Parametric Inference and Dynamic State Recovery from Option Panels

Transcription

1 Paramerc Inference and Dynamc Sae Recovery from Opon Panels orben G Andersen Ncola Fusar Vkor odorov Augus 23 Absrac We develop a new paramerc esmaon procedure for opon panels observed wh error We explo asympoc approxmaons assumng an ever ncreasng se of opon prces n he moneyness cross-seconal dmenson, bu wh a fxed me span We develop conssen esmaors for he parameers and he dynamc realzaon of he sae vecor governng he opon prce dynamcs he esmaors converge sably o a mxed-gaussan law and we develop feasble esmaors for he lmng varance We also provde semparamerc ess for he opon prce dynamcs based on he dsance beween he spo volaly exraced from he opons and one consruced nonparamercally from hgh-frequency daa on he underlyng asse Furhermore, we develop new ess for model f over specfc regons of he volaly surface and for he sably of he rsk-neural dynamcs over me A comprehensve Mone Carlo sudy ndcaes ha he nference procedures work well n emprcally realsc sengs In an emprcal applcaon o S&P 5 ndex opons, guded by he new dagnosc ess, we exend exsng asse prcng models by allowng for a flexble dynamc relaon beween volaly and prced jump al rsk Imporanly, we documen ha he prced jump al rsk ypcally responds n a more pronounced and perssen manner han volaly o large negave marke shocks Keywords: Opon Prcng, Inference, Rsk Prema, Jumps, Laen Sae Vecor, Sochasc Volaly, Specfcaon esng, Sable Convergence JEL classfcaon: C5, C52, G2 We wsh o hank Lars Peer Hansen he co-edor and anonymous referees for her dealed commens ha led o sgnfcan mprovemens of he paper We also hank Snehal Banerjee, Davd Baes, m Bollerslev, Federco Bugn, Peer Chrsoffersen, Rama Con, Drew Creal, Francs Debold, Dobrslav Dobrev, Krs Jacobs, Jean Jacod, Rav Jagannahan, Ja L, om McCurdy, Per Mykland, Por Orlowsk, Dmrs Papankolau, Jonahan Parker, Markus Ress, Erc Renaul, Andras Sal, Olver Scalle, Nel Shephard, Coss Skadas, George auchen, Rasmus Varneskov, Mark Wason and Dacheng Xu as well as many semnar parcpans Andersen graefully acknowledges suppor from CREAES funded by he Dansh Naonal Research Foundaon odorov s work was parally suppored by NSF Gran SES We are also graeful for suppor from he Zell Cener for Rsk a he Kellogg School Deparmen of Fnance, Kellogg School of Managemen, Norhwesern Unversy, Evanson, IL 628; NBER, Cambrdge, MA; and CREAES, Aarhus, Denmark; e-mal: -andersen@norhwesernedu Deparmen of Fnance, Kellogg School of Managemen, Norhwesern Unversy, Evanson, IL 628; e-mal: n-fusar@norhwesernedu Deparmen of Fnance, Kellogg School of Managemen, Norhwesern Unversy, Evanson, IL 628; e-mal: v-odorov@norhwesernedu

