Spectral analysis of restricted call and put operators and application to stable risk-neutral density recovery

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application to stable risk-neutral density recovery

Jean-Baptiste Monnier

To cite this version:

Jean-Baptiste Monnier. Spectral analysis of restricted call and put operators and application to stable

risk-neutral density recovery. 2011. �hal-00651239�

DENSITYRECOVERY

JEAN-BAPTISTE MONNIER

∗

Abstra t. Inthispaper,weproposeanewmethodforestimatingthe onditionalrisk-neutral

density(RND)dire tlyfroma ross-se tionofputoptionbid-askquotes. Morepre isely,wepropose

to view the RND re overy problem as an inverse problem. Werst show that it is possible to

denerestri tedputand alloperatorsthatadmitasingularvaluede omposition(SVD),whi hwe

omputeexpli itly.Wesubsequentlyshowthatthisnewframeworkallowstodeviseasimpleandfast

quadrati programmingmethodtore overthe smoothestRNDwhose orrespondingputpri eslie

insidethebid-askquotes. Thismethodistermedthespe tralre overymethod(SRM).Interestingly,

theSVDoftherestri tedputand alloperatorsshedssomenewlightontheRNDre overyproblem.

TheSRMimproveson otherRNDre overy methodsinthe sense that1) it isfast and simpleto

implementsin eit requiresto solveone singlequadrati program, yet being fullynonparametri ;

2)ittakesthebidaskquotesassoleinputanddoesnotrequireanysortof alibration,smoothingor

prepro essingofthedata;3)itisrobusttothepau ityofpri equotes;4)itreturnsthesmoothest

densitygiving risetopri esthat lieinsidethebidaskquotes. TheestimatedRNDisthereforeas

well-behavedas anbe;5)itreturnsa losedformestimateoftheRNDontheinterval

[0, B]

ofthe

positiverealline,where

B

isapositive onstantthat anbe hosenarbitrarily. Wethusobtainboth

themiddlepartoftheRNDtogetherwithitsfulllefttailandpartofitsrighttail. We onfrontthis

methodtobothrealandsimulateddataandobservethatitfareswellinpra ti e. TheSRMisthus

foundtobeapromisingalternativetootherRNDre overymethods.

Key words. Risk-neutral density; Nonparametri estimation;Singularvalue de omposition;

Spe tralanalysis;Quadrati programming.

AMSsubje t lassi ations.91G70,91G80,45Q05,62G05

1. Introdu tion.

1.1. The setting. Overthelast fourde ades,theno-arbitrageassumptionhas

proved to be a fruitful starting point that paved the way for the elaboration of a

ri h theoreti al framework for derivatives pri ing known today asarbitrage pri ing

∗

O e 5B01, LPMA, Université Paris 7, 175 rue du Chevaleret, 75013, Paris, Fran e. Tel:

theory. Amongitsnumerousa hievements,thearbitragepri ingtheoryhassetforth

twofundamental theorems. The FirstFundamental Theorem of Asset Pri ing (see

[21,p.72℄)provesthatamarketisarbitrage-freeifandonlyifthere existsameasure

Q

equivalentto thehistori al (orstatisti al) measure

P

,whi hturns theunderlying

pri epro ess into amartingale.

Q

is thereforereferred toas amartingalemeasure.

TheSe ondFundamentalTheoremofAssetPri ing(see[21,p.73℄)provesinturnthat

thismartingalemeasureisuniqueifandonlyifthemarketis omplete(see[21,p.300℄

forterminology). Letusdenoteby

S

τ

thepositivevaluedpri eoftheunderlyingata

deterministi futuredate

τ

andby

π(S

τ

)

thepayoofa ontingent laimmaturingat

time

τ

. Letusmoreoverdenoteby

q

themarginaldensityof

S

τ

under

Q

withrespe t

totheLebesguemeasureonthepositiverealline,assumingthatitexists. Asinitially

provedin [10℄, thearbitragepri eof thisderivativese urity writesasitsdis ounted

expe tedpayounder

Q

,that is,

e

−rτ

E

Q

π(S

τ

) = e

−rτ

Z

x≥0

π(x)Q(S

τ

∈ dx) = e

−rτ

Z

x≥0

π(x)q(x)dx,

where

r

standsforthe ontinuously ompoundedrisk-freerate. Itisawidely

a knowl-edgedfa tthat nan ialmarketsarein omplete,shallitonlybeduetothepresen e

ofjumps in theunderlying pri epro ess. Insu h asetting, and asdes ribedabove,

thereexisteventuallyverymany

q

s,andtherefore,verymany orrespondingsystems

of arbitrage-freepri es. Letus denoteby

Q

the orresponding set ofvalid densities

q

. The elements

q

of

Q

are mostoften referredto asrisk-neutraldensities(RNDs)

andwewillsti ktothis terminologyin thesequel.

RNDs are of ru ial interest for Central Banks and, in fa t, most institutions and

people on erned with nan ial markets sin e they representthe market sentiment

aboutagivenunderlying pri epro ess at afuture pointin time (see[3℄). They are

also of ru ial interest to the nan ial derivatives industry sin e the knowledge of

tionisveryextensive,thebulkofitdatingba ktothelate90'sand early2k's. Itis

notourpurposeheretopresentanexhaustivereviewofthisliterature. Ex ellentand

up-to-datereviews anin fa t befound in[14,17℄. Olderbutstill relevantones an

befoundin [9,3℄.

Among derivative se urities, all and put options play a very parti ular role sin e

theyarea tivelytradedin themarketandthusbelievedtobee ientlypri ed. Let

us re all that a allof strike

ξ

andmaturity

τ

givesitsholder the rightto buy the

underlying se urity at maturity time

τ

at pri e

ξ

. It is an insuran e againsta rise

in the pri e of the underlying. Its payo writes

π(S

τ

, ξ) = θ(S

τ

, ξ) = (S

τ

− ξ)

+

,

where wehavewritten

(x)

+

_{= max(x, 0)}

for

x

∈ R

. Conversely, aput option gives

theright to sellthe underlying se urity. It is aninsuran eagainst afall in the

un-derlyingpri eanditspayowrites

θ

∗

_(S

τ

, ξ) = (ξ

− S

τ

)

+

. Here andin whatfollows,

wedenotethestrikepri eby

ξ

andnotby

k

,whi hwillstandforarunningindexin

N

.

A ording to the elebratedBreeden-Litzenberger formula, the se ond derivativeof

put and allpri eswithrespe tto theirstrikepri ebothequalthedis ountedRND

e

−rτ

_q

(see [6℄). Therefore, ifa ontinuum of putor allpri es were available in the

market,wewouldhavedire ta esstotheRNDbythelatterformula. However,this

isnotthe aseandonlyafewstrikepri esaroundtheforwardpri earequotedand

a tivelytraded at ea h maturitydate. Depending onthe market,weoverall re kon

from

5

to

50

quotesatagivenmaturitydate

τ

. To ompli atethematterevenmore,

quotesdonotappearasasinglepri e. Dealersquoteinfa tabidpri e,atwhi hthey

oertobuythese urity,andanaskpri e,atwhi htheyoertosellthese urity. The

dieren ebetweenbothpri esisreferredtoasthebid-askspread. Foraninteresting

insight into the nature of option quotesand sour esof error in them, the readeris

referredto, say,[16, p.786℄.

