Elliptical Distributions

1. EllipticalDistributions VadymOmelchenko

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

3. Examples of the Elliptical Distributions

4. Examples of the Elliptical Distributions

5. Bivariate Normal Distribution

6. Multivariate Normal Distribution with correlation equal to 0.7

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

12. Laplace bivariate distribution

13. 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

14. Characteristic Function of the Symmetric Stable Distributions

15. 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.

16. Alternative Denoting of the Elliptical Distributions • X ̴ where is the density generator assuming that exists.

17. 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.

18. 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.

19. 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.

20. Convolutional Properties

21. 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.

22. 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.

23. 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

24. 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.

25. Examples of density generators

26. Construction of a density function with infinite variance

27. Construction of a density function with infinite variance

28. Financial ApplicationExpected Shortfall • The expected shortfall (or tail conditional expectation) is defined as follows: • and can be interpreted as the expected worse losses.

29. Expected Shortfall • For the familiar normal distribution N(μ, ), • with mean μ and variance , it was noticed by Panjer (2002) that:

30. 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 ̴

31. 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)

32. Generalization of the Formula for the Normal Law • The distribution function of Z has the form: • With mean 0 and variance equal to

33. Generalization of the Formula for the Normal Law • Define the function G(x) which we will call cumulative generator.

34. 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

35. Examples • 1. For Cauchy distribution the TCE doesn’t exist. Because it doesn’t satisfy conditions of the theorem

36. Logistic Distribution

37. Sums of Elliptical Risks • Suppose X ̴ is the vector of ones with dimension n. Define

38. 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.

39. 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:

40. 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:

41. Stable Distributions

42. 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