# Some Further Results on the Tempered Multistable Approach

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## Abstract

This article provides new results on the tempered multistable approach. After a preliminary section recalling the main definitions, we show the correspondence between a series representation and a characteristic function representation for asymmetrical field-based tempered multistable processes and for asymmetrical independent increments tempered multistable processes. We also show that both processes are semimartingales, which is a convenient property in finance. Next, we study the structure of autocorrelations that is conveyed by this approach. Finally, we provide an illustration showing the term structures of Value-at-Risk that can be obtained with this model.

## Keywords

Tempered multistable process Non-stationarity Dependence Asymmetry Kurtosis VaR Characteristic function## Mathematics Subject Classification

60G51 60G52## JEL Classification

G130## 1 Introduction

If we set aside the Brownian case, the first Lévy approach to modeling speculative prices is the stable one and dates back to the sixties (see, in particular, Mandelbrot 1963; Mandelbrot and Taylor 1967). This approach is not devoid of drawbacks: it predicts an infinite variance and distribution tails that are too heavy compared to empirical observations. The Variance Gamma process (see, e.g., Madan and Milne 1991) is one of the first non-stable Lévy processes constructed for financial applications that alleviates these drawbacks. This process, which has been of ample use in the past decades, is also the root of a more recent approach: the tempered stable or CGMY model of Carr et al. (2002), which made popular in finance processes originally introduced in Koponen (1995). The tails of the marginal distributions and of the Lévy measure of tempered stable processes can be cast in the form of a ratio of exponential and power functions, which justifies their name (see Rosiński 2007). The tails of these processes are therefore semi-heavy (in the sense that their analytical form can be represented by a ratio of exponential and power functions), more general than those of the Variance Gamma process, and better adapted to empirical data than their stable counterparts whose power tails are too thick to describe most stock market return behaviors. See the contributions of Küchler and Tappe (2013, 2014), of Imai and Kawai (2011), and the book of Rachev et al. (2011) for additional important results on the tempered stable approach.

The class of multistable processes is constructed as an extension of stable processes (see Samorodnitsky and Taqqu 1994 for a reference on these latter dynamics) by making the tail thickness parameter a function of time. Multistable processes have non-stationary (and potentially dependent) increments. This class of processes is introduced in Falconer and Lévy-Véhel (2009) and Falconer et al. (2009). The field-based multistable Lévy motion is an example of such processes. Its marginal distributions are those of a stable Lévy motion but it has non-stationary and dependent increments (see Guével and Lévy-Véhel 2012). The independent increments multistable Lévy motion has non-stationary but independent increments (see Falconer and Liu 2012). In Guével et al. (2015), it is shown that these two multistable processes are semimartingales.

In this article, we study and provide results for a class of processes that extends both tempered stable processs and multistable processes by combining the characteristic features of these two dynamics. These processes are called tempered multistable and are introduced (and studied in their symmetrical form) in Lévy-Véhel and Liu (2013). The first use of these dynamics in finance is found in Lévy-Véhel (2013), which is an empirical work numerically showing the importance of taking into account non-stationarities in the *Y* parameter of CGMY and similar models for computing VaRs, and in Lévy-Véhel and El Mekkedem (2013), where it is shown that stock indices can be represented by a self-stabilizing tempered process. Further studies are those of Fan and Lévy-Véhel (2013), which provides a detailed construction of self-stabilizing tempered processes and applications of these dynamics to the computation of VaRs, and of Lévy-Véhel and Lévy-Véhel (2013), which considers a tempered multistable process amenable to pricing whose jump intensity follows a CIR process. Specifically here, we consider the independent increments and field-based versions of tempered multistable processes. Our article generalizes from the symmmetrical case to the asymmetrical case results obtained in Lévy-Véhel and Liu (2013) and provides several additional results for these processes including the computation of autocorrelations.

The article is organized as follows. In a first section, we recall the main existing definitions and results available for multistable and tempered multistable processes. We concentrate on the independent increments and the field-based constructions of these dynamics. These two specific processes were originally defined by their marginal characteristic functions and by their tail thickness functions. A second section shows how these processes can in fact be defined via series representations when they are asymmetrical, generalizing results obtained in the symmetrical case. This section, which also shows that these processes are semimartingales, is not only important *per se*, but also because the proofs of its results are useful for obtaining the results of the next section. A third section is devoted to the study of the dependence of the increments of these processes. Specifically, the multivariate characteristic function of their increments is obtained, allowing us to confirm the independence of the increments of the first process and to study the correlation of the increments of the second one. A final section is dedicated to the computation of the moments and risk indicators related to these processes. We obtain the term structure of Value-at-Risk, which is markedly different from that obtained with more classic dynamics, namely Lévy processes. This result can have interesting applications in finance, where pension funds, e. g., put forward an argument of time diversification to justify the non-increase of risk with respect to time for high maturities.

