Copula (probability theory)
In probability theory and statistics, a copula is a multivariate probability distribution for which the marginal probability distribution of each variable is uniform. Copulas are used to describe the dependence between random variables. Their name comes from the Latin for "link" or "tie", similar but unrelated to grammatical copulas in linguistics^{[citation needed]}.
Sklar's Theorem states that any multivariate joint distribution can be written in terms of univariate marginal distribution functions and a copula which describes the dependence structure between the variables.
Copulas are popular in highdimensional statistical applications as they allow one to easily model and estimate the distribution of random vectors by estimating marginals and copulae separately. There are many parametric copula families available, which usually have parameters that control the strength of dependence. Some popular parametric copula models are outlined below.
Contents
Mathematical definition
Consider a random vector . Suppose its margins are continuous, i.e. the marginal CDFs are continuous functions. By applying the probability integral transform to each component, the random vector
has uniformly distributed marginals.
The copula of is defined as the joint cumulative distribution function of :
The copula C contains all information on the dependence structure between the components of whereas the marginal cumulative distribution functions contain all information on the marginal distributions.
The importance of the above is that the reverse of these steps can be used to generate pseudorandom samples from general classes of multivariate probability distributions. That is, given a procedure to generate a sample from the copula distribution, the required sample can be constructed as
The inverses are unproblematic as the were assumed to be continuous. The above formula for the copula function can be rewritten to correspond to this as:
Definition
In probabilistic terms, is a ddimensional copula if C is a joint cumulative distribution function of a ddimensional random vector on the unit cube with uniform marginals.^{[1]}
In analytic terms, is a ddimensional copula if

 , the copula is zero if one of the arguments is zero,
 , the copula is equal to u if one argument is u and all others 1,
 C is dincreasing, i.e., for each hyperrectangle the Cvolume of B is nonnegative:
 where the .
For instance, in the bivariate case, is a bivariate copula if , and for all and .
Sklar's theorem
Sklar's theorem,^{[2]} named after Abe Sklar, provides the theoretical foundation for the application of copulas. Sklar's theorem states that every multivariate cumulative distribution function
of a random vector can be expressed by involving only the marginals as
where is a copula. In case that the multivariate distributions has a density is available it holds further that where is the density of the copula.
The theorem also states that, given , the copula is unique on , which is the cartesian product of the ranges of the marginal cdf's. This implies that the copula is unique if the marginals are continuous.
The converse is also true: given a copula and margins then defines a ddimensional cumulative distribution function.
Fréchet–Hoeffding copula bounds
The Fréchet–Hoeffding Theorem (after Maurice René Fréchet and Wassily Hoeffding ^{[3]}) states that for any Copula and any the following bounds hold:
The function W is called lower Fréchet–Hoeffding bound and is defined as
The function M is called upper Fréchet–Hoeffding bound and is defined as
The upper bound is sharp: M is always a copula, it corresponds to comonotone random variables.
The lower bound is pointwise sharp, in the sense that for fixed u, there is a copula such that . However, W is a copula only in two dimensions, in which case it corresponds to countermonotonic random variables.
In two dimensions, i.e. the bivariate case, the Fréchet–Hoeffding Theorem states
Families of copulas
Several families of copulae have been described.
Gaussian copula
The Gaussian copula is a distribution over the unit cube . It is constructed from a multivariate normal distribution over by using the probability integral transform.
For a given correlation matrix , the Gaussian copula with parameter matrix can be written as
where is the inverse cumulative distribution function of a standard normal and is the joint cumulative distribution function of a multivariate normal distribution with mean vector zero and covariance matrix equal to the correlation matrix .
The density can be written as^{[4]}
where is the identity matrix.
Archimedean copulas
Archimedean copulas are an associative class of copulas. Most common Archimedean copulas admit an explicit formula, something not possible for instance for the Gaussian copula. In practice, Archimedean copulas are popular because they allow modeling dependence in arbitrarily high dimensions with only one parameter, governing the strength of dependence.
A copula C is called Archimedean if it admits the representation^{[5]}
where is a continuous, strictly decreasing and convex function such that . is a parameter within some parameter space . is the socalled generator function and is its pseudoinverse defined by
Moreover, the above formula for C yields a copula for if and only if is dmonotone on .^{[6]} That is, if it is times differentiable and the derivatives satisfy
for all and and is nonincreasing and convex.
Most important Archimedean copulas
The following tables highlight the most prominent bivariate Archimedean copulas, with their corresponding generator. Note that not all of them are completely monotone, i.e. dmonotone for all or dmonotone for certain only.
Name of Copula  Bivariate Copula  parameter 

