Extreme Distribution of a Generalized Stochastic Volatility Model,

(1)

HAL Id: halshs-00188535

https://halshs.archives-ouvertes.fr/halshs-00188535

Submitted on 17 Nov 2007

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

Extreme Distribution of a Generalized Stochastic

Volatility Model,

Aliou Diop, Dominique Guegan

To cite this version:

Aliou Diop, Dominique Guegan. Extreme Distribution of a Generalized Stochastic Volatility Model,.

South African Journal of Statistics„ 2003, 37, pp.127 - 148. �halshs-00188535�

(2)

stochastic volatility model AliouDIOP a;b y DominiqueGUEGAN c a

UFR de Mathematiques Appliquees et d'Informatique, B.P. 234

Universite Gaston Berger, Saint-Louis, Senegal,

e-mail : adiop@ugb.sn

and

b

Universite de Reims, Laboratoire de Mathematiques, UPRESA 6056

B.P. 1039, 51687 Reims Cedex2 France,

aliou.diop@univ-reims.fr

c

E.N.S. Cachan, GRID UMR CNRS C8534,

61 Avenuedu President Wilson, 94235, CachanCedex, France,

e-mail : dguegan@wanadoo.fr

Abstract

WegivetheasymptoticbehavioroftheextremevaluesofStochasticVolatility

model (Y

t )

t

when the noise follows a generalizederror distribution (GED). This

classofdistributionwhoseapresentationhasbeenstudiedinBoxandTiao(1973)

forinstanceincludesinparticulartheGaussianlaw. Inthispaper,weshowthatin

thegeneralcontext,thenormalizedextremesofalog-transformationof(Y

t )

t

con-verges in distribution to the double exponential distribution. Weinvestigate the

importance of the dierent assumptions using Monte Carlo simulations. We also

dealwith thenitedistancebehaviorofthenormalizedmaxima. Thein uence of

theparametersofthemodelsisdiscussed.

keywords: Extremevaluetheory,tailbehavior,stochasticvolatilitymodel,GED

distribution.

runningtitle: Extremesofstochasticvolatility

y

Correspondingauthor.

(3)

The class of stochastic volatility models has its roots and applications in

nance and nance econometrics. Indeed, volatilityplays a central role in

the analysis of a lot of phenomenon in these domains. There exists a lot

of versions of stochastic volatility models in the literature. Here we are

interested bya discrete time version (t2 Z)of volatilitymodelintroduced

rstbyTaylor(1986). ThismodelappearsasaparticularmodeloftheSARV

modelintroducedbyAndersenin1994. Theunivariatevolatilitymodel(Y

t )

t

thatwewillconsiderherecan be denedbythefollowingequation :

Y t =exp( t 2 )" t (1) with t = 1 X j=0 j Z t j (2)

where is a positive constant, ("

t )

t

a sequence of independent and

iden-tically distributed (iid) random variable (r.v.) and (Z

t )

t

a sequence of iid

Gaussianr.v. withmeanzero, variance

2

Z

andindependentofthesequence

(" t ) t . The parameters( j

) aresuchthat

P j0 j j j 2 <1. Inthe literature,

generally it is assumed that "

t

and Z

t

are not correlated with each other

forall t. Forsimplicityhere we willassume independencebetween thetwo

sequences. Themodelcanpickupthekindofasymmetricbehaviorwhichis

oftenfoundinstockprices,andanegative correlationbetween"

t

andZ

t

in-ducesaleverageeect. Thisexplainswhypractitionersoftenusethismodel.

Thebehavioroftheautocorrelationfunctionandthepower-momentsforthe

processdenedby(1)-(2) arewellknownwhen"

t

followsaGaussianlawor

aStudentlaw,see forinstanceHarvey(1993), Taylor(1986), Ghyselset al.

(1996) orShepard(1996).

Recently banks, insurance companies faced with questions concerning

ex-tremal rare events. In insurance, these extremal events may clearly

corre-spond to individualclaims which by far exceed thecapacity of a single

in-surance company. In nance these extremal events can present themselves

spectacularlywhenevermajorstockmarketcrashes like theonein1987

oc-curs forinstance. The rst preoccupation forthese institutionsis to dene

a well-functioningrisk management and controlsystemto cautionto these

problems. Thus, they need stochastic methodology forthe construction of

variouscomponentsofsuchtools.

In that context, it is important to know, for instance the extremal

distri-bution of the dierent process which are used, until now, to characterize

(4)

process(Y t

) t

denedby(1)-(2)whenthenoise("

t )

t

followsageneralized

er-rordistribution (GED). Thisclass ofdistribution whosea presentationhas

beenstudiedinBoxandTiao(1973)forinstance,includesinparticularthe

Gaussianlaw. Thislast casehas beenstudiedbyBreidt and Davis (1998).