## 2 Preliminary Presentation

In this preliminary section, we first present the class of multistable processes. Then, we concentrate on a derived class of processes that is more suitable to the modeling of economic and financial dynamics: tempered multistable processes. See Falconer and Liu (2012), Falconer and Lévy-Véhel (2009), Guével and Lévy-Véhel (2012) and Guével et al. (2015) for seminal definitions and results.

### 2.1 Multistable Processes

We first recall the definition of multistable Lévy processes, as stated for instance in Guével et al. (2015). The function \(\alpha : \mathbb {R} \rightarrow [c,d] \subset (0,2)\) is any continuously differentiable function. It is a generalization of the parameter \(\alpha \) that characterizes the heaviness of the tails of stable processes. Three sequences of random variables should also be considered. First, \((\Gamma _i)_{i \ge 1}\) is a sequence of arrival times of a Poisson process with unit arrival rate. It is therefore a sequence of dependent gamma random variables. Then, \((V_i)_{i \ge 1}\) is a sequence of i.i.d. random variables with uniform distribution on [0, *T*]. Finally, \((\gamma _i)_{i \ge 1}\) is a sequence of i.i.d. random variables with distribution \(P(\gamma _i=1)=P(\gamma _i=-\,1)=1/2\). The three sequences \((\Gamma _i)_{i \ge 1}\), \((V_i)_{i \ge 1}\), and \((\gamma _i)_{i \ge 1}\) are assumed jointly independent.

*t*, \(L_{FB}\) is simply the series representation of an \(\alpha -\)stable Lévy motion. The process \(L_{FB}\) is called

*multistable*because of the dependence of \(\alpha \) on time, and

*field-based*because at each fixed time

*t*its marginal law is that of a stable process. The joint characteristic function of \(L_{FB}\) equals

*t*, the field-based process has a stable marginal distribution of parameter \(\alpha _t\).

*t*, the independent increments process has a stable marginal distribution of parameter \(\int _0^t \alpha _s ds\). The field-based and the independent increments multistable processes are both semimartingales. The two processes are related at all times as follows:

The next subsection recalls the definition of the *tempered* (in the sense that large jumps are exponentially tempered) versions of the field-based and independent increments multistable processes.

### 2.2 Tempered Multistable Processes

*Z*) whose characteristic function is equal to

*Y*of the CGMY process has been replaced by a continuously differentiable function \(Y: [0,+\,\infty ) \rightarrow (0,1)\). This function is analogous to the function \(\alpha \) of multistable processes.

*t*, the marginal distribution of this process is that of a CGMY—or tempered stable—process. However, its increments are neither stationary nor independent: it is not a Lévy process.

Note that the field-based and independent increments tempered multistable processes were originally defined as processes having the above characteristic functions (10) and (11). We show in the next section how a definition via a series representation is able to give these characteristic functions as an output.

## 3 Results

In this section, we introduce series representations associated with both the independent increments and the field-based tempered multistable processes when these processes are asymmetrical. Specifically, the first and second propositions of this section extend the results of Lévy-Véhel and Liu (2013) that deal with the symmetrical case. We also show that both the independent increments and the field-based tempered multistable processes are semimartingales, extending the result of Guével et al. (2015) who show the semimartingale property for the independent increments and field-based non-tempered multistable processes.

The series representations of tempered multistable processes can be expressed in terms of five independent sequences of random variables. \(\{\Gamma \}_{j\ge 1}\) is a sequence of arrival times of a Poisson process with unit arrival rate. \(\{U\}_{j\ge 1}\) is a sequence of i.i.d random variables with uniform distribution on [0, *T*]. \(\{V\}_{j\ge 1}\) is a sequence of i.i.d. random variables with uniform distribution on [0, 1], \(\{e\}_{j\ge 1}\) is a sequence of i.i.d. exponential random variables with parameter 1, and \(\{\gamma \}_{j\ge 1}\) is a sequence of i.i.d random variables with distribution \(\mathbb {P}(\gamma _j=1)=\mathbb {P}(\gamma _j=-1)=1/2\).

