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Copula in Statistics

Copula is a tool used to describe the dependence between random variables in statistics and probability theory. Copulas got its name from linguistics. Random variable is a variable whose value results from a measurement on some type of random process. It is a function from a probability space to measurable real numbers. It is possible to write the cumulative distribution function of a random vector as distribution functions and a copula. The marginal components of each component of random vector can be described by marginal distribution functions. The dependence structure between the components of random vector is described by copula. Copulas help to model and calculate the distribution of random vectors very easily. So these tools are very popular in all statistical applications. A lot of parametric copula families are available. They usually have parameters that control the strength of dependence.

The basic Idea

Consider two random variables X & Y with cumulative distribution functions FX and FY. To these variables, probability integral transform has to be applied separately to define U = FX (X) and V = FY (Y). U & V have uniform distributions. But in general U & V are dependant if X & Y are dependant. The transformations are invertible. A copula is a multivariate joint distribution. It is based on n-dimensional unit cube in such a way that every marginal distribution is uniform on the interval.

C: [0, 1] n→ [0, 1] is an n-dimensional copula if and only if

• C (u) =  0 ; when u∈[0,1]n has at least one component equal to zero.
• C (u) = ui  when u∈[0,1]n has all components equal to one, except the ith one
• C is n : that is increasing for each hyper rectangle .

Applications of Copula

Copula functions are commonly used in financial risk assessment and actuarial analysis. Dependence modeling is done for a variety of applications using the copula functions. Copula is applied in credit and it has gained popularity. Copulas are very useful in pricing Collateralized debt Obligations (CDOs). Other asset classes also use copula as a tool to evaluate multi-asset derivative products. Copula is also used to construct implied basket volatility surface. Copula is very famous in pricing and risk management of many options in multi- assets. The pricing and risk management are done in the presence of volatility smile, in equity, foreign exchange and fixed income derivative business. Analyzing and pricing spread option is a best example of copula application.

Questions:

• What is copula?
• What are the major applications of copula?

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