1.2. The problemand briefliteraturereview. Asdetailedabove,iftraded

puts and alls at agivenmaturity

τ

are arbitragefree,they mustwrite astheir

thepau ityof quotedoptionpri esatagivenmaturity

τ

and thepresen eofa

bid-askspread,itis learthatmanyRNDs ouldinfa tbehiddenbehindquotedoption

pri es. Therefore, theRND questis notthat mu haboutestimatingthe trueRND

thatisusedbythemarketforpri ingpurpose,sin ethenatureofthequotesdoesnot

allowtoidentifyituniquely. Itisrathermoreaboutre overingavalidRND,meaning

ana tualdensityfun tion,to be hosena ordingtoa riterion typi allyrelatedto

itssmoothnessorinformation ontent. Histori ally,threemainrouteshavebeenused

to re over a RND from quoted option pri es: parametri methods, nonparametri

methods and models of theunderlying pri epro ess. Ea h of them havetheir pros

and ons. Parametri methodsarewelladaptedtosmalldatasetsandalwaysre over

adensity. However,they onstraintheRND tobelongtoagivenparametri family.

Onthe otherhand,modelsofthe underlyingpri epro esshavebeentherstgreat

su ess of arbitrage pri ing theory with the elebrated geometri Brownian motion

(see [4, 20℄). However, the limitation of the log-normal distribution is now widely

a knowledged and nosatisfying sto hasti pro esshas yet been proposed that both

reprodu ea uratelythedynami softheunderlyingpri epro essandbeanalyti ally

tra table. Nonparametri methods ir umventboth of these problems in the sense

thattheydonotrequireanystringentassumptiononthepro essgeneratingthedata

(they are model-free) and an re over all possible densities. As a main drawba k,

thesemethodsareoftendataintensive.

Letusbriey omeba konsome ontributionstothenonparametri literaturewhi h

arerelevanttothepresentpaper. We an lassifynonparametri methodsasfollows.

•

The expansion methods. It in ludes the Edgeworth (see [19℄) and

umu-lantexpansions(see [22℄), whi h allowto estimatea nite numberof RND

umulants. Italsoin ludesorthonormalbasismethodssu hasHermite

poly-nomials(see[1℄),whi hrelyonwellknownHilbertspa ete hniquesandgive

a esstothemiddlepartoftheRND.

performsestimationontheaveragequotedpri es(thatis,theaverageofthe

bid-askquotes)and requiresthereforeto pre-pro essthem in ordertomake

them arbitrage-free. Moreover, the returned RND depends on the kernel

hosenand itis not learhowitrelatesto theother validRNDs in termof

information ontentorsmoothness.

•

Themaximumentropymethod. Itisintrodu edin[8,23℄,wheretheRND

q

is

obtainedviathemaximizationofanentropy riterion. A ordingto[9,p.19℄,

this method oftengivesbumpy(multimodals)estimatessin eit imposes no

smoothnessrestri tionontheestimateddensity. Inaddition,itissaidin[18,

p.1620℄,thatthismethod presents onvergen eissues.

•

Other methods, whi h do not belong to any of the three ategories above.

Amongthem,we anrefertothepositive onvolutionapproximation(PCA)

of [5℄. In pra ti e, it ts a nite (but large) onvex linear ombination of

normaldensitiestotheaveragequotedputpri esandapproximatestheRND

by theweights of thelinear ombination. It thus presents similarities with

[18℄,sin eitultimatelytsadis retesetofprobabilitiestotheaveragequoted

pri es. We analso refer to the smoothedimplied volatility smile method

(SML)asin[14℄. ThismethodusestheBla k-S holesformulaasanon-linear

transform. It onsistsin ttingapolynomialthroughtheimpliedvolatilities

obtainedfromaveragequotedpri es,andusingthe ontinuumofoptionpri es

obtainedinthat wayto gettheRNDviatheBreeden-Litzenbergerformula.

[14℄ renes this method by taking the bid-ask quotes into a ount at the

impliedvolatilitytstage. TheSMLmethodgivesa esstothemiddlepart

of the RND. [14℄ proposes in addition a method for appending generalized

extremevalue(GEV)taildistributionstoit. TheSMLmethodis umbersome

and an seema bit odd sin e it requires going from pri e spa e to implied

volatilityspa e,ba kandforth. Itis laimedthatitisoutperformedinterm

ofa ura yandstabilitybysimplerparametri methodsin [7℄.

alloperators that admit asingularvaluede omposition (SVD), whi hwe ompute

expli itly. Wesubsequentlyshowthat thisnew framework allowsto deviseasimple

andfastquadrati programmingmethodto re overthesmoothestRND that is

on-sistentwithmarketbid-askquotes.

Tobemorepre ise,letusdenoteby

I

thesegment

[0, B]

ofthepositiverealline. We

denetherestri tedputand alloperators,denotedby

γ

∗

and

γ

,from

L

2

I

intoitself

(seeeq.(2.1)andeq.(2.2)below)andshowthattheyare onjugatesofoneanother.

Weprovethat theresultingself-adjointoperator

γ

∗

_γ

is ompa t. Asa onsequen e

of the spe tral theorem (see [15℄),

γ

∗

admits a singular value de omposition with

positivede reasingsingularvalues. Weprovethat the orrespondingsingularbases

are ompletein

L

2

I

(seeTheorem3.1,item3))and omputethemexpli itlytogether

withtheirsingularvalues(seeFigure1.1). Toxnotations,wewillwrite

(ϕ

k

)

k≥0

and

(ψ

k

)

k≥0

thetwoorthonormalfamiliesof

L

2

I

su hthat

γ

∗

_γϕ

k

= λ

2

k

ϕ

k

,

γγ

∗

_ψ

k

= λ

2

k

ψ

k

,

where

(λ

k

)

k≥0

isapositivede reasingsequen eofsingularvalues. Pre isely,weobtain

expli itly,

λ

k

=

 B

ρ

k



2

,

where

ρ

k

=

π

2

+ kπ + (

−1)

k

_β

k

,

k

∈ N,

and, for all

k

∈ N

,

β

k

is the smallest positive solution of the following xed point

equationin

u

,

exp(π/2 + kπ + (

−1)

k

_{u) =}

1 + cos(u)

sin(u)

.

Interestingly, thepositivesequen e

(β

k

)

de reases exponentially fasttowardzeroas

detailed in Lemma 6.8. Therefore, the sequen e of singular values

(λ

k

)

k≥0

tends

asymptoti ally toward zero at a rate of order

k

−2

0

20

40

60

80

100

120

140

160

180

200

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

phi0

phi1

phi2

phi3

0

20

40

60

80

100

120

140

160

180

200

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

psi0

psi1

psi2

psi3

Figure 1.1. Here,we plottherstfourelementsof bothsingular bases. At thetopwe plot

thereforesaidtobemildlyill-posedwithadegreeofill-posednessequalto

2

(see[13,

p.40℄). Furthermore,forall

ξ

∈ I

,weobtain,

ϕ

k

(ξ) =



a

k,1

e

ρ

k

ξ/B

+ a

k,2

e

−ρ

k

ξ/B



+



a

k,3

cos(ρ

k

t/B) + a

k,4

sin(ρ

k

ξ/B)



,

ψ

k

(ξ) =



a

k,1

e

ρ

k

ξ/B

+ a

k,2

e

−ρ

k

ξ/B



−



a

k,3

cos(ρ

k

t/B) + a

k,4

sin(ρ

k

ξ/B)



.

wherethe oe ients

a

k,i

, i = 1, . . . , 4

aresu hthat,

a

k,1

=

1

√

B

(

₋₁₎

k

e

ρ

k

+ (

−1)

k

,

a

k,2

= (

−1)

k

e

ρ

k

a

k,1

=

1

√

B

1

1 + (

₋₁₎

k

_e

−ρ

k

,

a

k,3

=

−

√

1

B

,

a

k,4

=

√

1

B

1

− (−1)

k

_e

−ρ

k

1 + (

−1)

k

_e

−ρ

k

.

