Elliptical Distributions24

8/18/2019 Elliptical Distributions24

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Normal Distribution

Laplace Distribution t-Student Distribution Cauchy Distribution Logistic Distribution Symmetric Stable Laws

Examples of the Elliptical Distributions


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Bivariate NormalDistribution

1

1,0

0 ρ

ρ N


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The further from !ero the more e"ident

ellipticity of the map# when obser"ing itfrom abo"e$ %hen &' then the map hasthe spherical form$


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Laplace bivariatedistribution


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The random "ector is said toha"e an elliptical distribution withparameters "ector and the matri(

if its characteristic function can bee(pressed as

for some scalar function and whereand ) are gi"en by

Denition of ellipticaldistributions

T n X X X ),...,( 1=

)1( ×n µ )( nn×Σ

[ ] ),tt()μtexp()Xtexp( Σ⋅= T T T ii E ψ ψ

),...,,(t 21 nT

t t t = T

AA=Σ


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!haracteristic "unction of the#$mmetric #table Distributions

)texp()μtexp()t( α α σ φ

−⋅= i

)()μtexp()Xt( 2t ii E ψ ⋅⋅=⋅

[ ]

⋅−⋅= 2/

2texp)μtexp()t(α

α σ φ i

[ ]( )2/exp)( α α σ ψ ⋅⋅−=⋅


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*f + has an elliptical distribution# we write+ , , , where is called

characteristic generator of + and hence# thecharacteristic generator of the multi"ariatenormal is gi"en by

The random "ector + does not# in general#

possess a density but if it does# itwill ha"e the form

or some non-negati"e function calleddensity generator and for some constantcalled normali!ing constant$

),,( ψ µ Σn E ψ

).2/exp()( uu −=ψ

)x( X f

[ ])()()( 1 µ µ −Σ−Σ

= − x x g c

x f T nn

X

)(⋅n g nc


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+ , where is thedensity generator assuming thate(ists$

%lternative Denotin& of theElliptical Distributions

),,( nn g E Σ µ (.)n g (.)n g


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.(amples of the distributions that don/tha"e mean nor "ariance0

1ll stable distributions whose inde( ofstability is lower than 2# e$g$ Cauchy orLe"y$

Mean and !ovariance 'roperties


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Let + , # let 3 be a matri( and $ Then

,

Corollary$ Let + , $ Then ,

,

4ence marginal distributions of elliptical distributions are elliptical distributions$

Mean and !ovariance 'roperties

),,( nn g E Σ µ nq×q Rb∈),,( q

T

q g B B Bb E Σ+ µ BX b +

),,( nn g E Σ µ r X ),,( 111 r r g E Σ µ r n X − ),,( 222 r nr n g E −− Σ µ


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4ence follows that the sum of elliptical

distribution is an elliptical distribution$ Thisproperty is "ery important when we dealwith portfolio of assets# represented bysum$

!onvolutional 'roperties


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2$ .lliptical distributions can be seen as ane(tension of the Normal distribution

5$ 1ny linear combination of ellipticaldistributions is an elliptical distribution

6$ 7ero correlation of two normal "ariables

implies independence only for Normaldistribution$ This implication does not holdfor any other elliptical distribution$

Basic 'roperties of the EllipticalDistributions


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8$ + , with rank9):&k if + hasthe same distribution as

%here 9radius : and is uniformlydistributed on unit sphere surface in and

1 is a 9k;p: matri( such that

Basic 'roperties of the EllipticalDistributions

),,( φ µ Σ p E

)(k T u Ar ⋅+

µ 0≥r )(k u

n R

Σ=⋅ A AT


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!onstruction of a densit$function with innite variance

.2/if

%on$t

)(2

11

1)(

eaninfiniteha$then"1,2/1(If .1if

%on$t

)(2

11

1)(

)(2

11

1)(

2

1)(

2/1if

)1()(,

)1(

1)(

22

2

122

2

12

2

1

2

12

11

1

0

2/1

0

1

0

12/11

>

∞

∞

∞


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The e(pected shortfall 9or tail conditionale(pectation: is de


