Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
Simulate
α-Stable
Random Variable and Estimate
Stable Parameters Based on Market Data
Author: Yiyang Yang
Advisor: Pr. Xiaolin Li, Pr. Zari Rachev
Department of Applied Mathematics and Statistics
State University of New York at Stony Brook
August 2012
Yiyang Yang
Simulate α-Stable Random Variable
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
Denition
Characteristic Function
Outline
1
Denition of
α-Stable
Random Variable
Denition
Characteristic Function
2
Weron's Algorithm for Generating Stable Random Variable
Properties of Stable Law
Integral Form of Stable Law
Generation of Stable Random Variable
3
Estimation of Stable Distribution Parameters
The Method of Moments
Time Regression Method
Yiyang Yang
Simulate α-Stable Random Variable
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
Denition
Characteristic Function
Denition
Given independent and identically distributed random variables
X1 , X2 , · · · , X and X , then random variable X is said to follow an
α-Stable distribution if there exists a positive constant C and a real
number D such that the following relation holds:
n
n
n
X1 + X2 + · · · + X = C X + D
n
n
n
1
where = denotes equality in distribution and constant C = n α
determines the stability property.
n
When
α = 2,
it is the Gaussian case; when 0
< α < 2,
have the non-Gaussian case.
There are three special cases with a closed form p.d.f.
the Gaussian case (α = 2)
the Cauchy case (α = 1, β = 0)
the Lévy case α = 21 , β = ±1
Yiyang Yang
Simulate α-Stable Random Variable
we
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
Denition
Characteristic Function
α-Stable distribution does not have closed form density function and
is expressed by characteristic function (form A):
φstable (t ; α, σ, β, µ) = E e itX
(
=
where
exp i µt − |σ t |α 1 − i β (signt ) tan πα
2 exp i µt − σ |t | 1 + i β π2 (signt ) ln |t |


 1,
signt =
0,


− 1,
t
t
t
α 6= 1
α=1
>0
=0
<0
Four related parameters are:
α: the index of stability or the shape parameter, α ∈ (0, 2)
β : the skewness parameter, β ∈ [−1, 1]
σ : the scale parameter, σ ∈ (0, +∞)
µ: the location parameter, µ ∈ (−∞, +∞)
Yiyang Yang
Simulate α-Stable Random Variable
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
Properties of Stable Law
Integral Form of Stable Law
Generation of Stable Random Variable
Outline
1
Denition of
α-Stable
Random Variable
Denition
Characteristic Function
2
Weron's Algorithm for Generating Stable Random Variable
Properties of Stable Law
Integral Form of Stable Law
Generation of Stable Random Variable
3
Estimation of Stable Distribution Parameters
The Method of Moments
Time Regression Method
Yiyang Yang
Simulate α-Stable Random Variable
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
Properties of Stable Law
Integral Form of Stable Law
Generation of Stable Random Variable
By transforming β, σ , there is an equivalent denition of α-stable
distribution and characteristic function.
Denition
A random variable X is α-stable with parameters (α, β2 , σ2 , µ) if and
only if its characteristic function (form B) is given by
ln φ (t ) =
(
i
µt − σ2α |t |α exp −i β2 (signt ) π2 K (α)
π
i µt − σ2 |t |
2 + i β2 (signt ) ln |t |
where
K
(α) = α − 1 + sign (1 − α) =
(
α
α−2
α 6= 1
α=1
α<1
;
α>1
and for α 6= 1
tan β2
π K (α)
2
= β tan
πα
2
πα 21α
, σ2 = σ 1 + β 2 tan2
;
2
and for α = 1
β2 = β, σ2 =
Yiyang Yang
2
π
σ.
Simulate α-Stable Random Variable
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
Properties of Stable Law
Integral Form of Stable Law
Generation of Stable Random Variable
Corollary
Any two admissible parameter quadruples (α, β, σ, µ) and (α, β, σ 0 , µ0 )
uniquely determine real numbers a > 0 and b such that
X
d
(α, β, σ, µ) = aX α, β, σ 0 , µ0 + b
where
σ
a =
,
σ0
(
b
=
µ − µ0 σσ0
µ − µ0 σσ0 + σβ π2 ln
σ
σ0
α 6= 1
.