Basedon this newframework,wepropose aspe tralapproa h to RND re overy. It

is fully nonparametri and an re overtherestri tion of any density to the interval

I

. To that end, we noti e that thesingular bases fun tions

ϕ

k

and

ψ

k

are in fa t

os illations

h

k,2

atfrequen y

ρ

k

/B

arriedbytheexponentialtrend

h

k,1

(seeeq.(6.2)

andeq.(6.1)for notations). Conveniently,smoothdensities arethereforeessentially

apturedbylowsingularspa es. Theideaofre overingthesmoothestdensityamong

thevalidoneswasinitiallysuggestedin[18℄. Subsequently,[9℄rightfullypointedout

that thesmoothness riterion anbedebated asitis di ulttogiveit ane onomi

oreveninformationtheoreti meaning. Ourspe tralapproa hshedssomenewlight

onthisissueand makesit learthatthesmoothness riterionisjustiedbythefa t

that therestri ted all andput operatorsbehaveaslow-passfrequen ylters. Itis

therefore illusoryto look for high frequen yinformation aboutthe RND in aset of

quoted optionspri es,sin e thisinformation hasbeendrasti ally attenuatedby the

operator. The smoothness riterion arises thereforeasa by-produ t of thespe tral

bynonparametri means.

Inwhat follows,weexploit theri hframeworkoered bytheSVD ofthe restri ted

putand alloperatorstore overthesmoothestRND thatis ompatiblewithmarket

quotes. Asdetailedineq.(7.1)below,thedis ountedrestri tedputoperator oin ides

withthe putpri efun tion (as afun tion ofthestrike)on

I

. Wethereforepropose

to re overthesmoothest RND su h that itsimage bythe dis ountedrestri tedput

operator

e

−rτ

_γ

∗

liesin-betweenthebid-askquotes (seeeq.(7.1)). Conveniently,the

singularbasespresentthepropertyofbeingimageofoneanotherbyse ondderivation

moduloamultipli ationbythe orrespondingsingularvalueof

γ

∗

(seeTheorem6.1).

Thisallowsusto hara terizethesmoothness oftheestimatedRNDdire tlyin term

of a quadrati form of the oe ients of the estimated put pri e fun tion, whi h

depends onthe singular valuesof the restri ted put operator (see Proposition 7.1).

This ru ialfeatureallowstore overthesmoothestRND asthesolutionof asimple

quadrati program, whi h takes the bid ask quotes as sole input. Our estimation

methodimprovesonexistingonesin severalways,whi hwesumuphere.

•

Itisfastandsimpletoimplementsin eitrequirestosolveonesinglequadrati

program,yetbeingfullynonparametri .

•

It is robust to the pau ity of pri e quotes sin e the smaller the numberof

quotes, the less onstrained the quadrati program and thus the easier to

solve.

•

It takes the bid ask quotes as sole input and does not require any sort of

smoothingorprepro essingofthedata.

•

Itreturnsthesmoothestdensitygivingrisetopri equotesthatlieinsidethe

bidaskquotes. TheestimatedRNDisthereforeaswell-behavedas anbe.

•

It returns a losed form estimate of the RND on

I

. We thus obtain both

the middle partof the RND together with its left tailand part of itsright

tail. Interestingly, the left tail ontains ru ial information about market

sentimentsrelativetoapotentialforth omingmarket rash.

singularvalue

λ

0

of

γ

and

γ

∗

lookthemselvesverymu hlike rossse tionsofputand

allpri es,respe tively(see Figure 1.1). Inthat sense,they willbeable to apture

the bulk of the shape of a ross se tion of option pri es, while thesubsequent

sin-gular ve torswill add orre tions tothis general behavior. Thisis a ru ial feature

of this SVD that leadsus to think that the singular bases of the restri ted pri ing

operatorsareappropriatetoolstore overtheRND

q

. Interestingly,theperforman e

ofourquadrati programmingalgorithmonreal dataisindeedquite onvin ing(see

Se tion8fordetails).

Readersinterested in appending afullrighttailto this estimatedRND arereferred

to [14℄, who proposes asimple method for smooth pastingof parametri GEV tail

distributionsto anestimated RND.

Here isthe paperlayout. Weintrodu etherestri ted alland putoperators,

γ

and

γ

∗

, andoperatorsderivedthereofinSe tion 2. Wedetailthepropertiesofoperators

γ

∗

_γ

and

γγ

∗

on the one hand, and

γ

and

γ

∗

on the other hand, in Se tion 3and

Se tion 4,respe tively. Otherresultsrelativetothese fouroperatorsare reportedin

Se tion5. Se tion6givesexpli itexpressionsforthe

(λ

k

)

,

(ϕ

k

)

and

(ψ

k

)

. TheSRM

isdetailedinSe tion7. Finally,werunasimulationstudyinSe tion8. AnAppendix

regroupssomeadditionalusefulresults.

2. Denitions and setting. Let us dene the restri ted all operator on the

interval

I = [0, B]

astheoperator

γ

from

L

2

I

into

L

2

I

su h that,

(γf )(ξ) =

Z

I

θ(ξ, x)f (x)dx,

ξ

_{∈ I, f ∈ L}

2

I,

(2.1)

θ(ξ, x) = (x

− ξ)

+

_.

Itisatrivialfa tthat

γf

belongsindeedto

L

2

I

. Let'sdenoteby

h., .i

theusuals alar

produ ton

L

2

I

andby

k.k

L

forall

ξ, x

∈ I

,

|θ(ξ, x)| ≤ B

and applyCau hy-S hwartzinequalitytoobtain,

kγfk

2

L

2

I

≤

Z

I

dξ

Z

I

dx

_{|θ(ξ, x)||f(x)|}



2

≤ B

4

kfk

2

L

2

I

<

∞.

Theadjointoperator

γ

∗

of

γ

issu hthat,forall

f, g

∈ L

2

I

,

hγ

∗

_{f, g}

i = hf, γgi

=

Z

I

duf (u)

Z

I

dxθ(u, x)g(x)

=

Z

I

dxg(x)

Z

I

duθ(u, x)f (u).

Hen e

γ

∗

f (ξ) =

Z

I

θ

∗

(ξ, x)f (x)dx,

ξ

_{∈ I, f ∈ L}

2

I,

(2.2)

θ

∗

(ξ, x) = θ(x, ξ).

Sothat

γ

∗

isnothingbuttherestri tedput operatorontheinterval

I

. Inparti ular,

we anwrite

γ

∗

γf (ξ) =

Z

I

ϑ

1

(ξ, x)f (x)dx,

ξ

∈ I, f ∈ L

2

I,

(2.3)

γγ

∗

f (ξ) =

Z

I

ϑ

2

(ξ, x)f (x)dx,

ξ

∈ I, f ∈ L

2

I,

(2.4) where

ϑ

1

(ξ, x) =

Z

I

duθ

∗

(ξ, u)θ(u, x)

=

Z

I

du(ξ

− u)

+

_(x

− u)

+

₌

Z

ξ∧x

0

du(ξ

− u)(x − u)

= ξx(ξ

_{∧ x) − (ξ + x)(ξ ∧ x)}

2

_{/2 + (ξ}

∧ x)

3

_/3,

and

ϑ

2

(ξ, x) =

Z

I

duθ(ξ, u)θ

∗

(u, x)

=

Z

I

du(u

− ξ)

+

_(u

− x)

+

₌

Z

B

ξ∨x

du(u

− ξ)(u − x)

= ξx(B

_{− ξ ∨ x) − (ξ + x)(B − ξ ∨ x)}

2

/2 + (B

_{− ξ ∨ x)}

3

/3.

