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In probability theory and statistics, a copula is a multivariate probability distribution for which the marginal probability of each variable is uniformly distributed. Copulas are used to describe the dependence between random variables. They are named for their resemblance to grammatical copulas in linguistics.
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 statistical applications as they allow one to easily model and estimate the distribution of random vectors by estimating marginals and copula 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. The formula was also adapted to Wall Street, where it took on a life of its own, used to estimate the probability distribution of losses on pools of loans or bonds. The formula was used to estimate risk, creating "evaluation cultures" that took the predictions of the formula as hard probabilities with which to make risk assessments.^{[1]}
Consider a random vector $(X\_1,X\_2,\backslash dots,X\_d)$. Suppose its margins are continuous, i.e. the marginal CDFs $F\_i(x)\; =\; \backslash mathbb\{P\}[X\_i\backslash leq\; x]$ are continuous functions. By applying the probability integral transform to each component, the random vector
has uniform margins.
The copula of $(X\_1,X\_2,\backslash dots,X\_d)$ is defined as the joint cumulative distribution function of $(U\_1,U\_2,\backslash dots,U\_d)$:
The copula C contains all information on the dependence structure between the components of $(X\_1,X\_2,\backslash dots,X\_d)$ whereas the marginal cumulative distribution functions $F\_i$ contain all information on the marginal distributions.
The importance of the above is that the reverse of these steps can be used to generate pseudo-random samples from general classes of multivariate probability distributions. That is, given a procedure to generate a sample $(U\_1,U\_2,\backslash dots,U\_d)$ from the copula distribution, the required sample can be constructed as
The inverses $F\_i^\{-1\}$ are unproblematic as the $F\_i$ were assumed to be continuous. The above formula for the copula function can be rewritten to correspond to this as:
In probabilistic terms, $C:[0,1]^d\backslash rightarrow\; [0,1]$ is a d-dimensional copula if C is a joint cumulative distribution function of a d-dimensional random vector on the unit cube $[0,1]^d$ with uniform marginals.^{[2]}
In analytic terms, $C:[0,1]^d\backslash rightarrow\; [0,1]$ is a d-dimensional copula if
For instance, in the bivariate case, $C:[0,1]\backslash times[0,1]\backslash rightarrow\; [0,1]$ is a bivariate copula if $C(0,u)\; =\; C(u,0)\; =\; 0$, $C(1,u)\; =\; C(u,1)\; =\; u$ and $C(u\_2,v\_2)-C(u\_2,v\_1)-C(u\_1,v\_2)+C(u\_1,v\_1)\; \backslash geq\; 0$ for all $0\; \backslash leq\; u\_1\; \backslash leq\; u\_2\; \backslash leq\; 1$ and $0\; \backslash leq\; v\_1\; \backslash leq\; v\_2\; \backslash leq\; 1$.
Sklar's theorem,^{[3]} named after Abe Sklar, provides the theoretical foundation for the application of copulas. Sklar's theorem states that a multivariate cumulative distribution function
of a random vector $(X\_1,X\_2,\backslash dots,X\_d)$ with marginals $F\_i(x)\; =\; \backslash mathbb\{P\}[X\_i\backslash leq\; x]$ can be written as
where $C$ is a copula.
The theorem also states that, given $H$, the copula is unique on $\backslash operatorname\{Ran\}(F\_1)\backslash times\backslash cdots\backslash times\; \backslash operatorname\{Ran\}(F\_d)$, which is the cartesian product of the ranges of the marginal cdf's. This implies that the copula is unique if the marginals $F\_i$ are continuous.
The converse is also true: given a copula $C:[0,1]^d\backslash rightarrow\; [0,1]$ and margins $F\_i(x)$ then $C\backslash left(F\_1(x\_1),\backslash dots,F\_d(x\_d)\; \backslash right)$ defines a d-dimensional cumulative distribution function.
The Fréchet–Hoeffding Theorem (after Maurice René Fréchet and Wassily Hoeffding ^{[4]}) states that for any Copula $C:[0,1]^d\backslash rightarrow\; [0,1]$ and any $(u\_1,\backslash dots,u\_d)\backslash in[0,1]^d$ 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 point-wise sharp, in the sense that for fixed u, there is a copula $\backslash tilde\{C\}$ such that $\backslash tilde\{C\}(u)\; =\; W(u)$. 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
Several families of copulae have been described.
The Gaussian copula is a distribution over the unit cube $[0,1]^d$. It is constructed from a multivariate normal distribution over $\backslash mathbb\{R\}^d$ by using the probability integral transform.
