NORMAL VARIANCE-MEAN MIXTURES (I) AN INEQUALITY BETWEEN SKEWNESS AND KURTOSIS

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1 Available online at Adv Inequal Appl 0 0: IN: NOMAL VAIANCE-MEAN MIXTUE I AN INEUALITY BETEEN ENE AND UTOI ENE HÜLIMANN olters luwer Financial ervices eefeldstrasse 69 CH-8008 Zürich witzerland Copyright 0 Hürlimann This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original wor is properly cited Abstract: It is shown that the ratio of squared sewness to urtosis for the normal variance-mean mixture model is bounded above by the same ratio for the mixing distribution Illustrations include the generalized hyperbolic distribution the normal tempered stable distribution and a mixture model with log-normal mixing distribution eywords: sewness and urtosis inequality; generalized hyperbolic; normal tempered stable; normal-inverse Gaussian; hyperbolic; variance-gamma; hyperbolic sew t 000 AM ubject Classification: 60E5 6E5 6P05 Introduction The topic of moment inequalities is prominent in Probability and tatistics For example such inequalities determine the conditions under which random variables on a given range with given moments exist For given moments up to the order four the solution to this existence problem is found in Jansen et al 986 Hürlimann 008 Theorem I A general proof for the existence of random variables with nown moments up to a given order is in De Vylder 996 II It is important to discuss the relationship between sewness and urtosis in more specific but still general classes of distributions From a practical point of view the application of such inequalities to statistical inference is useful because it yields eceived July 5 0

2 ENE HÜLIMANN necessary conditions under which a given statistical model can be fitted to data In the realm of uantitative Finance where sewness and urtosis play a ey role one is interested in large classes of non-gaussian distributions which are able to supersede the ubiquitous Blac-choles model A first choice is the normal variance-mean NVM mixture model which has even been proposed as theoretical foundation for a semi-parametric approach to financial modelling eg Bingham and iesel 00 In particular the NVM model includes two five parameter families of distributions namely the generalized hyperbolic GH distribution and the normal tempered stable NT distribution Important members of the GH distribution are the normal-inverse Gaussian NIG the hyperbolic HYP the variance-gamma VG and the hyperbolic sew t HT The NT family also includes the NIG distribution As a main result we show that the ratio of squared sewness to urtosis for the NVM model is bounded above by the same ratio for the mixing distribution A short account of the content follows ection recalls the normal variance-mean NVM mixture model and briefly discusses the moment equations associated to it ection derives the general inequality between sewness and urtosis for the NVM class ection is devoted to selected examples and a comparative study of such inequalities Moment equations for the normal variance-mean mixture model The normal variance-mean NVM random variable is defined to be the mixture of a normal random variable of the type eg Barndorff-Nielsen et al 98 X Z where Z ~ N0 is a standard normal random variable is a non-negative mixing random variable with cumulant generating function cgf C t and Z are independent e assume that the first four cumulants of exist and summarize them into a vector A short hand notation for the random variable is

3 NOMAL VAIANCE-MEAN INEUALITY X ~ NVM One nows that the cgf of the NVM model is given by eg Feller 97 ection II5 C X t t C t t The mean standard deviation sewness and excess urtosis of X are denoted throughout by Through differentiation of the cgf one obtains the moment equations 6 A concrete specification of the cgf C t and the vector depends upon a multi-parameter vector m and equivalently to the above one writes m X ~ NVM ince the degree of freedom of the system is m a solution to it will necessarily depend upon m of the parameters i say i i m For short-hand notation we set In applications one parameter say m 0 will be a scale parameter such that the distribution and cgf of / m is m independent of m e assume this and rename the remaining parameters as The moment problem for the NVM model consists to find necessary and m m sufficient conditions so that the system of equations with nown and arbitrary but fixed in the feasible parameter space has a unique solution In the following a simple general strategy to solve is presented First of all taing into account the mean equation it suffices to determine Then one has Let / 0 be an auxiliary unnown parameter chosen such that the variance equation reads Inserting into the

4 ENE HÜLIMANN squared sewness equation one finds the expression imilarly the urtosis equation can be rewritten as 5 Now consider the functions of the parameter vector defined by 6 ince is a scale parameter these functions depend only on and 5 is equivalent to the following system of two equations in the unnowns : 7 In general the second equation is quadratic in and it has the unique positive solution 9 8 Insert this into the first equation in 7 to get an implicit equation for by fixed Therefore the NVM moment problem reduces to find the necessary and sufficient conditions such that the latter non-linear equation has a unique solution In case has been found