Here we show, in a general context specied in the next section, that the

normalizedextremesofalog-transformationof(Y

t )

t

convergeindistribution

tothedoubleexponentialdistribution. Sinceweareinterested byafeasible

application of these results on real data, we investigate the importanceof

thedierentassumptionsusingMonteCarlo simulations. Thus,weareable

to showthatsome assumptionsarenotsoimportant intherealitycontext.

Nevertheless ourresultsobtainedinanasymptoticcontext can be verybad

at nite distance. This last situation is classical in extreme value theory,

indeed,someresultsobtainedinanasymptoticcontextarenotalways valid

withnitesamples. Obviously,this impliessome diÆcultiesto use directly

these resultsinview to constructa risk management theory.

Ourpaperisorganizedinthefollowingway: Sectiontwocontainsstatistical

propertiesoftheprocess(1)-(2)when ("

t )

t

followsaGEDdistribution. The

extremal behavior of a log-transformation of (Y

t )

t

is presented in Section

three. InSectionfourwefocusonnitesamplesbehaviorforthenormalized

maximum of thelog-transformation of (Y

t )

t

. Section ve is devoted to the

conclusion. The proofsarepostponedinanAppendix.

2 Stationarity of the process (Y

t )

t .

In this section we precise some properties of the autocorrelation function

and the power-moments of the process (Y

t ) t dened in (1)-(2) when (" t ) t

followsa GED distribution. TheGED densityisdenedby

f " (x)=c 0 exp( kjxj ); (3) withc 0

>0, >0 and k>0. Theexpression(3) canalso be written:

f " (x)= exp( 1 2 j x j ) 2 1+ 1 ( 1 ) ; >0 (4) with =[ 2 2 ( 1 ) ( 3 ) ] 1 2 :

Thisclass ofdensitycontainsthenormal density( =2),the Laplace

den-sity( =1) andhastheuniformdensityasalimit( !+1). Itwasrstly

(5)

t t

adensitydependsonthetail-thicknessparameter . Forinstance,if =2,

then"

t

N(0;1),whilefor <2thedistributionhasthickertailsthanthe

Gaussiandistribution. Nowweprecisesomeresultsconcerningthemoments

oftheprocess(1)-(2) assumingthat thedistributionof("

t )

t

isgiven by(3).

Proposition 1 (Covariance and strict stationarity)

If the process (Y

t )

t

follows the model (1)-(2) driven by a GED noise ("

t )

t

withindex then,

i)the power-moments of the process (Y

t ) t are givenby E(Y r t )= 8 < : r ( 1 ) r 2 1 ( r+1 ) ( 3 ) r 2 exp( r 2 8 2 ) if r iseven 0 if r isodd (5) and var(Y r t )= 8 > > > > < > > > > : 2r ( 1 ) r 2 ( r+1 ) 2 ( 3 ) r exp( r 2 4 2 )[K r exp( r 2 4 2 ) 1] if r is even 2r ( 1 ) r 1 ( 2r+1 ) ( 3 ) r exp( r 2 2 2 ) if r is odd (6) where K r = ( 1 ) ( 2r+1 ) ( r+1 ) 2 : (7)

ii)The excesskurtosis of Y

t is given by =3( K 2 3 exp( 2 ) 1); (8) where K 2 isgiven by (7) with r=2.

iii)The r-thautocorrelation function isequal to:

(r) h = 8 > > > < > > > : exp( r 2 4 (h)) 1 K r exp( r 2 4 (0)) 1 if r is even exp( r 2 4 ( (h) (0))) if r is odd (9) where

(:) is the autocovariance function of the stationary process (

t )

t .

Hence, the process (Y

r

t )

y

(6)

This propositionensures that the process (Y t

) t

dened by(1)- (2) and the

powers ofthisprocessare stationary. We aregoingto use thisresult inthe

nextsection.

3 Asymptotic behavior of the maxima of the

pro-cess (X

t )

t .

In this section we study the asymptotic behavior of the distribution of a

classof transformationsofthe process (Y

t )

t

. Firstof all,we dene 8t,

X t =ln( Y t ) 2 = t +ln" 2 t : (10) Nowwe put: t =ln" 2 t forall t2Z; (11)

thustheprocess (X

t )

t

becomes asum oftwo independentprocesses :

X t = t + t : (12)

Themodel(12)canbeconsideredasaregressionmodelwithlog-GEDnoise

( t

). The asymptotic behavior of the maxima for a regression model with

particular noises has been studied by Diop and Guegan (2000). We refer

alsotoHorowitz(1980),BalleriniandMcCormick(1989)andNiu(1997)for

regression models involving non stationarities. When ("

t )

t

ischaracterized

byaGEDdistribution,wedenoteF thedistributionfunctionof(X

t )

t

. First

ofall, we studytheasymptoticbehaviorof thedistributionF.

