The fractional multivariate normal tempered stable process

(1)

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Applied Mathematics Letters

journal homepage:www.elsevier.com/locate/aml

The fractional multivariate normal tempered stable process

Young Shin Kim

Department of Statistics, Econometrics and Mathematical Finance, School of Economics and Business Engineering, Karlsruhe Institute of Technology, Germany

a r t i c l e i n f o Article history:

Received 24 May 2012

Received in revised form 16 July 2012 Accepted 17 July 2012

Keywords:

Multivariate fractional normal tempered stable process

Lévy processes Tempered stable process Long-range dependence Fractional Brownian motion

a b s t r a c t

In this paper, the multivariate process having long-range dependency is presented. The process is defined by the time-changed fractional Brownian motion whose subordinator is given by the fractional tempered stable subordinator. The fractional tempered stable subordinator is a generalization of the non-decreasing tempered stable process with long-range dependence. The multivariate process allows for (1) the long-long-range dependence in the endogenous noise, (2) the long-range dependence in time or the volatility, (3) the fat-tailed marginal distribution, and (4) an asymmetric dependence structure between elements. Numerical methods to generating sample paths for the process are discussed.

1. Introduction

Fractional Brownian motion introduced by Mandelbrot and Ness [1] is a self-similar process with stationary increments. The process has short-range or long-range dependence controlled by the Hurst index. Fractional Brownian motion is a subclass of the fractional stable process (see [2]). Both fractional Brownian motion and fractional stable process have been applied in areas such as data communication, hydrology, and finance. In particular, the fractional stable process has been applied to the modeling of financial time series having long-range dependence (see [3,4]).

In finance, the modeling of asset prices is typically done using a stochastic process with time-varying variance (see [5–9]). In the model, the pricing process is driven by two driving noises: the endogenous noise and the exogenous noise from market volatility modeled by a positive stochastic process. The pricing process with stochastic market volatility is driven by the time-changed Lévy process, which captures the fat tail and the asymmetric properties in the empirical return distribution (see also [8,10,11]).

In this paper, a new multivariate process having the long-range dependence not only in the endogenous but also in exogenous noises is presented. The new process is defined by taking the time-changed fractional Brownian motion. That is, we take fractional multivariate Brownian motion and change the time variable to the stochastic process with long-range dependence. The stochastic process replacing the time variable is given by the fractional tempered stable subordinator. The fractional tempered stable subordinator is the totally right-skewed process with long-range or short-range dependence. Hence, the fractional Brownian motion captures the endogenous long-range dependence, while the exogenous is described by the fractional tempered stable subordinator. Moreover, the mutual dependence between two elements of the process is obtained by the correlation matrix of the fractional multivariate Brownian motion. The model captures the fat tail and asymmetric properties of the asset return distribution observed in financial markets as well as the non-linear asymmetric dependence structure between the returns of any pair of assets.

The remainder of this paper is organized as follows. In Section2, the fractional tempered stable subordinator is presented based on the result of [12]. The fractional univariate normal tempered stable process is defined by the time-changed

E-mail address:[email protected].

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fractional Brownian motion with the fractional tempered stable subordinator in Section3. The fractional multivariate normal tempered stable process is discussed in Section4. The numerical method to simulate the fractional multivariate normal tempered stable process is presented in Section5. Our principal findings are summarized in Section6.

2. Fractional tempered stable subordinator

To define a fractional tempered stable process, we use the Volterra kernel KH,α

: [

0

, ∞) × [

0

, ∞) → [

0

, ∞)

, given by

KH,α(t

,

s

) =

cH,α





t s



H−1_α

(

t

−

s

)

H−1α

−



H

−

1

α



sα1−H



t s uH−α1−1

₍

_u

₋

_s

₎

H−1_α du



1[0,t]

(

s

)

(1) with cH,α

=



GH,α

(

1

−

2GH,α)Γ



₁ 2

−

GH,α



Γ

(

2

−

2GH,α)Γ



GH,α

+

1₂





1₂

and GH,α

=

H

−

1_α

+

1₂. According to [12,13], we can find the following facts:

•

We have



t∧s 0 KH,α(t

,

u

)

KH,α(s

,

u

)

du

=

1 2



t2GH,α

₊

_s2GH,α

_{− |}

_t

₋

_s

_|

2GH,α



,

_and



t 0 KH,α(t

,

s

)

2ds

=

t 2  H−_α1  +1

.