5 NOMAL VAIANCE-MEAN INEUALITY 5 numerically the remaining parameters are obtained from the definition of the parameter and the variance equation as ~ sgn 9 This strategy yields even closed-form solutions for the variance-gamma VG distribution reformulation of Theorem in Hürlimann 0 as well as for the normal inverse Gaussian NIG distribution A more complex example for which this strategy is successful is the generalized sew t GT distribution see Proposition 5 in Ghysels and ang 0 A further discussion of the general moment problem is postponed to later The focus of the present note is solely on a simple general relationship between sewness and urtosis in the NVM model An inequality between squared sewness and urtosis Under the technical conditions of Lemma below the ratio of squared sewness to urtosis for the NVM model is bounded above by the same ratio for the mixing distribution To derive this upper bound let us divide the first and second equations in 7 to get the expression p f f g q g p q To derive the maximum of this ratio an analysis of the function f is required Lemma If the quantities A and B are non-negative then the function f is strictly monotone increasing for all 0 Proof The derivative of the function f is of the form

6 6 ENE HÜLIMANN f ' It follows that g g' p g' { q } p f ' q h h A p q { B } 5 A B Now if A B 0 then f ' 0 for all 0 because 0 non-degenerate random variable 0 Theorem NVM inequality between sewness and urtosis Let X ~ NVM be a normal variance-mean mixture with finite first four cumulants of the non-negative mixing random variable Assume the conditions of Lemma hold Then the sewness and urtosis pair of X satisfies the inequality where is the sewness and urtosis pair of the mixing distribution Proof The equation 8 with its restriction on the urtosis implies that 0 ith Lemma and 6 the result follows easily as follows: max lim f elected examples and comparisons The inequality between sewness and urtosis is illustrated at three NVM mixture models namely the NVM model with log-normal mixing distribution the normal tempered stable

7 NOMAL VAIANCE-MEAN INEUALITY 7 NT distribution and the generalized hyperbolic GH distribution The GH has been widely discussed eg Eberlein and eller 995 Prause 999 Eberlein 00 Eberlein and Prause 00 Bibby and orensen 00 Eberlein and Hammerstien 00 McNeil et al 005 etc In particular the variance-gamma VG subfamily of the GH is very popular in Finance It has been introduced by Madan and eneta 990 see also Madan and Milne 99 Madan et al 998 Madan 00 Carr et al 00 Geman 00 Fu et al 006 etc Example : Log-normal lnn mixing distribution Let ~ ln N m be a log-normal mixing random variable with sewness and excess urtosis A calculation shows that e e e e e 6 e e e 6e 6 The maximum of the ratio is determined by the function x f x x e x x 6x 6 ince ' x x x 6x 6 f x 0 for all x one obtains x x 6x 6 max 0 f On the other hand one has x x x x x x 6x It follows that A x x 0 x and B x x 6x x x 6

8 8 ENE HÜLIMANN is non-negative over a limited range of volatilities 0 max In this situation the inequality will hold A first application of this mixing distribution is Clar 97 A recent application of the generalized log-normal mixing distribution within the context of the semi-parametric multivariate NVM mixture model is found in Cui 0 Example : Classical tempered stable CT mixing distribution The classical tempered stable CT subordinator ~ CT is determined by the cgf C t { t } 0 0 The corresponding NVM mixture is called normal tempered stable NT model A calculation shows that C t t j j where an empty product is one It follows that j particular j in 6 The conditions of Lemma {8 } A 8 0 B 0 are fulfilled Therefore the NT distribution satisfies the inequality 5 6 The special case of the normal inverse Gaussian NIG is well-nown from the

9 NOMAL VAIANCE-MEAN INEUALITY 9 literature eg de Beus et al 00 Appendix Ghysels and ang 0 Proposition Hürlimann 0 Appendix see also Example Example : Generalized inverse Gaussian GIG mixing distribution An important class of NVM models is the generalized hyperbolic GH distribution It belongs to the generalized inverse Gaussian GIG mixing random variable ~ GIG with cgf C t ln ln t t where x is the modified Bessel function of the third ind The domain of variation of the parameters depends upon three cases Case : generic GH distribution with 0 0 Case : variance-gamma VG distribution with Case : sew hyperbolic t HT distribution with In the limiting Case the mixing distribution reduces to a gamma distribution and in Case one has an inverse gamma distribution e discuss the three cases separately Case : generic case It is convenient to re-parameterize the GIG by setting 0 Then the cgf rewrites as t C t ln ln 6 t The special case is the normal inverse Gaussian NIG and is the hyperbolic HYP distribution Let us first discuss the simple NIG case