AliMikhailHaq^{[7]}  
Clayton^{[8]}  
Frank  
Gumbel  
Independence  
Joe 
name  generator  generator inverse 

AliMikhailHaq^{[7]}  
Clayton^{[8]}  
Frank  
Gumbel  
Independence  
Joe 
Expectation for copula models and Monte Carlo integration
In statistical applications, many problems can be formulated in the following way. One is interested in the expectation of a response function applied to some random vector .^{[9]} If we denote the cdf of this random vector with , the quantity of interest can thus be written as
If is given by a copula model, i.e.,
this expectation can be rewritten as
In case the copula C is absolutely continuous, i.e. C has a density c, this equation can be written as
and if each marginal distribution has the density it holds further that
If copula and margins are known (or if they have been estimated), this expectation can be approximated through the following Monte Carlo algorithm:
 Draw a sample of size n from the copula C
 By applying the inverse marginal cdf's, produce a sample of by setting
 Approximate by its empirical value:
Empirical copulas
When studying multivariate data, one might want to investigate the underlying copula. Suppose we have observations
from a random vector with continuous margins. The corresponding "true" copula observations would be
However, the marginal distribution functions are usually not known. Therefore, one can construct pseudo copula observations by using the empirical distribution functions
instead. Then, the pseudo copula observations are defined as
The corresponding empirical copula is then defined as
The components of the pseudo copula samples can also be written as , where is the rank of the observation :
Therefore, the empirical copula can be seen as the empirical distribution of the rank transformed data.
Applications
Quantitative finance
In risk/portfolio management, copulas are used to perform stresstests and robustness checks that are especially important during “downside/crisis/panic regimes” where extreme downside events may occur (e.g., the global financial crisis of 2007–2008).
The formula was also adapted for financial markets and was used to estimate the probability distribution of losses on pools of loans or bonds. The users of the formula have been criticized for creating "evaluation cultures" that continued to use simple copulæ despite the simple versions being acknowledged as inadequate for that purpose.^{[10]} During a downside regime, a large number of investors who have held positions in riskier assets such as equities or real estate may seek refuge in ‘safer’ investments such as cash or bonds. This is also known as a flighttoquality effect and investors tend to exit their positions in riskier assets in large numbers in a short period of time. As a result, during downside regimes, correlations across equities are greater on the downside as opposed to the upside and this may have disastrous effects on the economy. ^{[11]} ^{[12]} For example, anecdotally, we often read financial news headlines reporting the loss of hundreds of millions of dollars on the stock exchange in a single day; however, we rarely read reports of positive stock market gains of the same magnitude and in the same short time frame.
Copulas are useful in portfolio/risk management and help us analyse the effects of downside regimes by allowing the modelling of the marginals and dependence structure of a multivariate probability model separately. For example, consider the stock exchange as a market consisting of a large number of traders each operating with his/her own strategies to maximize profits. The individualistic behaviour of each trader can be described by modelling the marginals. However, as all traders operate on the same exchange, each trader's actions have an interaction effect with other traders'. This interaction effect can be described by modelling the dependence structure. Therefore, copulas allow us to analyse the interaction effects which are of particular interest during downside regimes as investors tend to herd their trading behaviour and decisions.
Previously, scalable copula models for large dimensions only allowed the modelling of elliptical dependence structures (i.e., Gaussian and Studentt copulas) that do not allow for correlation asymmetries where correlations differ on the upside or downside regimes. However, the recent development of vine copulas^{[13]} (also known as pair copulas) enables the flexible modelling of the dependence structure for portfolios of large dimensions. ^{[14]} The Clayton canonical vine copula allows for the occurrence of extreme downside events and has been successfully applied in portfolio choice and risk management applications. The model is able to reduce the effects of extreme downside correlations and produces improved statistical and economic performance compared to scalable elliptical dependence copulas such as the Gaussian and Studentt copula. ^{[15]} Other models developed for risk management applications are panic copulas that are glued with market estimates of the marginal distributions to analyze the effects of panic regimes on the portfolio profit and loss distribution. Panic copulas are created by Monte Carlo simulation, mixed with a reweighting of the probability of each scenario.^{[16]}
As far as derivatives pricing is concerned, dependence modelling with copula functions is widely used in applications of financial risk assessment and actuarial analysis – for example in the pricing of collateralized debt obligations (CDOs).^{[17]} Some believe the methodology of applying the Gaussian copula to credit derivatives to be one of the reasons behind the global financial crisis of 2008–2009.^{[18]}^{[19]} Despite this perception, there are documented attempts of the financial industry, occurring before the crisis, to address the limitations of the Gaussian copula and of copula functions more generally, specifically the lack of dependence dynamics^{[clarification needed]} and the poor representation of extreme events.^{[20]} There have been attempts to propose models rectifying some of the copula limitations.^{[20]}^{[21]}^{[22]}
While the application of copulas in credit has gone through popularity as well as misfortune during the global financial crisis of 2008–2009,^{[23]} it is arguably an industry standard model for pricing CDOs. Copulas have also been applied to other asset classes as a flexible tool in analyzing multiasset derivative products. The first such application outside credit was to use a copula to construct an implied basket volatility surface,^{[24]} taking into account the volatility smile of basket components. Copulas have since gained popularity in pricing and risk management ^{[25]} of options on multiassets in the presence of volatility smile/skew, in equity, foreign exchange and fixed income derivative business. Some typical example applications of copulas are listed below:
