Elliptical Distributions

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# Elliptical Distributions - PowerPoint PPT Presentation

Elliptical Distributions. Vadym Omelchenko. Examples of the Elliptical Distributions. Normal Distribution Laplace Distribution t-Student Distribution Cauchy Distribution Logistic Distribution Symmetric Stable Laws. Examples of the Elliptical Distributions.

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### EllipticalDistributions

Examples of the Elliptical Distributions
• Normal Distribution
• Laplace Distribution
• t-Student Distribution
• Cauchy Distribution
• Logistic Distribution
• Symmetric Stable Laws

Thefurtherρfromzerothe more evidentellipticity of the map, when observing it from above. When ρ=0 then the map has the spherical form.

Definition of elliptical distributions
• The random vector is said to have an elliptical distribution with parameters vector and the matrix if its characteristic function can be expressed as
• for some scalar function and where

and Σ are given by

If X has an elliptical distribution, we write X ̴̴̴ where is called characteristic generator of X and hence, the characteristic generator of the multivariate normal is given by

• The random vector X does not, in general, possess a density but if it does, it will have the form

For some non-negative function called density generator and for some constant called normalizing constant.

Alternative Denoting of the Elliptical Distributions
• X ̴ where is the density generator assuming that exists.
Mean and Covariance Properties
• If X ̴̴̴ then if the mean exists then it will be
• If the variance matrix exists, it will be
• That is, the matrix Σ coincides with the covariance matrix up to the constant.
Mean and Covariance Properties
• Examples of the distributions that don’t have mean nor variance:
• All stable distributions whose index of stability is lower than 1, e.g. Cauchy or Levy.
Mean and Covariance Properties
• Let X ̴ , let B be a matrix and
• . Then

̴

Corollary. Let X ̴ . Then ̴

̴

Hence marginal distributions of elliptical distributions are elliptical distributions.

Convolutional Properties
• Hence followsthatthe sum ofelliptical distribution is an elliptical distribution. This property is very important when we deal with portfolio of assets, represented by sum.
Basic Properties of the Elliptical Distributions
• 1. Elliptical distributions can be seen as an extension of the Normal distribution
• 2. Any linear combination of elliptical distributions is an elliptical distribution
• 3. Zero correlation of two normal variables implies independence only for Normal distribution. This implication does not hold for any other elliptical distribution.
Basic Properties of the Elliptical Distributions
• 4. X ̴ with rank(Σ)=k if X has the same distribution as
• Where (radius ) and is uniformly distributed on unit sphere surface in and A is a (k×p) matrix such that
Basic Properties of the Elliptical Distributions
• As it was mentioned above, if the elliptically distributed function has a density then it is of the form:
• The condition

guarantees that is a density generator.

Financial ApplicationExpected Shortfall
• The expected shortfall (or tail conditional expectation) is defined as follows:
• and can be interpreted as the expected worse losses.
Expected Shortfall
• For the familiar normal distribution N(μ, ),
• with mean μ and variance , it was noticed by Panjer (2002) that:
Generalization of the Previous Formula
• Suppose that g(x) is a non-negative function for any positive number, satisfying the condition that:
• Then g(x) can be a density generator of a univariate elliptical distribution of a randomvariable X ̴
Generalization of the Formula for the Normal Law
• The density of this function has the form:
• where c is a normalizing constant.
• If X has an elliptical distribution then
• Has a standard elliptical distribution (spherical)
Generalization of the Formula for the Normal Law
• The distribution function of Z has the form:
• With mean 0 and variance equal to
Generalization of the Formula for the Normal Law
• Define the function G(x) which we will call cumulative generator.
Theorem 1
• Let X ̴ and G be the cumulative generator. Under condition (*), the tail conditional expectation of X is given by
• Where λ is expressed as
Examples
• 1. For Cauchy distribution the TCE doesn’t exist. Because it doesn’t satisfy conditions of the theorem
Sums of Elliptical Risks
• Suppose X ̴ is the vector of ones with dimension n. Define
Theorem 2
• The TCE can be expressed as
• This theorem holds as a result of convolution properties of the elliptical distributions and the previous theorem.
Sums of Elliptical Risks

Suppose X ̴ is the vector of ones with dimension n, and

• Then the contribution of to the overall risk can be expressed as:
Skewed Elliptical Distributions
• All elliptical distributions belong to this family.
• All stable distributions belong to this family.
• The density of the skewd Normal Distribution has a form:
Literatura
• 1. TAIL CONDITIONAL EXPECTATIONS FOR ELLIPTICAL
• DISTRIBUTIONS
• Zinoviy M. Landsman* andEmiliano A. Valdez†
• 2. CAPM and Option Pricing with Elliptical Distributions, Hamada M, Valdez.
• 3. Handbook of Heavy Tailed Distributions in Finance, Eds S.T. Rachev