Gaussian_copula_model

In statistics, a copula is a multivariate joint distribution defined on the n-dimensional unit cube [0, 1]ⁿ such that every marginal distribution is uniform on the interval [0, 1].

Specifically, $C:[0,1]^n\backslash to\; [0,1]$ is an n-dimensional copula (briefly, n-copula) if:

$C\backslash left(\backslash mathbf\; u\backslash right)=0$ whenever $\backslash mathbf\; u\backslash in\; [0,1]^n$ has at least one component equal to $0;$

$C\backslash left(\backslash mathbf\; u\backslash right)=u\_i$ whenever $\backslash mathbf\; u\backslash in\; [0,1]^n$ has all the components equal to $1$ except the i-th one, which is equal to $u\_i;$

$C\backslash left(\backslash mathbf\; u\backslash right)$ is n-increasing, i.e., for each $B=\backslash times\_\{i=1\}^\{n\}[x\_i,y\_i]\backslash subseteq\; [0,1]^n;$

$V\_\{C\}\backslash left(\; B\backslash right):=\backslash sum\_\{\backslash mathbf\; z\backslash in\; \backslash times\_\{i=1\}^\{n\}\backslash \{x\_i,y\_i\backslash \}\}\; (-1)^\{N(\backslash mathbf\; z)\}\; C(\backslash mathbf\; z)\backslash ge\; 0;$

where the $N(\backslash mathbf\; z)=\backslash operatorname\{card\}\backslash \{k\backslash mid\; z\_k=x\_k\backslash \}$.

Contents 1 Sklar\'s theorem 2 Fréchet-Hoeffding copula boundaries 3 Gaussian copula 4 Archimedean copulas 5 Applications 6 References 6.1 Notes 6.2 General 7 External links

Sklar\'s theorem

The theorem proposed by Sklar Sklar (1959) underlies most applications of the copula. Sklar\'s theorem states that given a joint distribution function H for p variables, and respective marginal distribution functions, there exists a copula C such that the copula binds the margins to give the joint distribution.

For the bivariate case, Sklar\'s theorem can be stated as follows. For any bivariate distribution function H(x, y), let F(x) = H(x, (−∞,∞)) and G(y) = H((−∞,∞), y) be the univariate marginal probability distribution functions. Then there exists a copula C such that

$H(x,y)=C(F(x),G(y))\backslash ,$

(where we have identified the distribution C with its cumulative distribution function). Moreover, if marginal distributions, say, F(x) and G(y), are continuous, the copula function C is unique. Otherwise, the copula C is unique on the range of values of the marginal distributions.

Fréchet-Hoeffding copula boundaries

Graphs of the Fréchet-Hoeffding copula limits and of the independence copula (in the middle).

Minimum copula: This is the lower bound for all copulas. In the bivariate case only, it represents perfect negative dependence between variates.

$W(u,v)\; =\; \backslash max(0,u+v-1).\backslash ,$

For $n$-variate copulas, the lower bound is given by

$W(u\_1,\backslash ldots,u\_n) :=\; \backslash max\backslash left\backslash \{1-n+\backslash sum\backslash limits\_\{i=1\}^n\; \{u\_i\}\; ,\; 0\; \backslash right\backslash \}\; \backslash leq\; C(u\_1,\backslash ldots,u\_n).$

Maximum copula: This is the upper bound for all copulas. It represents perfect positive dependence between variates:

$M(u,v)\; =\; \backslash min(u,v).\backslash ,$

For $n$-variate copulas, the upper bound is given by

$C(u\_1,\backslash ldots,u\_n)\backslash le\; \backslash min\_\{j\; \backslash in\; \backslash \{1,\backslash ldots,n\backslash \}\}\; u\_j\; =:\; M(u\_1,\backslash ldots,u\_n).$

Conclusion: For all copulas $C(u,v)$,

$W(u,v)\; \backslash le\; C(u,v)\; \backslash le\; M(u,v).$

In the multivariate case, the corresponding inequality is

$W(u\_1,\backslash ldots,u\_n)\; \backslash le\; C(u\_1,\backslash ldots,u\_n)\; \backslash le\; M(u\_1,\backslash ldots,u\_n).$

