Optimal calibration for exponential Levy models

fur Angewandte Analysis und Stohastik

imForshungsverbund Berline.V.

Preprint ISSN 0946 { 8633

Optimal alibration of exponential Levy models

Denis Belomestny 1

andMarkus Rei 2

submitted: 21thMarh2005

1

WeierstrassInstitute

forAppliedAnalysisandSto hastis,

Mohrenstr.39,10117

Berlin,Germany

E-Mail:belomestwias-berlin.de 2

WeierstrassInstitute

forAppliedAnalysisandSto hastis,

Mohrenstr.39,10117 Berlin,Germany E-Mail: mreisswias-berlin.de No. 1017 Berlin2005

W

I

A

S

2000MathematisSubjet Classiation. 60G51;62G20;91B28.

Keywordsand phrases. Europ eanoption,jumpdiusion,minimaxrates,severelyill-p osed,nonlinearinverse

Weierstra-InstitutfurAngewandte AnalysisundSto hastik(WIAS)

Mohrenstrae39

10117Berlin

Germany

Fax: +49302044975

E-Mail: preprintwias-b erlin.de

Basedonoptionsdataatthemarkettheproblemofalibratinganexp onentialLevymo del

fortheunderlying assetisinvestigated. Itisshownthatthis statistialinverseproblemisin

general severelyill-p osedandexatminimaxratesofonvergenearederived. Theestimation

pro edure we prop ose is based on the expliit inversion of the option prie formula in the

sp etral domainand aut-oshemeforhighfrequeniesas regularisation . Itsp erformane

is illustratedbynumerialsimulations.

1 Introdution

Already shortly after the intro dution of the Blak-Sholes mo del Merton (1976) argued that

basedonempirialevideneshareprie mo delsshould inorp orateajumpomp onent. Nowadays,

standard problems of mathematialnane like derivative priing have b een suessfully solved

for manygeneral Levy mo dels, as has b eomemanifest inthe monographby Cont and Tankov

(2004a). Ontheotherhand,theinvestigationofalibrationmetho ds forLevy mo delshasmainly

fo usedonertainparametrisationsoftheunderlyingLevypro ess. Sinetheharateristitriplet

ofaLevypro essisapriorianinnite-dimensionalobjet,thisapproahisalwaysexp osedtothe

problemof missp eiation, in partiularwhen there is no inherent eonomi foundationof the

parametersand theyareonlyusedtogenerate dierentshap es ofp ossiblejumpdistributions.

Thegoalofthispap eristoinvestigatemathematiallytheproblemofnonparametriinferenefor

theLevy tripletwhentheasset prie(S

t

)followsanexp onentialLevymo del

S

t =Se

r t+Xt

withaLevypro ess X

t

fort>0: (1.1)

We supp ose that at timet=0we disp ose ofpries for vanillaEurop eanallandput optionson

this asset with dierent strike pries andp ossiblydierentmaturities. Bybasing ourestimation

on optiondata we draw inferene onthe underlying risk neutral prie pro ess, whih in general

annotb e determinedfromhistorialpriedatadue totheinompletenessoftheLevymarket.

Theobservedoptionprieswillb eslightlyunpreiseduetobid-askspreadsorotherfritionsinthe

market. Itiswellknownthat intheidealaseofpreiseobservations forallp ossiblestrike pries

thestate prie density andhene theLevy tripletanb e uniquelyidentied,see e.g. At-Sahalia

andDuarte (2003). Under therealistimo delofnitely manynoisyobservations we annothop e

to determinethetriplet orretly, we shouldrather try to providean estimatorwhih is asgo o d

asp ossible forthegivenaurayofthedata. Thisoptimalityprop ertyisusuallyassessed bythe

minimaxparadigm,whih measurestheinherentomplexityofthestatistialproblemlass. One

mainresult ofthepresentpap erisalower b ound,showingthat alreadyinthesimpleexp onential

Levymo deltheestimationproblemisingeneralseverelyill-p osed,thatis,theestimationerrorfor

anypart of the Levy tripletas afuntionof theauray of theobservations willonly onverge

withalogarithmirateforanyoneivableestimationpro edure.

Onthe otherhand,we prop ose anexpliitonstrution of anestimatorthatattains thisoptimal

minimaxrate. Thepro edureis basedontheinversionoftheexpliitpriingformulaviaFourier

transformsbyCarrandMadan(1999)andaregularisationinthesp etraldomain. UsingtheFast

FourierTransformation,thepro edureiseasytoimplementandyieldsgo o dresultsinsimulations

asnonparametripart andthedriftanddiusiono eÆientas parametriparts.

The exp onential Levy mo delreets theassumption that thelog returns of the asset evolve

in-dep endently and withidential distributionforthesametimesteps, whih is plausiblefor liquid

marketsand notto olongtimehorizons. Thisbasi mo dellass hasb eenonsidered reently for

avariety ofpriing and optimisationproblems innane. Let us mentionhere Mordeki (2002)

forpriingAmerian-typ ep erp etual options,Contand Volthkova(2005)forpriingother

path-dep endent optionsand Eb erlein andPapapantoleon(2004)forago o dsurvey andgeneralisations

tothetime-inhomogeneousase. Kallsen(2000)andEmmerandKlupp elb erg(2004)studymarket

mo delsinamultidimensionalframework.

When no mo del for the prie pro ess is sp eied, alibration from option data an b e used to

estimate the state prie density, see At-Sahaliaand Duarte (2003). This density yields the

dis-tributionoftheasset prie atthetimesofmaturity,butdo es notprovideanyinformationonthe

evolutionofthe prie intime. Astrutural assumptionontheprie pro ess allowstond pries

for path-dep endent options or to p erform a dynamirisk management. Innanial engineering

informationab out the timeevolution exp eted at the market is obtained by smo othing implied

Blak-Sholes volatilities,f. Fengler, Hardle, and Mammen(2003). For the generalised

Blak-Sholes mo delDupire'sformulap ermitsthealibrationfromoptionpries,see e.g. Jakson,Suli,

andHowison(1999)foranumerialapproahandCrep ey(2003)foratheoretialstudy. The

al-ibrationofparametriexp onentialLevymo delshasb eenstudied forexamplebyEb erlein,Keller,

andPrause (1998)andCarr,Geman,Madan,andYor(2002).

The study by Cont and Tankov (2004b), also desrib ed in Cont and Tankov (2004a),is losest

to ournonparametri approahfor exp onentialLevy mo dels. Inorderto op ewith theinvolved

ill-p osedness,these authorsemployaleastsquaresmetho dp enalizedbytherelativeentropywith

resp et toanapriorihosen Levy triplet. Thistyp eofp enalisationhasertaingenuine features:

themetho dtakesintoaountpriorinformationandtheresultingfuntionalisonvex. However,

thevalueofthediusion o eÆientis thus xedinadvane,and theregularisingeet do es not

take plaeforindep endent randomerrors intheobservations,essentially b eausewhitenoise an

only b eonsidered as anelement inaSob olevspaeofnegative regularity, f. theHilb ert sales

approah in Engl, Hanke, and Neubauer (1996). In ontrast, we strive for a metho d that has

only few tuning parameters,p ermits the alibrationof thediusion o eÆient and is suited for

observationswithrandomerrors. Themetho dwe present b elowwillhaveallthese prop ertiesand

isinadditionprovablyrate-optimaloverstandardsmo othnesslasses. Insteadofminimizingsome

data-dep endent riterion,forwhihineah steptheoptionprie fortheurrenttripletvaluehas

tob eevaluated,wepreferusingtheexpliitnonlinearinversiondiretly. ThisresultsinaneÆient

straight-forward algorithm. Combiningthis metho dwith astage-wise aggregation pro edure, a

robust data-drivenmetho disobtained.

After intro duingthe nanialand statistialmo delinSetion 2, theestimationmetho dforthe

nite intensity ase is develop ed in Setion 3. The main theoretial results are formulatedin

Setion 4. A typialinniteintensityase is treated inSetion 5and we onlude inSetion 6.

Thepro ofsoftheupp er andlowerb oundsaredeferredtoSetions7and8,resp etively, whilethe

App endixprovidessomefurthertehnialresults.

2 The model

2.1 The exponential Levy model and option pries

Wesupp osethattheprieS

t

ofanassetattimetfollowstheLevymo del(1.1),whereS>0isthe

presentvalueoftheassetandr>0istherisklessinterestrate,whihisassumedtob eknownand

variationand absolutelyontinuousjump distribution. Its harateristi funtionis givenbythe Levy-Khinthinerepresentation ' T (u):=E[exp(iuX T )℄=exp T 2 2 u 2 +iu+ Z 1 1 (e iux 1)(x)dx : (2.1)

>0isalledvolatility,2Rdriftandthenon-negativefuntion,satisfying R

(jxj^1)(x)dx<

1, isthejump density. Itsjump intensityis dened as:=kk

L 1

(R)

. Theharateristi triplet

T := ( 2

;;)hasfor nite the intuitive explanationthat X isthe sum ofthree indep endent

lassial pro esses, namelya Wiener pro ess of volatility , a deterministi linear pro ess with

trend and a omp ound Poisson pro ess of intensity with jump distribution =. Pro esses

with inniteativityare obtainedbyalimitingpro edure and theirsamplepathshave innitely

manyjumps,butwithjumpsizesaumulatingatzero. ByexludingLevypro essesofunb ounded

variationweensureanintuitiveexplanationoftheparametersandweareinlinewiththeempirial

parametrindingsofCarr,Geman,Madan,andYor(2002),thoughsomeusefulparametrimo dels

likegeneralized hyp erb olidistributionsareexluded.

A Europ ean alloptionwith maturityT and strike K foran underlying asset grants theholder

therighttobuytheasset atthefuturetimeT fortheprieK. Ariskneutralprie attimet=0

forthisoptionisgivenby

C(K ;T)=e r T E Q [(S T K) + ℄; (2.2) where (A) +

:=max(A;0) andQisamartingalemeasureequivalenttothe realworldprobability

P. By onsidering optionpries we immediately draw inferene on this priing measure Qand

we assume from now on that S follows anexp onential Levy mo del (1.1) under Qand that the

disounted prie pro ess e r t

S

t

is amartingaleon the ltered probability spae (;F;Q;(F

t )),

xedthroughoutthepap er. Asisstandardinthealibrationliterature,themeasureQisassumed

tob e settledbythemarketandto b eidentialforalloptionstraded.