2 Inroducon A volumnous leraure spannng several decades has, unambguously, esablshed ha me-varyng volaly and jumps are nrnsc feaures of fnancal prces Moreover, here has been subsanal neres n lnkng reurn premums n he economy o he compensaon for such laen rsks In parallel, he radng of dervave conracs has grown explosvely, n par reflecng a desre among nvesors o manage her volaly and jump rsk exposures As a resul, ever more comprehensve daa for, n parcular, exchange-raded opons have become avalable over me hese opons span a varey of expraon daes enors and srke prces moneyness, effecvely provdng an opon or mpled volaly surface for each radng day, ndexed by moneyness and enor hs sequence of opon surfaces whch we label an opon panel provdes an deal npu for esmaon of dynamc asse prcng models Specfcally, n a frconless paramerc seng, he surface allows for perfec recovery of he rsk-neural parameers and he mulvarae sae vecor drvng he opon prce dynamcs Whn realsc asse prcng models, boh reduced form and srucural, he sae vecor ypcally conans much more nformaon han jus he curren spo volaly level For example, may nclude shor- and long-run volaly facors as well as componens governng he jump nensy Imporanly, hese sae varables are nrnscally laen from he perspecve of he underlyng asse prce, even f a connuous rajecory of he laer s observed In pracce, however, drawng nference from he opon panel s complcaed by he presence of non-rval observaon errors n he opon prces he sze of hose errors vary across srkes and enor and depend on general marke condons, ncludng possbly he sae vecor self he convoluon of he heorecal opon prce wh he observaon errors renders he sae vecor and parameers unobservable drecly from a nosy opon panel hs suaon resembles he recovery of volaly from hgh-frequency reurns, where he volaly s convolued wh Gaussan nnovaons as well as jumps and mcrosrucure nose In ha conex, an elegan soluon s o resor o n-fll asympocs n he me dmenson and average ou he Gaussan reurn nnovaons along wh he remanng confoundng facors hs effecvely renders volaly observable subjec o quanfable esmaon error Below, we follow an analogous sraegy n he spaal doman Consequenly, we develop rgorous nference echnques for he mpled laen sae vecor and rsk-neural parameers, whle avodng paramerc assumpons abou he acual measure governng he sae vecor dynamcs hs s feasble as we develop asympoc dsrbuonal approxmaons assumng only ha he number of opons underlyng each volaly surface s large, so we may rea he me dmenson as fxed We may also allow he observaon errors for he opon prces o exhb lmed dependence n he spaal across srkes and enors and me seres dmenson We

3 accommodae varaon n he number of opon quoes as well as he srke range and enor across me as n he daa and here s no requremen of saonary n he paern of maury and moneyness Smlarly, he observaon error may have a non-ergodc and me-varyng dsrbuon Our esmaon mehod s penalzed nonlnear leas squares NLS he objecve funcon has wo pars he prmary componen s he mean-square-error n fng he observed opon prces usng he paramerc opon prcng model he second pece of he objecve funcon penalzes esmaes dependng on how much he opon-mpled volaly sae devaes from a local nonparamerc esmae of spo volaly consruced from hgh-frequency daa on he underlyng asse hs consran sems from he no-arbrage condon ha he curren aggregae dffuson coeffcen mus be dencal under he acual and rsk-neural measures Assumng he opon prce errors average ou suffcenly when pooled n he objecve funcon, we can conssenly esmae boh he parameers of he rsk-neural densy and he realzed rajecory of he sae vecor We furher esablsh he asympoc properes of our esmaor he convergence s sable, e, holds jonly wh any bounded random varable defned on he probably space he lmng dsrbuon s mxed Gaussan wh an asympoc varance ha can depend on any random varable adaped o he flraon he lmng law reflecs he flexbly of he esmaon approach: we can accommodae opon errors ha depend n unknown ways on he volaly sae as well as opon characerscs such as moneyness and enor We also provde conssen esmaors for he asympoc varance, hus enablng feasble nference In analogy o sandard NLS, f he opon errors are heeroskedasc, we may enhance effcency by weghng he opon f appropraely for he dfferng degrees of moneyness and enor Consequenly, n conras o much earler work on opon prcng allowng for observaon error, eg, Baes 2, Jones 26, and Eraker 24, we do no mpose any paramerc assumpon on he prcng errors, and we allow hem o dsplay sgnfcan heeroskedascy As noed prevously, he recovery of he volaly sae from he opon surface has mporan feaures n common wh he realzed volaly esmaon of sochasc volaly or menegrals hereof based on hgh-frequency asse reurns, see, eg, Andersen and Bollerslev 998, Andersen e al 23, and Barndorff-Nelsen and Shephard 22, 26 In eher case, he volaly realzaon a specfc pons n me may be recovered pahwse Moreover, boh esmaors converge sably wh an asympoc varance ha depends on he observed rajecores of asse prces, bu do no requre saonary or ergodcy of he volaly process Whle he hgh-frequency jump-robus esmaor of volaly s based on averagng ou he nose n he hgh-frequency reurn daa, he opon-based volaly esmaor averages ou he observaon 2

