On non-linear dependence of multivariate subordinated Lévy processes

04/08/2020
by   Elvira Di Nardo, et al.
Politecnico di Torino
0

Multivariate subordinated Lévy processes are widely employed in finance for modeling multivariate asset returns. We propose to exploit non-linear dependence among financial assets through multivariate cumulants of these processes, for which we provide a closed form formula by using the multi-index generalized Bell polynomials. Using multivariate cumulants, we perform a sensitivity analysis, to investigate non-linear dependence as a function of the model parameters driving the dependence structure

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1 Introduction

Lévy processes are widely used in finance to model asset returns being more versatile than Gaussian driven processes as they can model skewness and excess kurtosis. Their characteristic function describes the distribution of each independent increment through the Lévy-Khintchine representation. In the following we focus our attention on the moment generating function (mgf) and its relation with the cumulant generating function (cgf). In particular, if

is a -valued Lévy process with mgf at each the Lévy-Khintchine representation allows us to work with where is the cgf of the time one distribution of the Lévy process.

Stochastically altering the clock on which the Lévy process is run models the economic time of the overall market activity: time runs fast when there are a lot of orders, while it slows down when trade is stale. The subordination of a Lévy process by a univariate subordinator , i.e. a Lévy process on with increasing trajectories, independent of , defines a new process by the composition Unfortunately, the resulting models exhibit several shortcomings including the lack of independence between asset returns and a limited span of linear correlations. Furthermore, there is empirical evidence that trading activity is different across assets (Harris (1986)). From the theoretical perspective, multivariate subordination allowing different assets to have different time-changes was introduced in the work of Barndorff-Nielsen et al. (2001). Given a -valued multiparameter Lévy process as defined in Barndorff-Nielsen et al. (2001), the -valued subordinated Lévy process is the composition where is a multivariate subordinator, i.e. a Lévy process on whose trajectories are increasing in each coordinate, independent of .

In Section we give a closed form formula of joint (or cross) cumulants of through the multi-index generalized (complete exponential) Bell polynomials introduced in Di Nardo et al. (2011). We use an umbral evaluation operator (Di Nardo (2015)) to recover the contribution of joint cumulants of the -dimensional subordinator. For multivariate subordinated Brownian motions, this closed form formula further simplifies by taking advantage of the well-known property that cumulants of Brownian motion are zero when their order is greater than two. The case of subordinated Brownian motions is of particular interest in finance for two reasons: first they link deviation of normality of asset returns to trade activity and second they often have analytical characteristic functions. A subclass of multivariate subordinated Brownian motions widely used in finance because of their economic interpretation is the so called -models, see Luciano and Semeraro (2010). These models exhibit a flexible dependence structure and allow to model also high correlations. They also incorporate non-linear dependence as discussed in Luciano and Semeraro (2010): however this feature has not be investigated so far.

In Section we propose to use joint cumulants to study higher order dependence and its behaviour in time. Indeed, as well known, the covariance matrix completely describes the dependence structure among components of a multivariate process only for Gaussian ones. Co-skewness and co-kurtosis measure extreme deviations or dispersions undergone by the components, as in the univariate case they measure asymmetry and fat-tailedness. Moreover, higher order cumulants play an important role in the analysis of non-Gaussian data and allow to detect higher order cross-correlations, a critical feature that exacerbates during financial turmoils, see Domino et al. (2018)

. If joint cumulants are asymptotically zero, central limit theorems can be investigated to forecast the market behavior or viceversa to choose among different models the one which better incorporates non-linear dependence. As case study, we investigate the non-linear dependence structure of

-models describing asset returns as a superposition of an idiosyncratic component, due to the asset specific trades, and a systematic one, due to the overall trade. We focus our attention on the Normal Inverse Gaussian (NIG) specification, whose one dimensional marginals are NIG processes and discuss the role played by the model parameters in driving non-linear dependence.