Bytheindep endene ofinrementsinX themartingaleonditionmayb e expliitlystatedas

8t>0: E[e Xt ℄=1 () 2 2 ++ Z 1 1 (e x 1)(x)dx=0: (2.3)

Observethatwehaveimp osedimpliitlytheexp onentialmomentondition R 1 0 (e x 1)(x)dx<1

to ensure the existene of E[S

t

℄. Another onsequene is that the harateristi funtion '

T is

denedonthewholestripfz 2C j Im(z)2[ 1;0℄gintheomplexplane,whihwillb eimp ortant

later. We reduethenumb erofparametersbyintro duingthenegative log-forwardmoneyness

x:=log(K =S) r T;

suh thattheallprie intermsofxisgivenby

C(x;T)=SE[(e X T e x ) + ℄:

Theanalogousformulafortheprieofaputoption,whihgivestheownertherighttosellanasset

at timeT fortheprie K,isP(x;T)=SE[(e x e X T ) +

℄. Thenthewell-knownput-allparityis

easilyestablished: C(x;T) P(x;T)=SE[e X T e x ℄=S(1 e x ): (2.4) 2.2 The observations

We fo us onthe alibrationfromoptionswithaxedmaturityT >0andmentionthe

straight-forward extension to several maturities in Setion 3.1. We observe the pries of N alloptions

(or by the put-all parity (2.4) alternatively put options) at dierent strikes K

j , j = 1;:::;N, orruptedbynoise Y j =C(K j ;T)+ j " j ; j=1;:::;N: (2.5)

j E[" 2 j ℄=1andsup j E[" 4 j

℄<1. Thenoise levels(

j

) areassumedto b ep ositive andknown. This

randomobservationmo delreetsthebid-askspread andotherfritionsatthemarket.

AsweneedtoemployFourier tehniques,we intro duethefuntion

O (x):= ( S 1 C(x;T); x>0; S 1 P(x;T); x<0 (2.6)

inthespiritofCarrandMadan(1999). Oreordsnormalisedallpries forx>0andnormalised

putpries forx60. Thefollowingimp ortantprop ertiesofO areprovedintheApp endix.

Prop osition2.1. (a) Wehave O (x)=S 1 C(x;T) (1 e x ) + forallx2R. (b) O (x)2[0;1^e x

℄holds forallx2R.

() If C

:=E[e

X

T

℄is niteforsome>1,thenO (x)6C

e

(1 )x

holdsforallx>0.

(d) Atanyx2Rnf0g,respetivelyx2R nf0;Tginthease=0and <1,thefuntion O

is twiedierentiablewith

Z Rnf0;Tg jO 00 (x)jdx63: The rst derivative O 0

has ajump of height 1atzeroand, in the ase=0and <1,

ajump of height+e T( )

oursin O 0

atT.

(e) The Fouriertransformof O satises

FO (v )= 1 ' T (v i) v (v i) ; v 2R : (2.7)

Thisidentityextendstoallomplexvaluesv withIm(v )2[0;1℄. Notethe properties'

T (0)=

1 and '

T

( i) = 1 derived from the general property of harateristi funtions and the

martingale ondition (2.3),respetively.

We transformourobservations(Y

j )andpreditors (K j )to O j :=Y j =S (1 K j e r T =S) + =O (x j )+Æ j " j ; (2.8) x j :=log(K j =S) r T; (2.9) where Æ j =S 1 j

. Inpratie, thedesign(x

j

)willb e ratherdense aroundx=0and sparsefor

optionsfurtheroutofthemoneyorinthemoney,f. Fengler, Hardle,andMammen(2003)fora

studyontheGermanDAX index.

In order to failitate the subsequent analysis we make a mildmomentassumptionon the prie

pro ess,whihguarantees byProp osition2.1(b,)theexp onentialdeayofO .

Assumption1. WeassumethatC

2 :=E[e

2XT

℄isnite. This isequivalenttopostulating forthe

assetprieanite seond moment: E[S 2

T ℄<1.

3 The estimation for bounded jump densities

LetusassumeherethattheLevypro esshasniteintensity. Laterweshallimp osealsoaertain

tehniqueneeds tob e employedandthepropagationoferrorsismoretransparent.

Sineourassetfollowsanexp onentialLevymo del,thejumpsintheLevypro essapp earexp

onen-tiallytransformedintheassetpriesanditisintuitivethatinfereneontheexp onentiallyweighted

jumpmeasure

(x):=e x

(x); x2R;

willleadtospatiallymorehomogeneousprop ertiesoftheestimatorthanforitself.Ouralibration

pro edure reliesessentiallyup ontheformula

(v ):= 1 T log 1+iv (1+iv )FO(v ) = 1 T log(' T (v i)) = 2 v 2 2 +i( 2 +)v+( 2 =2+ )+F(v ); (3.1)

whihis asimpleonsequene of theformulae(2.1)and(2.7). Note thatthe funtion is upto

a shift inthe argumentthe umulant-generating funtionof the Levy pro ess and a ontinuous

version of the logarithmmustb e taken suh that (0) = 0, whih is impliedby the martingale

ondition. Formula(3.1)showsthattheLevytripletisuniquelyidentiablegiventheobservation

of thewhole optionprie funtionO withoutnoise: F(v ) tends to zero as jv j !1due to the

Riemann-Leb esgueLemmaand 2

,, areidentiableaso eÆientsinthep olynomial,whihin

turnyieldsthefuntionF(v ). Aprop erlyrenedappliationofthis approahwillequipuswith

estimatorsfor thewhole tripletT =( 2

;;) (we parametrize Levytriplets equivalentlywith

or).

3.1 The basi proedure

Letusformulatethebasialgorithmtob eusedwhenaertainsmo othnessprop ertyisimp osedon

,thatisunderthepriorknowledge2G,whereGisasmo othnesslass. Thepro edureonsists

offoursteps: (a)webuildanapproximation ~

OofOfromthedata;(b)weobtainanapproximation

~

of byformula(3.1);()we estimatetheo eÆientsofthequadratip olynomialonthe

right-handsidein(3.1)from ~

underthepreseneofanoiseomp onentandthenonparametrinuisane

partF;(d)we obtainanestimatorforFbyonsideringtheremainder.

Themo del(3.1)hasasimilarstrutureasthewell-knownpartiallinearmo dels,butinfatthereis

onesubstantialdierene: thefuntionFisnotsupp osedtob esmo oth,butinsteaditisdeaying

forhigh frequeniesb eausewe workinthesp etraldomain. This isalsowhywe shallregularize

theproblembyuttingo frequeniesjv j higherthanaertain thresholdlevelU,whihdep ends

onthenoise levelandthesmo othnessassumptionsinG.

We nowgive adetailed desriptionofthedierentstepsinthepro edure.

(a) WeapproximatethefuntionObybuilding ~

Ofromtheobservations (O

j )intheform ~ O (x)= 0 (x)+ N X j=1 O j b j (x); x2R; and onsequentlyFO by F ~ O (u)=F 0 (u)+ N X j=1 O j Fb j (u); u2R; where (b j

)aresomebasisfuntions tob e hosen and thefuntion

0

isadded to take are

ofthejumpinthederivativeofOat zero: 0

0

(0+)

0

0

8x2R: b k (x)2[0;1℄; 8j;k=1;:::;N: b k (x j )=Æ jk ; lim juj!1 b k (u)=0:

Westress here that step (a)shouldnotb e understo o das asmo othingstep,but ratherasa

meansto ndareasonable approximationofFObased ondisrete data. As anb e seenin

thetheoretial analysisandthenumerialsimulationsb elow,itsuÆes to usesimplelinear

B-splines as basisfuntions. Theoretially, we need that theresults of Prop osition7.1 and

estimate(7.7)aresatised.

(b) For(v )2(0;1),sp eiedlaterin(4.1),wealulate

~ (v ):= 1 T log >(v ) 1+iv (1+iv )F ~ O(v ) ; v2R; (3.2)

where thefuntionlog

> :C nf0g!C isgivenby log > (z):= ( log(z); jzj log(z=jzj); jzj< (3.3)

and log()is takeninsuh away that ~

(v ) isontinuous with ~

(0)=0(almostsurelythe

argumentofthelogarithmin(3.2)do es notvanish). Ifweobserveoptionpriesfordierent

maturitiesT

k

,we p erformthesteps (a)and(b)foreahT

k

separatelyandaggregateatthis

p ointthedierentestimatorsfor toobtainoneestimatorwithless variane.

() Withanestimate ~

of athand,we obtainestimatorsfortheparametripart( 2

;;)by

an averagingpro edure takinginto aount thep olynomialstruture in(3.1). Up onxing

thesp etralut-ovalueU =U(G;(Æ

j );(x j )),weset ^ 2 := Z U U Re( ~ (u))w U (u)du; (3.4) ^ := ^ 2 + Z U U Im( ~ (u))w U (u)du; (3.5) ^ := ^ 2 2 +^ Z U U Re( ~ (u))w U (u)du; (3.6)

where theweightfuntionsw U ;w U and w U satisfy Z U U w U (u)du=0; Z U U u 2 w U (u)du= 2; Z U U uw U (u)du=1; (3.7) Z U U u 2 w U (u)du=0; Z U U w U (u)du=1: (3.8)

Forstandardsmo othnesslassesGasymptotiallyoptimalhoiesoftheut-ovalueU and

the weight funtionsare given in(4.9)and(4.2)-(4.4). Theestimateof theo eÆients an

b e understo o d as an orthogonal projetion estimate with resp et to an L 2

-salar pro dut

weightedaordingtothesupp oseddeayprop ertyofF.

(d) Finally,wedenetheestimateforastheinverseFouriertransformoftheremainder:

^ (u):=F 1 h ~ ()+ ^ 2 2 ( i) 2 i^( i)+ ^ 1 [ U;U℄ ( ) i (u); u2R : (3.9)

Notethat theomputationalomplexityofthis basiestimationpro edure isvery low. Theonly

funtions Fb

k

andthe funtionF

0

, whihwith ourhoie as linearB-splines an b e omputed

expliitly: Fb k (u):=u 2 e iux k e iux k 1 x k x k 1 e iux k+1 e iux k x k +1 x k ; F 0 (u)=u 2 1+ e iux j 0 x j0 1 e iux j 0 1 x j0 x j0 x j0 1 (3.10)

withk=1;:::;N,someextrap olateddesignp ointsx

0 andx N+1 ,whereweset ~ O (x 0 )= ~ O(x N+1 )=

0,andwiththeindexj

0 dened byx j0 1 <06x j0 .

