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
Characteristic Function of the Symmetric Stable Distributions
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