nacaskul, Ph.D. | lxx | . !ก#$ .%. &''(>>
completed), and our task is to estimate the parametric value Θ∈θ based on data, in other words calculate ( )d
ii 1ˆˆ
== xθθ .
For future references, we might as well also define statistical
sampling, denoted d
jjX1= for scalar/univariate case and d
jj 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 d
jjx1= for scalar/univariate case and d
jj 1=x for
vector/multivariate case:
(106)
n
ariatesvrandomdofsetsamplelstatistica
d
jjn
ji
j
d
jnj
ij
j
d
jjnrealisatio
ariablesvrandom
dofsetsampling
lstatistica
d
jj
d
jjnrealisatiod
jj
x
x
x
xxX
ℜ∈
=
= →
ℜ∈ →
=
==
=
=
=
==
x
x
x
x
xX ,
,
1,
,
,1
1
1
11
11
444444 3444444 21
M
M
M
M
321 [<]
One of the most popular parametric estimation methodologies is based on the so-called Maximum Likelihood Method (MLE) method.
...
<<2010 Poomjai Nacaskul, Ph.D. | lxxi | . !ก#$ .%. &''(>>
Wikipedia (2010), Maximum likelihood, [http://en.wikipedia.org/wiki/Maximum_likelihood].
Given one data point, the idea is to go with the distributional parameter which made the observed data point most likely.
Given 1>>d data points, the same thinking says one should go with the distributional parameter which made the 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()|()()(
)|()()(
θθθθθ
θθθ
xx
x
[<]
The maximum likelihood estimator is then found by way of optimisation:
...
(108) ( )( )
=≡ ∑=Θ∈Θ∈
d
jjdMLE fdatall
1
1 lnmaxarg)|(maxargˆ θθθθθ
x [<]
<<2010 Poomjai Nacaskul, Ph.D. | lxxii | . !ก#$ .%. &''(>>
Of course, the nicest thing about copula is that, by way of the canonical representation (53) (93), the log-likelihood also separates nicely:
... ['(] [(]
(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)(
,,,
θθθ
θθθ
θθθθθ
θθθθθθ
u
u
x
K
44 844 7644444444 844444444 76
K
444 8444 76
[<]
The fact that the log-likelihood separates into two parts, the first depending on both the copulavs parameter copulaθ and the marginalsv parameter rginalsmaθ , the second only on the marginalsv 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):
... [<']
<<2010 Poomjai Nacaskul, Ph.D. | lxxiii | . !ก#$ .%. &''(>>
(110) ( )( )
( ) ( )( )( )
=
=
∑
∑∑
=Θ∈
= =Θ∈
d
jcopularginalmanjnnrginalmajdcopula
d
j
n
irginalmaijiidrginalsma
thst
copulacopula
th
rginalsmarginalsma
xFxFcii
xfi
1,1,11
1
1 1,
1
ˆ,,ˆlnminargˆ)(
lnminargˆ)(
θθθθ
θθ
θ
θ
K
[<<]
But converting the marginally distributed statistical sample data d
jj 1=x into uniformly distributed points d
jj 1=u can be achieved,
as per non-parametric method, without relying on any modelling assumption whatsoever, i.e. by using the so-called empirical
distribution function (empirical c.d.f.), defined via the indicator
function 1,0'','': →FALSETRUE1 :
...
(111)
( )
( )
( )
444444444 3444444444 21
4444 84444 76
M
M
functionindicator
boolean
d
jnjndnnEmpirical
d
jijidiiEmpirical
d
jjdEmpirical
d
jj FALSETRUEbotw
TRUEbb
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, itvs also possible to perform the 2nd phase in parallel,
...
<<2010 Poomjai Nacaskul, Ph.D. | lxxiv | . !ก#$ .%. &''(>>
hence a 1-stage problem:
(112) ( ) ( )( )( )
= ∑=Θ∈
d
jcopulajnnEmpiricaljEmpiricaldcopula xFxFcii
copulacopula 1,,11
1|||| ,,lnminargˆ)( θθθ
K [<<&]
Of course, given any specific data set, we donvt expect the two estimates to be the same, but they ought to be fairly close:
...
(113) copulacopula θθ ˆˆ || ≅ [<<(]
Non-Parametric Copulas ;<=>($+,,I%2?$%$@/.J)%7
Taking the idea of non-parametric statistics even further, letvs pursue the idea of a non-parametric copula.
Just as the simplest of non-parametric univariate distribution, i.e. empirical c.d.f., is obtained by using the data points themselves, so too is the simplest of non-parametric copulas obtained in a similar manner, as follows.