3.2 A data-driven estimator for the jump density

Letusbrieydesrib etheonstrutionofadata-drivenpro edurewhihrequiresnopriorsmo

oth-nessassumptionsontoadjustthetuningparameters. Theideaistolterouttheparametripart

and to obtainastandard 'funtioninnoise'-estimationprobleminthesp etral domain. Instead

ofho osingoneut-ovalueU,wetakeageometrigridU

1 >U

2

>>U

J

ofut-ovaluesand

aggregatetheorresp ondingestimatorsadaptively.

Forlteringquadratip olynomialswe intro duetheonvolutionop erator

A g f(x):=f(x) 1 2 Z 1 1 f(y )Fg (x y )dy (3.11)

withasuÆientlyregularandnielydeayingfuntiong:R!Rsatisfying

Z 1 1 Fg (y )d y=1; Z 1 1 y k Fg (y )dy=0; k=1;2,thatisg (0)=1;g 0 (0)=g 00 (0)=0: (3.12)

Thersttwostepsofthedata-drivenpro edureareidential tothesteps(a)and(b)ofthebasi

pro edure. Thesubsequent stepsareas follows:

() Weapply theop eratorA

g to ~ andobtain ~ g (v ):=A g ~ (v ), whihby(3.1)isareasonable estimateof A g (v )=A g F(v )=F((1 g ))(v ).

(d) Weonsiderthefamilyofbasiestimators~ (j) g givenby ~ (j) g :=F 1 ~ g 1 [ U j ;U j ℄ (y ); j=1;:::;J:

(e) We onstrut the aggregated estimator ^

g as a onvex ombination of (~ (j) g ) j=1;:::;J with

data-dep endentweights. Theseweightsareobtainedbythefollowingalgorithm:

(i) Initialize^ (1) g =~ (1) g .

(ii) For j=2;:::;J sequentiallydene

^ (j) g := j ~ (j) g +(1 j )^ (j 1) g ; where j =K(m (j)

=)forsome>0,aompatlysupp orted kernel K and

m (j) := k~ (j) g ^ (j 1) g k 2 L2 kVar[~ (j) g ℄k L1 : (iii) Put^ g :=^ (J) g . AlthoughVar[~ (j) g

℄isnotknownexatly,itanb eeasilyestimatedfromab ove. The

param-eter istakeninaordanewiththesuggestionsgiveninBelomestnyandSp okoiny(2004),

−1.0

−0.5

0.0

0.5

1.0

0.00

0.02

0.04

0.06

0.08

0.10

0.12

−2

−1

0

1

2

0

5

10

15

data.driven

basic

0.16

0.17

0.18

0.19

0.20

0.21

Figure 1: Kou mo del. Left: Sample(O

j

) and true funtion O (dashed line). Center: True

(dashed) and estimated ^ (blak)mo diedLevy densities. Right: Box plot for the data-driven

andthebasipro edure basedon1000Monte-Carlosimulations.

(f) Thenalestimatorfor(x)isdenedas(x)^ =^

g

(x)=(1 g (x))forallx2Rwithg (x)6=1.

Example1. Apossiblefamilyoffuntionsgsatisfying(3.12)isgivenbyg

(x)=1 (1 e x 2 = 2 ) 2 ,

x 2 R, > 0, whih gives rise to the onvolution lter Fg

(u) = p e 2 u 2 =4 p 8 e 2 u 2 =8 . Observe that 1 g

only vanishes at zero and that for smaller values of the weight 1 g

is

losertooneoutsidethe origin, butthe lterFg

doesnot deaysorapidly.

3.3 Numerial Example

Twoempirialphenomenainnanialdatahaveattratedmuhattentionreently: theleptokurti

return distributionofassets withahigherp eakandtwo(asymmetri)heavier tailsthanthose of

thenormaldistribution,andtheimpliedvolatilitysmile. Toinorp oratethesefeatures,thedouble

exp onential jumpdiusion mo delwas prop osed by Kou (2002). Inhis mo del theLevy triplet is

sp eied bythejumpdensity

(x)= p + e +x 1 [0;1) (x)+(1 p) e x 1 ( 1;0) (x) ; x2R;

andtheparameters;;

+

; >0andp2[0;1℄,whileisuniquelydeterminedbythe

martin-galeondition. We simulatetheKou mo delwith parameters =0:1;=5; =4;

+

=8;p=

1=3and apply thenonparametri estimationpro edure giventheobservation ofnoisy Europ ean

optiondata withT =0:25,N =50,r=0:06and Æ

j =O (x

j )=10.

InFigure1(left)thesimulatedobservations(O

j

)andthetrueurve Oaredepited as funtions

ofthelog-forwardmoneyness. TheestimatedtransformedLevydensityintheenterisobtained

using thebasipro edure, assp eied inthemathematialanalysis,withahuman-drivenhoie

oftheut-oparameterU. Theparameterswereestimatedas^=0:035; ^

=7:56;^=0:556(=

0:423). Weobserve thattheestimatedtransformedLevydensityreoversthemainfeaturesofthe

Koumo dellikethemo deatzeroandtheskewness. Fromthefuntionalformoftheestimatorwe

aneasilyderiveestimatesforotherimp ortantquantities,e.g. fortheprop ortionofnegativejumps

byalulating ^ 1 R 0 1 ^ (x)dx= ^ 1 R 0 1 e x ^

(x)dx,whihinthesimulationexampleevaluates

to0:72(truevalue: 1 p=2=3).

IntherightpartofFigure1weomparethep erformaneofthedata-drivenaggregatedestimator

withtheoraleestimator(i.e,ho osingtheb estp ossibleU)obtainedfromthebasipro edurein

termsoftheempirialL 2

-loss. Ab oxplotisshownfor1000Monte-Carlorepliations. Inthisplot,

as provided by the statistial software pakage R, the b oxstrethes fromthe 25% p erentile to

the75%p erentile,rossed bythemedian,andthep ositionoftheremaining50%ofthevaluesis

basiestimators. Asp ointedoutbyCavalierandGolub ev(2004),standarddata-drivenestimation

pro eduresoftenp erformbadlyforinverseproblemssuhthatthemetho dofaggregationusedhere

anb e onsideredasomparativelyverystable.

4 Risk bounds for bounded jump densities

4.1 Mathematial results

We shall use throughout the notation A . B if A is b ounded by a onstant multiple of B,

indep endent of the parameters involved, that is, in the Landau notation A = O (B). Equally

A&B meansB.AandAsB standsforA.B andA&B simultaneously.

In order to assess the quality of the estimators, we quantify their risks under a Sob olev-typ e

smo othnessonditionofordersonthetransformedjumpdensity.

Denition4.1. For s 2Nand R;

max

>0 let G

s (R;

max

) denote the set of all Levy triplets

T =( 2

;;),satisfying the martingale onditionand Assumption 1withC

2

6R,suh that is

s-times (weakly)dierentiable and

2[0; max ℄; jj;2[0;R℄; max 06k 6s k (k ) k L 2 (R) 6R; k (s) k L 1 (R) 6R: We have enfored j ~ T

(v )j > log((v )) in (3.2) to prevent unb oundedness in the ase of large

sto hasti errors. For Levy triplets inG

s (R;

max

)a reasonable hoie for (v ) an b e obtained

fromthefollowingalulationusing theidentity

2

2

++F(0)=derivedfromthemartingale

ondition(2.3): 1 2 j' T (v i)j= 1 2 exp T 2 2 v 2 TF(0)+TRe(F(v )) > 1 2 exp T 2 max 2 v 2 4TR =:(v ): (4.1)

Theonlyreasonforthefator1=2isthemathematialtratabilitygivinglatertheb oundofLemma

7.2.

Conerning the hoie of the weight funtions, we take advantage of the smo othness s of by

takingfuntionswsuhthatFwhassvanishingmoments.Equivalentlyexpressedinthesp etral

domain,theweightfuntions w (u) growwithfrequenies juj like juj s

toprot fromthedeay of

jF(u)j. Hene, we dene for allU >0familiesofweightfuntions by resalingthose funtions

satisfyingrestritions(3.7)and(3.8)for U=1:

w U (u)=U 3 w 1 (U 1 u) withw 1 satisfying(3.7)andkF(w 1 (u)=u s )k L 1<1; (4.2) w U (u)=U 2 w 1 (U 1 u) withw 1 satisfying(3.7)andkF(w 1 (u)=u s )k L 1<1; (4.3) w U (u)=U 1 w 1 (U 1 u) withw 1 satisfying(3.8)andkF(w 1 (u)=u s )k L 1 <1: (4.4)

Inthesedenitionsitisundersto o dthatthesupp ortoftheweightfuntionsisontainedin[ U;U℄.

Notethat theprop erty F(w (u)=u s

)2L 1

(R)means inpartiular thatw (u)=u s

is ontinuousand

b oundedsuh that

jw U (u)j.U (s+3) juj s ; jw U (u)j.U (s+2) juj s and jw U (u)j.U (s+1) juj s : (4.5)

For the simulationswe have used symmetri weight funtions that are onstant multiplesof u 2

exept for three(at 0andU for)resp etivelyfour (atU 0 ;U with someU 0 <U for 2 ,)

theasymptotisofagrowingnumb erof observationswith := max j=2;:::;N (x j x j 1 )!0 and A:=min(x N ; x 1 )!1: (4.6)

WeuselinearB-splinesforthebasisfuntions(b

k

)andthefuntion

0

. Toeasethemathematial

treatmentof theextrap olation error,weassume that alldata isontainedintheinterval( A

;A+)andaddtheartiialobservationsx

0 = A ,x N+1 =A+withO 0 =O N+1 =0.

The reason why we ho ose a pieewise linear approximationis that this yields rate-optimal

in-terp olation errors for ~

O , knowing that O is twie dierentiable exept at nitely many p oints,

f. Prop osition 7.1 b elow. Of ourse, when assuming some p ositive regularity on or by some

adaptivemetho d,thenumerialapproximationratewithresp et toanb eaelerated,butthis

improvementisonlyvalidforavery smalldisretisation distanewhenthesto hasti

observa-tionerrorisusuallydominantanyway.Inontrasttostandardregressionestimatesweshallalways

trak expliitlythedep endene onthelevel(Æ

k

)of thenoiseintheobservations,whih isusually

rathersmallforobserved optionpries.

Thesubsequentanalysis an ertainly b e improvedforaonrete design(x

j

)and onrete noise

levels(Æ

j

),butforrevealingthemainfeaturesitismoretransparentandonisetostatetheresults

intermsoftheabstratnoise level

":= 3=2 + 1=2 kÆk l 1 ; (4.7)

omprisingthelevelofthenumerialinterp olationerrorandofthesto hastierrorsimultaneously.