4 errors across he opon surface he major dfference s ha he opon-based esmaor explos a paramerc prcng model whle he esmaor based on hgh-frequency reurns s fully nonparamerc If he opon prcng model s vald, he wo volaly esmaes should no dffer n a sascal sense We formalze and operaonalze hs observaon Under correc model specfcaon, we esablsh a jon sable convergence law for he wo esmaors, enablng us o devse a formal model specfcaon es based on he dsance beween he wo volaly measures We propose addonal new dagnosc ess for he opon prce dynamcs he frs explores he sably of he rsk-neural parameer esmaes over dsnc me perods If he model s msspecfed, he perod-by-perod esmaes wll, n general, converge o a pseudo-rue value, see, eg, Whe 982 and Goureroux e al 984 However, he laer changes over me as he rajecory of he sae vecor vares across esmaon nervals and, for ncorrec model specfcaon, hs canno be accommodaed by an nvaran parameer vecor Hence, we develop a es based on he dscrepancy beween he parameer esmaes over subsequen me perods Ye anoher dagnosc focuses on model performance over specfc pars of he mpled volaly surface he emprcal opon prcng leraure ypcally gauges performance based on he meaveraged f for a lmed se of opons In conras, we may es for adequacy of he model mpled opon prcng day-by-day hs dagnosc explos our feasble lm heory by quanfyng he sascal error over he relevan poron of he surface, and hen deermnes f he prcng errors are sgnfcan In essence, he approach dsenangles he mpac of observaon errors nose n he opon prces from he sysemac errors semmng from a msspecfed model We explore he fne-sample properes of he esmaors hrough an exensve Mone Carlo sudy usng he double-jump sochasc volaly model of Duffe e al 2, commonly used n he opon prcng leraure, as well as a wo-facor model he scale of hs smulaon sudy exceeds wha has been underaken prevously n he relaed leraure We fnd our nference echnque o perform well whn realscally calbraed sengs In he emprcal applcaon we propose a new hree-facor sochasc volaly model and esmae usng an exensve opon panel for he S&P 5 ndex he model generalzes he exsng wo-facor specfcaons by allowng he nensy of he jump al o depend on an addonal facor ha s no a componen of marke volaly alhough can depend on Our dagnosc ess reveal ha hs feaure s crucal for explanng he observed dynamc dependences beween shor maury ou-of-he-money pus and a-he-money opons he resuls mply a sgnfcan me varaon n he rsk-neural jump al rsk Furhermore, he lef and rgh jump als exhb very dfferen dynamcs wh he laer resemblng he dynamcs of marke volaly more 3

5 closely Fnally, we documen ha he response of he prced lef al rsk ofen s subsanally more pronounced and perssen han for he volaly process followng marke crses he res of he paper s organzed as follows Secon 2 nroduces our formal seup Secon 3 develops our esmaors and derves he feasble lm heory In Secon 4, we develop dagnosc ess for he opon prce dynamcs Secon 5 conans a Mone Carlo sudy of he proposed esmaors In Secon 6, we explo our new nference ools o analyze he opon prce dynamcs of he S&P 5 ndex Secon 7 concludes All proofs are deferred o he appendx In a supplemenary appendx we collec addonal resuls peranng o he Mone Carlo and he emprcal applcaon 2 he Basc Modelng Framework 2 Seup and Noaon We frs esablsh some noaon he underlyng unvarae asse prce process s denoed X and s defned on a flered probably space Ω, F, F, P over he calendar me nerval [, ], for > fxed I s assumed o be governed by he followng general dynamcs under P, dx = α d + V dw + x µd, dx, X where α and V are càdlàg; W s a P -Brownan moon; µ s an neger-valued random measure counng he jumps n X, wh compensaor ν P d, dx = a d ν P dx for some process a and Lévy measure ν P dx, and he assocaed marngale measure s µ = µ ν P Furhermore, we denoe he expecaons operaor under P by E[ ] We assume X sasfes he followng condon x> Assumpon A he process X n equaon sasfes: here exss a sequence of soppng mes r ncreasng o nfny, and for each neger r here exss a bounded { process V r sasfyng } V = V r for < r, and here are posve consans K r such ha E V r V s r 2 F s K r s for every s x> x β ν P dx <, for some β [, 2 nf [, ] V > and he processes α, V and a are locally bounded Assumpon A s que weak and sasfed for almos all sandard connuous-me asse prcng models A s sasfed f V s governed by a mulvarae sochasc dfferenal equaon Assumpon A resrcs he so-called Blumenhal-Geoor ndex of he jumps see, eg, Secon 32 n Jacod and Proer 22 o be below β Some of our resuls, such as heorem 5 below, depend on he value of hs coeffcen Fnally, assumpon A mples ha, a each pon 4