•

K1 α,α

(

t

,

s

) =

1[0,t]

(

s

)

.

•

KH,α(t

, ·) ∈

Lp

([

0

,

t

]

)

if and only if H

∈



1 α

−

1p

,

1 α

+

1p



. When KH,α(t

, ·) ∈

Lp

([

0

,

t

]

)

, we have



t 0 KH,α(t

,

s

)

pds

=

CH,α,pt p  H−_α1  +1

,

where CH,α,p

=

c p H,α



1 0

v

p_α1−H



(

1

−

v)

H−α1

−



H

−

1

α

 

1 v

w

H−1_α−1

_{(w − v)}

H−1_α d

w



p d

v.

The pure-jump Lévy process T

=

(

T

(

t

))

t≥0whose characteristic function

φ

T(t)is equal to

φ

T(t)(u

) =

exp



tk



∞ 0

(

eiux

−

1

)

e −θx xα/2+1dx



=

exp



tkΓ



−

α

2

 (θ −

iu

)

α 2

−

θ

α2

 ,

₍₂₎

is a subordinator and is referred to as the tempered stable subordinator with parameters

(α,

k

, θ)

. Let H

∈



1

α

−

12

,

1 α

+

12



, α ∈ (

0

,

2

)

. Consider the fractional Lévy process TH

=

(

TH

(

t

))

t≥0defined by TH

(

t

) =



t

0

KH,α(t

,

u

)

dT

(

u

),

(3)

where T is the tempered stable subordinator with parameter

(α,

k

, θ)

1_{and K}

H,αis the Volterra kernel. Then the process TH

is referred to as the fractional tempered stable subordinator with parameter

(

H

, α,

k

, θ)

. By Houdre and Kawai [12], we obtain the following proposition without proof. Proposition 1. Let t

,

s

≥

0. The covariance of TH

(

s

)

and TH

(

t

)

is given by

cov

(

TH

(

s

),

TH

(

t

)) =

1 2k

θ

α 2−2Γ



2

−

α

2





t2GH,α

₊

_s2GH,α

_{− |}

_t

₋

_s

_|

2GH,α



.

₍₄₎

By the same argument made for Proposition 3.4 in [12],

(

TH

(

t

))

t≥0exhibits long-range dependence when H

∈



₁

α

,

α1

+

12



and short-range dependence when H

∈



_α1

−

1 2

,

1 α



.

(3)

The characteristic function and cumulants are obtained as follows. Proposition 2. Suppose H

∈



1_α

−

1₂

,

1_α

+

1₂



and

α ∈ (

0

,

2

)

.

(a) The characteristic function of TH

(

t

)

is given by

φ

TH(t)(x

) =

E

[

exp

(

ixTH

(

t

))] =

exp



t 0 kΓ



−

α

2

 (θ −

ixKH,α(t

,

u

))

α 2

−

θ

α2



du



.

(b) Let n

∈

_{N and H}

∈



_α1

−

1 n

,

1 α

+

1n



. The cumulant cn

(

TH

(

t

))

of TH

(

t

)

is given by

cn

(

TH

(

t

)) =

1 in

∂

xlog

φ

TH(t)(x

)



x=0

=

kΓ



n

−

α

2

 θ

α 2−n_C_H_,α,_n_tn  H−1_α  +1

.

By the cumulants cn

(

TH

(

t

))

given inProposition 2, we obtain the mean and the variance of TH

(

t

)

as

E

[

TH

(

t

)] =

k

θ

α 2−1Γ



1

−

α

2



CH,α,1tH −1_α+1

_,

and Var

(

TH

(

t

)) =

k

θ

α 2−2Γ



2

−

α

2



t2  H−_α1  +1

.

The variance of TH

(

t

)

is also obtained by the particular case t

=

s of(4).

Proposition 3. Let H

∈



1 α

−

12

,

1 α

+

12



and

α ∈ (

0

,

2

)

. If x

≤

0, then the exponential moment E

[

exp

(

xTH

(

t

))]

of TH

(

t

)

is

well defined as E

[

exp

(

xTH

(

t

))] = φ

TH(t)(−ix

)

. Moreover, if a

∈

R and b

>

0 then

φ

TH(t)(a

+

ib

) =

E

[

exp

((

ia

−

b

)

TH

(

t

))]

is well defined.