α=1
X (α, β, 1, 0) and transform it to the general case
X (α, β, σ, µ) = aX (α, β, 1, 0) + b with
Consider the standard case
a = σ, b =
(
µ
µ + σβ π2
ln
σ
α 6= 1
.
α=1
There are three formulas regarding probability density function, cumulative
density function and characteristic function
f (−x , α, β) = f (x , α, −β)
F (−x , α, β) = 1 − F (x , α, −β)
φ (−t , α, β) = φ (t , α, −β)
Yiyang Yang
Simulate α-Stable Random Variable
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
Properties of Stable Law
Integral Form of Stable Law
Generation of Stable Random Variable
Assume f (x , α, β) and φ (t , α, β) are the density function and characteristic
function (form B) of stable random variable X (α, β, 1, 0), then according to
the inversion formula of characteristic function
f (x , α, β) =
=
1
ˆ
∞
0
π
2
1
ˆ
π
2
∞
e −itx φ (t , α, β) dt
−∞
e −itx φ (t , α, β) dt +
ˆ
∞
0
e itx φ (−t , α, β) dt
.
Since e −itx φ (t , α, β) = e itx φ (−t , α, β), then
f (x , α, β) = Re
1
=
1
π
Re
ˆ
π
∞
0
ˆ
∞
0
e −itx φ (t , α, β) dt
e itx φ (−t , α, β) dt = Re
1
ˆ
π
∞
0
e itx φ (t , α, −β) dt .
Finally, the integral form of the density function is
f (x , α, β) = Re
1
ˆ
∞
0
π
in the case α 6= 1, and
f (x , 1, β) = Re
1
π
in the case α = 1.
exp
ˆ
0
π
−itx − t α exp −i β K (α) dt
2
∞
exp
Yiyang Yang
−itx −
π
2
t − i β t ln t dt
Simulate α-Stable Random Variable
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
Properties of Stable Law
Integral Form of Stable Law
Generation of Stable Random Variable
Theorem
Let
π K (α)
,
(α) = sign (1 − α) , γ0 = − β2
2
α
1
K (α)
C (α, β2 ) = 1 − 1 + β
(1 + (α)) ,
4
α
α
1−α cos (γ − α (γ − γ0 ))
sin α (γ − γ0 )
Uα (γ, γ0 ) =
cos γ
cos γ
π
+ β2 γ
1
π
U1 (γ, β2 ) = 2
exp
+ β2 γ tan γ ,
cos γ
β2 2
then the cumulative function of a standard stable distribution can be written as:
F (x , α, β2 ) = C (α, β2 ) +
F (x , 1, β2 ) =
1
π
ˆ
(α)
π
π
2
−π
2
exp
ˆ
π
2
exp
γ0
− exp −
Yiyang Yang
α
−x 1−α Uα (γ, γ0 ) d γ, α 6= 1,
x
β2
x >0
U1 (γ, β2 ) d γ,
α = 1, β2 > 0.
Simulate α-Stable Random Variable
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
Properties of Stable Law
Integral Form of Stable Law
Generation of Stable Random Variable
Theorem
Let γ0 and Uα (γ, γ0 ) be dened as above, for α 6= 1,
Sα (1, β2 , 0) random variable i for x > 0
1
π
ˆ
π
2
γ0
exp −
α−1
x α Uα
(γ, γ0 )
(
d
(0 < X ≤ x )
P (X ≥ x )
P
γ=
γ0 < γ < π2 ,
X is a
α<1
.
α>1
Proof:
0
<α<1
F (x , α, β2 ) = P (X
≤ x) =
=
given that for
1
<α<2
α < 1,
1
1−β2 =
2
F (x , α, β2 ) = P (X
1
− β2
2
− β2
2
P (X
+
1
ˆ
π
π
2
exp
γ0
α−1
−x α Uα (γ, γ0 ) d γ
+ P (0 < X ≤ x )
≤ 0).
≤ x) = 1 −
1
π
ˆ
π
2
γ0
exp
−x
α−1
α
Uα (γ, γ0 ) d γ
= 1 − P (X ≥ x ) .
This completes the proof.