HereandinthesequelweusethenormskÆk

l 1 :=sup k Æ k andkÆk 2 l 2 := P k Æ 2 k .

We arenowinap ositionto statethemainresultsab outtheriskupp er b oundsof theestimators

obtainedbythebasipro edureandab outtherisklowerb oundsvalidforanyestimationpro edure

whatso ever. Thepro ofsaregiveninSetions7and8fortheupp erandlowerb ounds,resp etively.

Theorem4.2. Assume e A . 2 and kÆk 2 l2 .kÆk 2 l 1

. Choosing forsome>

max the ut-o U := 1 (2log(" 1 )=T) 1=2

,weobtain fortherisk of^ 2

the uniformonvergenerate

sup T=( 2 ;;)2G s (R; max ) E T [j^ 2 2 j 2 ℄ 1=2 . s+3 (log(" 1 )) (s+3)=2 : (4.8)

Theasymptotiriskintheestimationoftheotherunknownquantitiesshowsadihotomy. While

usuallyitislargerthantheriskfor^ 2

,itismuhsmallerifweknowthat=0holds,thatis,for

theomp oundPoissonase.

Theorem4.3. Assume e A . 2 andkÆk 2 l 2 .kÆk 2 l 1 . For any> max wehoose U := 1 2log(" 1 )=T 1=2 ; U 0 :=" 2=(2s+5) ; (4.9) inthe ases max >0and max

=0, respetively. Then the riskbounds for^and ^ are sup T=( 2 ;;)2Gs(R;max) E T [j^ j 2 ℄ 1=2 . ( s+2 (log(" 1 )) (s+2)=2 ; 2[0; max ℄unknown ; " (2s+4)=(2s+5) ; = max =0; (4.10) sup T=( 2 ;;)2G s (R; max ) E T [j ^ j 2 ℄ 1=2 . ( s+1 (log(" 1 )) (s+1)=2 ; 2[0; max ℄unknown, " (2s+2)=(2s+5) ; = max =0: (4.11) Theorem4.4. Assume e A . 2 and kÆk 2 l 2 .kÆk 2 l 1. For some> max we hooseU and U 0

asin (4.9)toobtainthe followingrisk estimatesfor:^

sup T=( 2 ;;)2Gs(R;max) E T h Z 1 1 j(x)^ (x)j 2 dx i 1=2 . ( s (log(" 1 )) s=2 ; 2[0; max ℄ unknown, " 2s=(2s+5) ; = max =0: (4.12)

thewidth A of thedesignonly needs to growlogarithmiallyandthe error levels (Æ

k

) needonly

b e square summableafter renormalisation. Thelatter onditionan ertainlyb e further relaxed

sine thistermisausedbyarough b oundonthequadratiremainderterm.

Forthelowerb oundsweapp ealtotheequivaleneb etweentheregressionandtheGaussianwhite

noisemo del,asestablishedbyBrownandLow(1996),andonsidermerelytheidealizedobservation

mo del

dZ(x)=O (x)dx+"dW(x); x2R; (4.13)

with the noise level asymptotis " ! 0, a two-sided Brownian motion W and with O = O

T

denotingthe optionprie funtion from(2.6)for thegiventriplet T. This simpliation avoids

tedious numerialapproximationsinthepro ofs.

Theorem 4.5. Lets 2N, R>0and

max

>0be given. For the observationmodel (4.13)and

any quantityq2f 2

;;;gthe following asymptotirisklowerbounds hold:

inf ^ q sup T2Gs(R;max) E T [k^q q k 2 ℄ 1=2 &v q ;max ;

wherekkdenotes theabsolutevalueforq2f 2

;;gand theL 2

(R)-normforq=,theinmum

isalways takenover allestimators, thatisallmeasurable funtions of the observationZ,and the

rate v

q ;

max

isgiven inthe followingtable:

2 max >0 log(" 1 ) (s+3)=2 log(" 1 ) (s+2)=2 log(" 1 ) (s+1)=2 log(" 1 ) s=2 max =0 0 " (2s+4)=(2s+5) " (2s+2)=(2s+5) " 2s=(2s+5) 4.2 Disussion

We have seen that for > 0 the rate orresp onds to a severely ill-p osed problem (f. Engl,

Hanke,andNeubauer (1996)and thereferenes there),whileforknown=0therates aremuh

b etter, but still ill-p osedompared to those obtainedinlassial nonparametriregression. The

reason forthesevereill-p osedness for>0isthat we fae anunderlyingdeonvolutionproblem

with a Gaussian distribution: thelaw of the diusion part of X

T

is onvolvedwith that of the

omp ound Poisson part to give the density of X

T

. This typ e of estimation problem has b een

studied thoroughlybyButuea andMatias(2005)inanidealizeddensity estimationsetup. Note

thegeneral orderinwhih the(asymptoti)qualityof estimationdereases: 2

, , and nally

,whihisrelatedtothedominationprop ertyformulatedinAt-SahaliaandJao d(2004). Inthe

upp er b oundswehave kepttrak ofthedep endene onb eauseforsmallvaluesof andnite

samplesthep erformaneisnotsobad,omparethesimulationsinSetion3.3;itjustneeds alot

moreobservationstoimproveonthat.

At rst sight the rates for the parametri estimation part in the ase = 0 are astonishing.

They are worse than in usual semi-parametri problems whih also indiates that missp eied

parametrimo delswillgiveunreliableestimatesforthevolatilityandjumpintensity. Theserates

are, however,easilyundersto o d whenemployingthelanguageofdistributions. WithÆ

0

denoting

theDiradistributioninzero andÆ 0

0

itsderivative we have

log(' T (u))=TF Æ 0 0 + Æ 0 (u):

EstimatingthedensityofX

T

andsimilarlyitsharateristi funtionfromthenoisyobservations

of O amountsroughly to dierentiate the observed funtion twie, f. At-Sahalia and Duarte

(2003)andtheremarkafterequation(7.6)b elow. Thisgivestheminimaxrateforandasthat

ofestimatingtheseondderivative ofaregression funtionofregularitys+2. For theparameter

it suÆes to estimate the jump in the antiderivative of F 1

(log('

T

analogyis the estimationof the regression funtionitself at zero. This explains also why inthe

lass G

s

we have measured theregularitynotonly in L 2

, but also uniformly. In fat, ifwe only

assumeanL 2

-Sob olevondition,thenthesamelowerb oundtehniques willyieldslower ratesfor

the parameters,as is typial for p ointwise estimationproblems. An interesting way to estimate

diretly andissuggestedbyProp osition2.1(d): ahangep ointdetetion algorithmforjumps

inthederivativeof O ,as prop osed by Goldenshluger,Tsybakov, andZeevi (2004), an equip us

with anestimateof and asubsequent estimateof thejumpsize yieldsan estimateof, whih

gives thesameminimaxrates.

Asusual,theestimationpro edureneedsertaintuningparameters.Theapproximatesizeof

max

and thenoise levelis ingeneral knownto thepratitioner. The stabilisationofthe logarithmby

the funtion(v ) was enfored mainlyfor theoretial reasons to prevent explosions due to large

deviations. Theusuallyunknownordersofsmo othnessofthetransformedjumpdensity,however,

is needed to determine ago o dhoie of theut-o frequeny U andalso app ears inthe weights

w 1 ;w 1 ;w 1

. Yet, for thelatter it suÆes to useweight funtions satisfying (4.2)-(4.4)for some

larges

max

likeinstandardnonparametriswheretheorderofthekernelmustonlyb esuÆiently

large. WearethusleftwithonlyonetuningparameterU,whihisthesameforallfourestimation

problems. Thedata-drivenpro edurepresentedinSetion3.2isonewaytoop ewiththisproblem

forthejumpdensity. Note,however,thataprop er mathematialanalysisforthegeneralproblem

seems hallengingduetotheunderlying nonlinear'hangep ointdetetion'-struture, forwhiha

data-drivenalgorithmeveninthe idealizedlinear settingof Goldenshluger,Tsybakov,andZeevi

(2004)isnotyetavailable. Finally,observethattheestimationofthejumpdensityatzeroisonly

p ossiblebyimp osingaertainregularitythere,otherwiseitislearlynotp ossibletodetetjumps

ofheightzero.

5 Estimation for unbounded jump densities

Letusnowdisussthease that isajumpdensitywithasingularityatzero. For simpliitywe

restritthepresentationtothease=0,whihisalsoinagreementwiththeempirialparametri

ndings by Carr, Geman, Madan,and Yor (2002). We then dedue as b efore for(x) =e x

(x)

using(2.3),(2.7)andthedenition(3.1)of intermsofO

(v )=iv+ Z 1 1 (e iv x 1)(x)dx=iv+ Z v 0 F(ix(x))(w )dw : UnderAssumption1x(x)2L 1

(R)holds. Bytakingderivativeswe nd

0 (v )= (i 2v )FO (v ) (v iv 2 )F(xO (x))(v ) T(1+(iv v 2 )FO(v )) =i+F(ix(x))(v ):

We rst onsider the problem of estimating in someweighted L 2

-loss with a weight funtion

vanishing inzero. More preisely, we aimatestimating

g

(x)=(x)(1 g (x)) inL 2

(R)-loss for

somedierentiableniely deaying funtiong :R![0;1℄with g (0)=1. We obtain

g 2L 1 (R) and F g (v )= 1 2 i F(ix(x))F((1 g (x))=x)( v ) =g 0 (0)+ 1 2 i (i 2)FO (+i 2 )F(xO (x)) T(1+(i 2 )FO ) F((1 g (x))=x) ( v ) (5.1)

The onvolution kernel F((1 g (x))=x) deays rapidly for smo oth funtions g suh that for a

go o d approximationof F

g

(v ) it suÆes to know the funtions FO and F(xO (x)) in a lose

g

~

O of O into formula(5.1) and using someg with g 0

(0)=0. Thenoise levelfor frequenies v in

theempirialounterpart of (5.1)willb eoforderv 2

intheniteintensityase =kk

L 1

(R) <1

exatly asintheprevious analysisfor=0. For=1theharateristi funtiontendstozero

andtheestimationerror willdeteriorate signiantly,seethedisussion b elow.