2 Cumulants of multivariate subordinated Lévy processes

Let us consider the multiparameter Lévy process where is a multiparameter Lévy process with independent components and According to Barndorff-Nielsen et al. (2001), the subordinated process

(2.1)

is a Lévy process. It’s a straightforward consequence of Theorem 4.7 in Barndorff-Nielsen et al. (2001) to prove that

(2.2)

Indeed if with Kronecker’s then has mgf

(2.3)

Since is the cgf of and then (2.2) follows by plugging (2.3) in In order to recover the -th cumulant of we need to expand in formal power series the cgf given in (2.2). To this aim, let us recall the notion of multi-index partition introduced in Di Nardo et al. (2011).

Definition 2.1.

A partition of a multi-index is a matrix of non-negative integers with rows and no zero columns in lexicographic order, such that for

As for integer partitions, let us fix some notation:
is the sum of all components of
denotes the multi-index partition of with columns equal to
columns equal to and so on, with

is the vector of multiplicities of


is the number of columns of and
given the multi-indexed sequence the product is said associated to the
sequence through in particular if
The -th coefficient of with is the -th generalized (complete exponential) Bell polynomial (Di Nardo et al. (2011) )

(2.4)

where is associated to the sequence through for

Theorem 2.1.

The -th cumulant of is where

i)

is an umbral evaluation linear operator (Di Nardo et al. (2011)) such the -th joint cumulant of

ii)

is associated to with where are cumulants of and is the -th column of

Proof.

From (2.2) the cgf of is a composition with and Then the -th coefficient of is obtained by expanding in formal power series and then by applying the evaluation linear operator see Di Nardo et al. (2011) for the details. Thus we recover (2.4) with the product replaced by the -th coefficient of the formal power series that is the joint cumulant By further expanding in formal power series, we have where is the -th cumulant of and is the -th column of Thus the result follows from (2.4). ∎

2.1 Multivariate subordinated Brownian motion

As multiparameter Lévy process in (2.1) let us consider obtained from a -valued Brownian motion with independent components, drift and covariance matrix and the multiparameter Lévy process

(2.5)

Note that has drift and covariance matrix and that, if and

is the identity matrix, we recover the subcase of independent Brownian motions. Suppose

independent of and consider Theorem 2.1 allows us to compute the -th cumulant by taking advantage of the well-known property that cumulants of a Brownian motion are zero when their order is greater than Indeed in (2.4), the products are not zero if and only if that is the set of all multi-index partitions whose columns have sum of components not greater than Thus the summation in (2.4) reduces to Moreover the sequence is given by

(2.6)

with Indeed in (2.2) the cgf reduces to as

are Gaussian distributed with mean

and variance

Note that when in Theorem 2.1 reduces to

(2.7)

2.2 The -model

In this section, we consider a class of processes used in finance to model asset returns: the -models. Jevtić et al. (2018) proved that they belong to the class of multivariate subordinated Brownian motions (2.5), by properly choosing the matrix and the subordinator

A -model (Luciano and Semeraro (2010)) is constructed by subordinating independent Brownian motions with independent subordinators and by subordinating a multidimensional Brownian motion with a unique subordinator More in details, let be a multivariate Brownian motion, with correlations and Lévy triplet , where with and , with and for Assume independent of obtained from a -valued Brownian motion with independent components, drift and covariance matrix The -model is the -valued subordinated process

(2.8)

where and are independent subordinators, independent of and If , the process is said an -model. As has independent margins, from the additivity property of cumulants, we have where

(2.9)

and joint cumulants

(2.10)

with and the -th coefficient of and in (2.9) respectively. By using (2.7) we have

(2.11)

where and are the -th cumulants of and respectively, and are the coefficients of the inner power series of and respectively, as given in (2.9). To discuss non-linear dependence for the bivariate case in Section we consider the normalized cumulants

(2.12)

as a function of the time scale. For it’s straightforward to get

(2.13)

where and are functions of the marginal parameters . Cumulants of increase linearly in so that co-skewness measures are proportional to while co-kurtosis measures are proportional to converging to Gaussian values asymptotically. Notice that is a scale parameter for all cross-cumulants, driving the general level of dependence, both linear and non-linear. Furthermore, the Brownian motion correlation providing an extra-term in (2.13), affects non only asset correlation measured by but also non-linear dependence measured by the other cross-cumulants. Thus the -models not only span a wider range of linear dependence compared to the -models, but they can also incorporate higher non-linear dependencies.