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or the familiar normal distribution N(μ, ), with mean μ and "ariance # it was noticed

by =an>er 95''5: that0

Expected #hortfall

2σ 2σ

2

1

1

)( σ

σ

µ

σ µ ϕ

σ µ

−Φ−

−

+=q

q

q X x

x

xTCE


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Suppose that g9(: is a non-negati"e functionfor any positi"e number# satisfying thecondition that0

Then g9(: can be a density generator of a

uni"ariate elliptical distribution of a random"ariable + ,

(enerali)ation of the'revious "ormula

∞


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The density of this function has the form0

where c is a normali!ing constant$ *f + has an elliptical distribution then

4as a standard elliptical distribution9spherical:

(enerali)ation of the "ormulafor the Normal Law

−=

2

2

1)(

σ

µ

σ

x g

c x f X

σ

µ −= X

!


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The distribution function of 7 has the form0

%ith mean ' and "ariance e?ual to

(enerali)ation of the "ormulafor the Normal Law

( ) ,duu2/1)( 2∫ ∞−

⋅= "

! g cu

).0()2/1(2 -

0

22ψ σ =⋅= ∫

∞

duu g uc !


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De


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Let + , and @ be the cumulati"egenerator$ Ander condition 9B:# the tailconditional e(pectation of + is gi"en by

%here is e(pressed as

*heorem +

),,( 2 g E n σ µ

,)( 2σ λ µ ⋅+=q X xTCE

)(2

1

)(2

1 22

q !

q

q X

q

"

" #

x

" #

=

=λ


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2$ or Cauchy distribution the TC. doesn/te(ist$ 3ecause it doesn/t satisfy conditionsof the theorem

Examples


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Lo&istic Distribution

( )

+

+

=

+=

+=

+=

=

+−=

−

−

−−

2tanh1

)(

2

1

)(1

2tanh

21

21)(

)(2

1

)(1

2

1

2

1

2

12

1

2

1

112

1

2

101

2

2

2

)2/1(

)2/1(

1)2/1(2

q

q

q

q

q

q !

q

q

"

"

"

q

"

"

"

TCE

" "

"

"

e

e

e " #

q

q

q

ϕ

π

ϕ σ

ϕ π

ϕ σ

π π

π

σ

σ σ


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Suppose + , is the"ector of ones with dimension n$ De


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The TC. can be e(pressed as

This theorem holds as a result ofcon"olution properties of the ellipticaldistributions and the pre"ious theorem$

*heorem

S S qq s

q s !

q s

S S

n

k $ k $

T

S

n

k k s

S S sqS

s "

"

" #

sTCE

σ µ

σ λ

σ σ µ µ µ

σ λ µ

/)(&ith

,)(

2

11

and

ee,e&here

)(

,

,

2

,

1,1 ,

2

1

T

2

−=

=

=Σ===

⋅+=

∑∑ ===


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Suppose + , is the"ector of ones with dimension n# and

Then the contribution of to theo"erall risk can be e(pressed as0

#ums of Elliptical ,is-s

T

nn g E )1,...,1,1(eand),,( =Σ µ

Xe...1

21

T n

k

k nn X X X X S ==+++= ∑=nk X k ≤≤1,

S k

S k

S k

S k S k S k qS X

nk

sTCE k

σ σ

σ ρ

ρ σ σ λ µ

,

,

,

&here

,...,2,1for

,)(

=

=⋅+=


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Skewed

#tableDistributions


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2$ *%/L !ND/*/N%L E1'E!*%*/N#", ELL/'*/!%L

D/#*,/B2*/N# 7ino"iy $ LandsmanB and .miliano 1$

Valde!E 5$ C1= and Option =ricing with .lliptical

Distributions# 4amada # Valde!$

6$ 4andbook of 4ea"y Tailed Distributions ininance# .ds S$T$ Fache"

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Elliptical Distributions24

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