3. Fractional univariate normal tempered stable process

Assume

(

BH

(

t

))

t≥0is the (univariate) fractional Brownian motion with Hurst index H

∈

(

0

,

1

]

. Then it is known that BH

(

t

) =



t 0

KH,2

(

t

,

s

)

dB

(

s

),

where

(

B

(

t

))

t≥0is a standard Brownian motion and KH,2is a special case of the Volterra kernel defined in(1). It is well known

that

(

BH

(

t

))

t≥0exhibits long-range dependence when H

∈



0

,

1₂



and short-range dependence when H

∈



1 2

,

1



.

Let

(

BH1

(

t

))

t≥0 be the fractional Brownian motion with Hurst index H1

∈

(

0

,

1

]

and

(

TH2

(

t

))

t≥0 be the univariate

fractional tempered stable subordinator with parameters

(

H2

, α,

k

, θ)

. Suppose that

(

BH1

(

t

))

t≥0 and

(

TH2

(

t

))

t≥0 are

independent. Consider a process X

=

(

X

(

t

))

t≥0defined by X

(

t

) = β(

TH2

(

t

))

2H1

₊

_B

H1

(

TH2

(

t

)),

(5)

where

β ∈

_{R. Then we refer to X as the fractional (univariate) normal tempered stable process. The}

(

BH1

(

t

))

t≥0is the Brownian

motion if H1

=

1₂ and

(

TH2

(

t

))

t≥0is the tempered stable subordinator if H2

=

1

α since K1

α,α

(

s

) =

1(0,t)(s

)

. Therefore, the

fractional normal tempered stable process becomes the normal tempered stable process when H1

=

1₂and H2

=

_α1.

The characteristic function of X

(

t

)

is equal to

φ

X(t)(x

) = φ

₍T_H2(t))2H1



β

x

+

ix2

_/

₂



where

φ

₍_T

H2(t))2H1is the characteristic function of the random variable

(

TH2

(

t

))

2H1_{. Unfortunately,}

φ

(T_H2(t))2H1is not given by a general closed form for all H1

∈

(

0

,

1

]

.

However, if H1

=

1₂then we have

φ

X(t)(x

) =

exp





t 0 kΓ



−

α

2







θ −



i

β

x

−

x 2 2



KH2,α(t

,

u

)



α2

−

θ

α2



du



.

The mean of X

(

t

)

is equal to E

[

X

(

t

)] = β

E

[



TH2

(

t

)

2H1

]

.

In particular, H1

=

12, E

[

X

(

t

)] = β

k

θ

α2−1Γ



1

−

α

2



CH2,α,1t H2−1_α+1

.

Let 0

≤

s

≤

t. Since we have E



BH1

(

TH2

(

s

))

BH1

(

TH2

(

t

)) =

1 2E





TH2

(

t

)

2H1

₊



TH2

(

s

)

2H1

₋



TH2

(

t

) −

TH2

(

s

)

2H1

 ,

the covariance of X

(

s

)

and X

(

t

)

is given by cov

(

X

(

s

),

X

(

t

)) =

1 2

β

2_cov





TH2

(

s

)

2H1

_{, }

TH2

(

t

)

2H1



+

1 2E





TH2

(

t

)

2H1

₊



TH2

(

s

)

2H1

₋



TH2

(

t

) −

TH2

(

s

)

2H1

 .

(4)

In the case, H1

=

1₂, by(4)we obtain cov

(

X

(

s

),

X

(

t

)) =

1 2

β

2_k

_θ

α2−2Γ



2

−

α

2





t2GH2,α

₊

_s2G_H2_,α

₋

₍

t

−

s

)

2GH2,α



+

k

θ

α2−1Γ



1

−

α

2



CH2,α,1s H2−1_α+1

.

4. Fractional multivariate normal tempered stable process

Consider a multivariate fractional Brownian motion BH1

=

(

BH1

(

t

))

t≥0such that BH1

(

t

) = (

BH1,1

(

t

),

BH1,2

(

t

), . . . ,

BH1,N

(

t

))

T_{, and suppose that}

cov

(

BH1,m

(

t

),

BH1,n

(

t

)) = σ

m,nt 2H1

for all m

,

n

∈ {

1

,

2

, . . . ,

N

}

. We know

σ

n,n

=

1 for all n

∈ {

1

,

2

, . . . ,

N

}

. LetΣbe the covariance matrix for BH1

(

1

)

, that is,

Σ

= [

σ

_m_,_n

]