Proposition 2 Let (X

t )

t

be the process dened by (12), assume that the

distributionof ("

t )

t

is (3), then the asymptotic behavior of F is :

F(x) = PfX t >xg (13) p exp n x 2 2 2 + axlng(x) 2 2x(1+R ) 2 +Alng(x) bxlng(x) 2 g(x) 2ln 2 g(x) 2 2 cx 2 g(x) + 1 8 2 ( 1 2 1) 2 )+ 1 2 ( 1 2 1 )+C+O( ln 2 g(x) g(x) ) o where a= 2 ; b=( 2 ) 3 ; c= ( 2 ) 3 R 2 2 2 + 2 ; R=lnk 2 ; A= 4 2 2 (R+1)+ 1 3 2 ;

(7)

C = 2 2 2 R 2 +( 1 2 1 4 2 2 )R 1 + 1 2 +ln 4 ( 2 )c 0 p k 2 ; and g(x)= 2x+1 1=2:

Proof : SeetheAppendix.

Remark : When the noise ("

t )

t

follows a Gaussian distribution, then

c 0 = 1 p 2 , =2, k = 1 2

and we ndthe resultsof Breidt and Davis (1998,

p.667). We dene now M n = max(X 1 ;X 2 ;:::;X n

), n 2, the maxima of the

process (X

t )

t

. Here,we investigate the asymptoticdistributionof M

n . Theorem 1 Let(X t ) t

beaprocessdenedin(12). Assumethatthe density

of ("

t )

t

is given by (3). Assume also that

(h)lnh ! 0, as h ! +1,

where

(h)denotesthe autocorrelation functionofthe process(

t ) t dened in(2) and 1 2

. Thenthereexist normalizingconstants (a

n >0) and (b n ) such that P[a n (M n b n )x] ! exp( e x ); (14) where a n = (2lnn) 1 2 ; d n =(lnn) 1 2 and b n =c 1 d n +c 2 lnd n +c 3 +c 4 lnd n d n + c 5 d n ; with c 1 =(2 2 ) 1 2 ; c 2 =a; c 3 =a[ 3 2 ln2+ln 2 ln 2 2 1]; c 4 =( 1 3 2 ) p 2 ; c 5 = 1 2(2 2 ) 1 2 n a 2 +(1 R+ 1 2 ln2 2 ) 2 (1 a)+( 2 4 ( 1 2 1 )+2+2 2 )( 1 1 2 ) +(3 a) 2 lna 2 2 ln( a 2 c 0 p p k )+ 2 ln(2) o :

Proof : SeetheAppendix.

Remark : We note that the coeÆcient a

n

in (3.12) of Breidt and Davis

(1998) obtained with a Gaussian law for ("

t )

t

seems to contain a

compu-tational error which is also highlighted by simulation results in the next

(8)

(X t

) t

using nite samples.

Inthissectionwe willstudythebehaviorof thedistributionofthemaxima

of (X

t )

t

given by (11)-(12) for dierent versions of the process (

t )

t

. We

precise now these versions. First we consider the iid stochastic volatility

modelwith aGED noise("

t ) t : theprocess( t ) t isdenedby: t =Z t ; fZ t gisan iidN(0; 2 ): (15)

Secondly we studythe rst-orderautoregressive stochastic volatility model

(ARSV) witha GEDnoise("

t ) t : the process ( t ) t isdened by: t = t 1 +Z t ; fZ t giidN(0; (1 2 ) 2 ); (16)

with jj < 1. Finally, we consider the long memory stochastic volatility

model (LMSV), (see Breidt et al. (1998)) with a GED noise("

t ) t and the process ( t ) t is denedby: (1 B) d t =Z t ; fZ t giidN(0; 2 2 (1 d) (1 2d) ); (17)

withjdj<1=2. Inthefollowing,wehighlightthein uenceoftheparameters

on the limiting distribution for these dierent models. We study also the

importanceofthe dierentassumptionsmade inthetheorem 1.

4.1 Assumptions

Theresult statedin thetheorem 1 requiresthe condition

(h)lnh!0 as

h !1. We illustrate thisconvergence for nitesamples. The model (15)

corresponds to an iid process, thus

(h) = 0, h 6= 0. The model (16) is

an AR(1) process, thus

(h) decreases with the rate of

jhj

. The model

(17)isaFARIMA(0 ,d, 0)process: itisknownthat

(h)C(d)h

2d 1

,as

h ! 1, where C(d) is some constant which depends on d, thus the speed

ofconvergence of

(h)lnh towards 0 isvery slow.

Now we investigate empirically the behavior of

(h)lnh, as h ! 1 for

the model (16) and (17). In Figure 1, the graphes (a), (b), (c), (d)

cor-respond to the model (16) for dierent values of the parameter ( =

0:2; 0:5; 0:95; 0:99). The graphes(e), (f), (g),(h) correspond tothe model

(17)fordierent valuesoftheparameter d (d=0:1; 0:2; 0:3; 0:4). Forthe

model (16), when we are far from the non stationary case ( = 0:2; 0:5),

the decreasing of

(h)lnh to zero is very fast. Now when d is small

(d = 0:1; 0:2) the decreasing of

(h)lnh is not so fast than the one

ob-tainedwiththeAR(1) model. Butthisdecreasingbecomesquickerassoon

asd=0:3. Ford=0:4, whichcorrespondsto astronglongmemory

behav-ior,

(9)

In view to compare our results with those of Breidt and Davis (1998),

we use in our simulations the same value as them for

2 ( 2 = 2:3976,

0.6933; 0:0953), for(=0:95) and ford (d=0:4).