3 A case study

To show the role played by cumulants in analyzing non-linear dependence over time, let us consider a bivariate price process such that where is the drift term, not affecting the dependence structure, and

is a bivariate Lévy process. Since we are in the class of multivariate Lévy models, the centered asymptotic distribution of daily logreturns is a bivariate Normal distribution

where is the constant covariance matrix of the process (see Jammalamadaka et al. (2004)). This analysis is performed considering the NIG specification of the -model.

3.1 NIG specification

Recall that a NIG process has no Gaussian component, is of infinite variation and can be constructed by subordination as where is independent of the standard Brownian motion In particular the time one distribution has a NIG distribution with parameters and .

Now, let us consider the -model (2.8), with and the time one distributions of the subordinators and respectively. If we choose the parameters of and of the subordinators so that:

(3.1)

with and for , the subordinated process in (2.8) has the remarkable property that its one dimensional marginal processes have NIG distributions NIG for The process is named -NIG process. Thus is given in (2.10) for suitable replacements of the parameters in (2.11), taking into account that if its cumulants are and for

3.2 Numerical results

In this section, we use the cross-cumulants (2.13) to study the evolution in time of non-linear dependence in a bivariate -NIG model as a function of the common parameters and . In fact, dependence is driven by the common subordinator representing the systematic component in two ways: first, through the common parameter which defines the distribution of the common time change; second, through the action of the Brownian motions’ correlation on . The reference parameter set is given in Jevtić et al. (2018) to which we refer for further details111The -NIG specification has been calibrated by the generalized method of moments (GMM) to a bivariate basket composed by Goldman Sachs and Morgan Stanley US daily logreturns from January 3, 2011 to December 31st, 2015. Marginal parameters are: . The marginal drifts not involved in the dependence structure are and . We plot the normalized cumulants in (2.12) up to the fourth order as a function of the Brownian correlation . We consider two scenarios corresponding to different values of the scaling parameter . Since and the limit value corresponds to independence, we choose the intermediate value and the higher boundary value . For each scenario, we plot the evolution of cross-cumulants for three levels of time to maturity. Comparing the two scenarios, it is evident that drives both linear and non-linear dependence and it allows to reach maximal correlation. Nevertheless, also in the first scenario, where the maximal attainable asset correlation is moving we are able to incorporate non-linear dependence. The evolution in time confirms that higher order cumulants go to zero according to the rates in equation (2.13).

Figure 1: Scenario 1. Normalized cross-cumulants.
Figure 2: Scenario 2. Normalized cross-cumulants.

4 Conclusions

Cumulants are a powerful tool to measure non-linear dependence. We take advantage of the Lévy-Kinthchin representation of multiparameter subordinated Lévy processes to find their cumulants in closed form. We use them to study non-linear dependence captured by some class of processes widely used in finance to model asset returns. Indeed higher order statistics and suitable tests of hypotheses have been employed aiming to identify nonlinear processes Masson (2001)

. Nevertheless, the closed formula we found has other possible uses, as for example the estimate of the process parameters since the maximum likelihood estimation is computationally cumbersome within a multivariate framework. For large samples or high frequency data, the closed formulae of multivariate cumulants can be matched to multivariate

-statistics, unbiased estimators with minimum variance

Di Nardo (2015), and thus estimates of parameters can be recovered by an analogous of GMM. This is in the agenda of our future research.

References

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