_m_,_n∈{1,2,...,N}. Let TH2

=

(

TH2

(

t

))

t≥0be the uni-variate fractional tempered stable subordinator with parameter

(

H2

, α,

k

, θ)

. Suppose that TH2is independent of BH1. Let X

=

(

X

(

t

))

t≥0with X

(

t

) = (

X1

(

t

),

X2

(

t

), . . . ,

XN

(

t

))

T_{be a process}

of the random vector defined by X

(

t

) = (

T_H₂

(

t

))

2H1

β +

_B

H1

(

TH2

(

t

)),

(6)

where

β = (β

1

, β

2

, . . . , β

n

)

T

∈

RN. Then X is referred to as the fractional multivariate normal tempered stable process. This process becomes the multivariate normal tempered stable process (see [14]), when H1

=

1₂and H2

=

_α1.

The characteristic function of X

(

t

)

is given by

φ

X(t)(z

) =

E

[

exp

(

izTXt

)] = φ

₍T_H2(t))2H1



β

T_z

₊

i 2z T_Σ_z



,

where z

=

(

z1

,

z2

, . . . ,

zN

)

T

∈

RNand

(

TH2

(

t

))

2H1_{is the characteristic function of}

(

_T

H2

(

t

))

2H1_{. Moreover, if}

β

T_z

₋

1 2z

T_Σ_z

_>

_0,

then we have the exponential moment for X

(

t

)

as E

[

exp

(

zX

(

t

))] = φ

₍_T H2(t))2H1



i

β

T_z

₋

i 2z T_Σ_z



< ∞.

Let t

≥

0

,

m

,

n

∈ {

1

,

2

, . . . ,

N

}

, and Xm

(

t

)

and Xn

(

t

)

be the m-th and the n-th elements of the vector Xt, respectively.

Then the covariance of Xm

(

t

)

and Xn

(

t

)

is given by

cov

(

Xm

(

t

),

Xn

(

t

)) = β

m

β

nVar



(

TH2

(

t

))

2H1

 +

σ

m,nE



(

TH2

(

t

))

2H1



.

In the case, H1

=

1₂, we have

cov

(

Xm

(

t

),

Xn

(

t

)) = β

m

β

nk

θ

α 2−2Γ



2

−

α

2



t2  H2−_α1  +1

+

σ

_m_,_nk

θ

α2−1Γ



1

−

α

2



CH2,α,1t H2−_α1+1

.

5. Simulation

In this section, we consider the numerical method to generate the sample path for the fractional multivariate normal tempered stable process defined in(6).

By the same argument made for Proposition 4.1 in [12], we obtain the series representation for TH2

(

t

)

as follows:

TH2

(

t

) =

lim M→∞ M



j=1 KH2,α(t

, τ

j

)





αξ

j 2kT

−

2/α

∧



eju2j/α

θ



,

t

∈ [

0

,

T

]

,

(7) where

• {

uj

}

is an i.i.d. sequence of random variables on

(

0

,

1

)

,

• {

ej

}

and

{

e′j

}

are i.i.d. sequences of exponential random variables with parameter 1,

• ξ

_j

=

e′ 1

+

e ′ 2

+ · · · +

e ′ j,

• {

τ

_j

}

be an i.i.d. uniform random variable in

[

0

,

T

]

,

• {

uj

}

, {

ej

}

, {

e′j

}

and

{

τ

i

}

are assumed to be independent,

(5)

Fig. 1. (a) Sample paths for the tempered stable subordinator and the fractional tempered stable subordinator. (b) Sample paths for the normal tempered stable process and the three fractional normal tempered stable processes. (c) Sample paths for the fractional bivariate normal tempered stable process.

Let L_Σ be the lower triangular matrix obtained by the Cholesky decomposition forΣwithΣ

=

L_ΣLT

Σ, whereΣis

the correlation matrix in(6). Then we have BH1

(

t

) =

LΣB

¯

H1

(

t

)

whereB

¯

H1

(

t

) = (¯

BH1,1

(

t

), ¯

BH1,2

(

t

), . . . , ¯

BH1,N

(

t

))

T _{is a}

mutually independent vector of fractional Brownian motions. For a given partition

{

t0

,

t1

, . . . ,

tM

}

of the interval

[

0

,

T

]

with t0

=

0

,

tM

=

T and tj

<

tkfor j

<

k, we have

¯

BH1,n

(

TH2

(

tj

)) − ¯

BH1,n

(

TH2

(

tj−1

)) = (

TH2

(

tj

) −

TH2

(

tj−1

))