In Figures 2-6, we give in solid lines the empirical distribution functions

for1000 normalizedmaxima[a

n (M

n b

n

)]obtainedfrom 1000 replications

of samples of size 1000 and in dotted lines the limiting double

exponen-tial distribution. The quality of the convergence depends on

and a n ( the coeÆcient a n is given by a n = (2lnn) 1 2

). The in uence of the

tail-thickness parameter of the GED distribution is also studied in Figures

2-4. Wepresentthelimitingdistributionwhen =2(Gaussiancase), =1

(Laplace distribution which is thicker tail than the Gaussian distribution)

and = 3 (thinner tail than the Gaussian distribution). The asymptotic

behavior of the normalized maxima is closely related to the value of

as

shown in Figures 2-4. We also study in Figures 5-6 the sensitivity of the

convergence obtained in (14), with respect to the parameters and d

re-spectively when (

t )

t

follows the models (16)-(17). We detail now these

results.

4.2.1 Gaussian noise ( =2)

The behavior of the asymptotic distribution of a

n (M n b n ) when we use

a Gaussian driven noise ("

t )

t

is given in Figure 2. In each row,

2 is con-stant ( 2

=2:3976, 0.6933; 0:0953). First column corresponds to the iid

stochasticvolatility modelwith (

t )

t

denedin (15), second columnto the

ARSVmodelwith (

t )

t

dened in(16), thirdcolumn to the LMSV model

with ( t ) t dened in (17). When 2

= 2:3976, the approximation to the

empiricaldistributionfunctionofa

n (M

n b

n

); n=1000 bythedouble

ex-ponentialdistributionisverypoorwhateverthedependencestructuretaken

for (

t )

t

. For nite samples, the approximation gives better results when

2 =0:6933 and 2

=0:0953 asshowninthe lowerpanelsof Figure 2.

4.2.2 GED noise with =3

We turnnowto the caseof thestochastic volatilitymodel(12) driven with

a noise("

t )

t

characterized by tail-thicknessparameter =3, see Figure 3:

thesolidlinesrepresenttheempiricaldistributionofthenormalizedmaxima

a n (M n b n

), with n = 1000 and the dotted lines represent the double

exponential distribution. Looking at the nine graphes of the gure 3, we

see that for nite samples the approximation of the empirical distribution

with the double exponential distribution is better when (

t )

t

follows the

model (17) with d =0:4 and

2

=0:6933. This reveals that thecondition

(10)

althoughinthat lastcase, thecondition

(h)lnh!0 ash!1,fails.

4.2.3 Laplace noise =1

We now abstract from the thinner tailed case and consider a stochastic

volatilitymodeldrivenbyLaplacenoise =1. TheresultisgiveninFigure

4. By reading across the rows of this picture, the panels again show the

in uenceof thevalue of the parameter

2

. More

2

is small, better is the

rateofconvergenceofthenormalizedmaximaof(X

t )

t

denedin(12)tothe

double exponential distribution. The dependence structure in (

t )

t seems

notto have a great in uence on thisconvergence. A comparaison between

Figure2 andFigure 4shows clearlythattheconvergenceof thenormalized

maximaisbetterwhenthedrivennoise("

t )

t

isGaussian. Despitethechange

ofthescaleinthex-axisofFigure4,thecomparaisonbetweenFigure3and

Figure 4 corresponding respectively to = 3 and = 1 reveals a better

approximation when =3.

4.2.4 In uence of the autoregressive parameter

Here we assume for simplicitythat the noise ("

t )

t

follows a Gaussian law

and we investigate the in uence of the autoregressive parameter to the

asymptotic behavior of a n (M n b n

) for the process (X

t )

t

dened in (12)

whentheprocess(

t )

t

follows themodel(16). Todeterminethesensitivity

oftheresultswithrespectto thisparameter,dierentalternative valuesfor

have been used : =0:2, =0:95 and =0:99. The results are given

inFigure5. By lookingacrosstherows of Figure5,wenotethat more is

small,better istherateofconvergence. Whentheautoregressiveparameter

is equal to 0.99, the approximation is dramatically poor whatever the

valueof

asshowninthelastcolumnofFigure5. Apossibleexplanation

isthefactthattheunderlyingprocess tendsto become nonstationarywhen

the parameter is close to 1. It seems that this situation needs other

investigations. Thiscasehasbeenalreadypartiallystudied,seeforinstance,

Horowitz (1980),Balleriniand Mc Cormick(1989) and Niu (1997).

