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**1. Introduction**

Copula is the joint distribution function of a collection of random variables U_{1}, ..., U_{d} such that each of then is uniformly distributed on [0,1]. Even though the marginal distributions are fixed, the copula can take a variety of forms because variables U_{1}, ..., U_{d} may have strong codependence or no codependence at all, they may be connected in a continuous or discrete fashion, they may exhibit stronger codependence in the tails or stronger dependence in the middle of the distribution. By itself copulas would represent little interest if not for the following result.

**Theorem (Sklar)**: Let X_{1}, ..., X_{d} be random variables with any marginal distribution functions F_{1}(x), ..., F_{d}(x). Then H(x_{1},...,x_{d}) is the joint distribution function of X_{1}, ..., X_{d} if and only if there exists a copula C(u_{1},...,u_{d}) such that

If X

This result makes copulas popular in modeling correlated phenomena because they provide a nice

Or the context of copula usage may be somewhat less polished. Rarely does the researcher sit at the table at the beginning of the modeling process and plans all the stages ahead, in a careful and consistent manner. Research groups in the industry have notoriously little human resources to address all the aspects of the problem most accurately even if exploiting the know-how in the public domain (let alone any proprietary efforts). If some parts of the system have already been modeled by other desks or outside vendors, it may be a blessing to just use their infrastructure, tweaking it a bit perhaps. The infrastructure can be combined with original research of the remaining parts of the system. All in one copula.

In other cases, there are very good reasons for using the output of other professionals, because they may be true experts in certain segments which are only components in your product. Or they may have access to the information that you do not have. So it is best to absorb their modeling insights at the fullest and combine with whatever you have managed. Say, an investment bank needs to price an interest-rates-commodity hybrid. This a derivative which is sensitive to movements in the interest rates term structure as well as the future price of a certain commodity. Separately, the IRP desk has already modeled the interest rate term structure quite well. And their traders tune the parameters daily. Separately, the commodities desk has already modeled the price dynamics of the commodity. And their traders tune the parameters several times a day. All that know-how can and must be combined, in one hybrid model. A copula is not the only way to do it (and often not the best one) but in many situations it is a competitive method, especially when considering speed and numerical stability of the implementation.

For all the reasons mentioned above, copulas have gained prominence in actuarial science, financial derivatives pricing, engineering and bioinformatics... Since codependence of

respectively. If random variables X

Copulas are directly related to two prominent measures of nonlinear association (nonlinear dependence) between two variables. If random variables X and Y have continuous marginal distributions and copula C(u,v), then their Kendall's tau and Spearman's rho are given by

When working with copulas, simplicity and numerical tractability may prove to be important. For that reason the so-called

where function

There are many well-researched families of copulas, with different properties and character. Below I am listing the most popular ones. As you will notice, they are not ideal, quite detached from reality, in fact. All of them have symmetric nature in the sense that each two random variables have the same shape of the joint distribution, albeit governed by different parameters (generally speaking). Nonetheless, the listed copulas are utilized frequently in statistical modeling because they are relatively transparent and computationally tractable. Whenever researchers can afford to look into more complex structures, they play with copulas where the conditional distribution of X

1]

where

2]

where

3]

This is an Archimedean copula with generator

4]

This is an Archimedean copula with generator

5]

This is an Archimedean copula with generator

6]

Let

- Simulate
_{}independent random variables_{}from U([0,1]). This gives us one random variable per each non-empty subset of_{} - Set
- Set
- Set

The Marshall-Olkin copula arises naturally in the study of systems reliability. Consider a d-component system where each non-empty subset of components receives a fatal shock according to an independent Poisson process with intensity

All the aforementioned copula families have the property that any two variables are characterized by a bivariate copula from the same copula family. Therefore, without any ambiguity we can summarize the relevant bivariate association measures in the table below. To remind the reader, the most important bivariate association measures are Kendall's tau, Spearman's rho, lower tail dependence and upper tail dependence.

COPULA |
PLOTS |
||||

Gaussian | 0 | 0 | 2D 3D | ||

Student | 2D 2D 3D | ||||

Gumbel-Hougaard | 0 | 2D 3D | |||

Clayton | 0 | 2D 3D | |||

Frank | 0 | 0 | 2D 3D | ||

Marshall-Olkin | 0 | 2D 2D 3D |

Nelsen, R. B. (2006). An Introduction to Copulas (2nd ed). New York: Springer.

Salvadori, G., De Michele, C., Kottegoda, N. T., & Rosso, R. (2007). Extremes in Nature: An Approach Using Copulas. Springer.

Embrechts, P., Lindskog, F., & McNeil, E. J. (2001). Modelling Dependence With Copulas and Applications to Risk Management. Working paper.

Jaworski, P., Durante, F., Härdle, W.K., & Rychlik, T. (2010). Copula Theory and Its Applications. Proceedings of the Workshop Held in Warsaw, 25-26 September 2009.

Cherubini, U., Gobbi, F., Mulinacci, S., & Romagnoli, S. (2011). Dynamic Copula Methods in Finance. Wiley.

Joe, H. (2014). Dependence Modeling with Copulas. Chapman & Hall / CRC. -

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