A Derivation of higher order moments

# Multivariate Mixed Tempered Stable Distribution

## Abstract

The multivariate version of the Mixed Tempered Stable is proposed. It is a generalization of the Normal Variance Mean Mixtures. Characteristics of this new distribution and its capacity in fitting tails and capturing dependence structure between components are investigated. We discuss a random number generating procedure and introduce an estimation methodology based on the minimization of a distance between empirical and theoretical characteristic functions. Asymptotic tail behavior of the univariate Mixed Tempered Stable is exploited in the estimation procedure in order to obtain a better model fitting. Advantages of the multivariate Mixed Tempered Stable distribution are discussed and illustrated via simulation study.

Keywords: MixedTS distribution, MixedTS Tails, MixedTS Lévy process, Multivariate MixedTS.

## 1 Introduction

The Mixed Tempered Stable (MixedTS from now on) distribution has been introduced in Rroji and Mercuri (2015) and used for portfolio selection in Hitaj et al. (2015) and for option pricing in Mercuri and Rroji (2016). It is a generalization of the Normal Variance Mean Mixtures (see Barndorff-Nielsen et al., 1982) since the structure is similar but its definition generates a dependence of higher moments on the parameters of the standardized Classical Tempered Stable (see Küchler and Tappe, 2013; Kim et al., 2008) that replaces the Normal distribution.
Recently different multivariate distributions have been introduced in literature for modeling the joint dynamics of financial time series. For instance, Kaishev (2013) considers the LG distribution defined as a linear combination of independent Gammas for the construction of a multivariate model whose properties are investigated based on its relation with multivariate splines. Another model is based on the multivariate Normal Tempered Stable distribution, defined in Bianchi et al. (2016) as a Normal Mean Variance Mixture with a univariate Tempered Stable distributed mixing random variable that is shown to capture the main stylized facts of multivariate financial time series of equity returns.
In this paper, we present the multivariate MixedTS distribution and discuss its main features. The dependence structure in the multivariate MixedTS is controlled by the components of the mixing random vector. A similar approach has been used in Semeraro (2008) for the construction of a multivariate Variance Gamma distribution starting from the idea that the components in the mixing random vector are Gamma distributed. However, as observed in Hitaj and Mercuri (2013a, b), Semeraro’s model seems to be too restrictive for describing the joint distribution of asset returns. In particular, the sign of the skewness of the marginal distributions determines the sign of the covariance. This means, for instance, if two marginals of a multivariate Variance Gamma have negative skewness, their correlation can never be negative. The additional parameters in the multivariate MixedTS introduce more flexibility in the dependence structure and overcome these limits. Indeed, we compute higher moments for the multivariate MixedTS and show how the tempering parameters break off the bond between skewness and covariance signs.
We discuss a simulation method and propose an estimation procedure for the multivariate MixedTS. In particular the structure of univariate and multivariate MixedTS allows us to generate trajectories of the process using algorithms that already exist in literature on the simulation of the Tempered Stable distribution (see Kim et al. (2008)).
The proposed estimation procedure is based on the minimization of a distance between empirical and theoretical characteristic functions. As explained for instance in Yu (2004), in absence of an analytical density function, estimation based on the characteristic function is a good alternative to the maximum likelihood approach. An estimation procedure can be based on the determination of a discrete grid for the transform variable and on the Generalized Method of Moments (GMM) as for instance in Feuerverger and McDunnough (1981). The main advantage of this approach is the possibility of obtaining the standard error for estimators whose efficiency increases as the grid grows finer. However, the covariance matrix of moment conditions becomes singular when the number of points in the grid exceeds the sample size. The GMM objective function explodes thus the efficient GMM estimators can not be computed. To overcome this problem Carrasco and Florens (2000) developed an alternative approach, called Continuum GMM (henceforth CGMM), that uses the whole continuum of moment conditions associated to the difference between theoretical and empirical characteristic functions.
Starting from the general structure of the CGMM approach, we propose a constrained estimation procedure that involves the whole continuum of moment conditions. Results on asymptotic tail behavior of marginals are used as constraints in order to improve fitting on tails. An analytical distribution that captures the dependence of extreme events is helpful in many areas such as in portfolio risk management, in reinsurance or in modeling catastrophe risk related to climate change. The proposed estimation procedure is illustrated via numerical analysis on simulated data from a bivariate and trivariate MixedTS distributions. We estimate parameters on bootstrapped samples and investigate their empirical distribution.
The paper is structured as follows. In Section 2 we give a brief review of the univariate MixedTS, study its asymptotic tail behavior and discuss the MixedTS Lévy process. The definition and main features of the multivariate MixedTS distribution are given in Section 3. In Section 4 we explain the estimation procedure and present some numerical results. Section 5 draws some conclusions.