We assume that *Y* takes values in (0, 1) and we allow the processes to be asymmetrical (\(G \ne M\)). We obtain representations that produce the characteristic functions given in Eqs. (10) and (11). We start with the independent increments tempered multistable process.

### Proposition 3.1

### Proof

See “Appendix”. \(\square \)

Using a similar method, we define the field-based tempered multistable process via a series representation.

### Proposition 3.2

### Proof

See “Appendix”. \(\square \)

### Remark 3.1

(Truncation) An important question for the simulation of the series (13) is that of the order of truncation. It appears that a truncation at \(j_\text {max} = 10^6\) yields a reasonable degree of precision. This is confirmed by Table 1, which shows the convergence of one simulation of \(Z_{FB}(T)\) with respect to \(j_\text {max}\) when \(T=20\), \(C=1\), \(G=30\), \(M=30\), and \(Y(T)=0.5\) (similar conclusions are reached with other parameter sets).

Although \(10^6\) does not seem a small number, it takes only approximately a tenth of a second to simulate \(Z_{FB}(T)\) when the series is truncated at this order. Also, one should note a big advantage of the method. The random numbers in the series need to be simulated only once for obtaining all the trajectory. This is particularly useful for problems solvable by Monte-Carlo simulations.

Series truncated at various orders

\(j_\text {max}\) | \(10^3\) | \(10^4\) | \(10^5\) | \(10^6\) | \(10^7\) |
---|---|---|---|---|---|

\(Z_{FB}(T)\) | 0.83353 | 0.90376 | 0.89712 | 0.89714 | 0.89714 |

We now come to the semimartingale property of the independent increments tempered multistable process presented in this article. This result is not *a priori* obvious: for instance, not all independent increments processes are semimartingales, contrary to Lévy processes.

We first concentrate on the field-based tempered multistable process.

### Proposition 3.3

Assume that *Y* is differentiable and that its derivative is bounded. Further assume that *Y* takes values in \([c,d] \in (0,1)\). Then, the field-based tempered multistable process is a semimartingale.

### Proof

See “Appendix”. \(\square \)

We obtain a similar result in the case of the independent increments tempered multistable process.

### Proposition 3.4

The independent increments tempered multistable process is a semimartingale.

### Proof

See “Appendix”. \(\square \)

These results guarantee the tractability of tempered multistable models as a class of financial models. The next section uses the general results obtained in this section to derive more applied results concerning the dependence structure of these models.

## 4 Dependence

In order to study the dependence of increments of tempered multistable processes, we compute in this section their multivariate characteristic functions. We first confirm that \(Z_{II}\) is an independent increments process, and then we concentrate on \(Z_{FB}\). We are particularly interested in the dependence structure that this latter process conveys. Specifically, we examine how the autocorrelation that it bears decreases with the passage of time.

To start with, we set out the following proposition.

### Proposition 4.1

*u*equal to

*T*until the end of that proof, and substituting \(\theta \mathbb {1}_{(U_j \le t)}\) by \(\sum \nolimits _{k=1}^K \theta _k \mathbb {1}_{(U_j \le t_k)}\). The proofs of the results shown in this section are simple and are thus only sketched for brevity reasons.

Based on this result, we compute the characteristic function associated with an increment of \(Z_{II}\) by setting \(K=2\), \(\theta _1=\theta \), \(\theta _2=-\theta \), \(t_1=t\), and \(t_2 = s\). We obtain:

### Corollary 4.1

### Proposition 4.2

We can now derive the characteristic function associated with an increment of the process \(Z_{FB}\). For this purpose, we set \(K=2\), \(\theta _1=\theta \), \(\theta _2=-\,\theta \), \(t_1=t\), and \(t_2 = s\) in Proposition 4.2.

### Corollary 4.2

Then, we give the characteristic function associated with two increments of the process \(Z_{FB}\), which is useful for the study of the autocorrelation of this process. For this purpose, we set \(K=4\), \(\theta _1=\theta \), \(\theta _2=-\,\theta \), \(\theta _3=\eta \), \(\theta _4=-\,\eta \) , \(t_1=t\), \(t_2 = s\), \(t_3=\tau \), and \(t_4 = \zeta \) in Proposition 4.2.