Letusnowturnto thedetailed inspe tionoftheseoperators.

3. Results relative to

γ

∗

_γ

and

γγ

∗

. Let us denote by

R(κ)

the range of

an operator

κ

of

L

2

I

and by

N (κ)

its null spa e (see [12, p.23℄). Obviously both

γ

∗

_γ

and

γγ

∗

are self-adjoint. This translates into the fa t that their kernels are

symmetri (meaning

ϑ

i

(ξ, x) = ϑ

i

(x, ξ)

). Inaddition,both

ϑ

1

and

ϑ

2

are ontinuous

ontheboundedsquare

I×I

. Therefore,theasso iatedoperatorsare ompa t(see[12,

Ex.4.8.4,p.172℄). Assu h,theyverifythespe traltheorem(see[12,Th.4.10.1,4.10.2,

p.187-189℄).

Theorem 3.1. Given the operators

γ

∗

_γ

and

γγ

∗

dened in eq. (2.3) and eq. (2.4)

above,wehave thefollowing results.

1) The operators

γ

∗

_γ

and

γγ

∗

are ompa t and self-adjoint. As su h, they admit

ountable families of orthonormal eigenve tors

(ϕ

k

)

and

(ψ

k

)

asso iated to the

samepositive de reasingsequen eofeigenvalues

λ

2

k

,whi hare ompletein

R(γ

∗

_γ)

and

R(γγ

∗

₎

, respe tively. 2) Besides, wehave

R(γ

∗

γ)

_{⊂ L}

2

I ∩ C

4

I,

R(γ

∗

_γ)

⊂ L

2

I ∩ C

4

I,

where

C

4

I

stands forthe set offour timesdierentiablefun tionson

I

.

other words, theyareboth orthonormalbases of

L

2

I

. Infa t, we an write

L

2

I = R(γ

∗

γ) =

Span

{ϕ

k

, k

∈ N},

=

R(γγ

∗

_{) =}

Span

{ψ

k

, k

∈ N},

where

R(γ

∗

_γ)

standsforthe losureof

R(γ

∗

_γ)

in

L

2

I

(see[12,p.16℄)andSpan

{ϕ

k

, k

∈

N

_}

for theset of(potentiallyinnite) linear ombinations ofelements

ϕ

k

.

4) Therefore,

γ

∗

_γ

and

γγ

∗

arebothinvertibleandadmit the fourthorderdierential

operator

∂

4

ξ

as an inverse (see [12, p.155℄ for terminology). More pre isely, we

have got

∂

ξ

4

γ

∗

γf = f,

∀f ∈ L

2

I,

γ

∗

γ∂

ξ

4

f = f,

∀f ∈ R(γ

∗

γ),

andidem for

γγ

∗

.

5) Finally, wehave the following spe tral de ompositions,

f =

X

k≥0

hf, ϕ

k

iϕ

k

,

f

∈ L

2

I,

γ

∗

γf =

X

k≥0

λ

2

k

hf, ϕ

k

iϕ

k

,

f

∈ L

2

I,

and

f =

X

k≥0

hf, ψ

k

iψ

k

,

f

∈ L

2

I,

γγ

∗

f =

X

k≥0

λ

2

k

hf, ψ

k

iψ

k

,

f

∈ L

2

I.

Proof. As detailed above,1) follows dire tlyfrom the spe tral theorem. 2)follows

dire tly from the kernel representations in eq. (2.3) and eq. (2.4). It an also be

seenfrom the fa t that, for any

f

∈ L

2

I

, both

γf

and

γ

∗

_f

whi hfollowsbysimpleinspe tionof eq.(2.1)andeq.(2.2). 3)followsdire tlyfrom

Proposition 5.1 below. 4)is adire t onsequen eof Lemma 8.3 below. Finally, 5)

followsdire tly from1)and3).

4. Resultsrelative to

γ

and

γ

∗

. Thefollowingtheoremdetailstheproperties

oftherestri tedput and alloperators. ItbuildsuponTheorem3.1above.

Theorem4.1. Givenoperators

γ

and

γ

∗

denedineq. (2.1)andeq. (2.2)above,we

havethe following results.

1) Considerthesequen eofpositivede reasingsingularvalues

λ

k

andsingularve tors

(ϕ

k

)

and

(ψ

k

)

denedinTheorem 3.1above. Therestri tedputand alloperators

γ

∗

and

γ

aresu hthat,for all

k

≥ 0

,

γϕ

k

= λ

k

ψ

k

,

γ

∗

ψ

k

= λ

k

ϕ

k

.

2) Besides, wehave

R(γ

∗

)

_{⊂ L}

2

I ∩ C

2

I,

R(γ) ⊂ L

2

I ∩ C

2

I,

where

C

2

I

stands forthe set oftwotimesdierentiablefun tionson

I

.

3) Inaddition,wehave

L

2

I = R(γ

∗

_{) =}

_R(γ)

. Sothatboth

γ

and

γ

∗

areinvertibleand

admitthese ondorderpartialdierentialoperator

∂

2

ξ

asaninverse. Inparti ular,

weobtain

∂

ξ

2

γf (ξ) = ∂

ξ

2

γ

∗

f (ξ) = f (ξ),

∀f ∈ L

2

I.

(4.1)

So that the knowledge of

γf

or/and

γ

∗

_f

allows tore over

f

dire tly astheir

se -ond derivative. This is nothing but the so- alled Breeden-Litzenberger formula

4) Wehave furthermorethe following spe tralde ompositions,

f =

X

k≥0

hf, ϕ

k

iϕ

k

,

f

∈ L

2

I,

γf =

X

k≥0

λ

k

hf, ϕ

k

iψ

k

,

f

∈ L

2

I,

and

f =

X

k≥0

hf, ψ

k

iψ

k

,

f

∈ L

2

I,

γ

∗

f =

X

k≥0

λ

k

hf, ψ

k

iϕ

k

,

f

∈ L

2

I.

5) Finally, wehave aput- allparityonthe interval that anbewritten asfollows

(γ

_{− γ}

∗

)f (ξ) = ¯

m

1

(f )

− ξ ¯

m

0

(f ),

wherewe havedened

m

¯

k

(f ) :=

R

I

x

k

_{f (x)dx}

.

Proof. Theproofof1)followsdire tlyfrom[13,p.37℄. 2)followsbysimpleinspe tion

ofeq.(2.2)andeq.(2.1). Therstpartof3)followsfromthefa tsthat

R(γ) = R(γγ

∗

₎

and

R(γ

∗

_{) =}

_R(γ

∗

_γ)

(see1)above) and Theorem 3.1, item 3). These ond partof

3)followspartlyfrom Lemma8.3below(seeAppendix)andpartlyfromtheobvious

fa t that

f = γ

∗

_∂

2

ξ

f

for all

f

∈ R(γ

∗

₎

(idem for

γ

). 4)followsdire tly from 1)and

3). Finally,5)followsimmediatelyfrom thefollowingobvious omputations,

(γ

_{− γ}

∗

)f (ξ) = γf (ξ)

_{− γ}

∗

f (ξ)

=

Z

I

[θ(ξ, x)

_{− θ}

∗

_{(ξ, x)]f (x)dx}

=

Z

I

(x

_{− ξ)f(x)dx}

= ¯

m

1

(f )

− ξ ¯

m

0

(f ).

Weregroupotherresultsrelativetotheaboveoperatorsinthefollowingse tion.