## 2 Univariate Mixed Tempered Stable

Let us recall the definition of a univariate Mixed Tempered Stable distribution.

###### Definition 1.

A random variable is Mixed Tempered Stable distributed if:

 Y=μ+βV+√VX (1)

where parameters , and conditioned on the positive r.v. , follows a standardized Classical Tempered Stable distribution with parameters i.e.:

 X|V∼stdCTS(α,λ+√V,λ−√V) (2)

or equivalently

 Y|V∼CTS(α,λ+,λ−,VΓ(2−α)(λα−2++λα−2−),VΓ(2−α)(λα−2++λα−2−),μ+βV) (3)

where and (see Kim et al., 2008, for more details on CTS).

For this distribution it is possible to obtain the first four moments which are reported in the following proposition.

###### Proposition 2.

The first four moments of the MixedTS have an analytic expression since:

 Missing or unrecognized delimiter for \left (4)

where and are the third and fourth central moments respectively.

We observe that and depend on the mixing random variable and on the tempering parameters and . Indeed, we are able to obtain an asymmetric distribution even if we fix . It is worth to note that parameters and may have an economic interpretation. In particular, can be thought as the risk free rate and as the risk premium of the unit variance process . In the Normal Variance Mean Mixtures is not possible to have negatively skewed distribution with . From an economic point of view, it is not possible to have a positive risk premium for unit variance for negatively skewed distributions. This is a drawback of the Normal Variance Mean Mixture model since negative skewness is frequently observed in financial time series.
The mixture representation becomes very transparent for cumulant generating functions. Let

 ΦY(u)=logE[euY],ΦV(u)=logE[euV], (5)

and

 ΦH(u)=(λ+−u)α−λα++(λ−+u)α−λα−α(α−1)(λα−2++λα−2−)+(λα−1+−λα−1−)u(α−1)(λα−2++λα−2−), (6)

where is the cumulant generating function of a random variable . Then we have

 ΦY(u)=μu+ΦV(βu+ΦH(u)). (7)

As shown in Rroji and Mercuri (2015), if , we get some well-known distributions used for modeling financial returns as special cases. For instance if the Variance Gamma introduced in Madan and Seneta (1990) is obtained. Fixing and letting go to infinity leads to the Standardized Classical Tempered Stable Kim et al. (2008). Choosing:

 λ+ = λ−=λ a = 1 b = λα−2γα∣∣ ∣ ∣∣α(α−1)cos(απ2)∣∣ ∣ ∣∣

and computing the limit for we obtain the Geometric Stable distribution (see Kozubowski et al. (1997)).

### 2.1 Fundamental strip and moment explosion

Laplace transform theory tells us that given a random variable the set of where:

 E[|euX|]<∞.

is a strip, which is called the fundamental strip of . Depending on the tails of the strip can be the entire set of complex numbers , a left or right half-plane, a proper strip of finite width or degenerate to the imaginary axis if both tails are heavy. From here on we neglect the case since the MixedTS becomes a Normal Variance Mean Mixture and we refer to Barndorff-Nielsen et al. (1982) for the fundamental strip and tail behavior in this special case. When , with cumulant generating function in (6) has fundamental strip .

###### Theorem 3.

Suppose now has fundamental strip for some . A concrete example would be . Then we have:

1. If then has fundamental strip .

2. If then has fundamental strip , where is the unique real solution to .

3. If then has fundamental strip , where is the unique real solution to .

4. If then has fundamental strip where are the two real solutions of .

###### Proof.

First of all we prove point 1 where has fundamental strip . Any point can be written as:

 u⋆=γ(−λ−)+(1−γ)(λ+),  0≤γ≤1.