For a given univariate statistical sample d
jjx1=, letvs define
(univariate) order statistics d
jjx1)( = and (univariate) rank statistics
d
jjr1= thus:
...
<<2010 Poomjai Nacaskul, Ph.D. | lxxv | . !ก#$ .%. &''(>>
(114)
4444444 84444444 76
K
4444444444 34444444444 21K
LL
,,1,,
,,1
)(
1
)()()1(1)(
11
dtjtrxx
statisticsrankdr
xxrrr
statisticsorderxxxx
xX
jtj
j
lklkd
jj
djd
jjd
jjnrealisatiod
jj
∈≤⇔≤
⇓
∈
≤⇔≤∋
≤≤≤≤∋
⇒ →=
=
==
[<<l]
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 K∈ .
For a given multivariate statistical sample d
jj 1=x , define order
statistics and rank statistics that achieve ordering/ranking within each of the dimension ,,1 ni K∈ .
Still, we can define (multivariate) order statistics d
jj 1)( =x and
(multivariate) multivariate rank d
jj 1=r statistics thus:
...
<<2010 Poomjai Nacaskul, Ph.D. | lxxvi | . !ก#$ .%. &''(>>
(115)
444444444 8444444444 76
K
44444444444 344444444444 21K
K
KLL
M
M
,,1,,
,,1
,,1,
,,1,
)(,
1
)()()1(1)(
1,
,
,1
11
dtjtx
d
ni
ni
x
x
x
ijitji
nj
ilikilikd
jj
idiji
d
jj
d
jjn
ji
j
d
jjnrealisatiod
jj
∈≤⇔≤
⇓
∈
=≤⇔≤∋
=≤≤≤≤∋
⇒
= →
=
=
=
==
rx
r
xxrrr
xxxx
xX
[<<']
[CLV: 161] Then define DeheuvelsU empirical copula thus: ...
(116) ( ) ( )
1,,,,0,
,,,,
1
1 1
1
1 1
1
1 1,
1
1 1,
11
≤≤≤=
≤=
≤=
≤=
∑∏∑
∑∏∑
= == =
= == =
d
t
d
t
d
ttt
xxd
t
d
t
d
tC
nid
j
n
iiijd
d
j
n
iiijd
d
j
n
iitjid
d
j
n
iitjid
niEmpirical ii
KK
KK
I
I
r1rI
x1xI [<<]
Goodness-of-Fit Tests for Copulas ก$%-1B),|$3GB$%>=1KB'$E%9,;<=>($
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
...
<<2010 Poomjai Nacaskul, Ph.D. | lxxvii | . !ก#$ .%. &''(>>
Anderson-Darling distances as their distributional test metrics.
Meanwhile, Mashal & Zeevi(2002) and Chen, Fan, Patton (2004) exploited the fact that the Studentvs t distribution is a 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 wevre 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 wevre 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
<<2010 Poomjai Nacaskul, Ph.D. | lxxviii | . !ก#$ .%. &''(>>
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)
V-
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)
V-
Arnold, Helen (2006), Dependence Modelling via the Copula Method, available online via
<<2010 Poomjai Nacaskul, Ph.D. | lxxix | . !ก#$ .%. &''(>>
[www.cmis.csiro.au/techreports/docs/x0000lcd.pdf].
Chen, Fan, Patton (2004), HSimple Tests for Models of Dependence Between Multiple Financial Time Series, with Applications to U.S. Equity Returns and Exchange Rate, SSRN: [http://papers.ssrn.com/sol3/papers.cfm?abstract_id=513024].
Fermanian, Jean-David (2005), Goodness-of-Fit Tests for Copulas, Journal of Multivariate Analysis, vol. 95, no.1, pp.119-152.
Malevergne, Y. & Sornette, D. (2003), Testing the Gaussian Copula Hypothesis for Financial Assets Dependences, Quantitative Finance, vol. 3, pp. 231-250.
Mashal, R. & Zeevi, A. (2002), Beyond Correlation: Extreme Co-Movements between Financial Asset, Working Paper, Columbia University, SSRN: [http://papers.ssrn.com/sol3/papers.cfm?abstract_id=317122].
Nacaskul, P. & Sabborriboon W. (2009), Gaussian Slug + Simple Nonlinearity Enhancement to the 1-Factor and Gaussian Copula Models in Finance, with Parametric Estimation and Goodness-of-Fit Tests on US and Thai Equity Data, 22nd
Australasian Finance and Banking Conference (16th -18th December), Sydney, SSRN: [http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1460576].