For a given correlation matrix $R\backslash in\backslash mathbb\{R\}^\{d\backslash times\; d\}$, the Gaussian copula with parameter matrix $R$ can be written as
where $\backslash Phi^\{-1\}$ is the inverse cumulative distribution function of a standard normal and $\backslash Phi\_R$ is the joint cumulative distribution function of a multivariate normal distribution with mean vector zero and covariance matrix equal to the correlation matrix $R$.
The density can be written as^{[5]}
= \frac{1}{\sqrt{\det{R}}}\exp\left(-\frac{1}{2} \begin{pmatrix}\Phi^{-1}(u_1)\\ \vdots \\ \Phi^{-1}(u_d)\end{pmatrix}^T \cdot \left(R^{-1}-\mathbf{I}\right) \cdot \begin{pmatrix}\Phi^{-1}(u_1)\\ \vdots \\ \Phi^{-1}(u_d)\end{pmatrix} \right), where $\backslash mathbf\{I\}$ is the identity matrix.
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^{[6]}
where $\backslash psi\backslash !:[0,1]\backslash times\backslash Theta\; \backslash rightarrow\; [0,\backslash infty)$ is a continuous, strictly decreasing and convex function such that $\backslash psi(1;\backslash theta)=0$. $\backslash theta$ is a parameter within some parameter space $\backslash Theta$. $\backslash psi$ is the so-called generator function and $\backslash psi^$ is its pseudo-inverse defined by
Moreover, the above formula for C yields a copula for $\backslash psi^\{-1\}\backslash ,$ if and only if $\backslash psi^\{-1\}\backslash ,$ is d-monotone on $[0,\backslash infty)$.^{[7]} That is, if it is $d-2$ times differentiable and the derivatives satisfy
for all $t\backslash geq\; 0$ and $k=0,1,\backslash dots,d-2$ and $(-1)^\{d-2\}\backslash psi^\{-1,(d-2)\}(t;\backslash theta)$ is nonincreasing and convex.
The following table highlights the most prominent bivariate Archimedean copulas with their corresponding generator. Note that not all of them are completely monotone, i.e. d-monotone for all $d\backslash in\backslash mathbb\{N\}$ or d-monotone for certain $\backslash theta\; \backslash in\; \backslash Theta$ only.
When studying multivariate data, one might want to investigate the underlying copula. Suppose we have observations
from a random vector $(X\_1,X\_2,\backslash dots,X\_d)$ with continuous margins. The corresponding "true" copula observations would be
However, the marginal distribution functions $F\_i$ 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 $\backslash tilde\{U\}\_k^i=R\_k^i/n$, where $R\_k^i$ is the rank of the observation $X\_k^i$:
Therefore, the empirical copula can be seen as the empirical distribution of the rank transformed data.
In statistical applications, many problems can be formulated in the following way. One is interested in the expectation of a response function $g:\backslash mathbb\{R\}^d\backslash rightarrow\backslash mathbb\{R\}$ applied to some random vector $(X\_1,\backslash dots,X\_d)$.^{[10]} If we denote the cdf of this random vector with $H$, the quantity of interest can thus be written as
If $H$ 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
If copula and margins are known (or if they have been estimated), this expectation can be approximated through the following Monte Carlo algorithm:
In risk/portfolio management, copulas are used to perform stress-tests and robustness checks that are especially important during “downside/crisis/panic regimes” where extreme downside events may occur (i.e., the global financial crisis of 2008–2009) 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 flight-to-quality 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 traders’ 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 interesting 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 Student-t copulas) that do not allow for correlation asymmetries where correlations differ on the upside or downside regimes. However, the recent development of vine copulas (also known as pair copulas) enables the flexible modelling of the dependence structure for portfolios of large dimensions. ^{[13]} 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 Student-t copula. ^{[14]} 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 re-weighting of the probability of each scenario.^{[15]}
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).^{[16]} 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.^{[17]}^{[18]} 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 and the poor representation of extreme events.^{[19]} There have been attempts to propose models rectifying some of the copula limitations.^{[19]}^{[20]}^{[21]}
While the application of copulas in credit has gone through popularity as well as misfortune during the global financial crisis of 2008–2009,^{[22]} 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 multi-asset derivative products. The first such application outside credit was to use a copula to construct an implied basket volatility surface,^{[23]} taking into account the volatility smile of basket components. Copulas have since gained popularity in pricing and risk management ^{[24]} of options on multi-assets 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:
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,^{[25]} mechanical and offshore engineering.
Copula functions have been successfully applied to the analysis of spike counts in neuroscience. ^{[26]}
Copulas have been extensively used in climate and weather related research.^{[27]}
Large synthetic traces of vectors and stationary time series can be generated using empirical copula while preserving the entire dependence structure of small datasets.^{[28]} Such empirical traces are useful in various simulation-based performance studies.^{[29]}
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