We start from

 βu⋆+ΦH(u⋆)=β[γ(−λ−)+(1−γ)(λ+)]+ΦH(γ(−λ−)+(1−γ)(λ+)).

Since is a convex function, we have:

 βu⋆+ΦH(u⋆)≤β[γ(−λ−)+(1−γ)(λ+)]+γΦH(−λ−)+(1−γ)ΦH(λ+).

Collecting terms with and we get:

 βu⋆+ΦH(u⋆)≤γ[β(−λ−)+ΦH(−λ−)]+(1−γ)[β(λ+)+ΦH(λ+)]. (9)

If the right hand side in (9) is less than .
To prove the second point, it is enough to observe that

 G(u)=βu+ΦH(u)−b

is a convex continuous function. Moreover, the condition implies that:

 G(−λ−)<0,  G(λ+)>0.

As observed in Giaquinta and Modica (2003), a convex continuous function in the compact interval , and has a unique zero . In our case, this result ensures that is the unique real solution of the equation Following the same steps in point 1 we get the result in point 2.
Point 3 is the same of point 2.
Point 4. It is sufficient to observe that is a continuous convex function such that . For any , there exists a neighborhood of zero such that . Continuity and convexity ensure the existence of two zeros . The remaining part of the proof arises from the same steps in point 1. ∎

### 2.2 Tail Behavior of a Mixed Tempered Stable distribution

In order to study the tail behavior of a , that denotes a MixedTS distribution with Gamma mixing r.v., we need first to recall the structure of its moment generating function where without loss of generality we require .
If , the moment generating function is defined as:

 MY(u)=E(euY)=[bb−(βu+ΦH(u))]a. (10)

We recall some useful results on the study of asymptotic tail behavior given in Benaim and Friz (2008). Given the moment generating function of a r.v. with cumulative distribution function defined as:

 M(u):=∫euxdF(x)

we consider and defined respectively as:

 −q⋆:=inf{u:M(u)<∞}, (11)

and

 r⋆:=sup{u:M(u)<∞} (12)

where . Criterion I in Benaim and Friz (2008) for asymptotic study of tails states:

###### Proposition 4.
1. If for some , for some , as then

 logF((−∞,−x])∼−q⋆x.
2. If for some , for some , as then

 logF((x,∞))∼−r⋆x.

We remark that stands for regularly varying functions of order , i.e. set of slowly varying functions and the derivative of order of the moment generating function .

Before studying the tail behavior of the , let us study first the tail behavior of a . The fundamental strip is and the moment generating function is:

 MCTS(u)=exp[(λ+−u)α−λα++(λ−+u)α−λα−α(α−1)(λα−2++λα−2−)+(λα−1+−λα−1−)u(α−1)(λα−2++λα−2−)] (13)

We consider separately two cases:

1. ,

2. .

Considering the right tail of a CTS we have and the converges to constant as .
CTS case - 1: Under the assumption that , we apply criterion 1 in Benaim and Friz (2008) checking that the first derivative of satisfies for some , as . The first derivative of in (13) is:

 M(1)CTS(u) = MCTS(u)[(λ+−u)α−1−λα−1+−(λ−+u)α−1+λα−1−(1−α)(λα−2++λα−2−)].

Evaluating at point and computing the limit for , we obtain:

 lims→0+M(1)CTS(λ+−s) ∼ MCTS(λ+)(1−α)(λα−2++λα−2−)s−(1−α)  for  α(0,1),

where the term is a constant. Therefore we have shown that the first order derivative of satisfies criterion 1 in Benaim and Friz (2008) when .

CTS case - 2 Let us consider now the right tail behavior for where both and converge to some constants as . We compute the second order derivative of the and show that criterion 1 in Benaim and Friz (2008) is verified for .