Panchenko, Valentyn (2005), Goodness-of-Fit Test for Copulas, Physica A, vol. 355. pp. 176-182.
Prokhorov, Artem (2008), A Goodness-of-Fit Test for Copulas, available online via [http://alcor.concordia.ca/~aprokhor/papers/start.pdf].
QUIZ 14 Does the Goodness-of-Fit test methodology described by Nacaskul & Sabborriboon (2009) and/or Arnold (2006) require
...
<<2010 Poomjai Nacaskul, Ph.D. | lxxx | . !ก#$ .%. &''(>>
additional assumptions regarding the marginal distributions? Why/why not?
Monte Carlo Simulation with Copulas ก$%&'$()*B~$0ก$%p7+,,@)0.J<$%7;(1234;<=>($
Recall how probability is concerned with the distributional and expectation properties of random variables, such as those comprising our statistical sampling d
jj 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 d
jj 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 d
jj 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 distribution best accounts for the appearance of observed data, while Monte Carlo simulation generates numerical examples consistent with whatever
...
<<2010 Poomjai Nacaskul, Ph.D. | lxxxi | . !ก#$ .%. &''(>>
so-and-so distribution was specified.
In univariate setting, the task involves two steps: (1) generating a generic sequence of pseudo-random numbers d
jju1= (random in
the sense that one cannot predict the next number) and (2) transforming these pseudo-random numbers into random variates d
jjx1= with the specified distributional and expectation properties.
The first step involves the pseudo-random number generator
(PRNG), whose generated 1>>d i.i.d. pseudo-random numbers lie uniformly distributed between zero and one:
(117)
( ) ( )
4444444 84444444 76K
444444444444444444444 3444444444444444444444 21KKa
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)
≤=≤=
≤=≤ −−
321321321xXuU
uFUFuUxFXFxX )()(Pr)Pr()()(Pr)Pr( 11 [<<]
<<2010 Poomjai Nacaskul, Ph.D. | lxxxii | . !ก#$ .%. &''(>>
So that a sequence of non-uniform pseudo-random variates can be generated and stored as artificial data:
...
(119)
( ) ( )
444444 8444444 76K
444444444444444444444 3444444444444444444444 21KKa
,1,)(~...
,1,)(~...,PrPr,1,)(
=
⇓
=≤≅≤∋=
jDistrXdii
jDistrXdiixXxXjxuii
jpseudo
jjjpseudojpseudojpseudo
θ
θ
[<<]
Altogether: ...
(120) ( ) Kaa ,1,1
)()(== − juFxuPRNG jpseudojpseudo
iijpseudo
i [<&]
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 1>n 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-
...
<<2010 Poomjai Nacaskul, Ph.D. | lxxxiii | . !ก#$ .%. &''(>>
random vectors, as opposed to mere scalars, are generated:
(121)
( )
( )
( )
K
M
M
a
M
M
M
M
a ,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
jpseudoi
xu [<&<]
The key feature here is that kpseudoU and klpseudoU ≠ are allowed to not be independent of one another.
...
(122) ( ) ( ) ( ) lknlkuUuUuUuU llpseudokkpseudollpseudokkpseudo ≠∈≤⋅≤≠≤∩≤ ,,,1,,PrPrPr K [<&&]
This is done inside the first step, essentially separating it into two sub-routine steps:
...
(123) Kaaa ,1,)()2()1(
=−−
jPRNG jpseudoii
jpseudoi
jpseudoi
xuv [<&(]
This additional step jpseudojpseudo uv a 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
...
<<2010 Poomjai Nacaskul, Ph.D. | lxxxiv | . !ก#$ .%. &''(>>
triangular matrix L and its transpose ΤL .
Given a p.d. matrix nnM ×ℜ∈ in general, Cholesky decomposition is accomplished as follows:
(124)
( )
( )
444444444444444444 8444444444444444444 76
KLL
L
KL
KL
44444444444444444 344444444444444444 21LL
MOMM
LL
MMOM
LL
LL
MOMM
LL
MMOM
LL
LL
MOMM
LL
MMOM
LL
niill
llmllllllllm
lmlllllllm
nil
llmllllllm
lmlllllm
nil
mllllm
mlllm
LL
l
ll
lll
lll
ll
l
mmm
mmm
mmm
M
ii
i
k kilkli
liiliiliililli
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
1
22211
22
1212232221212
22122222222122122
11
1121111
1111111111
1111
1
1
11
1
1
1111
+=
⋅−=⇒+⋅+⋅++⋅+⋅=
−=⇒⋅++⋅+⋅=
=
⋅−=⇒+⋅+⋅+⋅=
−=⇒⋅+⋅=
=
=⇒+⋅+⋅=
=⇒⋅=
⇓
=
=
=
∑∑
−
=+
−
=
Τ
[<&l]
Summarised thus: ...