5. Other results relative to

γ

∗

_γ

,

γγ

∗

,

γ

∗

and

γ

. We proveherethat both

orthonormalfamilies

(ϕ

k

)

and

(ψ

k

)

are ompletein

L

2

I

. Otherinterestingresultsare

to befound in theAppendix. Someof themare purelyte hni al,while someothers

areofmoregeneralinterest.

Proposition5.1. Wehave got,

L

2

I = R(γ

∗

γ) =

Span

{ϕ

k

, k

≥ 0},

=

_R(γγ

∗

_{) =}

Span

{ψ

k

, k

≥ 0},

where

R(γ

∗

_γ)

standsforthe losureof

R(γ

∗

_γ)

in

L

2

I

(see[12,p.16℄)andSpan

{ϕ

k

, k

∈

N

_}

for theset of(potentiallyinnite) linear ombinations ofelements

ϕ

k

.

Proof. Weknowfrom [13,Ÿ2.3.℄ that,

L

2

I = R(γ

∗

γ)

⊕

⊥

N (γ

∗

γ),

=

_R(γγ

∗

₎

_⊕

⊥

N (γγ

∗

_).

Therefore,itisenoughtoshowthatbothnull-spa esredu etothezeroelement. The

kernel

N (γ

∗

_γ)

of

γ

∗

_γ

is onstitutedbythefun tions

f

∈ L

2

I

that aresolutionsof

0 = γ

∗

γf (ξ),

∀ξ ∈ I.

Derivingfourtimeswith respe tto

ξ

andapplyingLemma8.3 (seeAppendix)leads

to

f (ξ) = 0, ξ

∈ I

. Sothat

N (γ

∗

_{γ) =}

{0}

. Nowitisenoughtonoti ethat

N (γ

∗

_{γ) =}

N (γ)

. However,weknowfrom Lemma 8.4 that

f

∈ N (γ)

if andonly if

f

˘

∈ N (γ

∗

₎

(see eq.(8.1) for notation). Therefore

N (γγ

∗

_{) =}

N (γ

∗

_{) = ˘}

N (γ) = ˘

N (γ

∗

_{γ) =}

{0}

,

whereby

N

˘

,wemean

{ ˘

f , f

∈ N }

.

6. Expli it omputationof

(λ

k

)

,

(ϕ

k

)

and

(ψ

k

)

.

gatheredbelowinTheorem6.1. Letuswrite

f

k,1

(ξ) = e

ρ

k

ξ/B

,

f

k,2

(ξ) = e

−ρ

k

ξ/B

,

f

k,3

(ξ) = cos(ρ

k

t/B),

f

k,4

(ξ) = sin(ρ

k

ξ/B),

where

ρ

k

=

π

2

+ kπ + (

−1)

k

_β

k

,

k

∈ N,

(6.1)

and, for all

k

∈ N

,

β

k

is the smallest positive solution of the following xed point

equationin

u

,

exp(π/2 + kπ + (

₋₁₎

k

_{u) =}

1 + cos(u)

sin(u)

.

Interestingly, thepositivesequen e

(β

k

)

de reases exponentially fasttowardzeroas

detailed inLemma6.8. Inaddition,wewrite,

h

k,1

= a

k,1

f

k,1

+ a

k,2

f

k,2

,

h

k,2

= a

k,3

f

k,3

+ a

k,4

f

k,4

,

(6.2)

wherethe oe ients

a

k,i

, i = 1, . . . , 4

aresu hthat,

a

k,1

=

√

1

B

(

−1)

k

e

ρ

k

+ (

−1)

k

,

a

k,2

= (

−1)

k

e

ρ

k

a

k,1

=

1

√

B

1

1 + (

₋₁₎

k

_e

−ρ

k

,

a

k,3

=

−

1

√

B

,

a

k,4

=

1

√

B

1

_{− (−1)}

k

_e

−ρ

k

1 + (

₋₁₎

k

_e

−ρ

k

.

Then,wehavethefollowingtheorem.

Theorem 6.1. The eigenve tors

(ϕ

k

)

of

γ

∗

_γ

and

(ψ

k

)

of

γγ

∗

aresu hthat

They arerelatedby thefollowing relationships,

γϕ

k

= λ

k

ψ

k

,

γ

∗

ψ

k

= λ

k

ϕ

k

,

(6.4)

wherewehave written

λ

k

=

 B

ρ

k



2

,

(6.5)

and

ρ

k

is dened in eq. (6.1). They verify

kϕ

k

k

L

2

I

=

kψ

k

k

L

2

I

= 1

. Moreover, we have

ψ

k

(B) = ψ

k

′

(B) = 0,

ϕ

k

(0) = ϕ

′

k

(0) = 0,

(6.6) together with

˘

ψ

k

= (

−1)

k

ϕ

k

,

ϕ

˘

k

= (

−1)

k

ψ

k

,

(6.7)

wherewe have written

ψ

˘

k

(ξ) = ψ

k

(B

− ξ)

. Andnally, weobtain asa dire t

onse-quen eof eq. (4.1)above that

λ

k

∂

ξ

2

ψ

k

= ∂

ξ

2

γϕ

k

= ϕ

k

,

λ

k

∂

ξ

2

ϕ

k

= ∂

ξ

2

γ

∗

ψ

k

= ψ

k

.

Proof. Noti e readilythat eq.(6.6), eq.(6.7)and thefa t that both

ϕ

k

and

ψ

k

are

unit normedarestraightforward onsequen esofeq.(6.3). Inaddition,eq.(6.4)isa

repetition ofTheorem4.1, item1). Sothat wearein fa tleft withprovingeq.(6.3)

andeq.(6.5). Ea heigenve tor

f

of

γ

∗

_γ

asso iatedtotheeigenvalue

r

4

issolutionof

theproblem,

r

4

_{f = γ}

∗

_γf,

for some

r

6= 0

and

f

∈ L

2

I

. After dierentiating four times the latter equation

withrespe tto

ξ

(assumingthat

f

∈ L

2

I ∩ C

4

I

)andapplyingLemma8.3,weobtain

thatthesolutionsofeq.(6.8)arealsosolutionsofthefollowingfourthorderordinary

dierentialequation,

r

4

d

4

ξ

f

− f = 0,

where

d

4

ξ

standsforthefourth orderordinarydierentialoperator. Its hara teristi

polynomialadmits four roots

±r

−1

and

±ir

−1

. Consequently, the real solutionsof

theaboveordinarydierentialequationareoftheform

f (ξ) = b

1

e

ξ/r

+ b

2

e

−ξ/r

+ b

3

cos(ξ/r) + b

4

sin(ξ/r).

(6.9)

The

ϕ

k

sarethusofthisform. Pluggingthisgeneri solutionba kintoeq.(6.8)leads

inturn, aftertediousbutstraightforward omputations, to

M b = 0,

(6.10)

where

b

isa

4

× 1

ve torsu hthat

b

T

₌



b

1

b

2

b

3

b

4



and

M

isthe

4

× 4

matrix

denedby

M (r, B) =



















r

−1

_e

B/r

_−r

−1

_e

−B/r

_r

−1

_{sin (B/r)}

_−r

−1

_{cos (B/r)}

−r

−2

_e

B/r

_−r

−2

_e

−B/r

_r

−2

_{cos (B/r)}

_r

−2

_{sin (B/r)}

r

−3

_−r

−3

₀

_r

−3

r

−4

_r

−4

_r

−4

₀



















.