 M(2)CTS(u) = M(1)CTS(u)[(λ+−u)α−1−λα−1+−(λ−+u)α−1+λα−1−(1−α)(λα−2++λα−2−)] + MCTS(u)[(α−1)−(λ+−u)α−2−(λ−+u)α−2(1−α)(λα−2++λα−2−)] = MCTS(u)[(λ+−u)α−1−λα−1+−(λ−+u)α−1+λα−1−(1−α)(λα−2++λα−2−)]2 + MCTS(u)[(λ+−u)α−2+(λ−+u)α−2(λα−2++λα−2−)]

We evaluate at point and for we obtain the following result:

 lims→0+M(2)CTS(λ+−s) ∼ MCTS(λ+)[λα−1−−λα−1+−(λ−+λ+)α−1(1−α)(λα−2++λα−2−)]2 + MCTS(λ+)[(s)α−2+(λ−+λ+)α−2(λα−2++λα−2−)] ∼ MCTS(λ+)(λα−2++λα−2−)s−(2−α).

Now we study the right tail behavior of the . From Theorem 3 we have that can be or , therefore in order to study the behavior of the moment generating function in (10) we consider separately two cases:

• that refers to points 1 and 3 in Theorem 3 where .

• that refers to points 2 and 4 in Theorem 3 where .

case 1: covers case 1 and 3 in Theorem 3. The moment generating function of the , defined in (10), at the critical point is finite. We compute the first order derivative of and verify if criterion 1 in Benaim and Friz (2008) is satisfied. We consider separately the two cases and .

• and

 M(1)Y(u) = a[bb−(βu+ΦH(u))]a−1−b(b−(βu+ΦH(u)))2(−β−Φ′H(u)) = [bb−(βu+ΦH(u))]aa(β+Φ′H(u))(b−(βu+ΦH(u))) = MY(u)a(β+Φ′H(u))(b−(βu+ΦH(u)))

Observing from (6) that

 Φ′H(u)=(λ+−u)α−1−λα−1+−(λ−+u)α−1+λα−1−(1−α)(λα−2++λα−2−)

and

 lims→0+Φ′H(λ+−s)∼s−(1−α)(1−α)(λα−2++λα−2−), α∈(0,1)

we obtain:

 lims→0+M(1)Y(λ+−s) = Missing or unrecognized delimiter for \right ∼ MY(λ+)b−(βλ++ΦH(λ+))a(1−α)(λα−2++λα−2−)s−(1−α)  for  α∈(0,1)

Observe that since the term is a positive constant, we can conclude that the moment generating function of the , in case of and , satisfies criterion 1 in Benaim and Friz (2008) for .

• and . In this particular case both the moment generating function of the and and its first derivative are constants, therefore we compute the second derivative of the m.g.f. of the .

 M(2)Y(u) = ∂M(1)Y(u)∂u = M(1)Y(u)a(β+Φ′H(u))(b−(βu+ΦH(u))) + Missing or unrecognized delimiter for \left
 M(2)Y(u) = MY(u)(β+Φ′H(u))2a2(b−(βu+ΦH(u)))2 + Missing or unrecognized delimiter for \left = MY(u)(β+Φ′H(u))2(a2+a)(b−(βu+ΦH(u)))2 + aMY(u)(b−(βu+ΦH(u)))Φ′′H(u).

Since , we have that the following limit converges to a positive constant as :

 lims→0+Φ′H(λ+−s)=λα−1++(λ−+λ+)α−1−λα−1−(α−1)(λα−2++λα−2−)>0.

The term is a positive constant term in case 1 and 3 of Theorem 3 and the same holds for the positive constant term . Now we study the asymptotic behavior of as

 lims→0+Φ′′H(λ+−s) = Missing or unrecognized delimiter for \right ∼ s−(2−α)(λα−2++λα−2−)

Combining these results together we conclude that the moment generating function of the in case of and satisfies criterion 1 in Benaim and Friz (2008) for i.e.:

 lims→0+M(2)Y(λ+−s) ∼aMY(λ+)(λα−2++λα−2−)(b−(βλ++ΦH(λ+))) s−(2−α).

case - 2 At this point we are left with the case when which covers cases 2 and 4 in Theorem 3. We recall that the m.g.f of the at point is:

 M0Y(r⋆−s)=[bb−(β(r⋆−s)+ΦH(r⋆−s))]a. (14)

Multiplying and dividing by in (14) we have:

 M0(r⋆−s)=[bsb−(β(r⋆−s)+ΦH(r⋆−s))]as−a.