Yiyang Yang
Simulate α-Stable Random Variable
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
Properties of Stable Law
Integral Form of Stable Law
Generation of Stable Random Variable
Theorem
Let γ0 be dened as above, γ be uniformly distributed on − π2 , π2 and
W be an independent exponential randome variable with mean 1. Then
sin α (γ − γ0 ) cos (γ − α (γ − γ0 ))
X=
1
W
(cos γ) α
1−α
α
(1)
is Sα (1, β2 , 0) for α 6= 1.
X=
W cos γ
+ β2 γ tan γ − β2 log π
2
2 + β2 γ
π
is S1 (1, β2 , 0).
Yiyang Yang
Simulate α-Stable Random Variable
(2)
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
Properties of Stable Law
Integral Form of Stable Law
Generation of Stable Random Variable
Proof.
α
1−α
1−α
α
cos(γ−α(γ−γ0 ))
0)
Assume a (γ) = sin α(γ−γ
, then (1) is α(γ)
cos γ
cos γ
When 0 < α < 1, (1) implies that X > 0 i γ > γ0 . Since 1−α
α >0
W
P (0 < X
=P
=P
0
<
W
α (γ)
1−α
α
W
.
≤ x ) = P (0 < X ≤ x , γ > γ0 )
≤ x , γ > γ0
!
=P
W
≥ x α−1 a (γ) , γ > γ0
α
α
α
≥ x α−1 a (γ) P (γ > γ0 ) = Eγ exp −x α−1 a (γ) 1{γ>γ0 }
ˆ
=
π
2
γ0
exp
α
−x α−1 a (γ)
1
π
dγ
Given a (γ) = Uα (γ, γ0 ) , we have proved X ∼ Sα (1, β2 , 0).
And with the similar method, we can prove the case 1
the caseα
<α<2
= 1.
Yiyang Yang
Simulate α-Stable Random Variable
and
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
Properties of Stable Law
Integral Form of Stable Law
Generation of Stable Random Variable
Algorithm:
Generate a random variableU uniformly distributed on − π2 , π2 and an
independent exponential random variable E with mean 1.
For α 6= 1, compute
sin (α (U + Bα,β )) cos (U − α (U + Bα,beta ))
1
E
(cos (U )) α
X
= Sα,β
arctan
β tan πα
2 )
(
where Bα,β =
α
For α = 1, compute
X
, and Sα,β = 1 + β 2 tan2
πα
2
1
2α
1−α
α
,
.
π
E cos U
2 π
2
=
+ β U tan U − β log π
.
π
2
2 + βU
Generalize scale and location Y
Yiyang Yang
(
σ X + µ,
=
σ X + π2 βσ log σ + µ,
α 6= 1
.
α=1
Simulate α-Stable Random Variable
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
The Method of Moments
Time Regression Method
Outline
1
Denition of
α-Stable
Random Variable
Denition
Characteristic Function
2
Weron's Algorithm for Generating Stable Random Variable
Properties of Stable Law
Integral Form of Stable Law
Generation of Stable Random Variable
3
Estimation of Stable Distribution Parameters
The Method of Moments
Time Regression Method
Yiyang Yang
Simulate α-Stable Random Variable
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
The Method of Moments
Time Regression Method
Given sample of observed data X = {x1 , x2 , · · · , xN } and assume it is directed
by α-stable distribution.Then, dene the sample characteristic function
φ̂X (u ) =
N
1 X iuxj
e
.
N
j =1
By the law of large number, φ̂X (u ) is a consistent estimator of the charateristic
function φX (u ).
By simple transformation, we have for all α
|φX (u )| = exp (−σ α |u |α ) .
Thus
− log |φX (u )| = σ α |u |α .
Assume α 6= 1, choose two dierent nonzero values uk ,
k
= 1, 2
− log φ̂X (uk ) = σ α |uk |α .
Yiyang Yang
Simulate α-Stable Random Variable
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
The Method of Moments
Time Regression Method
Solve these two equations and get α̂, σ̂
log
α̂ =
u
log log||φ̂(u21 )||
φ̂(
)
log uu12
log |u1 | log − log φ̂ (u2 ) − log |u2 | log − log φ̂ (u1 )
log σ̂ =
.
log uu12
The estimation of β̂ and µ̂ based on the imaginary and real parts of the
characteristic function
Re
Im
πα ,
(φX (u )) = exp (− |σµ|α ) cos µu + |σ u |α β (signu ) tan
2
(φX (u )) = exp (− |σµ|α ) sin µu + |σ u |α β (signu ) tan
πα Then, we have
arctan
(φX (u ))
Re (φX (u ))
Im
= µu + |σ u |α β (signu ) tan
Yiyang Yang
πα
2
2
.