4.2.5 Importance of the long memory parameterd.

Figure6showsthein uenceofthelongmemoryparameterdto the

conver-gence of the normalized maxima of the process (X

t )

t

dened in (12). For

sake of concisness, we assume that the noise ("

t )

t

follows a Gaussian

dis-tribution. We compare theempirical distribution function of a

n (M n b n )

whentheprocess(X

t )

t

followstheLMSVmodelandthedoubleexponential

distribution when d = 0:1, d = 0:2 and d = 0:4. It seems that the values

of the parameter d have small in uence on the convergence (14). With

-nite samples, when d = 0:4 the condition

(11)

other cases when

is equalto 0.0953. We notethat when

is small(so

doesa

n

),the asymptoticbehaviorof thenormalized maximumis thesame

whateverthevaluesofd.

4.2.6 Tail comparaison

Wegivehereabriefcomparisonbetweenthetailbehaviorofthetwo

compo-nentsoftheprocess(X

t )

t

denedin(12). Thedensityofther. v.

t

dened

in (11) when the r.v. "

t

follows a GED distribution with shape parameter

isgiven by h(x)=c 0 exp( x 2 ke 2 x ):

Figure 7 shows the tails of the distribution of the Gaussian linear process

( t

) t

dened in (2) and the noise (

t )

t

dened in (11). The three panels

correspond respectively to 2 = 2:3976, 2 = 0:6933 and 2 = 0:0953. When 2

= 2:3976, the Gaussian upper tail dominates the tail of

t for

=0:5; 1; 2; 3: When

2

=0:0953, theGaussian uppertailis dominated

bytheuppertailof

t

. Finallywhen

2

=0:6933, all thetails aretendency

to have the same behavior as the Gaussian upper tail. It is evident by

lookingaccrossthepanelsofFigure 7thatthelowertailof

t

dominate the

Gaussiantailwhateverthevalue of

2

.

5 Conclusion

In this paper, we get the asymptotic distribution of the maxima of a

log-transformationofaprocess(X

t )

t

drivenbyastochasticvolatilitymodeland

we study theempirical behavior of its asymptotic distribution . We point

outthein uenceof thetail-thicknessparameter of thedrivennoise("

t )

t .

Our ndings are that thechoice of ,

2

and the dependence structureof

( t

) t

doesaect thegoodness-of-tof thelimitingdistribution.

In practice, the results stated in theorem 1 allow us to calculate quantile

riskmeasuresusingthemaximumblockmethod,see forinstance Embrecht

et al. (1997). However innite samplescare must be taken because of the

poornessoftheapproximation. Thisisoneofthelimitationstoevaluatefor

instancetheValue atRiskforreal datausingthispartoftheextremevalue

theory. It wouldbe interesting to investigatedtheasymptotic behaviorfor

more complicatednonstationary models, this willbe done in a companion

(12)

Proof of Proposition 1 : We can establish usingDevroye (1986) p. 175

thattheGED noise("

t ) t denedin(3) satises " t d =k 1 VG 1 (18) where d

= denote the equality in distribution, V a uniform r.v. on [ 1; 1]

and G a r.v. following a Gamma distribution (1+

1

; 1), independent of

V. The identity (18) is used in Section 4 to draw samples from a GED

distribution. From(18), it can beshown that

E(" r t )= 8 > > < > > : k r r+1 ( r+1 +1) ( 1 +1) ; ifr iseven 0 ifr isodd. (19) i) Since (" t ) t

has a symmetric distribution and (

t )

t

dened in (2) is a

Gaussian linear process independent of ("

t )

t

, (5) and (6) are obtained

di-rectlyfromthe propertiesof thelognormal distribution.

ii) The excess kurtosis of (Y

t ) t dened by E[Y 4 t ] E[Y 2 t ] 2

3 follows directly from

(5).

iii)The r-thautocorrelationfunctionis denedby

(r) h = E[Y r t Y r t+h ] E[Y r t ] 2 E[Y 2r t ] E[Y r t ] 2 : (20)

From (1)and usingagainthe properties oflognormal distribution,wehave

forallr : E(Y r t Y r t+h )= 2r exp( r 2 4 ( 2 + (h)))E(" 2r t ): (21)

Theremainder ofthe prooffollows directly from(5) and (21).

Proof of Proposition 2: We avoid here to go throughthe details of the

computations which need some great eorts. We focus only on the

im-portant steps. Breidt and Davis (1998) use a Tauberian argument in the

Gaussian case to express the asymptotic approximation to the tail

distri-bution of (X

t )

t

, we use it again. In step 1, we give some expansions of

the two derivatives m() and S() of the log-moment generating function

of (X

t )

t

dened in (12). In the second step, we dene an inverse function

m 1

(:)ofm()andwejustifytheinversenotation. Someusefulexpansions

of functions in terms of m

1

(x) are also provided. In the third step, we

(13)

Step 1 :

We give here theexpansions of the two rst derivatives of the log-moment

generatingofthe process (X

t )

t

denedin(12).