<<2010 Poomjai Nacaskul, Ph.D. | lxxxv | . !ก#$ .%. &''(>>
(125)
( )
⋅−=+∈∀
−=
∈∀ ∑
∑−
=
−
=
ii
i
k kilkli
li
i
k ikiiii
l
llmlnil
lml
ni 1
1
1
1
2
,,,1
,
,,,1K
K [<&']
Or in the form of a double loop algorithm: ...
(126) ( )
⋅−=+=−==
∑∑−
=−
=ii
i
k kilkli
li
i
k ikiiii l
llmlnilforlmlnifor
1
11
1
2 ,,,1;,,,1 KK [<&]
Wikipedia (2010), Cholesky decomposition, http://en.wikipedia.org/wiki/Cholesky_decomposition].
<<2010 Poomjai Nacaskul, Ph.D. | lxxxvi | . !ก#$ .%. &''(>>
(127)
( )
( )
( )
( )
( )
( )
444444444444444444 8444444444444444444 76
K
4444444 34444444 21
M
M
M
M
a
444444 3444444 21
M
M
a
44444444 344444444 21
M
M
M
M
a
4444 34444 21
M
M
a
4444444 34444444 21
,1,L
0L,,LLR
11
T
1
1
1
11
11
=
Φ
Φ
Φ
=
=
=
=
Φ
Φ
Φ
=
=
=
⇓
=⇒>∀∋=
−
−
−
Τ
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 independent standard normal random variates at a time (efficiency comes from the simple analytical expression, not from the fact that two random variates
...
<<2010 Poomjai Nacaskul, Ph.D. | lxxxvii | . !ก#$ .%. &''(>>
are generated together):
(128)
( ) ( )( ) ( )
44444 844444 76
444444444444444 3444444444444444 21444444444 3444444444 21
4444 84444 76
a
43421
a
Ν
⇓
⋅⋅−
⋅⋅−=
=
−
10
01,
0
0~...
2sinln2
2cosln2
21
21
2
1
2
1
Z
z
pseudo
normalandardsttindependen
transformMullerBox
pseudo
pseudo
pseudo
uniformtindependen
pseudo
pseudo
dii
uu
uu
z
z
u
uPRNG
π
π
[<&]
Roy, Ritabrata (2002), Comparison of Different Techniques to Generate Normal Random Variables, available online via [www.winlab.rutgers.edu/~rito/ece541p1.pdf].
Wikipedia (2010), Box-Muller transform, [http://en.wikipedia.org/wiki/Box-Muller_transform].
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 ก$%&'$()*+,,<[email protected]:4*12$0ก$%.*/01234;<=>($
First up, letvs go over some backgrounds before we bring in ...
<<2010 Poomjai Nacaskul, Ph.D. | lxxxviii | . !ก#$ .%. &''(>>
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 ℘Ω, , which itself comprises of the set, i.e. the sample space, Ω , representing the (infinite/uncountable) universe of possible outcomes in all its infinite details, and the 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 itvs 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, letvs not use this term here.
<<2010 Poomjai Nacaskul, Ph.D. | lxxxix | . !ก#$ .%. &''(>>
By probability, we mean there is to be a function, called probability
measure, Ρ , 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 Ρ℘Ω ,, is referred to in mathematics as a probability space.
By utility, we mean there exists a kind of preference structure, f , 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 Ρ .
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
<<2010 Poomjai Nacaskul, Ph.D. | xc | . !ก#$ .%. &''(>>
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 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.
<<2010 Poomjai Nacaskul, Ph.D. | xci | . !ก#$ .%. &''(>>
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, itvs 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 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
<<2010 Poomjai Nacaskul, Ph.D. | xcii | . !ก#$ .%. &''(>>
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.