Gaussian copula

Cumulative distribution and probability density functions of Gaussian copula with $\backslash rho=0.4$

One example of a copula often used for modelling in finance is the Gaussian Copula, which is constructed from the bivariate normal distribution via Sklar\'s theorem. For X and Y distributed as standard bivariate normal with correlation ρ the Gaussian copula function is

$C\_\backslash rho(u,v)\; =\; \backslash Phi\_\{U,V,\; \backslash rho\}\; \backslash left(\backslash Phi^\{-1\}(u),\; \backslash Phi^\{-1\}(v)\; \backslash right)$

where the marginals of U and V are N(0,1) distributions and Φ denotes the cumulative normal density. Differentiating this yields

$c\_\backslash rho(u,v)\; =\; \backslash frac\{\backslash phi\_\{U,V,\; \backslash rho\}\; (\backslash Phi\_X^\{-1\}(u),\; \backslash Phi\_Y^\{-1\}(v)\; )\}$

{\phi(\Phi^{-1}(u)) \phi(\Phi^{-1}(v))}

where

$\backslash phi\_\{X,Y,\; \backslash rho\}(x,y)\; =\; \backslash frac\{1\}\{2\; \backslash pi\backslash sqrt\{1-\backslash rho^2\}\}\; \backslash exp\; \backslash left\; (-$

\frac{1}{2(1-\rho^2)} \left [{x^2+y^2} -2\rho xy \right ] \right )

is the density function for the bivariate normal variate with Pearson\'s product moment correlation coefficient $\backslash rho$, φ is the density of the N(0,1) distribution (the marginal density).

Archimedean copulas

One particularly simple form of a copula is

$H(x,y)\; =\; \backslash Psi^\{-1\}(\backslash Psi(F(x))+\backslash Psi(G(y)))\backslash ,$

where $\backslash Psi$ is known as a generator function. Such copulas are known as Archimedean. Any generator function which satisfies the properties below is the basis for a valid copula:

(x) > 0.

Product copula: Also called the independent copula, this copula has no dependence between variates. Its density function is unity everywhere.

$\backslash Psi(x)\; =\; -\backslash ln(x);\; \backslash qquad\; H(x,y)\; =\; F(x)G(y).$

Where the generator function is indexed by a parameter, a whole family of copulas may be Archimedean. For example:

Clayton copula:

$\backslash Psi(x)\; =\; x^\{\backslash theta\}\; -1;\backslash qquad\; \backslash theta\; \backslash le\; 0;\; \backslash qquad\; H(x,y)\; =\; (F(x)^\backslash theta+G(y)^\backslash theta-1)^\{1/\backslash theta\}.$

For θ = 0 in the Clayton copula, the random variables are statistically independent. The generator function approach can be extended to create multivariate copulas, by simply including more additive terms.

Applications

Copulas are used in the pricing of collateralized debt obligations.

References

Notes

General

• David G. Clayton (1978), "A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence", Biometrika 65, 141-151. JSTOR (subscription)
• Frees, E.W., Valdez, E.A. (1998), "Understanding Relationships Using Copulas", North American Actuarial Journal 2, 1-25. Link to NAAJ copy
• Roger B. Nelsen (1999), An Introduction to Copulas. ISBN 0-387-98623-5.
• S. Rachev, C. Menn, F. Fabozzi (2005), Fat-Tailed and Skewed Asset Return Distributions. ISBN 0-471-71886-6.
• A. Sklar (1959), "Fonctions de répartition à n dimensions et leurs marges", Publications de l\'Institut de Statistique de L\'Université de Paris 8, 229-231.
• W.T. Shaw, K.T.A. Lee (2006), "Copula Methods vs Canonical Multivariate Distributions: The Multivariate Student T Distibution with General Degrees of Freedom". PDF

External links

• MathWorld Eric W. Weisstein. "Sklar\'s Theorem." From MathWorld--A Wolfram Web Resource

• Copula Wiki: community portal for researchers with interest in copulas

• FinancialMathematics.com Matlab code for copula estimation and simulation.

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