Thelog ofthe moment generatingfunctionof (

t ) t isgiven by: lnEexp( t )=ln2c 0 ln 2+1 lnk+ln ( 2+1 ):

Thus, thelogof themoment-generating functionof (X

t ) t is equalto lnC() = 2 2 2 +ln2c 0 ln 2+1 lnk+ln ( 2+1 ) (22) = 2 2 2 g()lnk g()+g()lng()+ln 2 p 2 ( 1 2 )c 0 p k +O( 1 ): where g()= 2+1 1 2 :

Therst two derivativesoflnC() are

m() = d d lnC()= 2 2 lnk+ 2 ( 2+1 ) = 2 2 lnk+ 2 lng()+O( 1 2 ) (23) and S 2 ()= d 2 d 2 lnC() = 2 +( 2 ) 2 0 ( 2+1 ) = 2 +O( 1 ); (24) where (:)and 0

(:)arethedigammaandtrigammafunctionsrespectively.

Step 2 : We set m 1 (x)= g(x) 2 2 alng(x) 2 + blng(x) 2 g(x) + c 2 g(x) d 2 ; (25) where d= 2 ln(k 2 ): Itfollows that lng(m 1 (x))=ln( g(x) 2 ) 2alng(x) g(x) + D 1 g(x) +O( ln 2 g(x) g(x) 2 ); (26)

(14)

D 1 = 2d+ 2 2 2 :

After somecomputations, using(23), (25)and (26), weget

m(m 1 (x))=x+O( ln 2 g(x) g(x) 2 ) (27)

which justiestheinverse notation.

Step 3 :

WeestablishtheasymptoticnormalityforthenormalizedtransformEsscher

ofthe distributionF(see Feiginand Yashchin (1983)). Wewrite

F(m()) exp( m())C() (2) 1 2 S() : (28)

Usingtheinverse of m denedin(25), we have

F(x) exp( xm 1 (x))C(m 1 (x)) (2) 1 2 m 1 (x)S(m 1 (x)) : (29)

Theremainder ofthe proofneeds thefollowingexpansions.

2 2 (m 1 (x)) 2 = 2 g(x) 2 8 2 + a 2 ln 2 g(x) 2 2 ag(x)lng(x) 2 2 + ( b+2ad)lng(x) 2 2 dg(x) 2 2 + d 2 + c 2 2 +O( ln 2 g(x) g(x) 2 ); and g(m 1 (x))lng(m 1 (x)) = ln( 2 ) 2 g(x)+D 2 lng(x)+ g(x)ln(g(x)) 2 2aln 2 g(x) 2 + D 1 2 (1 ln 2 )+O( ln 2 g(x) g(x) ); where D 2 = ad+a 2 ln 2 2a 2 :

We concludetheproof byusing(24), (25), (27), (29) andthese expansions.

Proof of theorem1: Usingtheproposition2,weget

F n (u n ) !exp( e x ); x2IR (30) whereu n =a 1 n x+b n

. Now, we needto show that

jP(M n u n ) F n (u n )j !0 as n !+1: (31)

(15)

(1983, page 81)) and follows the great lines of Breidt and Davis (1998). First,weestablish jP(M n u n ) F n (u n )jnK n X i=1 j (i)j(Eexp (u n 1 ) 2 2 2 (1+j (i)j) ) 2 (32)

where K is a positive constant,

(:) the autocorrelation function of the

process ( t ) and t =ln" 2 t

. Using the asymptotic relationship for the tail

probabilityof theGED distributionfor

1 2 ,givenby F " (x) c 0 k x 1 e kx as x !+1;

then, itcan beshownthat

Eexp (u n 1 ) 2 2 2 (1+j (i)j) K 0 (lnn) [ 1 2(1+j(i)j) ] E( F 1 1+j(i)j " ( 1 (u n 1 )))+o(n 1 ); (33)

where[:]standsfortheintegerpartandK

0

apositiveconstant. ByJensens's

inequalityandthe relation(30), we have

(Eexp (u n 1 ) 2 2 2 (1+j (i)j) ) 2 K 00 (lnn) 1 1+j(i)j n 2 1+j(i)j (34) whereK 00

isapositiveconstant. Theremainderoftheproofissimilartothe

one of Breidt and Davis(1998, page671); see also Leadbetter et al. (1983,

page86).

References

Andersen,T. G.,Stochasticautoregressive volatility: aframework for

volatilitymodelling,Mathematical nance, 4(2), (1994) 75-102.

Ballerini,R.and Mc Cormick,W.P., Extremevaluetheory forprocesses

withperiodicvariances. Comm. Statist. Stochastic Models, 5,(1989),

45-51.

Box, G.E.P.and Tiao, G.C., BayesianInferenceinStatistical Analysis.

Reading, MA:Addison-Wesley. (1973).