<<2010 Poomjai Nacaskul, Ph.D. | xciii | . !ก#$ .%. &''(>>
(129)
⇒
⇒
=⇒
=
=
+
Ρ℘Ω=
)(
)(/
/
//)(
,//)(
,//)(
,//)(
)(
,,,
,,
sderivativefinancialviacontingent
sinstrumentltraditionaviadirectpositionsxposureserisk
latent
observablefactorsdynamicsrisk
driversriskModelsRisk
reportanalysereviewiv
managecontrolmitigateiii
monitorassessmeasureii
indicatedefineidentifyi
rocessPManagementRisk
RiskslOperationaRisksCreditRisksMarketRiskFinancial
Risk
StructurereferenceP
Utility
SpacerobabilityP
robabilityPyPossibilit
K
f
44 344 21
876
[<&]
With that, now letvs bring in copulas.
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.
Letvs 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
...
<<2010 Poomjai Nacaskul, Ph.D. | xciv | . !ก#$ .%. &''(>>
risk drivers are essentially added together, such as a portfolio of return-correlated assets.
And perhaps itvs useful to make the forth and final distinction between when portfolio risk drivers are simply multivariate normal random variables, or otherwise.
(130)
( )
⇒
ΣΝ⇒
⇒
⇒⇒
⇒
copulaotwdriversriskadditive
copuladriversrisktivemultiplica
driversriskdependent
driversrisktindependen
driversriskaggregatemultiple
driverriskindividualingles
ModelsRisk
.
,~/
/
µX
[<(]
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 ก$%&'$()*+,,<[email protected]:4*12$0.<%1/J1234;<=>($
First up, letvs go over some backgrounds before we bring in copulas.
Recall that with single-borrower loans, primary focus in given to
...
<<2010 Poomjai Nacaskul, Ph.D. | xcv | . !ก#$ .%. &''(>>
assessing the probability that that particular loan will 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 Pearsonvs 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) ( )( ) ( )
( )4444 84444 76 ncorrelatiodefault
Defaultii DDppppp
pppipBernoulliD 11Pr,
112,1,~ 2112
2211
2112 =∩=≡−⋅−
⋅−≡⇒= ρ [<(<]
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
...
<<2010 Poomjai Nacaskul, Ph.D. | xcvi | . !ก#$ .%. &''(>>
practical needs for more general multi-default dependency structures.
Again, given a basket of 2 loans:
(132) ( ) 212112 1Pr ppDDpncorrelatiodefault ⋅≠===⇒ [<(&]
For basket of 2≥n 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 variables):
...
(133)
=⇒/=⇒ ∑=
),(~,,1,)(~1
pnBinXDnipBernoulliDncorrelatiodefaultn
iii K [<((]
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, itvs better to think of a default process in general before recapitulating back into simple ¥yesv/vnov default event (a Boolean random variable).
...
<<2010 Poomjai Nacaskul, Ph.D. | xcvii | . !ก#$ .%. &''(>>
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, letvs talk in term of a positive continuous default time or time-to-default or time-until-default random variable, 0>T , 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 ≤≡== [<(l]
The flipped side to the default-time c.d.f. is called the survival
function: ...
(135) ( ) [ ]tTtFtTtS >Ε=−=>≡ 1)(1Pr)( [<(']
From which it follows that the default time p.d.f. can be written in terms of either:
...
<<2010 Poomjai Nacaskul, Ph.D. | xcviii | . !ก#$ .%. &''(>>
(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
>≤≡⇒≤≥≥
θθ thresholddefaultXTtimestoppingtX tt
t [<(]
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)( >= λλ t .
...
<<2010 Poomjai Nacaskul, Ph.D. | xcix | . !ก#$ .%. &''(>>
(138) )(
)(')exp()(')exp()(
tS
tSttSttS −=⇒⋅−⋅−=⇒⋅−= λλλλ [<(]
From which it then follows that default arrival is a homogeneous
Poisson process, in turn, implying that default time is exponentially distributed, thus:
...
(139) )(~)exp(1)()exp()(1)Pr(1)Pr()( λλλ 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
dsstStS
tSt
0
)(exp)()(
)(')( λλ [<l]
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 0>t .
To see that, first rewrite )(tλ as follows:
...
<<2010 Poomjai Nacaskul, Ph.D. | c | . !ก#$ .%. &''(>>
(141) )(1
)(
)(1
)('
)(
)(')(
tF
tf
tF
tF
tS
tSt
−=
−=−=λ [<l<]
Recalling that, by definition: ...
(142) ( ) ( )t
tTttT
t
tFttFtFtf
tt ∆≤−∆+≤
=∆
−∆+==
→∆→∆
PrPrlim
)()(lim)(')(
00 [<l&]
Use Bayes theorem to arrive at the instantaneous default probability, conditional on having survived up to time 0>t :
...