Whendrawinginfereneontheb ehaviourofnearzero,wehavetosp eifythekindofsingularity

andsmo othnesswe exp etthere. Letustherefore p ostulatethat

(x)= (x) jxj with2(0;2)andjF (u)j.(1+juj) (s+1) ; s2N: (5.2)

To avoid additional onsiderations we assume that 6= 1. Note that this mo del inludes for

example temp ered stable pro esses with regularity index s = 1, when their transformed jump

densityisgivenby (x)=e x (x)=C e (1+ )x jxj 1 ( 1;0) (x)+ e (1 + )x jxj 1 (0;1) (x) ; C ; >0; + >1; x2R;

f. Chapter4 inCont andTankov(2004a) whih also gives further examples. Under themo del

(5.2) an interesting information on the b ehaviour of near zero is given by the value

(0),

for whih we now want to derive an estimationpro edure. Beause of x(x) = x 1 (x) (we understand always x := jxj sgn (x)) we an draw inferene on F(x 1

(x)), but this Fourier

transformdeaysslowlyduetoitsnon-dierentiableargumentandwillnotyieldawellp erforming

estimator. Consequently, we have to use more rened frational dierentiation results for the

preisestruture ofthis Fouriertransform. Thefollowingresult isderived intheApp endix.

Prop osition5.1. The followingasymptotiestimate holdsforjuj!1:

F(ix 1 (x))(u) 2 (2 )sin( =2) s 1 X k =0 2 k (k ) (0)i k u 2 k .juj s min(1;2 ) ;

where denotes the EulerGamma funtion.

Hene, we anexpand 0

inanon-integerp ower series:

0 (u)=i+2 (2 )sin( =2) s 1 X k =0 2 k (k ) (0)i k u 2 k +R(u)

with the remaindersatisfyingjR(u)j .juj

s max(1;2 )

. Exatly as inthe expansion (3.1), this

p ermitstoestimate (0)based onanestimator ~ by^ (0):= R U U ~ 0 (u)w U

(u)duwithaweight

funtionw U satisfying Z U U u 2 w U (u)du= 1 2 (2 )sin( =2) ; Z U U u w U (u)du=0for=0;= 2 k

with k 2 f1;:::;s 1g. For this estimator a similar analysis as in the ase of b ounded jump

densities an b ep erformed. Themaindigressionisthat for>1,theinniteintensityase,the

harateristi funtionis not b ounded away fromzero anymore and the risk willb e essentially

determinedbythegrowthofj'

T (u i)j

1

withjuj!1,whih isusuallyexp onential(e juj

1

in

thetemp eredstablease)and thusyields againaseverelyill-p osedproblem.

6 Conlusion

Wehavedevelop edanestimationpro edureforthenonparametrialibrationofexp onentialLevy

thatthealibrationisingeneralahardproblemtosolve,atleastifveryhigh aurayisdesired.

Nevertheless the estimationpro edure is well suited to gain general insight into the size of the

parametersandthestruture ofthejumpdensity. Evenifreasonableparametrimo delsexistthat

anb e b ettertted,ago o dness-of-ttestbasedonournonparametriapproahshouldalwaysb e

usedto hekagainstmissp eiation.

As already seen inthe ase of unb ounded jumpdensities, our pro edure an b e adapted to

dif-ferent mo delsas long as the inverse transformationfrom the optionpries to the harateristi

funtionan b ealulated and theunknown quantities an b e determined fromthestruture of

theharateristifuntion. Asempirialoptiondatasuggests,theriskneutralpriepro ess isnot

homogeneousintimeandtheexp onentialLevymo delshouldb eextendedinthatdiretion. A

suit-ablemo dellassisforinstane givenbytheaÆnemo delsofDuÆe,Filip ovi,andShahermayer

(2003). Web elievethat thequestion ofalibrationformo delsinnanialmathematisshouldb e

addressedwiththesamerigourandintensityasotherprimaryquestionslikepriing,hedgingand

riskmanagement.

7 Proof of the upper bounds

Allalulationstake plaeinthesettingofSetion 4. Asgeneral referene forFourier tehniques

like the Planherel identity and normestimates we reommend Rudin (1991). To failitate the

alulationswe intro duetheexp onentiallyinreasingfuntion

E(x):= e x 1 x ; x>0; andsetE(0):=1: (7.1)

UsinglinearB-splines(f. Setion 3.1)weenounter thefollowinglinearinterp olationofO

O l (x):=E[ ~ O (x)℄= N X j=1 O (x j )b j (x)+ 0 (x); x2R : (7.2)

7.1 A numerial approximation result

Prop osition 7.1. Under the hypothesis e A

.

2

we obtain uniformly over all Levy triplets

satisfying Assumption1

sup

u2R jE[F

~

O(u) FO (u)℄j=sup

u2R jFO l (u) FO (u)j. 2 : (7.3)

Proof. BystandardFourierestimatestheassertionfollowsonewehaveprovedkO

l O k L 1 . 2 . Note that O 0

is twie dierentiable exept at the p oints x

j 0 1 ;0;x j 0 and p ossibly T by

Prop osition2.1(d). Whilethedisontinuitiesof(O

l

0 )

0

attheknotp ointsdonotdoanyharm,

O

0

hasaderivativenearzerowhihisuniformlyb oundedbyaonstantC

0

aordingto(9.1).

Starting with the ase > 0, that is without a jump at T, we obtain using the mean value

theoremwithsuitable

j 2(x j 1 ;x j ): Z x N x1 j ~ O l (x) O (x)jdx = N X j=2 Z xj x j 1 (O 0 )(x j ) x x j 1 x j x j 1 +(O 0 )(x j 1 ) x j x x j x j 1 + 0 (x) O (x) dx

= N+1 X j=1 xj x j 1 x x j 1 ((O 0 ) 0 ( j ) (O 0 ) 0 (y ))d y dx 6 X j2f2;:::;Ngnfj 0 g Z x j x j 1 Z x x j 1 Z x j x j 1 jO 00 (z)jdzdydx+2C 0 2 6kO 00 k L 1 2 +2C 0 2 :

ByAssumption1andProp osition2.1(b,)theextrap olationerror isb oundedby

Z [x0;x1℄[[xN;xN+1℄ jE[ ~ O(x) O (x)℄jdx64C 2 e (A ) :

An appliationofProp osition2.1(d)therefore showsfor>0

Z 1 1 jE[ ~ O (x) O (x)℄jdx6e A +3 2 +2C 0 2 +4C 2 e (A ) . 2 :

Inthe ase =0we onsider the index j

with x

j 1

6 T <x

j

andfae an additionalerror

estimatedby Z xj xj 1 jE[ ~ O (x) O (x)℄jdx6 Z xj xj 1 k(O 0 ) 0 k L 1 2(x x j 1 )(x j x) x j x j 1 dx 6k(O 0 ) 0 k L 1 (x j x j 1 ) 2

Withalo okat(9.1)we inferthatthiserror termisalsooforder 2

andthusdo esnotenlargethe

onvergenerate.

7.2 Proof of Theorem 4.2

Theassertedrate(4.8)followsonethegeneralriskestimate

E[j^ 2 2 j 2 ℄.U 2(s+3) +E(T 2 U 2 )U 1 " 2 +E(T 2 max U 2 ) 2 U 4 " 4 (7.4)

hasb een shown for U . 1

uniformlyover G

s (R;

max

), sine theexpliit hoie of U renders

theseondandthirdtermasymptotiallynegligible.

Consider inthedenition(3.2)of ~

separatelythelinearisationL,negletingthestabilisationby

,andtheremaindertermR:

L(u):=T 1 ' T (u i) 1 (u i)uF( ~ O O )(u); (7.5) R(u):= ~

(u) (u) L(u): (7.6)

Whennegletingtheremainderterm,we mayview ~

(u) asobservation of (u)inadditivenoise,

whoseintensitygrowslikej'

T (u i)j 1 j(u i)ujsu 2 e T 2 u 2

forjuj!1. Thisheteroskedastiity

reets thedegreeofill-p osednessoftheestimationproblem.

Lemma 7.2. Forallu2Rtheremainder termsatises

jR(u)j6T 1 (u) 2 (u 4 +u 2 )jF( ~ O O )(u)j 2 :

Proof. Letusset'~

T

(u i):=1 u(u i)F ~

O (u)whihequalse T

~

(u)

ifj'~

T

(u i)j>(u). Using

je T

~

(u)

j>(u),u2R,weobtainbyaseond-orderexpansion ofthelogarithm

jT ~ (u) log(' T (u i))) ' T (u i) 1 (e T ~ (u) ' T (u i))j6 1 2 (u) 2 je T ~ (u) ' T (u i)j 2 :

T T (u)6j' T (u i)j=2toinfer j' T (u i) 1 (e T ~ (u) ~ ' T (u i))j6 1 2 (u) 1 je T ~ (u) ~ ' T (u i)j j'~ T (u i) ' T (u i)j(u) 1 6 1 2 (u) 2 j'~ T (u i) ' T (u i)j 2 = 1 2 (u) 2 (u 4 +u 2 )jF( ~ O O )(u)j 2 :

Togetherwiththeprevious result thisgivesfor allu2Rtheassertion ofthelemma.

We shall frequently use the following norm b ounds for the B-splines (b

k

), whih follow from

kb k k 1 =1andjx k +1 x k 1 j62: kFb k k L 2 = p 2 kb k k L 26(4 ) 1=2 ; kFb k k 1 6kb k k L 1 62: (7.7) We deomp ose^ 2

intermsofLandRfrom(7.5)and(7.6):

^ 2 = Z U U 2 2 (u 2

1)++Re(F(u)) +Re(L(u)+R(u)) w U (u)du = 2 + Z U U

Re F(u)+L(u)+R(u) w U (u)du; (7.8) whihyields E[j^ 2 2 j 2 ℄63 Z U U F(u)w U (u)du 2 +3E h Z U U L(u)w U (u)du 2 i +3E h Z U U R(u)w U (u)du 2 i :

Letusonsiderthethreetermsinthesumseparately. ThenuisaneofFauses adeterministi

error whih anb eb oundedusing(iu) s

F(u)=F (s)

(u)and thePlanherelisometry:

Z U U F(u)w U (u)du =2 Z 1 1 (s) (x)F 1 (w U (u)=(iu) s )(x)d x 6 k (s) k 1 kF(w 1 (u)=u s )k L 1 U s+3 : (7.9)

Thelinearerror termanb e splitintoabiasandavarianepart (Var[Z℄:=E[jZ E[Z℄j 2 ℄): E h Z U U L(u)w U (u)du 2 i = Z U U ' T (u i) 1 (u i)uE[F( ~ O O )(u)℄w U (u)du 2 +Var h Z U U ' T (u i) 1 (u i)uF ~ O (u)w U (u)du i =:L 2 b +L v :