Breidt,F. J., Crato, N.and DeLima, P., Thedetection and estimationof

long memoryinstochastic volatility. J.Econometrics,83, (1998)

325-348.

Breidt,F.J. and Davis,R.A. Extremes of Stochastic volatilitymodel. The

Annals of Applied Probability ,8 (3)(1998) 664-675.

(16)

Devroye, L., Non Uniform Random Variate Generation, Springer, New

York,(1986).

Diop A.,GueganD. Asymptotoc behavior for the extreme values of a

regression model, Preprint00-11, Universityof Reims,France,(2000).

Embrecht P,KluppelbergC.,Mikosch T.,Modelling extremal events for

insuranceandnance,Springer,ApplicationsofMathematics,33,(1997).

Feigin, P.D. and Yashchin,E., Ona strongTauberianresult. Z. Wahrsch.

Verw. Gebiete 65,(1983) 35-48.

Ghysels,E.,Harvey,A.andRenault, E.Stochasticvolatility in Handbookof

Statistics, Vol. 14, "Statistical Methodsinnance",J. Wiley,(1996).

Harvey,A.C.Longmemoryinstochasticvolatility,Discussionpaper,L.S.E.,

London,(1993).

Horowitz,J., Extremevaluesfora nonstationarystochastic process : an

application to airqualityanalysis. Technometrics, 22, (1980), 469-478.

Leadbetter, M. R.Lindgren,G. andRootzen, H., Extremes and Related

Properties of Random sequences and Processes,Springer, NewYork,

(1983).

Niu,X-F.Extremevaluetheoryforaclassofnonstationarytimeserieswith

applications. J.of Appl. Proba.,7, (1988), 508-522.

Shepard, N. Statistical aspect of ARCH and stochastic volatility, in Time

SeriesModelsinEconometrics,Financeandotherelds,eds. Cox,

Hink-ley,Barndor-Nielsen, Chapmanand Hall, Monographeson Statistics

and appliedprobability,65, (1993) 2-67.

Subbotin, M.T., On the law of frequency errors. Matematicheskii Sbornik,

31, (1923) ,296-300.

(17)

0

200

400

600

800 1000

a

0.000

0.005

0.010

0.015

0.020

0.025 phi=0.2

0

200

400

600

800 1000

b

0.00

0.05

0.10

0.15 phi=0.5

0

200

400

600

800 1000

c

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4 phi=0.95

0

200

400

600

800 1000

d

0.0

0.5

1.0

1.5

2.0

2.5 phi=0.99

0

200

400

600

800 1000

e

0.1

0.2

0.3 Your Text

d = 0.1

0

200

400

600

800 1000

f

0.1

0.2

0.3

0.4

0.5 d = 0.2

0

200

400

600

800 1000

g

0.000

0.225

0.450

0.675

0.900 d = 0.3

0

200

400

600

800 1000

h

0.0

0.5

1.0

1.5 d = 0.4

Figure 1: Rate of convergence of

(h)lnh to 0 as h ! 1 for dierent

valuesof theautoregressive parameter (= 0:2; 0:5; 0:95; 0:99) (upper

plots : a, b, c, d) and dierent values for the long memory parameter d

(d=0:1; 0:2; 0:3; 0:4) (lower plots: e, f, g, h). The graph (h)shows the

failure of theconvergence for nitesamples in the long memory case with

(18)

-10

-5

0

5

10 a

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

11

1

111

11

111 1111111

11111111111111111111111111

-10

-5

0

5

10 d

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

11

111 1111111

11111111111111111111111111

-10

-5

0

5

10 g

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

11

111 1111111

11111111111111111111111111

-10

-5

0

5

10 b

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

11

111 111111111111111111111111111111111

-10

-5

0

5

10 e

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

11

11 1111111111

11111111111111111111111111

-10

-5

0

5

10 h

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

11

111 111111111111111111111111111111111

-10

-5

0

5

10 c

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

111

11

111 111111111111111111111111111111111

-10

-5

0

5

10 f

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

111

11

111 111111111111111111111111111111111

-10

-5

0

5

10 i

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

111

11

111 111111111111111111111111111111111

Figure 2: Comparaison between the empirical distribution function (solid

lines)for1000 normalized maximaand thedoubleexponentialdistribution

(dotted lines) for the process (X

t )

t

dened in (12) when the driven noise

(" t

) t

is Gaussian. In rst line for the three graphes

2 = 2.3976, in the second line 2

=0.6933,in thethirdline

2 =0.0953. a, b,c: t IID; d,e,f : t AR (1),=0:95; g,h,i: t FAR IMA(0;d;0), d=0:4.