### Corollary 4.3

We can now examine the autocorrelation, or dependence of increments, that is borne by the field-based tempered multistable process.

### Proposition 4.3

*Y*as follows:

*T*is given for the sake of exhaustiveness but has no impact on the results.

Data

| | | | | | |
---|---|---|---|---|---|---|

0 | 1 | 30 | 30 | 1 | 0.8 | 20 |

We observe negative autocorrelations that are equal to approximately \(-80\%\) for consecutive returns, so when \(\zeta =1\). These negative autocorrelations vanish out when the time lag between increments becomes large. This experiment is useful from an empirical viewpoint because the logarithmic returns of stock indexes often display negative autocorrelations that decrease in absolute value with the time lag.

## 5 Illustration

This section is devoted to the computation of the moments of tempered multistable processes and of a key risk indicator: Value-at-Risk. For a first empirical study on VaR and tempered multistable processes, see Lévy-Véhel and El Mekkedem (2013).

### 5.1 Moments

We start with the moments of the independent increments tempered multistable process. We readily obtain:

### Proposition 5.1

By construction, the moments of the field-based tempered multistable process are equal to those of a CGMY process (see, e.g., Küchler and Tappe 2013) at each time *t*, whereas those of the independent increments tempered multistable process involve integrals with respect to *Y*(*v*). Indeed, in the field-based case, the characteristic functions are identical at each time *t*, without the field-based tempered multistable process being a Lévy process. Thus, we have:

### Proposition 5.2

*b*, set respectively to 0.1 and 1. We observe that, excepting a hump at short times, kurtosis decreases with time for both values of

*b*. Kurtosis can display large values at small times, but also quite important values at intermediate times and the asymptotic decrease to the value of three can be quite slow. Hence, even for large values of

*t*, the process’ marginal distributions do not become identical to those of a Gaussian process.

We see that higher values of *b* are equivalent to a slower decrease of kurtosis with respect to the time horizon. So, for higher values of *b*, the process exhibits a higher persistence of extreme risks.

### 5.2 VaR

*F*under study. This latter quantity can be obtained from the characteristic function \(\Phi \) as follows:

*Y*as follows:

*Y*converges to zero with respect to time

*t*when \(b \ne 0\), so that the process marginals are asymptotically distributed as variance gamma random variables for both the dashed and dotted lines.

We now freeze the following values: \(a=0.8\) and \(b=0.1\), and we allow *G* and *M* to vary. We show in Fig. 4 the term structure of VaR for various levels of *G* and *M*. Note that high values of the *G* and *M* parameters correspond to small proportions of jumps of negative and positive size, respectively. Therefore, the plain curve, associated with \(G=20\) and \(M=40\), corresponds to an important proportion of negative jumps and a small proportion of positive jumps. Accordingly, Value-at-Risk is high in such a scenario. When we move to the dashed and dotted curves, we increase *G* and we decrease *M*, so we decrease the proportion of negative jumps in the process and we increase the proportion of positive jumps. Consistently, we see that the VaR term structure is decreased in these scenarios.

## 6 Conclusion

We have shown the correspondence between a series representation and a characteristic function representation—and the semimartingale property—for both field-based and independent increments asymmetrical tempered multistable processes. The form of these characteristic functions have allowed us to compute and study the autocorrelation function that is attached to the increments of such processes. Interestingly, these autocorrelation structures could be useful for the modeling of actual dependence properties for stock data. We have also computed the moments and the term structure of Value-at-Risk that are associated with tempered multistable processes. The features observed could be useful in risk management for explaining, for instance, the time diversification effect that is put forward by pension funds for massively investing in stocks in the long run.

As a natural extension of this article, more applied research should be conducted. For example, a statistical study on tempered multistable processes could be conducted and an algorithm for the estimation of the process parameters could be elaborated. It could also be interesting to compare the behaviors of various risk indicators, including, e.g., the tail conditional expectation, when such models are postulated.

## Notes

### Acknowledgements

The author wishes to thank Jacques Lévy-Véhel for many fruitful discussions on the topic of the paper and for introducing him to this field of research. The author also thanks Philippe Desurmont, Abdou Kelani, Mohamed Majri, François Quittard-Pinon, Andrea Roncoroni, Hubert Rodarie, Bertrand Tavin, and Christian Walter for their useful comments on the paper, which is the outcome of a collaboration with the SMA Group.

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