Thebiastermiseasilyb ounded byProp osition7.1,usingtheuniformb oundonU s+3 w U (u)=u s : jL b j6kF(O l O )k 1 Z U U j' T (u i)j 1 (u 4 +u 2 ) 1=2 jw U (u)jdu . 2 U (s+3) Z U U e T 2 2 u 2 +2Tkk L 1 juj s+2 du: Makinguseof R U 0 2ue u 2 du= e U 2 1 =E(U 2 )U 2

forany>0,we estimatethelast integralby

Z U U e T 2 2 u 2 +2Tkk L 1 juj s+2 du6e 2Tkk L 1 U s+3 E(T 2 2 U 2 )

L jL b j. 2 E(T 2 2 U 2 ): (7.10)

Forthe variane partof thelinearerror term we usethesupp ort prop erties supp (w U )2[ U;U℄ and supp(b k )=[x k 1 ;x k +1

℄. Severalappliations ofthePlanherel identity,the Cauhy-Shwarz

inequalityandestimate(7.7)thenyield

L v = Z U U Z U U Cov ' T (u i) 1 (u i)uF ~ O(u);' T (v i) 1 (v i)v F ~ O(v ) w U (u)w U (v )d udv = N X k =1 Æ 2 k Z U U ' T (u i) 1 (u i)uFb k (u)w U (u)du 2 =2 N X k =1 Æ 2 k Z 1 1 F 1 ' T (u i) 1 (u i)uw U (u) (x)b k ( x)dx 2 62 N X k =1 Æ 2 k Z x k+1 xk 1 F 1 ' T (u i) 1 (u i)uw U (u) ( x) 2 dxkb k k 2 L 2 .kÆk 2 l 1 Z 1 1 F 1 ' T (u i) 1 (u i)uw U (u) ( x) 2 dx skÆk 2 l 1 Z U U j' T (u i)j 2 (u 4 +u 2 )w U (u) 2 du .U 1 E(T 2 U 2 )kÆk 2 l 1 :

Altogether we obtainforthelinearerror term

E h Z U U L(u)w U (u)du 2 i .E(T 2 U 2 ) 4 +U 1 kÆk 2 l 1 : (7.11)

It remains to estimate the quadrati remainder term. We use Lemma 7.2, Prop osition7.1, the

indep endene of("

k

),thenitenessoftheirfourthorder momentsandestimates(4.5),(7.7):

E h Z U U R(u)w U (u)du 2 i . Z U U Z U U E h F( ~ O O )(u)F( ~ O O )(v ) 2 i u 4 w U (u)v 4 w U (v ) (u) 2 (v ) 2 dudv . Z U U Z U U kF(O l O )k 4 1 +E[jF( ~ O O l )(u)F( ~ O O l )(v )j 2 ℄ u 4 w U (u)v 4 w U (v ) (u) 2 (v ) 2 dudv . Z U U Z U U 8 +E h N X k ;l=1 Æ k Æ l " k " l Fb k (u)Fb l (v ) 2 i u 4 w U (u)v 4 w U (v ) (u) 2 (v ) 2 dudv . Z U U Z U U 8 + N X k ;l=1 Æ 2 k Æ 2 l jFb k (u)j 2 jFb l (v )j 2 u 4 w U (u)v 4 w U (v ) (u) 2 (v ) 2 du = 4 Z U U u 4 w U (u) (u) 2 du 2 + Z U U N X k =1 Æ 2 k jFb k (u)j 2 u 4 w U (u) (u) 2 du 2 . 8 U 4 + 4 U 4 kÆk 2 l 2 E(T 2 max U 2 ) 2 :

Thisgivestheresultthat thetotalriskof^ 2 is oforder E[j^ 2 2 j 2 ℄.U 2(s+3) + 4 +U 1 kÆk 2 l 1 E(T 2 U 2 )+ 8 U 4 + 4 U 4 kÆk 2 l 2 E(T 2 max U 2 ) 2 :

Beause ofU . and kÆk l 2 .kÆk l 1

theb oundsimpliesto(7.4).

7.3 Proof of Theorem 4.3

The rates (4.10)and (4.11)follow fromtherate-optimal hoie (4.9) of U and the G

s (R;

max

)-uniformriskestimates

E[j^ j 2 ℄.U 2(s+2) +E(T 2 U 2 )U" 2 +E(T 2 max U 2 ) 2 U 6 " 4 ; (7.12) E[j ^ j 2 ℄.U 2(s+1) +E(T 2 U 2 )U 3 " 2 +E(T 2 max U 2 ) 2 U 8 " 4 ; (7.13)

wheninserting=0inthease

max =0.

Sinethelaimedriskb oundfor^islargerthanfor^ 2

,we onlyneedtoestimatetheriskof^+ ^ 2

2

instead ofthat for^. Equally,wean restrit to ^ ^ 2 2 ^ instead of ^

. Then thepro of follows

exatly thelines of the pro of for ^ 2

, theonly dierene b eing the dierent norminginestimate

(4.5)givingrise toafatorU for andafatorU 2

for. Itremainsto notethat we obtainthe

b ounds inthe omp ound Poisson ase by setting =

max

=0and onsidering the ontinuous

extensionoftheb oundsforthatase: For^we obtainasbias

Z U U F(u)w U (u)du .U (s+2) : (7.14)

Thelinearerror termisestimatedby

E h Z U U L(u)w U (u)du 2 i . ( E(T 2 U 2 ) U 2 4 +UkÆk 2 l 1 ; 2[0; max ℄unknown, U 2 4 +UkÆk 2 l 1 ; = max =0: (7.15)

andtheremaindersatises

E h Z U U R(u)w U (u)du 2 i . ( 8 U 6 + 4 U 6 kÆk 2 l 2 E(T 2 max U 2 ) 2 ; 2[0; max ℄unknown, 8 U 6 + 4 U 6 kÆk 2 l 2 ; = max =0: (7.16)

Altogether we obtaintheriskestimate(7.12).

For ^

weobtainthesameasymptotierror b oundsas for,^ but multipliedbyU whenregarding

thero otmeansquareerror. Thisgives(7.13)and(4.11).

7.4 Proof of Theorem 4.4

Theassertionfollowsasso onasthefollowingG

s (R;

max

)-uniformriskb oundforgeneralU holds:

E h Z 1 1 j(x)^ (x)j 2 dx i .U 2s +E(T 2 U 2 )U 5 " 2 +E(2T 2 max U 2 )U 9 " 4 : (7.17)

Thebiasinestimatingdueto theutoatU an b eestimatedby

Z 1 1 jF(u)(1 1 [ U;U℄ )j 2 du6U 2s Z 1 1 juj 2s jF(u)j 2 d u=U 2s k (s) k 2 L 2: (7.18)

Thevariane terman b e split upaording to thedierent riskontributions. For u2[ U;U℄

we obtain E[jF(^ )(u)j 2 ℄64E[j ~ (u) (u))j 2 ℄+4(u 2 +1) 2 E[j^ 2 2 j 2 ℄ +4(u 2 +1)E[j^ j 2 ℄+4E[j ^ j 2 ℄ .E[jL(u)j 2 ℄+E[jR(u)j 2 ℄+U 4 E[j^ 2 2 j 2 ℄+U 2 E[j^ j 2 ℄+E[j ^ j 2 ℄ .E[jL(u)j 2 ℄+E[jR(u)j 2 ℄+U 2(s+1) +E(T 2 U 2 )U 3 " 2 +E(T 2 max U 2 ) 2 U 8 " 4 :

E[jL(u)j 2 ℄6j' T (u i)j 2 (u 4 +u 2 )(kF(O O l )k 2 1 +Var[F ~ O (u)℄).e T 2 u 2 u 4 4 + 2 kÆk 2 l 2 :

Withalo okatLemma7.2weestimatetheremainderby

E[jR(u)j 2 ℄616(u) 4 (u 4 +u 2 ) 2 E[jF(O l O )(u)j 4 +jF( ~ O O l )(u)j 4 ℄ .e 2T 2 max u 2 u 8 8 + 4 kÆk 4 l 2 :

ThePlanherel identityandtheseestimatesyieldtogether(7.17)via

Z 1 1 E[j^(x) (x)j 2 ℄dx.U 2s +E(T 2 U 2 )U 5 " 2 +E(2T 2 max U 2 )U 9 " 4 +E(T 2 U 2 )U 4 " 2 +E(T 2 max U 2 ) 2 U 9 " 4 sU 2s +E( 2 U 2 )U 5 " 2 +E(2T 2 max U 2 )U 9 " 4 :

8 Proof of the lower bounds

WefollowtheusualBayespriortehnique,seee.g.KorostelevandTsybakov(1993),andp erturba

xedLevytripletT

0 =(0; 0 ; 0 )intheinteriorofG s (R; max

)suhthatthep erturbationsremain

inG

s (R;

max ).

8.1 Lower bound for in the ase =0

Fix a p ositive integer j. Let (j)

2 C 1

(R) b e some funtion with supp ort in [0;1℄ satisfying

k (j) k L 2 =1, R (j) (x)e 2 j x d x=0and R jF (j) (u)u 2 j 2

du<1. Certainly,thereareinnitely

manyfuntions (j)

fulllingtheserequirements;thelastprop ertyfollowsforinstane if isthe

seondderivativeofanL 2

-funtion. Intro due thewavelet-likenotation

jk (x):=2 j=2 (j) (2 j x k ); j>0;k=0;:::;2 j 1:

Considerforanyr=(r

k

)2f 1;+1g 2

j

andsome>0thep erturb edLevytripletsT

r =(0; 0 ; r ) with r (x)= 0 (x)+2 j(s+1=2) 2 j X k =1 r k jk (x); x2R :

We note that due to F

jk (0)=0and R e x jk (x)dx=0thetriplet T r

satises themartingale

onditionsuhthat T

r 2G

s

(R;0)holdsforasuÆientlysmallhoieoftheonstant>0.

TheGaussianlikeliho o dratiooftheobservationsundertheprobabilitiesorresp ondingtoT

r 0

and

T

r

underthelawofT

r forsomer;r 0 withr k =r 0 k

forallk exept onek

0 isgivenby (r 0 ;r )=exp Z 1 1 (O r 0 O r )(x)" 1 dW(x) 1 2 Z 1 1 jO r 0 O r )(x)j 2 " 2 dx :

Hene, the Kullbak-Leibler divergene (relative entropy) b etween the two observation mo dels

equals KL(T r 0 jT r )= 1 2 Z 1 1 j(O r 0 O r )(x)j 2 " 2 dx:

ThestandardAssouadLemma(KorostelevandTsybakov1993,Thm. 2.6.4)nowyieldsthelower

b oundfortheriskofanyestimator^of

inf ^ sup T=(0;;)2G s (R;0) E T h Z j^(x) (x)j 2 dx i &2 j k r r 0k 2 L 2 s2 2js ;

providedtheKullbak-LeiblerdivergeneKL(T

r jT

r

)staysuniformlyb oundedbyasmallonstant.

