Transcript Elliptical Distributions - Univerzita Karlova v Praze

Vadym Omelchenko
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Normal Distribution
Laplace Distribution
t-Student Distribution
Cauchy Distribution
Logistic Distribution
Symmetric Stable Laws
0  1  
 
N   , 
0

1

  

The further ρ from zero the more evident
ellipticity of the map, when observing it from
above. When ρ=0 then the map has the
spherical form.

The random vector X  ( X1,..., X n )T is said to
have an elliptical distribution with parameters
vector  (n 1) and the matrix (n  n) if its
characteristic function can be expressed as


E exp(it T X)  exp(itT μ)  (t T t ),

for some scalar function and where
t T  (t1 , t2 ,...,tn ) and Σ are given by   AAT
E(it  X)  exp(it  μ)  (t )
2


 ( t )  exp( it μ )  exp(  t )
 
 ( t )  exp( it  μ ) exp      t 2


 ()  exp     

 /2

 /2




If X has an elliptical distribution, we write
X En (, , ) where is called characteristic
generator of X and hence, the characteristic
generator of the multivariate normal is given
by  (u)  exp(u / 2).
The random vector X does not, in general,
possess a density f X ( x ) but if it does, it will
have the form
f X ( x) 
cn


g n ( x   )T  1 ( x   )

For some non-negative function g n () called
density generator and for some constant cn
called normalizing constant.

En ( , , g n ) where g n (.) is the density
X
generator assuming that g n (.) exists.
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If X En (, , ) then if the mean exists then
it will be E (X )  
If the variance matrix exists, it will be
 (0)
Cov ( X )  

t

That is, the matrix Σ coincides with the
covariance matrix up to the constant.
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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.
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
En ( , , g n ) , let B be a q  n matrix and
b  R q . Then
Let X
b  BX
Eq (b  B, BB , gq )
T
Corollary. Let X En ( , , g n ). Then X r Er (1 , 11 , g r )
X nr Enr (2 , 22 , gnr )
Hence marginal distributions of elliptical distributions are elliptical
distributions.
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If S n  X 1  X 2  ...  X n
and X  ( X 1 , X 2 ,..., X n )
L ( X )  E n (  , , g n )
T hen
L(S)  E1 (e  , e e, g1 )
T
T
T

Hence follows that the sum of elliptical
distribution is an elliptical distribution. This
property is very important when we deal with
portfolio of assets, represented by sum.
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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.
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4. X
E p (, ,  ) with rank(Σ)=k if X has
the same distribution as
 rA u
T

(k )
Where r  0 (radius ) and u
is uniformly
distributed on unit sphere surface in R n and
A is a (k×p) matrix such that
(k )
A A
T

As it was mentioned above, if the elliptically
distributed function has a density then it is of the
form:
cn
f X ( x) 
g n ( x   )T  1 ( x   )



The condition


n / 2 1
x
g n ( x)dx  

0
guarantees that g n (.) is a density generator.
(n / 2) 1
cn 
D ;
n/2
(2 )

D   x n / 21 g n ( x)dx
0
Cauchy
g n (u )  (1  u )  ( n 1) / 2
Exponentia
l power
g n (u )  exp[ r  u s ], r , s  0
Laplace
g n (u )  exp( u )
Logistic
g n (u ) 
Normal
g n (u )  exp(u / 2)
Student
 u
g n (u )  1  
 m
t
exp(u )
[1  exp(u )]2
( n  m ) / 2
,m  0
an
integer
1
g1 (u ) 
,
(1  u ) a
if

x
1 / 2 1
0

dx
g1 ( x)dx  
a
0 (1  x )

x

x
dx
a 1 / 2
 c1  
0
a  1/ 2
1
 1
 c
g1 
( x   )2   1
a
  2
  
1
2
(x  ) 
1 
2



c
1
dx
EX   x f X ( x)dx   x 1
dx

const

a
2 a 1

 
x
1

( x   )2 
1 
2


f X ( x) 
if
c1
a  1. If
2
a  (1 / 2, 1] t hen X
EX 2   x f X ( x)dx   x 2
if
a  3 / 2.
c1
has infinit e mean
1
 