Simulate α-Stable Random Variable
.
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
The Method of Moments
Time Regression Method
Based on α̂, σ̂ and two dierent nonzero values u , k = 3, 4, we can
solve the system of equations to obtain estimation of β̂ and µ̂.
k
µ̂ =
β̂ =
u4α̂ arctan
Im
Re
(φX (u3 ))
(φX (u3 ))
α̂
u3 u4 − u4 u3α̂
(φX (u3 ))
(φX (u3 ))
π α̂
σ̂ α̂
u4 arctan
Im(φX (u4 ))
− u3α̂ arctan Re
(φX (u4 ))
Im
Re
tan
2
Im(φX (u4 ))
− u3 arctan Re
(φ (u ))
X 4 .
u4 u3α̂ − u3 u4α̂
In this estimation, the values are u1 = 0.2, u2 = 0.8, u3 = 0.1 and
u4 = 0.4 are proposed in the simulation study.
Yiyang Yang
Simulate α-Stable Random Variable
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
The Method of Moments
Time Regression Method
Based on the equations
log − log |φ (u )|2 = log 2σ α + α log |u |
X
πα
Im (φ (u ))
α
= µu + σ α |u | β (signu ) tan
.
arctan
Re (φ (u ))
2
X
X
By regression y = log − log |φ (u )|2 and w = log |u | in the model
k
X
k
k
y = aw + b + k
propose u =
k
πk
25 ,
k
k
k
k = 1, 2, · · · , K , then
α̂ = a, σ̂ =
Yiyang Yang
1
e
2
b
1a
.
Simulate α-Stable Random Variable
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
By regression
the model
yl =
Im(φX (ul )) 1 and
Re (φX (ul )) ul
arctan
y = aw + b + l
propose u =
l
The Method of Moments
Time Regression Method
πl
50 ,
l
wl = |uul l|
α
signul in
l
l = 1, 2, · · · , L, then
µ̂ = b , β̂ =
a
σ̂ α̂ tan π2α̂
.
A very important step is normalization with initial estimation
σ0 =
x0.72 − x0.28
, µ0 = E [X0.25−0.75 ]
1.654
where x is the f sample quantile. Then the normalized data is
f
x =
0
j
Yiyang Yang
x − µ0
j
σ0
.
Simulate α-Stable Random Variable
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
The Method of Moments
Time Regression Method
Assume the estimated four paramters from regression of x are
0
j
0
0
0
0
α, β, σ, µ.
Then the stable parameters of original data are
α=α
β=β
0
0
0
σ = σ σ0
(
µ=
0
µ σ0 + µ0
µ σ0 + µ0 − σβ π2 ln σ0
0
Yiyang Yang
α 6= 1
α=1
Simulate α-Stable Random Variable
Denition of α-Stable Random Variable
Weron's Algorithm for Generating Stable Random Variable
Estimation of Stable Distribution Parameters
The Method of Moments
Time Regression Method
References
J.M. Chambers, C.L. Mallows and B.W. Stuck, A Method for
Simulating Stable Random Variables,
Jounarl of the American
Statistical Association 1976 Vol. 71, No. 354, pages 340-344
Rafel Weron, On the Chambers-Mallows-Stuck method for
simulating skewed stable random variables,
Probability Letters 28 (1996) 165-1771
Statistics &
Rafel Weron, Correction to: On the Chambers-Mallows-Stuck
method for simulating skewed stable random variables
Ioannis A. Koutrouvelis, Regression-Type Estimation of the
Parameters of Stable Laws,
Journal of the American
Statistical Association, December 1980
Svetlozar T. Rachev, Young Shin Kim, Michele Leonardo
Bianchi, Frank J. Fabozzi, Financial Models with Levy Process
and Volatility Clustering.
Yiyang Yang
Simulate α-Stable Random Variable