Itremainstodetermineaminimalratefor2 j

!1suhthatthisholdswhenthenoise leveltends

tozero.

Arguinginthesp etraldomainandusingthegeneralestimateje z

1j62jzj,forjzj6Æ andsome

smallÆ>0,togetherwithk'

T;r 0=' T;r k 1 !1for2 j

!1,we obtainforallsuÆientlylargej

KL(T r 0 jT r )= 1 4 " 2 Z 1 1 jF(O r 0 O r )(u)j 2 du 6" 2 Z 1 1 ' T;r (u i) ' T;r 0 (u i) u(u i) 2 du 64" 2 Z 1 1 j' T;r (u i)j 2 T 2 jF( r r 0)(u)j 2 (u 4 +u 2 ) 1 d u ." 2 2 j(2s+1) Z 1 1 jF jk0 (u)j 2 u 4 du =" 2 2 j(2s+5) Z 1 1 jF (j) (v )j 2 v 4 dv : Hene,for2 j(2s+5) s" 2

withasuÆientlylargeonstanttheKullbak-Leiblerdivergeneremains

b oundedandtheasymptotilowerb oundforfollows.

8.2 Lower bound for and in the ase =0

Letusstartwiththelowerb oundfor. Wepro eedasb eforebyp erturbingatripletT

0 =(0; 0 ; 0 )

fromthe interiorof G

s

(R;0),but this timewe only onsider one alternative T

1 =(0; 1 ; 1 ) and

ho ose thep erturbation insuh awaythattheharateristi funtion'

T

(u i)do es nothange

forsmallvalues ofjuj.ForanyÆ>0andU >0put

1 := 0 +Æ; F 1 (u):=F 0

(u) Æi(u i)e u

2

=U 2

; u2R :

Thenthefuntion

1

isreal-valued. Moreover,themartingaleondition(2.3)issatised:

1 +F 1 (0) F 1 (i)= 0 +Æ+F 0 (0) Æ F 0 (i)+0=0: Beause of k (s) 1 (s) 0 k 1 62 Z 1 1 juj s jF( 1 0 )(u)jdu.Æ Z 1 1 juj s+1 e u 2 =U 2 dusÆU s+2

andevenb etter b oundsfork (k ) 1 (k ) 0 k L 2 ,k=0;:::;s,itsuÆes toho ose UsÆ 1=(s+2) small

enough to ensure that T

1

still lies in our nonparametri lass G

s

(R;0). The basi lower b ound

result (KorostelevandTsybakov1993,Prop. 2.2.2)thenyields

inf ^ sup (0;;)2Gs(R;0) E ; [j^ j 2 ℄&Æ 2 ;

providedtheKullbak-Leiblerdivergeneb etweenT

1 andT

0

remainsasymptotiallyb ounded. As

inthelowerb ound pro offorweobtainasymptotially

KL(T 1 jT 0 ) 64" 2 Z 1 1 j' 0;T (u i)j 2 T 2 ji( 1 0 )(u i)+F( 1 0 )(u) F( 1 0 )(i)j 2 (u 4 +u 2 ) 1 du ." 2 Æ 2 Z 1 1 ji(u i)(1 e u 2 =U 2 )j 2 (u 4 +u 2 ) 1 du =" 2 Æ 2 Z 1 1 (1 e v 2 ) 2 U 2 v 2 Udv ." 2 Æ 2 U 1 s" 2 Æ (2s+5)=(s+2) :

ThelatterremainssmallforÆ s" withasmallonstant,whihgivestheasymptoti

lowerb oundfor.

Forthelowerb oundofwep erturb thetripletT

0

leaving

0 and

0

=0xedandputting

F 1 (u):=F 0 (u)+Æe u(u i)=U 2 : Then 1 isreal-valued, 1 0 =F( 1 0

)(i) =Æ andthetriplet T

1 =(0; 0 ; 1 ) satisesthe

martingaleondition. ForU sÆ

1=(s+1)

withasuÆientlysmallonstantthep erturbation

1 lies inG s (R;0)dueto k (s) 1 (s) 0 k 1 .Æ Z 1 1 juj s e u 2 =U 2 dusÆU s+1

and even b etter b ounds for k (k ) 1 (k ) 0 k L 2,

k = 0;:::;s. The Kullbak-Leibler divergene is

asymptotiallyb ounded by KL(T 1 jT 0 )64" 2 Z 1 1 j' 0;T (u i)j 2 T 2 jF( 1 0 )(u) F( 1 0 )(i)j 2 (u 4 +u 2 ) 1 d u ." 2 Æ 2 Z 1 1 j1 e u(u i)=U 2 j 2 (u 4 +u 2 ) 1 du =" 2 Æ 2 Z 1 1 j1 e v 2 +iv =U j 2 (U 4 v 4 +U 2 v 2 ) 1 Udv ." 2 Æ 2 U 3 s" 2 Æ (2s+5)=(s+1)

andweobtaintheasymptotilowerb ound for.

8.3 Lower bound for in the ase >0

The interesting deviation from standard pro ofs of lower b ounds (see e.g. Butuea and Matias

(2005)) for severely ill-p osed problems is that we fae the restrition that F is analyti in a

strip parallelto thereal line andis uniquelyidentiablefromitsvalueson any op enset. So,let

T 0 = ( 2 0 ; 0 ; 0 ) with 0

> 0 b e a Levy triplet fromthe interiorof G

s (R; max ). Consider the p erturbation T 1 =( 2 0 ; 0 ; 1 )with F 1 (u):=F 0 (u)+Æm 1=4 e (T 2 0 u 2 =m) m =2 (T 2 0 =m) m u m (u i) m ; u2R :

form2N,Æ >0. Thenwe haveuniformlyform!1 andÆ !0

k 1 0 k 2 L 2 =2 kF( 1 0 )k 2 L 2 = 2 Æ 2 p T 2 0 Z 1 0 e v v (1+2m)=2m (1+m 1 v 1=m ) m dvsÆ 2 :

Similarly,fork=1;:::;swederiveuniformlyinmandÆ

k (k ) 1 (k ) 0 k L 2 = p 2 ku k F( 1 0 )(u)k L 2 sÆm k =2 ; k (s) 1 (s) 0 k 1 6ku s F( 1 0 )(u)k L 16Æm s=2 1=4 : Thereforeho osingÆsm s=2

withasmallonstantyieldsT

1 2G

s (R;

max

)b eausewethenalso

have that

1

isreal-valuedandT

1

satisesthemartingaleonditionandAssumption1.

By thesameargumentsas b eforeandbyStirling'sformulato estimatethe Gammafuntion,the

Kullbak-Leibler divergene b etween the observations under T

0

and under T

1

KL(T 1 jT 0 )64" 2 Z 1 1 j' 0;T (u i)j 2 T 2 jF( 1 0 )(u)j 2 (u 4 +u 2 ) 1 du ." 2 Æ 2 Z 1 1 e T 2 0 u 2 m 1=2 e (T 2 0 u 2 =m) m (T 2 0 =m) 2m u 2m 2 ju ij 2m 2 du =" 2 Æ 2 m 7=2 (T 2 0 m) 1=2 Z 1 0 e mv 1=m e v v (2m 1)=2m (1+m 1 v 1=m ) m 1 dv ." 2 Æ 2 m 4 Z 1 0 e mv 1=m dv =" 2 Æ 2 m 4 Z 1 0 e z z m 1 m 1 m dz =" 2 Æ 2 m m 3 (m)." 2 Æ 2 m m 3 (m 1) m 1=2 e 1 m s" 2 m 3 s e m

Consequently, theKullbak-Leiblerdivergene remains smallwhen ho osingm>2log(" 1

), but

m . log(" 1

), whih gives Æ s log(" 1

) s=2

. From the basi general lower b ound result we

therefore obtaintheasymptotilowerb oundfor.

8.4 Lower bound for 2

, and in the ase >0

Letusstartwiththelowerb oundfor. Wepro eedasinthease =0byp erturbingthetriplet

T 0 =( 0 ; 0 ; 0 ) with 0

> 0in suh awaythat the harateristi funtion '

T

(u i) do es not

hangemuhforsmallvalues ofjuj. ForanyÆ put

1 := 0 +Æ; F 1 (u):=F 0

(u) Æi(u i)e u 2m =U 2m : Then 1

isreal-valuedandthemartingaleondition(2.3)issatised. Beause of

k (s) 1 (s) 0 k 1 6 Z juj s jF( 1 0 )(u)jdu.Æ Z 1 1 juj s+1 e u 2m =U 2m dusÆU s+2

and smaller b ounds for k (k ) 1 (k ) 0 k L 2, k =0;:::;s, we ho ose U s Æ 1=(s+2) smallenough to

ensure that thep erturb ed tripletT

1

still lies inG

s (R;

max

). In thesamemanner as b efore and

usingj1 e x

j6jxj,x>0,aswellasStirling'sformula,weobtain

KL(T 1 jT 0 )64" 2 Z 1 1 j' 0;T (u i)j 2 T 2 ji( 1 0 )(u i)+ +F( 1 0 )(u) F( 1 0 )(i)j 2 (u 4 +u 2 ) 1 du ." 2 Æ 2 Z 1 1 e T 2 0 u 2 ji(u i)(1 e u 2m =(2U 2m ) )j 2 (u 4 +u 2 ) 1 du ." 2 Æ 2 Z 1 1 e T 2 0 u 2 u 4m U 4m u 2 du s" 2 Æ 2 U 4m (2m 1 2 ) s" 2 Æ 2+4m=(s+2) (2m) 2m e 2m

TokeeptheKullbak-Leiblerdivergenesmall,weho ose

Æ (2s+4m+4)=(s+2) s" 2 (2m) 2m e 2m

andthusobtainuniformlyoverm theb ound

inf ^ sup T=( 2 ;;)2G s (R; max ) E T [j^ j 2 ℄ 1=2 & " 2 (2m) 2m e 2m (s+2)=(2s+4m+4) :

Themaximizerof thisexpressionm slog(" )then yieldstheasymptotilowerb oundfor.

Forwe p erturbthetripletT

0

leaving

0 and

0

xedandputtingforanevenintegerm

F 1 (u):=F 0 (u)+Æe u m (u i) m =U 2m : Then 1 0 =F( 1 0

)(i)=ÆandthetripletT

1 =( 0 ; 0 ; 1

)satisesthemartingaleondition.