H1

(¯

_B

H1,n

(

j

) − ¯

BH1,n

(

j

−

1

)),

where n

∈ {

1

,

2

, . . . ,

N

}

. Therefore, the X

(

tm

) = (

TH2

(

tm

))

2H1

β +

m



j=1

(

TH2

(

tj

) −

TH2

(

tj−1

))

H1_L ΣZj

,

(8) where

β = (β

1

, β

2

, . . . , β

N

)

Tand Zj

=



_¯

BH1,1

(

j

) − ¯

BH1,1

(

j

−

1

), ¯

BH1,2

(

j

) − ¯

BH1,2

(

j

−

1

), . . . , ¯

BH1,N

(

j

) − ¯

BH1,N

(

j

−

1

)

T . Panel (a) ofFig. 1exhibits one simulated sample path for the fractional tempered stable subordinator (the thick line) for parameters

α =

1

.

1970

,

k

=

0

.

3024

, θ =

0

.

3664, and H2

=

1

/α+

0

.

1 using the series representation(7)with M

=

10

,

000.

In this simulation, the long-range dependence process is considered by selecting H2larger than 1

/α

. In order to show the

effect of long-range dependence, the sample path for the non-fractional tempered stable subordinator (the thin line) is presented together by substituting H2

=

1

/α

in the series representation(7). The parameters

α,

k,

θ

of the tempered stable

sample path are the same as the simulation of the fractional tempered stable sample path. Panel (b) ofFig. 1shows sample paths for the univariate fractional normal tempered stable processes defined by(5)with the parameter

β = −

0

.

2626. The four sample paths are generated by(8)with N

=

1. The two sample paths given in panel (a) are used as the subordinator for the processes in panel (b). The thin solid line and the dotted line use the non-fractional tempered stable subordinator which is the thin line in panel (a). The thick solid line and the dash line use the fractional tempered stable subordinator which is the thick line in panel (a). The dash line and dot line mix their subordinator with the non-fractional Brownian motion with H1

=

1

/

2, while the thin and thick lines mix their subordinator with the fractional Brownian motion with H1

=

0

.

65. In

panel (c) ofFig. 1, the two-dimensional plot for the bivariate fractional normal tempered stable process is presented using (8)with H1

=

0

.

65

, β = (−

0

.

2626

, −

0

.

2593

)

T, and cov

(

BH1,1

(

1

),

BH1,2

(

1

)) =

0

.

7.

2

(6)

6. Conclusion and further research

In this paper, the fractional multivariate normal tempered stable process is introduced. The process is defined by the time-changed fractional multivariate Brownian motion with the fractional tempered stable subordinator. The process is characterized by long-range dependence in volatility, and long-range dependence in noise itself. Moreover, the multivariate process captures the non-linear asymmetric dependence structure for the joint distribution and the fat tail and asymmetric properties for the marginal distribution.

We define a new fractional Lévy process SH

=

(

SH

(

t

))

t≥0as SH

(

t

) =



t 0

KH,α

(

t

,

u

)

dS

(

u

),

(9)

where S

=

(

S

(

t

))

t≥0is the rapidly decreasing tempered stable subordinator,3whose characteristic function

φ

S(t)is equal to

φ

S(t)(u

) =

exp



tk



∞ 0

(

eiux

−

1

)

e −θ2_x2_/₂ xα/2+1 dx



.

Then the exponential moment E

[

exp

(

xSH

(

t

))]

of SH

(

t

)

is well defined on entire real numbers, while the exponential moment

of the fractional tempered stable subordinator is defined on negative real number (Proposition 3). Moreover, using Theorem 4.3 of [15], we can clarify that the series representation for SH

(

t

)

forms as follows:

SH

(

t

) =

lim M→∞ M



j=1 KH,α(t

, τ

j

)





αξ

j 2kT

−

2/α

∧

 

2eju2j/α

θ



,

t

∈ [

0

,

T

]

where

{

uj

}

, {

ej

}

, {

ej′

}

, {ξ

j

}

and

{

τ

i

}

are the same sequences defined in Section5. Details remain for further research on account

of space considerations. Acknowledgments

The author thanks Professor Svetlozar T. Rachev, Professor Frank J. Fabozzi, and anonymous referees for their helpful comments and suggestions.

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