 Analyzing and pricing volatility smile/skew of exotic baskets, e.g. best/worst of;
 Analyzing and pricing volatility smile/skew of less liquid FX^{[clarification needed]} cross, which is effectively a basket: C = S_{1}/S_{2} or C = S_{1}·S_{2};
 Analyzing and pricing spread options, in particular in fixed income constant maturity swap spread options.
Civil engineering
Recently, copula functions have been successfully applied to the database formulation for the reliability analysis of highway bridges, and to various multivariate simulation studies in civil,^{[26]} mechanical and offshore engineering.^{[citation needed]}Researchers are also trying these functions in field of transportation to understand interaction of individual driver behavior components which in totality shapes up the nature of an entire traffic flow.
Reliability engineering
Copulas are being used for reliability analysis of complex systems of machine components with competing failure modes. ^{[27]}
Warranty data analysis
Copulas are being used for warranty data analysis in which the tail dependence is analysed ^{[28]}
Turbulent combustion
Copulas are used in modelling turbulent partially premixed combustion, which is common in practical combustors. ^{[29]} ^{[30]}
Medicine
Copula functions have been successfully applied to the analysis of neuronal dependencies ^{[31]} and spike counts in neuroscience ^{[32]} .
Weather research
Copulas have been extensively used in climate and weatherrelated research.^{[33]}
Random vector generation
Large synthetic traces of vectors and stationary time series can be generated using empirical copula while preserving the entire dependence structure of small datasets.^{[34]} Such empirical traces are useful in various simulationbased performance studies.^{[35]}
References
 ↑ Nelsen, Roger B. (1999), An Introduction to Copulas, New York: Springer, ISBN 0387986235<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Sklar, A. (1959), "Fonctions de répartition à n dimensions et leurs marges", Publ. Inst. Statist. Univ. Paris, 8: 229–231<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ "J J O'Connor and E F Robertson" (March 2011). "Biography of Wassily Hoeffding". School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved 8 November 2011.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
 ↑ ^{5.0} ^{5.1} ^{5.2} Nelsen, R. B. (2006). An Introduction to Copulas, Second Edition. New York, NY 10013, USA: Springer Science+Business Media Inc. ISBN 9781441921093.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
 ↑ ^{7.0} ^{7.1} Ali, M.M., Mikhail, N.N. and Haq, M.S. (1978). A class of bivariate distributions including the bivariate logistic. J. Multivariate Anal. 8, 405412
 ↑ ^{8.0} ^{8.1} Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
 ↑ Alexander J. McNeil, Rudiger Frey and Paul Embrechts (2005) "Quantitative Risk Management: Concepts, Techniques, and Tools", Princeton Series in Finance
 ↑ MacKenzie, Donald; Spears, Taylor (June 2012). "The Formula That Killed Wall Street"? The Gaussian Copula and the Material Cultures of Modelling (pdf) (Technical report). University of Edinburgh School of Social and Political Sciences.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
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 ↑ Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
 ↑ Cooke, R.M.; Joe, H.; Aas, K. (January 2011). Kurowicka, D.; Joe, H., eds. Dependence Modeling Vine Copula Handbook (PDF). World Scientific. pp. 37–72. ISBN 9789814299879.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
 ↑ Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
 ↑ Meucci, Attilio (2011), "A New Breed of Copulas for Risk and Portfolio Management", Risk, 24 (9): 122–126<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
 ↑ Recipe for Disaster: The Formula That Killed Wall Street Wired, 2/23/2009
 ↑ MacKenzie, Donald (2008), "EndoftheWorld Trade", London Review of Books (published 20080508), retrieved 20090727<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ ^{20.0} ^{20.1} Lipton, Alexander; Rennie, Andrew. Credit Correlation: Life After Copulas. World Scientific. ISBN 9789812709493.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Donnelly, C; Embrechts, P, (2010). "The devil is in the tails: actuarial mathematics and the subprime mortgage crisis". ASTIN Bulletin 40(1), 1–33.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Brigo, D; Pallavicini, A; Torresetti, R (2010). Credit Models and the Crisis: A Journey into CDOs, Copulas, Correlations and dynamic Models. Wiley and Sons.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Jones, Sam (April 24, 2009), "The formula that felled Wall St", Financial Times<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Qu, Dong, (2001). "Basket Implied Volatility Surface". Derivatives Week, (4 June.). <templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Qu, Dong, (2005). "Pricing Basket Options With Skew". Wilmott Magazine (July.). <templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
 ↑ Pham, Hong (2003), Handbook of Reliability Engineering, Springer, pp. 150–151<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
 ↑ Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
 ↑ Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
 ↑ Eban, E; Rothschild, R; Mizrahi, A; Nelken, I; Elidan, G (2013), Carvalho, C; Ravikumar, P, eds., "Dynamic Copula Networks for Modeling Realvalued Time Series" (PDF), Journal of Machine Learning Research, 31<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
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 ↑ Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
 ↑ Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
 ↑ Bandara, H. M. N. D.; Jayasumana, A. P. (Dec 2011). "On Characteristics and Modeling of P2P Resources with Correlated Static and Dynamic Attributes". IEEE GLOBECOM: 1–6.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
Further reading
 The standard reference for an introduction to copulas. Covers all fundamental aspects, summarizes the most popular copula classes, and provides proofs for the important theorems related to copulas