(143) ( ) ( )( )
( )( ) )(1
)()(lim
Pr
Prlim
Pr
PrlimPrlim
0000 tF
tFttF
tT
ttTt
tT
tTttTtTttT
tttt −−∆+
=>
∆+≤<=
>>∩∆+≤
=>∆+≤→∆→∆→∆→∆
[<l(]
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 )(tλ :
... [<l(][<l&]
(144) ( )( )
)()(1
)(
)(1
)()(lim
Prlim
00t
tF
tf
tFt
tFttF
t
tTttTtt
λ=−
=−⋅∆−∆+
=∆
>∆+≤→∆→∆
[<ll]
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
...
<<2010 Poomjai Nacaskul, Ph.D. | ci | . !ก#$ .%. &''(>>
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)
⋅
+−Ε=
⋅
−Ε ∫∫ eBondDefaultFredsssreBondDefaultabldssrt
ratediscount
odifiedm
ratehazard
t
rateshort 00
)()(exp)(exp48476λ [<l']
Finally, itvs 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.
...
Duffie, D.J. & Singleton, K.J. (1999), Modeling Term Structure of Defaultable Bonds, Review of Financial Studies, vol. 12, pp. 687-720.
Lando, David (1998), On Cox Processes and Credit Risky Securities, Review of Derivatives Research, vol. 2-3, pp. 99-120.
With that, now letvs bring in copulas.
Given the individual default processes in terms of individual default-time c.d.f.vs, alternatively in terms of survival functions, we
... ['][l]
<<2010 Poomjai Nacaskul, Ph.D. | cii | . !ก#$ .%. &''(>>
can then employ a copula, alternatively a survival copula (57)(94), to construct a joint default-time c.d.f., alternatively a joint survival
function:
(146) ( ) ( )( ) ( )nnn
nnn
tTtTttSfunctionsurvivalntjoi
tTtTttFfdcdefaultntjoi
>∩∩>≡
≤∩∩≤≡
KK
KK
111
111
Pr,,
Pr,,... [<l]
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.vs/survival functions:
...
(147)
( ) ( ) ( )( )( ) ( ) ( )( )
( ) ( ) ( )( )
( ) ( ) ( )( )
( ) ( ) ban
n
iii
nnb
n
iii
nnb
n
iii
nnn
nnan
CCiffttFtS
tFtTtTtStFtFCtS
CCtStSCtS
copulasurvivalaingustStSCttS
copulaaingustFtFCttF
=+−=
=≤=>−=−+−=
−+−=−−+−=
=
=
∑
∑
∑
=
=
Τ
=
,,1
)()Pr()Pr(1)(1,,1
)(1)(1,,11
)(,,,,
)(,,,,
11
111
111
111
111
K
QK
QK
KK
KK
u1u1u [<l]
In particular, Li (2000) proposed using a Gaussian copula construction of joint default-time c.d.f. where, in a bivariate case:
...
<<2010 Poomjai Nacaskul, Ph.D. | ciii | . !ก#$ .%. &''(>>
(148) ( ) ( ) ( ) 11,,,
...
...'
1
...'
1
...
≤≤−
Φ
ΦΦ=−
−
−
−
−
ρρ
444444 8444444 76
4484476
321
4484476
32143421
fdcnormalariatebiv
ariatevrandomuniform
fdctimedefaultsB
BB
ariatevrandomuniform
fdctimedefaultsA
AA
fdctimedefaultntjoi
BA tFtFttF [<l]
Then by invoking the 1-year horizon, ρ 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
⋅=ΦΦΦ= −− ,
,)1(,)1()1,1( 11 ρρ [<l]
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
... [<&]
<<2010 Poomjai Nacaskul, Ph.D. | civ | . !ก#$ .%. &''(>>
generating correlated multivariate standard normal random variates, as per (127), from which dependent default times can then be imputed.
(150)
( )( )
( )( )
( )( )
L
444444444 3444444444 21
M
M
M
M
4444 34444 21
M
M
,1,
1
1
11
111
=
Φ
Φ
Φ
=
=⇒
=
−
−
−
j
xF
xF
xF
t
t
t
x
x
x
timesdefaultdependent
jnsimulatedn
isimulatedi
simulated
jnsimulated
isimulated
simulated
jsimulated
normalandardstcorrelated
jnpseudo
ipseudo
pseudo
jpseudo tx [<']
Li, David X. (2000), On Default Correlation: A Copula Function Approach, Journal of Fixed Income, vol. 9, pp. 43-54.