(6.11)

There exists a non-trivial solution to eq. (6.10) if and only if

r

is su h that the

determinantof

M

an els,that isDet

(r, M ) = 0

. Asdetailed inProposition 6.2,the

rootsofDet

(r, M ) = 0

areexa tlythe

r

m

= B/ν

m

where

ν

m

isdened ineq.(6.16).

Inaddition, weprovein Proposition 6.3that thesystem

M (r

m

, B)b = 0

admits the

unique solution

b

m

. Reading o eq. (6.9), we obtain that the eigenve tor of

γ

asso iatedtoeigenvalue

r

4

m

writesas

α

m

= η

m,1

+ η

m

2

whereboth

η

m,1

and

η

m,2

are

dened in eq. (6.15). Now, it is enoughto noti e that, given the properties of the

sequen e

(ν

m

)

detailedin Proposition6.5,

r

4

2k+1

= r

4

2k

and

r

4

2k+2

< r

4

2k+1

,

k

∈ N

. In

addition,weknowfromLemma6.4that

α

2k+1

= α

2k

. Thisallowsusto on ludethat

the eigenvaluesof

γ

∗

_γ

are, withoutredundan y, the

λ

2

k

,

k

∈ N

, dened in eq. (6.5)

andtheasso iatedeigenspa esareunit-dimensionalandrespe tivelyspannedbythe

eigenve tors

ϕ

k

,

k

∈ N

,dened ineq.(6.3).

Computing

ψ

k

= λ

−1

k

γϕ

k

leads, after tedious but straightforward omputations to

ψ

k

= h

k,1

− h

k,2

and on ludes theproof.

6.2. Additionalresults. Thisse tion ontainsaseriesofresultsthat areused

throughouttheproofof Theorem6.1above. Inthisse tionwemakeuseofthemap

E : N

_{7→ N}

su hthat

E(2k + 1) = E(2k) = k

forall

k

∈ N

.

Proposition 6.2. Let

M (r, B)

be the

4

× 4

matrix dened in eq. (6.11). The set

of solutions

r

to the problem Det

M (r, B) = 0

is ountable. Let us denote them by

r

m

, m

∈ N

. Forany

m

∈ N

,the solution

r

m

anbe writtenas

r

m

=

B

ν

m

,

where

ν

m

isdenedineq. (6.16). Weobtainin fa tthat,

Det

M (r

m

, B) = 0

⇔

e

ν

m

=

−

1 + (

−1)

E(m)

_sin(ν

m

)

cos(ν

m

)

.

Besides, the following relationships hold true

cos ν

m

:=

−

2

e

ν

m

+ e

−ν

m

=

−

1

cosh ν

m

,

(6.12)

sin ν

m

:=

−(−1)

E(m)

+ (

−1)

E(m)

2

1 + e

−2ν

m

.

Proof. Itfollowsfromstraightforward omputationsthat,

Det

M (r, B) = 2e

−B/r



_{cos (B/r)}



_e

B/r



2

_{+ 2e}

B/r

_{+ cos (B/r)}



_.

(6.14)

Let us write

ν := B/r

and noti e that if

cos(ν) = 0

, then Det

M (r, B) = 2

6= 0

so

thatwemusthave

cos ν

6= 0

foreq.(6.10)toadmitanon-trivialsolution. Tobemore

spe i Det

M (r, B) = 0

redu esto

P (e

ν

_{) = 0}

where

P (x) := cos (ν) x

2

_{+ 2x + cos (ν)}

.

Howevertherootsof

P

aregivenby

δ

±

(ν) :=

−1 ± sin(ν)

cos(ν)

.

Hen eforth,

r = B/ν

an elsDet

M (r, B)

ifandonlyif

ν

issolutionofanyoneofthe

twofollowingxedpointequations,

e

ν

=

−1 + sin(ν)

cos(ν)

,

e

ν

₌

−1 − sin(ν)

cos(ν)

.

Theprooffollowsnowdire tlyfrom Proposition6.5.

Proposition6.3. Forany

r

m

solutionoftheequationDet

M (r

m

, B) = 0

(see

Propo-sition6.2above),the nullspa eof

M (r

m

, B)

isofdimension

1

andisspannedbythe

ve tor

b

T

m

=



b

m,1

b

m,2

b

m,3

b

m

4



,

wherewehave written,

b

m,1

=

1

√

B

(

₋₁₎

E(m)

e

ν

m

+ (

−1)

E(m)

,

b

m,2

= (

−1)

E(m)

e

ν

m

a

m,1

=

√

1

B

1

1 + (

₋₁₎

E(m)

_e

−ν

m

,

b

m,3

=

−

√

1

B

,

b

m,4

=

1

√

B

1

_{− (−1)}

E(m)

_e

−ν

m

1 + (

−1)

E(m)

_e

−ν

m

,

and

ν

m

isdenedineq. (6.16).

Proof. It is a matter of straightforward linear algebra and thus left to the reader.

Noti ehowever,that itreliesontheuseofbotheq.(6.12)andeq.(6.13).

Lemma 6.4. Letuswrite

ζ

m,1

(ξ) = e

ν

m

ξ/B

,

ζ

m,2

(ξ) = e

−ν

m

ξ/B

,

ζ

m,3

(ξ) = cos(ν

m

ξ/B),

ζ

m,4

(ξ) = sin(ν

m

ξ/B),

where

ν

m

isdenedineq. (6.16). In addition, wewrite,

η

m,1

= b

m,1

ζ

m,1

+ b

m,2

ζ

m,2

,

η

m,2

= b

m,3

ζ

m,3

+ b

m,4

ζ

m,4

,

(6.15)

wherethe oe ients

b

m,i

, i = 1, . . . , 4

are denedin Proposition6.3. Forall

k

∈ N

,

wehave thefollowing relationships

η

2k+1,1

= η

2k,1

,

η

2k+1,2

= η

2k,2

Proof. It follows from straightforward omputations using the fa t that

ν

2m+1

=

−ν

2m

.

Proposition6.5. Letusdenethemap

E : N

7→ N

su hthat

E(2k) = E(2k +1) = k

for

k

∈ N

. Letus write

g(ν) =

−1 + sin ν

cos ν

,

h(ν) =

−1 − sin ν

cos ν

,

and onsider the xed point equations

e

ν

_{= g(ν)}

and

e

ν

_{= h(ν)}

. The set of

orre-sponding solutionsis exhaustedbythe sequen e

ν

m

= (

−1)

m



π

2

+ E(m)π + (

−1)

E(m)

_β

E(m)



,

m

_{∈ N.}

(6.16)

where

(β

m

)

is dened in Lemma 6.8. In parti ular, noti e that

ν

2k+1

=

−ν

2k

and

|ν

m

1

| < |ν

m

2

|

for all

m

1

, m

2

∈ N

su hthat

E(m

1

) < E(m

2

)

. Noti einaddition that,

by onstru tion,

ν

m

is solutionof

e

ν

m

=

₋

1 + (

−1)

E(m)

_{sin ν}

m

cos ν

m

.

This latter result, together with the fa t that Det

M (B/ν

m

, B) = 0

(see eq. (6.14)),

leads straightforwardlytothe following relationships,

cos ν

m

:=

−

2

e

ν

m

+ e

−ν

m

=

−

1

cosh ν

m

,

sin ν

m

:=

−(−1)

E(m)

+ (

−1)

E(m)

2

1 + e

−2ν

m

.

Proof. Considerthexedpointequation

g(ν) = e

ν

. Giventhepropertiesof

g

detailed

in Proposition 6.6, two ases arise depending whether

ν

is positiveor negative. In

the ase where

ν

is positive, the exponential map meets

g

at points of the form

p

m

=

3π

₂

+ 2mπ

− u

m

for

m

∈ N = {0, 1, 2, . . .}

and somesmall but positive

u

m

s.