ForU sÆ

1=(s+1)

with asuÆientlysmallonstant thep erturbation

1 liesinG s (R; max ). As

b eforewe provethat theKullbak-Leiblerdivergeneremainsb oundedwhenever

Æ (2s+4m+2)=(s+1) s" 2 (2m) 2m+1 e 2m : Cho osingm slog(" 1

)as b eforegivestheasymptotilower b oundfor.

For 2

we p erturbthetripletT

0

leaving

0

invariantandputting

2 1 := 2 0 +2Æ; F 1 (u):=F 0 (u)+Æ(u i) 2 e u 2m =U 2m :

Thenthemartingaleondition(2.3)issatisedandforU sÆ

1=(s+3)

suÆientlysmallweremain

inG

s (R;

max

). ItisagainroutinetoprovethattheKullbak-Leiblerdivergeneremainsb ounded

whenever Æ (2s+4m+6)=(s+3) s" 2 (2m) 2m 1 e 2m : Cho osingm

as b eforegivestheasymptotilowerb oundfor 2

.

9 Appendix

9.1 Proof of Proposition 2.1

(a) This followsfromtheput-allparity(2.4).

(b) O (x)>0followsdiretly from(2.6)whileO (x)6E[e XT ℄ (1 e x ) + =1^e x followsfrom

(a) andthemartingaleondition.

() Weonlude byHolder'sandMarkov'sinequalityforx>0

O (x)6E[e X 1 fX>xg ℄6C 1= P(X >x) ( 1)= 6C 1= C e x ( 1)= =C e (1 )x : (d) Letusdenotebyf T

thedensityoftheabsolutelyontinuouspartofthedistributionofX

T .

TheonlyatominthedistributionofX

T

ano ur atT, namelyintheomp oundPoisson

ase whennojumpuntilT hastakenplae. For x6=0wehave

O 0 (x)= E[(e XT e x )1 fX T >xg ℄ 0 +e x 1 fx<0g =e x P(X T >x)+1 fx<0g : (9.1) This yieldsO 0 (0+) O 0

(0 )= 1andinthease =0,<1,6=0also

O 0 (T+) O 0 (T )=e T P(X T =T)=e ( )T :

Atallp ointsx6=0wherethelawof X

T

hasnoatomwe obtain

O 00 (x)= e x P(X T >x) 0 +e x 1 fx<0g =e x P(X T <x)+f T (x) 1 fx>0g :

Consequently,bypartialintegrationandusingE[e XT ℄=1we arriveat Z Rnf0;Tg jO 00 (x)jdx=O 0 (0 )+ Z 1 0 e x P(X T <x) 1+f T (x) dx 6P(X T <0)+ Z 1 0 e x (1 P(X T <x))dx+E 1 fXT>0g e XT =2P(X T <0) 1+2E 1 fX T >0g e XT 61+2E[e XT ℄=3:

FO (v )=S 1 Z 0 1 e iv x P T (x)dx+ Z 1 0 e iv x C T (x)dx = Z 0 1 e iv x E 1 fX T 6xg (e x e XT ) dx+ Z 1 0 e iv x E 1 fX T >xg (e XT e x ) dx:

By partialintegrationweobtain

Z 0 1 e (iv +1)x P(X T 6x)dx= 1 1+iv P(X T 60) 1 1+iv E 1 fX T 60g e (1+iv )XT ; Z 0 1 e iv x E 1 fXT6xg e XT dx= 1 iv E 1 fXT60g e XT 1 iv E 1 fXT60g e (1+iv )XT and onsequently Z 0 1 e iv x E 1 fXT6xg (e x e XT ) dx= 1 1+iv P(X T 60) 1 1+iv E 1 fXT60g e (1+iv )XT 1 iv E 1 fX T 60g e X T + 1 iv E 1 fX T 60g e (1+iv )X T :

Inthesamewaywe derive

Z 1 0 e iv x E 1 fXT>xg (e X T e x ) dx= 1 iv E 1 fXT>0g e X T + 1 iv E 1 fXT>0g e (1+iv )X T + 1 1+iv P(X T >0) 1 1+iv E 1 fXT>0g e (1+iv )XT :

TakingintoaountE[e X

T

℄=1,weobtainformula(2.7).

9.2 Proof of Proposition 5.1

We only sketh the main steps in the pro of, the reasoning b eing similar to that for frational

derivatives, f. Samko,Kilbas,and Marihev (1993). The followingformulais easilyestablished

andloselyrelatedtoequation(5.8)inSamko,Kilbas,andMarihev(1993):

F( (x)x 1 )(u)= (2 )sin( =2) i Z 1 0 z 2 F (u+z) F (u z) dz:

Letusonlyonsidertheaseu>0andset

G u (x):= 1 (s 1)! Z x 0 (x ) s 1 F (u+)d +2 1 ( 1; u℄ (x) s 1 X k =0 (k ) (0) ( i) k (u+x) s k 1 (s k 1)!k !

Thenwehaveinadistributionalsense

G (k ) u (0)=0; k=0;:::;s 1; G (s) u (x)=F (u+x)+2 s 1 X k =0 (k ) (0) ( i) k Æ (k ) u (x) k ! :

Hene, bys-foldpartialintegrationweobtain

Z 1 0 z 2 F (u+z) F (u z) dz 2 s 1 X k =0 (k ) (0)i k 2 k u 2 k = Z 1 0 z 2 G (s) u (z) G (s) u ( z) dz= s Y k =1 (k+1 ) Z 1 0 G u (z) ( 1) s G u ( z) z s+2 dz:

It therefore suÆes to show that the last integral is of order juj

s min(1;2 )

, whih is

aom-plishedbysplitting theintegrationintervalintotheparts [0;1℄,[1;u℄and[u;1)andmakinguse

ofjF

(u)j.(1+juj) s 1

and oftheprop erties ofG

u

A 

t-Sahalia, Y., and J. Duarte (2003): \Nonparametri optionpriing under shap e

restri-tions.,"J.Eonom.,116(1-2),9{47.

A 

t-Sahalia, Y., andJ. Jaod(2004): \Fisher'sinformationfordisretelysampledLevy

pro-esses,"Prepubliation950,Lab oratoire deProbabilitesetMo deles Aleatoires,Paris.

Belomestny, D., andV.Spokoiny (2004): \Lo al-likeliho o dmo delingviastage-wise

aggrega-tion,"Preprint1000,Weierstra InstituteforAppliedAnalysisandSto hastis,Berlin.

Brown, L. D., and M. G. Low (1996): \Asymptotiequivaleneof nonparametri regression

andwhitenoise.,"Ann.Stat.,24(6),2384{2398.

Butuea, C., andC. Matias(2005): \Minimaxestimationofthenoise leveland ofthe

deon-volutiondensity inasemiparametrionvolutionmo del,"Bernoulli,to app ear.

Carr, P., H. Geman, D. B. Madan, and M. Yor (2002): \The Fine Struture of Asset

Returns: AnEmpirialInvestigation,"J.Business,75(2),305{332.

Carr,P.,and D. Madan(1999): \OptionvaluationusingthefastFouriertransform,"J.

Com-put.Finane,2,61{73.

Cavalier, L., and Y. Golubev (2004): \Riskhullmetho dandregularizationbyprojetions of

ill-p osedinverse problems,"Preprint,Universite Aix-Marseille1.

Cont, R., and P. Tankov (2004a): Finanial modelling with jump proesses, Chapman &

Hall/CRCFinanialMathematisSeries. Chapman&Hall/CRC,Bo a Raton,FL.

(2004b): \Nonparametri alibrationof jump-diusionoptionpriingmo dels,"Journal

of ComputationalFinane, 7(3),1{49.

Cont,R., and E.Volthkova(2005): \Integro-dierentialequationsforoptionpriesinexp

o-nentialLevymo dels,"Finane Stoh., toapp ear.

Cr

epey,S.(2003): \Calibrationofthelo alvolatilityinageneralizedBlak{Sholesmo delusing

Tikhonovregularization.,"SIAMJ.Math.Anal.,34(5),1183{1206.

Duffie, D.,D.Filipovi,andW.Shahermayer (2003): \AÆnepro essesandappliations

innane.,"Ann.Appl.Probab.,13(3),984{1053.

Eberlein, E., U. Keller, and K. Prause (1998): \New insightsinto smile,mispriing, and

valueat risk: thehyp erb olimo del,"Journalof Business,71(3),371{405.

Eberlein, E., and A. Papapantoleo n (2004): \Symmetriesand priing of exoti optionsin

Levy mo dels,"Preprint,UniversitatFreiburg.

Emmer, S., and C. Kl

uppelberg (2004): \Optimal p ortfolios when sto k pries follow an

exp onentialLevypro ess," FinaneStoh.,8(1),17{44.

Engl, H.W.,M. Hanke,andA.Neubauer(1996): Regularizationofinverseproblems.

Math-ematisanditsAppliations(Dordreht).375.Dordreht: KluwerAademiPublishers.

Fengler, M., W. Hardle, and E. Mammen (2003): \Implied Volatility String

Dy-namis," Disussion Pap er 54, Humb oldt University Berlin, Sonderforshungsb ereih 373,

http://sfb.wiwi.hu-berlin.de.

Goldenshluger, A., A. Tsybakov, and A. Zeevi (2004): \Optimal

hange-p oint estimation from indiret observations," Preprint, Columbia University,

Jakson, N., E. uli, and S. Howison (1999): \Computationofdeterministi volatility

sur-faes,"JournalComp.Finane, 2(2),5{32.

Kallsen,J.(2000): \Optimalp ortfoliosforexp onentialLevypro esses,"Math.Meth.Operations

Res.,51(3),357{374.

Korostelev, A., and A. Tsybakov (1993): Minimax theory of image reonstrution. Leture

NotesinStatistis(Springer).82.New York:Springer-Verlag.

Kou, S. (2002): \Ajumpdiusionmo delforoptionpriing,"Management Siene,48(4),1086{

1101.

Merton, R. (1976): \Option Priing WhenUnderlying Sto k Returns Are Disontinuous," J.

FinanialEonomis,3(1).

Mordeki, E. (2002): \Optimalstopping and p erp etual optionsfor Levy pro esses.," Finane

Stoh.,6(4),473{493.

Rudin,W.(1991): Funtional analysis.2nd ed.InternationalSeries inPureandApplied

Mathe-matis.New York,NY:MGraw-Hill.

Samko, S., A. Kilbas, and O. Marihev (1993): Frational integrals and derivatives: theory