QUIZ 17 What is the difference between default correlation
( ) ( )2211
2112
11 pppp
pppDefault
−⋅−
⋅−≡ρ and the expression
( )( ) ( )21
21,
TVarTVar
TTCov
⋅ (where 1T and 2T represent time-to-defaults)?
...
Detour in Credit Derivatives & Derivatives Pricing +3GI=.%:)*J%$B$%)0C?90D7.<%1/Jก9,ก$%JK%$<$J%$B$%)0C?90D7
First up, letvs go over some backgrounds before we bring in copulas.
...
<<2010 Poomjai Nacaskul, Ph.D. | cv | . !ก#$ .%. &''(>>
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
<<2010 Poomjai Nacaskul, Ph.D. | cvi | . !ก#$ .%. &''(>>
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 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)
( ) 321
44 344 2144444444444 344444444444 21
problem
t
assumption
marketfreearbitrage
conditionarbitrageno
upset
TTTT
T
t
t
C
sSnrealisatioonlconditionaknownSCC
unknownS
knownS
processstochastictS
?
0,
/
= →
===
=
=
=≥
−
− [<'<]
<<2010 Poomjai Nacaskul, Ph.D. | cvii | . !ก#$ .%. &''(>>
Whereas the prices of underlying assets are subject principally by financial marketsv 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.
Letvs 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 pay-offs discounted at the risk-free rate of return, taken against a so-called risk-neutral (probability) measure:
...
<<2010 Poomjai Nacaskul, Ph.D. | cviii | . !ก#$ .%. &''(>>
(152) [ ]
−−=
−
−=
=
Ε⋅=
−
)(exp),(..
)(..
,)(),(
,),(
/
Q
tTrTtBei
billtreasurycouponzeroviage
discountfreeriskcreditTtB
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 K 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:
...
<<2010 Poomjai Nacaskul, Ph.D. | cix | . !ก#$ .%. &''(>>
(153) ( ) ( ) ( ) ( )tTtSremiumPCDSNotionaltStDiscountremiumsPPVm
i
paymentpremium
fractionyear
annuallyxpressede
protectionofvalueface
functionsurvival
neutralrisk
ii >≡∆××××=∑=
−
11
Q1 Pr)(,'
444444 8444444 76
44 344 214342143421 [<'(]
Conversely, the present value of the protection leg is given, in terms of risk-neutral default c.d.f., by:
...
(154) ( ) ( ) ( ) ( ) ( )tTtFoveryRateecRNotionaltFTDiscountrotectionPPV
paymentprotection
defaultgivenlossprotectionof
valueface
fdcdefault
neutralrisk
m ≤≡−×××=−
%
...
Q Pr)(,144444 844444 76
444 3444 214342143421 [<'l]
Equating (153) with (154) yields the CDS premium the start of the contract:
... [<'(][<'l]
(155) ( ) ( ) ( )
( ) ( )∑=
−− ∆××
−××=
m
iii
m
tStDiscount
overyRateecRtFTDiscountremiumPCDS
11
Q1
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:
...
<<2010 Poomjai Nacaskul, Ph.D. | cx | . !ก#$ .%. &''(>>
(156) ( ) ( ) ( ) ( )( ) ( )∑=
−
− −××−×=m
i
paymentprotection
defaultgivenlossprotectionof
valueface
yprobabilitdefault
neutralrisk
iii overyRateecRNotionaltStStDiscountrotectionPPV1 %
Q1
Q 144444 844444 76
444 3444 2143421444 3444 21 [<']
Equating (153) with (156) in lieu of (154) then yields the CDS premium the start of the contract:
... [<'(][<'][<'l]
(157) ( ) ( ) ( )( ) ( )
( ) ( )∑
∑
=−−
=−
∆××
−×−×=
m
iii
m
iiii
tStDiscount
overyRateecRtStStDiscountremiumPCDS
11
Q1
1
Q1
Q 1' [<']
This CDS premium is fixed at the start of the contract, and subsequently the marked-to-market (M2M) value of the 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 ก$%JK%$<$J%$B$%)0C?90D7.<%1/J1234;<=>($
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
... [l][<] [<(<]
<<2010 Poomjai Nacaskul, Ph.D. | cxi | . !ก#$ .%. &''(>>
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.
Letvs 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 Sklarvs 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
<<2010 Poomjai Nacaskul, Ph.D. | cxii | . !ก#$ .%. &''(>>
with risk-neutral copula density, and so on.