A dire tappli ation ofLemma 6.7showsthat thenegativesolutionsareexa tlythe

−p

m

, m

∈ N

.

The se ond xed point equation

h(ν) = e

ν

an be rewritten as

g(

−ν) = e

ν

. The

positivesolutionsareoftheform

q

m

=

π

2

+ 2mπ + v

m

, m

∈ N

. And,fromLemma6.7

again,the orrespondingnegativesolutionsarethe

−q

m

, m

∈ N

.

Letuswrite

t

m

=

π

2

+ mπ + (

−1)

m

β

m

, m

∈ N

. Itis learthat

t

2k

= q

k

and

t

2k+1

= p

k

for

k

∈ N

. Inparti ular,

t

m

issolutionof

e

t

m

=

−

1 + (

−1)

m

_{sin t}

m

cos t

m

(6.17)

Let us denethe map

E : N

7→ N

su h that

E(2k + 1) = E(2k) = k

for all

k

∈ N

.

Wedene

ν

m

, m

∈ N

su h that

ν

m

= (

−1)

m

_t

E(m)

,thatis

ν

2k

= t

k

and

ν

2k+1

=

−t

k

,

e

ν

_{= g(ν)}

and

e

ν

_{= h(ν)}

. Infa t,

ν

m

issolutionof

e

ν

m

=

₋

1 + (

−1)

E(m)

_{sin ν}

m

cos ν

m

Proposition6.6. Noti ereadilythat

h(ν) = g(

−ν)

,sothatitisenoughtostudythe

propertiesof

g

alone. Wehave thefollowing results,

1.

g

isdenedon the domain

D

g

= R

\{

3π

2

+ 2mπ, m

∈ Z}

;

2.

g

is

2π

periodi andsu hthat,forall

ν

∈ S

g

= (

−

π

2

,

3π

2

)

,

g(ν + 2mπ) = g(ν)

;

3. Finally,

g

isstri tlyin reasing on

S

g

andsu hthat,

lim

ν→

⊕

−

π

2

g(ν) =

−∞,

g(

π

2

) = 0,

ν→

lim

⊖

3π

2

g(ν) = +

∞.

where we write

→

⊕

(resp.

→

⊖

) to mean the limit from the above (resp.

below).

4. Noti ethat

R

\D

g

(resp.

R

\D

h

) orrespondsexa tlytothe setofallthezeros

of

h

(resp.

g

). Thus

D

g

∩ D

h

is the subset of

R

ontaining all the points

whereboth

g

and

h

arewelldenedanddierent fromzero.

Proof. Letus rst fo uson thedomain of

g

. Itis dened on

R

\{

π

2

+ mπ, m

∈ Z}

.

However,

g

anbeextendedby ontinuitytobeworthzeroatpoints

π

2

+ 2mπ, m

∈ Z

.

Noti eindeedthat foranysmallpositive

u

and

ℓ

∈ N

,onehasgot

g(

π

2

+ (

−1)

ℓ

_{u) =}

−1 + cos u

−(−1)

ℓ

_{sin u}

=

−

u

2

2

+ O(u

4

)

−(−1)

ℓ

_{u + O(u}

3

₎

= (

−1)

ℓ

u

2

+ O(u

3

_).

With a slight abuseof notations, wedenote the latterextension by

g

. So that

g

is

a tuallydenedon

R

\{

3π

2

+ 2mπ, m

∈ Z}

. Theotherpropertiesarestraightforward.

Lemma 6.7. Re allthat

D

g

and

D

h

aredened inProposition 6.6. Noti e rstthat

ν

_{∈ D}

g

∩ D

h

,wehavethe following results,

1. If

ν

is solution of the xed point equation

e

ν

_{= g(ν)}

, then

−ν

is also a

solution.

2. If

ν

is solution of the xed point equation

e

ν

_{= h(ν)}

, then

−ν

is also a

solution.

Proof. Noti e rstthat wehavetheidentity

h(ν)g(ν) = 1

forany

ν

∈ D

g

∩ D

h

. Its

proof is immediate. And therefore, for any

ν

∈ D

g

∩ D

h

solution of

e

ν

_{= g(ν)}

, we

obtain

g(

−ν) = h(ν) = g(ν)

−1

_{= e}

−ν

. Andidemforthesolutionsof

e

ν

_{= h(ν)}

.

Lemma6.8. Thesequen e

(β

k

)

issu hthat,forall

k

∈ N

,

β

k

isthe smallestpositive

solution ofthe following xedpointequationin

u

,

exp(π/2 + kπ + (

−1)

k

_{u) =}

1 + cos(u)

sin(u)

.

In addition, the approximation

β

k

≈ 2e

−

π

2

−kπ

holdstruewith alargedegree of

a u-ra yfrom

k = 1

onward.

Proof. Letuswrite

t

k

=

π

2

+ kπ + (

−1)

k

_u

,forsomesmallbut positive

u

su hthat

t

k

issolutionof eq.(6.17). Noti ethat

cos



π

2

+ kπ + (

−1)

k

_u



₌

− sin(u) = −u + O(u

3

),

sin



π

2

+ kπ + (

−1)

k

_u



_{= (}

−1)

k

cos(u) = (

₋₁₎

k

+ O(u

2

),

exp



π

2

+ kπ + (

−1)

k

_u



_{= e}

π

2

+kπ

(1 + (

−1)

k

u + O(u

2

)).

Sothateq.(6.17)redu esto

exp(π/2 + kπ + (

−1)

k

_{u) =}

1 + cos(u)

sin(u)

.

Plugging-intheTaylorexpansionsabove,weobtain

e

π

2

+kπ

(1 + (

−1)

k

u + O(u

2

)) =

2 + O(u

2

₎

u + O(u

3

₎

=

1

u

(2 + O(u

2

_)),

whi h anberewrittenas

u = e

−

π

2

−kπ

(2 + O(u)).

(6.18)

It an be veriednumeri ally that

2e

−

π

2

−kπ

is averygood approximationof

β

k

as

soon as

k

≥ 1

in the sense that eq. (6.17) holds true with a very large degree of

a ura y.

7. The spe tral re overymethod(SRM). InthisSe tion,werstdes ribe

how

γ

and

γ

∗

relate to the bid-ask quotes. We then show that the SVD of the

restri tedpri ingoperatorsdes ribedabove anbeusedtodesignasimplequadrati

programthatre oversthesmoothestRND ompatiblewithmarketquotes.

7.1. From

γ

and

γ

∗

to alland putpri es. Letusdenoteby

P (ξ)

and

C(ξ)

the put and all pri es at strike

ξ

and by

q

the orresponding riskneutral density.

Letusfurthermorewrite

I = R

¯

+

\I = (B, ∞)

. Weassumethat therestri tion

q

|I

to

theinterval

I

of

q

isin

L

2

I

. Forall

ξ

∈ I

,thefollowingrelationshipsareimmediate.

e

rτ

P (ξ) = γ

∗

q(ξ),

(7.1)

e

rτ

C(ξ) = γq(ξ) +

Z

∞

B

(x

_{− ξ)q(x)dx}

= γq(ξ) + m

1

(q)

− ξm

0

(q),

(7.2)

wherewehavedened,

m

k

(f ) =

Z

¯

I

x

k

_{f (x)dx.}

Noti einparti ularthat

m

0

(q) = Q(S

τ

≥ B) = 1 − ¯

m

0

(q),

Eq.(7.1)showsthatputpri esdire tlyrelatetotherestri tedputoperator.Froman

estimationperspe tive,thisisa ru ialfeature thatwillallowustore overtheRND

dire tlyfrommarket putquotes. Unfortunately,thesituationisslightlydierentfor

allpri es. As shown fromeq. (7.2), all pri esrelatetotherestri ted alloperator

via

m

1

(q)

and

m

0

(q)

, whi h arebothunknown. Although, they ouldbeestimated

andgiverisetoanestimatoroftheRNDbasedonquoted allpri es,wewontpursue

thisroutehere,butratherfo usonthesimplerrelationgivenbyeq.(7.1).