Substituting with we obtain:

 M0(r⋆−1t)=⎡⎢ ⎢⎣btb−(β(tr⋆−1)+tΦH(r⋆−1t))⎤⎥ ⎥⎦ata=g(t)ta. (15)

We can show that is a slowly varying function that implies is a regularly varying function.
Let us study the following limit:

 limt→+∞g(kt)g(t) = limt→+∞⎡⎢ ⎢⎣tb−[β(r⋆t−1)+tΦH(r⋆−1t)]tbk−[β(r⋆kt−1)+tkΦH(r⋆−1kt)]⎤⎥ ⎥⎦a. (16)

When we have , is a slowly varying function, since applying de l’Hôpital theorem to (16) we get:

 Missing or unrecognized delimiter for \right (17)

Concluding we can say that in case the criterion one in Benaim and Friz (2008) is satisfied for .

The study of the left tail behavior follows the same steps as above.

###### Remark 5.

In the CTS distribution and influence both higher moments and tail behavior. The singularities in Theorem 3 are helpful in describing the asymptotic behavior of the MixedTS tails based on the result:

 P(Y>y)∼e−u⋆y. (18)

From point 1 in Theorem 3 we get for the MixedTS the same asymptotic tail behavior as in the CTS, i.e. exponentially decaying, while in the other points of Theorem 3 the ’s satisfy the additional condition . Singularities in point 2 and 3 describe respectively right and left asymptotic tail behavior. In point 4 asymptotic of both tails are deduced. The scale parameter of the mixing r.v. allows us to have more flexibility in capturing tails once skewness and kurtosis, which depend on and , are computed. Consider for example point 2 where that implies, for fixed , from where we deduce that a higher weight is given to the right tail of the MixedTS than in the CTS case.

We conclude this section by investigating numerically the implications of Proposition 4. Results on the behavior of tails can be used for the identification of and in (11) and in (12). Indeed, for , we have:

 log[F(x)]=−q⋆x+o(x), (19)

while, for , we obtain:

 log[1−F(x)]=−r⋆x+o(x). (20)

Figure 1 refers to the behavior of and of for the MixedTS- with parameters , , , and .

Considering relations in (19) and in (20), we estimate and as the slope of two linear regressions following four steps: Given a sample composed by observations, we determine the empirical cumulative distribution function . Then we determine and as the empirical quantiles at level and , i.e.:

 ^xζ:=inf{xi:ˆF~n(xi)≥ζ}

and

 ^x1−ζ:=inf{xi:ˆF~n(xi)≥1−ζ}.

The set [, , ,] refers to the sorted values from the smallest to the largest .
We introduce the sets and defined as:

 L~n(ζ):={(xi,ˆF~n(xi)):xi∈[x(1),^xζ]} U~n(ζ):={(xi,ˆF~n(xi)):xi∈[^x1−ζ,x(~n)]}.

We use the elements in the set to estimate as the slope of the linear regression:

 log[ˆF~n(xi)]=−q⋆xi+ϵi,  (xi,ˆF~n(xi))∈Ln(ζ).

while the elements in the set are used for the estimation of the coefficient as the slope of the following regression:

 log[1−ˆF~n(xi)]=−r⋆xi+ϵi,  (xi,ˆF~n(xi))∈Un(ζ),

where is an error term. In Figure 2 we show the behavior of the estimated and for varying if true values are and . This result is useful in estimation of a MixedTS- since it can be used as a constraint in the optimization routine when we require the empirical and to be equal to the corresponding counterpart.

### 2.3 MixedTS Lévy process

Suppose is an infinitely divisible distribution on with cumulant function . Then there is a convolution semigroup of probability measures on and a Lévy process such that for and .
In Rroji and Mercuri (2015) it is shown that the distribution is infinitely divisible. According to the general theory, see for example Prop.3.1, p.69 in Cont and Tankov (2004), there exists a Lévy process such that . We have

 ΦYt(u)=μtu+ΦVt(βu+ΦH(u)), (22)

thus if is with mixing distribution , then is with mixing distribution . In the case is with mixing distribution then is with mixing distribution , since implies .

###### Definition 6.

A Lévy process such that