10 0 ENE HÜLIMANN Normal inverse Gaussian NIG The cgf 6 simplifies to t t C from which one gets the first four cumulants as well as the quantities 5 The conditions of Lemma 0 7 A 0 9 B are fulfilled Therefore the NIG distribution satisfies the inequality nown to be strict 5 General case For fixed the moments of the GIG mixing distribution are eg Paolella 007 ] [ E 7 Applying the relationships between moments and cumulants one obtains

11 NOMAL VAIANCE-MEAN INEUALITY } 6 { } { } { 8 6 The associated quantities 6 are independent from the scale parameter and given by } { 6 } { In general not much is nown about the analytical properties of the preceding quantities The HYP special case might illustrate what can happen Hyperbolic distribution HYP A numerical calculation with shows that the functions A and B are non-negative hence the conditions of Lemma are fulfilled An application of Theorem and a further numerical evaluation shows the HYP inequality max max max Case : variance-gamma VG The cumulants of the gamma distributed mixing random variable ~ are and one has

12 ENE HÜLIMANN The conditions A 0 B 0 are fulfilled Therefore the VG distribution satisfies the inequality 9 This inequality is sharp eg Hürlimann 0 Case of Theorem The present simple proof of this inequality is new A different derivation which is valid for the more general bilateral gamma BG convolution is Hürlimann 0 Theorem A Case : sew hyperbolic t HT The cumulants of the inverse gamma distributed mixing random variable ~ I exist only for and are given by It follows that as well as 5 9 A 0 B 0 ince the conditions of Theorem are fulfilled one obtains the inequalities max 8 max max Let us conclude this note with a brief comparison of some sewness and urtosis inequalities Comparison with the domain of maximum size ecall the general inequality between sewness and urtosis for arbitrary distributions on

13 NOMAL VAIANCE-MEAN INEUALITY namely which is sharp and attained at a biatomic random variable with support where Pearson 96 ilins 9 Guiard 980 Hürlimann 008 Theorem I A family of distributions which is able to model any admissible pair is the Johnson system introduced in Johnson 99 see also Johnson et al 99 George 007 among others Note that for distributions with a finite range A B A B the inequality extends to a two-sided inequality further information is found in Hürlimann 008 ChapI Clearly the domain of variation of sewness and urtosis for the selected examples is more restricted than the domain of maximum size prescribed by the inequality The selected examples have a maximum ratio of squared sewness to urtosis equal to / for the gamma mixing distribution VG special case of the GH This ratio is also closely approximated by a CT mixing distribution with 0 The maximum ratio of / coincides with the corresponding ratio for the five parameter bilateral gamma BG convolution in Hürlimann 0 Theorem A In fact the BG bound is sharp and attained for limiting left- and right-tail gamma distributions opcit Comparison of the VG HYP NIG and HT boundaries The domain of variation between sewness and urtosis is larger for the VG/HYP than for the NIG Indeed the NIG domain 5 is contained in the VG/HYP domain For the VG this result is also found in Ghysels and ang 0 p8 These authors also show that the NIG domain contains the feasible domain of the sew hyperbolic t HT called generalized sew t by them applications of the HT are found in Freca and Hopwood 98 Theodossiu 998 Aas and Haff 006 Hürlimann 009 etc

14 ENE HÜLIMANN Comparison of the VG with Hansen s generalized t Hansen 99 considers another generalization of the tudent t distribution simply called generalized t GT by Jondeau and ocinger 00 The boundaries of maximum sewness by given urtosis for the VG are delimited by the two curves In this situation let VG denote the maximum squared sewness as a function of The GT sewness and urtosis boundary has been determined in Jondeau and ocinger 00 ection Fig 5 note that the excess urtosis is obtained by subtracting the constant from the expression in ection Let GT denote the corresponding maximum squared sewness The GT domain is contained in the VG domain for urtosis higher than some relatively moderate value: VG GT 0 VG GT 0 77 EFEENCE [] Aas and H Haff The generalized hyperbolic sew tudent's t-distribution Journal of Financial Econometrics [] OE Barndorff-Nielsen J ent and M orensen Normal variance-mean mixtures and z distributions Int tatist eviews [] P de Beus M Bressers and T de Graaf Alternative investments and ris measurement 00 UL: wwwactuariesorg/afi/colloquia/maastricht/debeus_bressers_degraafpdf [] BM Bibby and M orensen Hyperbolic processes in finance In: achev Ed Handboo of Heavy Tailed Distributions in Finance 00-8 Elsevier [5] NH Bingham and iesel emi-parametric modelling in finance: theoretical foundations uantitative Finance 00-0 [6] P Carr H Geman DB Madan and M Yor The fine structure of asset returns: an empirical investigation Journal of Business [7] P Clar A subordinated stochastic process model with finite variance for speculative prices Econometrica [8] Cui emiparametric Gaussian variance-mean mixtures for heavy-tailed and sewed data IN

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