7.2. Arefresheronno-arbitrage onstraints. Foradetailedreviewof

model-freeno-arbitrage onstraints,thereaderisreferredto[21,p.32,Ÿ1.8℄and[11℄. Letus

denoteby

S

0

thepri etodayoftheunderlyingsto k. Letusmoreoverassumethatit

paysa ontinuousdividendyield

δ

. Letusdenoteby

r

the ontinuously ompounded

shortrateandby

τ

thetimetomaturity. Letusre allrstthat,byno-arbitrage,put

and allpri esarerelatedbytheput- allparity.

C(ξ)

_{− P (ξ) = S}

0

e

−δτ

− ξe

−rτ

.

(7.3)

Besides

C(0) = S

0

and

P (0) = 0

. Letus nowfo usonputpri es. Wehave,

max(0, ξe

−rτ

− S

0

e

−δτ

)

≤ P (ξ) ≤ ξe

−rτ

,

(7.4)

0

_{≤ ∂}

ξ

P (ξ)

≤ e

−rτ

,

(7.5)

0

_{≤ ∂}

2

ξ

P (ξ).

(7.6)

Assumewearegivenanin reasingsequen eof

n

strikes

ξ

1

< ξ

2

< ... < ξ

n

and aset

of orrespondingput pri es

m

1

, . . . , m

n

. Asdes ribedin [2℄, theaboveno-arbitrage

relationshipstranslateinto a nite set ofane onstraintson thelatter put pri es.

These onstraints aninfa tbewrittenin matrixform as

Am

≤ b

p

, where

A

stands

fora

2n

× n

matrix,

m

isthe

n

× 1

ve torsu hthat

m

T

₌



m

1

. . . m

n



and

b

p

is

a

2n

× 1

ve tor. Morepre isely,eq.(7.6)translatesinto

n

− 2

onstraintsas,

[Am]

i

:=

m

i+1

− m

i

ξ

i+1

− ξ

i

−

m

i+2

− m

i+1

Moreover, the left-hand-side of eq. (7.4) is fully aptured in-sample by adding the

followingadditional

n

onstraints,

[Am]

i+n−2

:=

−m

i

≤ − max(0, ξ

i

e

−rτ

− S

0

e

−δτ

) := [b

p

]

i+n−2

,

i = 1, . . . , n

(7.7)

The right-hand-side of eq. (7.4) neednot betaken into a ount at this stage. It is

indeed less stringent than the upper-bound onstraints we will impose in the next

se tion. Finally, giventherst

n

− 2

onstraints,eq.(7.5)redu esto twoadditional

onstraints,

[Am]

2n−1

:=

m

n

− m

n−1

ξ

n

− ξ

n−1

≤ e

−rT

_{:= [b}

p

]

2n−1

,

[Am]

2n

:= m

1

− m

2

≤ 0 := [b

p

]

2n

.

Finally,letusre allthatiftheforwardpri e

F

0

oftheunderlyingsto kisobservable

today,then,byno-arbitrage,itmustbeequalto

S

0

e

(r−δ)τ

.

7.3. Bid-ask spread onstraints. Letus assumethat themarketprovidesus

with an in reasing sequen e of strike pri es

ξ

1

< ξ

2

< . . . < ξ

s

, where

s

is small.

Typi ally

s

rangesfrom

5

to

50

dependingontheunderlying. Inaddition,themarket

providesus witha orrespondingsequen e ofbidask quotesforput options. Letus

denotethemby

y

Ask

1

, . . . , y

s

Ask

and

y

Bid

1

, . . . , y

Bid

s

. Wewantthe orrespondingtted

putpri es

(m

i

)

tolieinsidethebidaskquotes. This orrespondstothefollowing

2s

ane onstraints,

m

i

≤ y

Ask

i

,

−m

i

≤ −y

i

Bid

,

i = 1, . . . s.

(7.8)

The quoted strikesmighteventuallyspan averysmall portion of thesegment

I

on

whi hwewanttore overtheRND.Inordertoimprovethequalityofourestimator,

we an onstrain it to verify the above no-arbitrage onstraints on a denser set of

strikes than thequoted ones. Letus denote by

ξ

1

< ξ

2

< . . . < ξ

n

this new set of

ξ

P (ξ)

S

0

e

−δτ

y

Ask

i

y

Bid

i

ξ

n

= B

(ξ

i

e

−

rτ

−

S

0

e

−

δτ

)

+

y

Ask

1

ξ

0

= 0

y

Ask

n

Figure7.1. Thisgraphsumsupthesetof onstraintsveried byestimated putpri es,whi h

are solutions of the quadrati optimization problem des ribed in eq. (P1). Estimated put pri es

m

1

, . . . , m

n

on thedense grid

ξ

1

, . . . , ξ

n

are displayed asbla k dots. Theymust lie in-between

thebid-askquotes,whi harerepresented bythi kred dotsrangingoverquotedstrikes

ξ

i

1

, . . . , ξ

i

s

,

whi h orrespond toasparse subset of theunderlyingdense grid

ξ

1

, . . . , ξ

n

. Inaddition, extreme

put pri es

m

1

and

m

n

are bounded above by

y

Ask

1

= 0

and

y

Ask

n

, respe tively, where the value

of

y

Ask

n

is givenin Se tion 7.3. Both

y

Ask

1

and

y

Ask

n

appearasthi kblue dotsatstrikes

ξ

1

= 0

and

ξ

n

= B

, respe tively.

m

1

, . . . , m

n

mustalso verifythein-sample onstraints des ribedby the

lhs ofeq.(7.4). Inparti ular, thelhs of eq.(7.4) ensures thatthe

m

i

s arelower-bounded by the

(ξ

i

e

−rτ

− S

0

e

−δτ

)

+

s,whi happearasthi kbluedots. Sin ethislower-boundisworth

0

for

i = 1

,

this,togetherwith theupper-bound

y

Ask

1

= 0

a tuallyimpose

m

1

= 0

. Finally,

m

1

, . . . , m

n

verify

botheq.(7.5)andeq.(7.6)above. Thelatter onstraintimposesin-sample onvexity.

laterreferen e, we denoteby

I =

{i

1

, . . . , i

s

}

thesubset of

{1, . . . , n}

orresponding

to the indexes of the initial quoted strikes. We know that, in any ase, we must

have

0 = P (0) = m

1

, so that we andene

y

Ask

1

= 0

. Furthermore, weknowfrom

eq.(7.5)that

P (ξ)

annotgrowataratefasterthan

e

−rτ

,sothatwe andene

y

Ask

n

to be the orresponding linear extrapolation of the right-most market quote

y

Ask

i

s

, meaning

y

Ask

n

= y

i

Ask

s

+ e

−rτ

_(ξ

n

− ξ

i

s

)

. Insummary,therequirementthatthe

m

i

sfall

in-betweenthebid-askquotestranslatesinto

2s + 2

additional onstraints,whi hwe

anwrite asfollows

m

i

≤ y

i

Ask

,

i

∈ I ∪ {1, n},

(7.9)