1

( x   )2 
1 
2


a
dx  const
dx
x
2a 2

if a  (0.5, 1] then X has an infinite mean
if a  (1, 1.5] then X has a finite mean and infinite variance
if a  (1.5, ) then both momentsare finite

The expected shortfall (or tail conditional
expectation) is defined as follows:
TCE X ( xq )  E ( X X  xq )

and can be interpreted as the expected worse
losses.
F ( xq )  1  q



For the familiar normal distribution N(μ,  ),
with mean μ and variance  2, it was noticed
by Panjer (2002) that:
2
 1  xq    
 
  
    2

TCE X ( xq )   


 xq    
 
1  
  


Suppose that g(x) is a non-negative function
for any positive number, satisfying the
condition that:


0

g ( x)
dx  
x
Then g(x) can be a density generator of a
univariate elliptical distribution of a random
variable X E1 ( ,  2 , g )

The density of this function has the form:
2

c 1 x  
f X ( x)  g  
 
  2    




where c is a normalizing constant.
If X has an elliptical distribution then
X 
Z

Has a standard elliptical distribution
(spherical)

The distribution function of Z has the form:
z

FZ (u)  c  g 1 / 2  u
2
 du,


With mean 0 and variance equal to

 Z  2c  u g (1 / 2  u )du   (0).
0
2
2
'

Define the function G(x) which we will call
cumulative generator.
x
G ( x)  c  g (u ) du
0
G (  )  ,
Define
~
G(x)  G( ) - G(x)

Let X En ( ,  2 , g ) and G be the cumulative
generator. Under condition (*), the tail
conditional expectation of X is given by
TCEX ( xq )      ,
2


Where λ is expressed as
~ 1 2 
~ 1 2 
G zq  G zq 
2
2






FX ( xq )
FZ ( z q )
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1. For Cauchy distribution the TCE doesn’t
exist. Because it doesn’t satisfy conditions of
the theorem



1
1 ~ 1 2 
1 
 (1 / 2 ) z q2

G zq  
1

1

e

 2
2 



1
 (1 / 2 ) z q2
1
e
 ( zq )
1
1
2




2
1
1
1

(
1
/
2
)
z
2
2
q

e
  ( zq )
2
2
2
 zq 
1
1
FZ ( z q ) 

t anh
 2 

2
2




1


 ( zq )



 1



(
z
)
q


2


TCE q 
 zq 
1  t anh
 2 




Suppose X
vector of ones with dimension n. Define
En ( , , g n )
and e  (1,1,...,1)T is the
n
S n  X 1  X 2  ...  X n   X k  eT X
k 1

The TCE can be expressed as
TCE S ( sq )   s  S   S2
where  s  e T   k 1  k ,  S2  eT e   j 1,k 1
n
n
j ,k
and
1 ~ 1 2 
G  z s ,q 
S  2

S 
,
FZ ( z s ,q )
wit h z s ,q  ( sq   S ) /  S

This theorem holds as a result of convolution
properties of the elliptical distributions and
the previous theorem.
Suppose X En ( , , g n ) and e  (1,1,...,1)T is the
vector of ones with dimension n, and
n
S n  X 1  X 2  ...  X n   X k  eT X
k 1


Then the contribution of X k ,1  k  n to the
overall risk can be expressed as:
TCE X k S ( sq )   k  S   k S  k , S ,
for
where
k  1,2,...,n
 k ,S 
 k ,S
 k S



All elliptical distributions belong to this
family.
All stable distributions belong to this family.
The density of the skewd Normal Distribution
has a form:
f ( x)  2 ( x)   (  x)
 (.) is density function of normal law
 (.) is the distribution function of normal law
 is the skewness parameter
λ  0  positivelyskewed
λ  0  negatively skewed
λ  0  symmetric
Skewed Elliptic
Elliptic
Distributions
Skewed
stable
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1. TAIL CONDITIONAL EXPECTATIONS FOR
ELLIPTICAL
DISTRIBUTIONS
Zinoviy M. Landsman* and Emiliano 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