(19)

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

111

11

111 11111111

1111111111111111111111111

a

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8 111111111111111111111111111

1111111111111111111111111111111111111111111

1

11

111 111111

d

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

11

11 1111111111

11111111111111111111111111

g

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

111

11

111 111111111111111111111111111111111

b

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

11

111 1111111

11111111111111111111111111

e

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

11

111 111111111111111111111111111111111

h

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

111

11

111 11111111

1111111111111111111111111

c

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

11

111 11111111

1111111111111111111111111

f

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

111

11

111 1111111

11111111111111111111111111

i

Figure 3: Comparisonbetween the empiricaldistribution function and the

doubleexponentialdistributionforthe process (X

t )

t

dened in(12)with a

GED driven noise("

t )

t

with =3. Inrst lineforthe three graphes

2

=

2.3976,inthesecondline

2

=0.6933, inthethirdline

2 =0.0953. a,b,c: t IID;d,e,f: t AR (1),=0:95; g,h,i: t FAR IMA(0;d;0), d=0:4.

(20)

-20

-10

0

10

20 a

0.0

0.2

0.4

0.6

0.8

1.0 11111111111111111111111111111111111111111111111

1

11 11111111111111111111111111111111111111111

-20

-10

0

10

20 d

0.0

0.2

0.4

0.6

0.8

1.0 11111111111111111111111111111111111111111111111

1

11 1111111111

1111111111111111111111111111111

-20

-10

0

10

20 g

0.2

0.4

0.6

0.8

1.0 11111111111111111111111111111111111111111111111

1

111 11111111111111111111111111111111111111111

-20

-10

0

10

20 b

0.0

0.2

0.4

0.6

0.8

1.0 11111111111111111111111111111111111111111111111

1

11 11111111111111111111111111111111111111111

-20

-10

0

10

20 e

0.0

0.2

0.4

0.6

0.8

1.0 11111111111111111111111111111111111111111111111

1

111 111111111111

11111111111111111111111111111

-20

-10

0

10

20 h

0.0

0.2

0.4

0.6

0.8

1.0 11111111111111111111111111111111111111111111111

1

11 11111111111111111111111111111111111111111

-20

-10

0

10

20 c

0.0

0.2

0.4

0.6

0.8

1.0 11111111111111111111111111111111111111111111111

1

111 11111111111111111111111111111111111111111

-20

-10

0

10

20 f

0.0

0.2

0.4

0.6

0.8

1.0 11111111111111111111111111111111111111111111111

1

111 11111111111111111111111111111111111111111

-20

-10

0

10

20 i

0.0

0.2

0.4

0.6

0.8

1.0 11111111111111111111111111111111111111111111111

1

111 11111111111111111111111111111111111111111

Figure 4: Comparison between the empirical distribution function (solid

lines)for 1000 normalized maxima of (X

t )

t

dened in(12) and thedouble

exponential distribution (dotted lines)with a GED driven noise ("

t )

t with

= 1. In rst line for the three graphes

2

= 2.3976, in the second line

2

= 0.6933, in the third line

2 = 0.0953. a, b, c : t IID; d, e, f : t AR (1), =0:95; g,h, i: t FAR IMA(0;d;0), d=0:4.

(21)

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

1 1 1

1 1 1 1 1 1 1

111111111

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

1 11 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

1 11 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

1 11 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

1 11 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

t )

t

exponentialdistribution(dottedlines)forARSVwhenthedrivennoise("

t )

t

is Gaussian. Foreach line

2

is constant, thevaluesof

2

are the same as

in Figure 2. The columns correspond to = 0:2, = 0:95 and = 0:99

(22)

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8 1111111111111111111111111111111111111111111

1

11

111 1111111

1111111111111

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

11

111 1111111

11111111111111111111111111

-10

-5

0

5

10

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

11

1

1 -10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

11

1

11 11111

11111111

1111111111111111111111111

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

11

1

11 11111

11111111

1111111111111111111111111

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

11

1

11 11111

11111111

1111111111111111111111111

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

11

11 1111111111

11111111111111111111111111

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

11

11 1111111111

11111111111111111111111111

-10

-5

0

5

10

0.0

0.2

0.4

0.6

0.8

1.0 1111111111111111111111111111111111111111111

1

11

11 1111111111

11111111111111111111111111

t )

t

exponentialdistribution(dottedlines)forLMSVwhenthedrivennoise("

t )

t

is Gaussian. Foreach line

2

is constant, thevaluesof

2

are the same as

in Figure 2. The columns correspond to d = 0:1, d = 0:2 and d = 0:4

(23)

−6

−4

−2

0

2

4

6

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35 Normal density

gamma = 0.5

gamma = 1

gamma = 2

gamma = 3

−6

−4

−2

0

2

4

6

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5 Normal density

gamma = 0.5

gamma = 1

gamma = 2

gamma = 3

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4 Normal density

gamma = 0.5

gamma = 1

gamma = 2

gamma = 3

Figure7: Comparaisonbetweenthedensitiesof

t

denedin(2)(solidline)

and

t

dened in (11) for dierent values of ( = 0:5; 1; 2; 3). The

three graphes correspond respectively to

2 = 2:3976, 2 = 0:6933 and 2 =0:0953.