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 1>k assets, whose default times are thus designated by the following random vector.
(158) [ ] ( ) kiExpT
T
T
T
ii
k
k
i ,,1,~
1
K
M
M
=∋ℜ∈
= + λT [<']
First, letvs start with another random vector, one whose components are independent and exponentially distributed:
...
<<2010 Poomjai Nacaskul, Ph.D. | cxiii | . !ก#$ .%. &''(>>
(159) ( )
( )( ) ( )
kji
jisSsS
sSsS
ExpS
S
S
S
jjii
jjii
ii
k
k
i ,,1,,
,PrPr
Pr
~
)(
1
K
M
M
=
≠≤≤=
≤∩≤∋ℜ∈
= +
λ
S [<']
Now use individual exponential c.d.f.vs 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
K
M
M
M
M
M
M
=⇒
−−
−−
−−
=
=
=
λ
λ
λ
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
K
M
M
M
M
M
M
M
M
=Ν⇒
−−Φ
−−Φ
−−Φ
=
Φ
Φ
Φ
=
Φ
Φ
Φ
=
=
−
−
−
−
−
−
−
−
−
λ
λ
λ
Z [<<]
<<2010 Poomjai Nacaskul, Ph.D. | cxiv | . !ก#$ .%. &''(>>
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.
...
(163)
( )
( )
( )
kiUnifV
X
X
X
V
V
V
copulaGaussianaeistructuredependenceawithendowednow
i
k
i
k
i ,,1,)1,0(~
)..(
11
K4484476
M
M
M
M
=⇒
Φ
Φ
Φ
=
=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
T
copulaGaussianaeistructuredependenceawithendowednow
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
K4484476
M
M
M
M
M
M
M
M
=⇒
Φ−−
Φ−−
Φ−−
=
−−
−−
−−
=
=
=
−
−
−
λ
λ
λ
λ
λ
λ
λ
T [<l]
<<2010 Poomjai Nacaskul, Ph.D. | cxv | . !ก#$ .%. &''(>>
When pricing via EMM, our task then involves Monte Carlo simulation to generate default-time random variates, essentially starting from (160):
... [<]
(165)
( )
( )
( )
sim
rrpseudorpseudo
sim
rrkpseudo
ipseudo
pseudo
rkpseudo
ipseudo
pseudo
rpseudo
sim
rrkpseudo
ipseudo
pseudo
u
u
u
z
z
z
u
u
u
#
1
T
#
1
1
1
11
1
#
1
1
L=
=
−
−
−
=
=⇒
Φ
Φ
Φ
=
=⇒
zxz
M
M
M
M
M
M
[<']
For each simulation run simr #,,1K= , 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
=
−
−
−
Φ−−
Φ−−
Φ−−
=
−−
−−
−−
=
=
=
λ
λ
λ
λ
λ
λ
M
M
M
M
M
M
M
M
t [<]
For instance, for pricing 1tD-CDS, register, for each simulation run, the minimum default time.
...
<<2010 Poomjai Nacaskul, Ph.D. | cxvi | . !ก#$ .%. &''(>>
(167) sim
rkrpseudoirpseudorpseudoi
rpseudo t#
11
min ,,,,min=
= ttt KK [<]
We can now use (153) to calculate the present value of the premium leg (in an expression which involves an indicator function):
... [<'(]
(168) ( ) ( )
sim
r
m
i
paymentpremium
fractionyear
annuallyxpressede
protectionofvalueface
ttirsim remiumPCDStDNotionaltDiscountremiumsPPVirpseudo
#
1
11 '1
1min
=
=≥−
∆×−×××=∑−
4444444 84444444 76
444 3444 21434211 [<]
Then use (156), which for simulation purpose is somewhat preferable to (154), to calculate the present value of the protection leg thus:
... [<'l][<']
(169) ( ) ( ) ( )
sim
r
m
i
paymentprotection
defaultgivenlossprotectionof
valuefacetttirsim overyRateecRNotionaltDiscountrotectionPPVirpseudoi
#
1
1 %
11
min
=
=≥>
−×××=∑−
44444 844444 76
444 3444 21434211 [<]
The (M2M) present value of the 1tD-CDS, as a function of premium is given by:
...
(170) ( )[ ] ( ) ( )( )∑=
−=−Εsim
rrsimrsimsimsim rotectionPPVremiumsPPVCDStDPV
#
1#
11 [<]
<<2010 Poomjai Nacaskul, Ph.D. | cxvii | . !ก#$ .%. &''(>>
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?
...
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