 Roger B. Nelsen (1999), "An Introduction to Copulas", Springer. ISBN 9780387986234
 A book covering current topics in mathematical research on copulas:

 Piotr Jaworski, Fabrizio Durante, Wolfgang Karl Härdle, Tomasz Rychlik (Editors): (2010): "Copula Theory and Its Applications" Lecture Notes in Statistics, Springer. ISBN 9783642124648
 A reference for sampling applications and stochastic models related to copulas is

 JanFrederik Mai, Matthias Scherer (2012): Simulating Copulas (Stochastic Models, Sampling Algorithms and Applications). World Scientific. ISBN 9781848168749
 A paper covering the historic development of copula theory, by the person associated with the "invention" of copulas, Abe Sklar.

 Abe Sklar (1997): "Random variables, distribution functions, and copulas – a personal look backward and forward" in Rüschendorf, L., Schweizer, B. und Taylor, M. (eds) Distributions With Fixed Marginals & Related Topics (Lecture Notes – Monograph Series Number 28). ISBN 9780940600409
 The standard reference for multivariate models and copula theory in the context of financial and insurance models

 Alexander J. McNeil, Rudiger Frey and Paul Embrechts (2005) "Quantitative Risk Management: Concepts, Techniques, and Tools", Princeton Series in Finance. ISBN 9780691122557
External links
 Hazewinkel, Michiel, ed. (2001), "Copula", Encyclopedia of Mathematics, Springer, ISBN 9781556080104<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Copula Wiki: community portal for researchers with interest in copulas
 A collection of Copula simulation and estimation codes
 Thorsten Schmidt (2006) "Coping with Copulas"
 Copulas & Correlation using Excel Simulation Articles
 Chapter 1 of JanFrederik Mai, Matthias Scherer (2012) "Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications"
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