6 n me, he prce process has a non-vanshng connuous marngale componen We noe ha assumpon A does no nvolve any negrably or saonary condons for he model he rsk-neural probably measure, Q, s guaraneed o exs by no-arbrage resrcons on he prce process, see, eg, secon 6K n Duffe 2, and s locally equvalen o P I ransforms dscouned asse prces no local marngales In parcular, for X under Q, we have, dx = r δ d + V dw + x µd, dx, 2 X where r s he nsananeous rsk-free neres rae and δ s he nsananeous dvdend yeld Moreover, wh slgh abuse of noaon, W now denoes a Q-Brownan moon and he jump x> marngale measure s defned wh respec o he rsk-neural compensaor ν Q d, dx We furher assume he dffusve volaly and jump processes are governed by a laen sae vecor, so ha V = ξ S and ν Q d, dx = ξ 2 S ν Q dx, where ν Q dx s a Lévy measure ; ξ and ξ 2 are known funcons n C 2, and S denoes he p sae vecor Moreover, r and δ are smooh funcons of S, and he laer follows a jump-dffusve Markov process under Q hs specfcaon ness mos connuous-me models used n emprcal work, ncludng he affne jumpdffuson class of Duffe e al 2 he seng allows for volaly processes whose dynamcs closely approxmae long-memory ype dependence, bu snce S s fne dmensonal and follows a Markov process, we do rule ou genune long-memory volaly processes We sress ha we do no mpose any resrcon on he dependence beween he laen sae vecor S and eher W or he jump measure µ ha s, he so-called leverage effec, workng hrough eher he dffusve or he jump componen of X, or boh, s allowed for We denoe European-syle ou-of-he-money opon prces for he asse X a me by O,k,τ Assumng frconless radng n he opons marke, he opon prces are gven as, O,k,τ = E Q E Q [ e ] +τ r s ds X +τ K +, f K > F,+τ, [ e ] +τ r s ds K X +τ +, f K F,+τ, where τ s he enor, K he srke prce, F,+τ he fuures prce of he underlyng asse a me for he fuure dae + τ, and k = lnk/f,+τ he log-moneyness he Markovan assumpon on he sae vecor, S, mples ha er,+τ O,k,τ F,+τ s a funcon only of he enor, sae vecor, and moneyness as well as, f S s no saonary under Q, where r,+τ s he rsk-free neres rae for he perod he separably of he Lévy measure n a me-nvaran jump measure on he jump sze and a sochasc process s a nonrval resrcon I essenally amouns o resrcng he me-varaon of jumps of dfferen szes o be he same Neverheless, hs assumpon s sasfed n mos paramerc jump models n emprcal applcaons used o dae, eg, holds for he whole affne jump-dffuson class of models 3 5

7 [, + τ] 2 We denoe he Black-Scholes mpled volaly BSIV correspondng o O,k,τ by κ,k,τ hs merely represens an alernave, and convenen, prcng convenon for he opons, as he BSIV s a deermnsc and srcly monoone ransformaon of he rao e r,+τ O,k,τ / F,+τ 22 he Paramerc Opon Prcng Framework Henceforh, we assume a paramerc model for he rsk-neural dsrbuon, characerzed by he q parameer vecor θ, wh θ sgnfyng he rue value, whle we do no resrc he objecve dsrbuon for he underlyng asse beyond wha s mpled by he equvalence of he wo probably measures 3 For exposonal convenence we assume ha he funcons ξ and ξ 2 do no depend on he parameer vecor 4 he opon panel has a fxed me span, [, ], and we observe he opon surface a gven mes =,, We have a large cross-secon of k values, spannng a sgnfcan srke range, avalable each dae for several dfferen enors, τ hs s a naural assumpon for acve and lqud opon markes In hs secon, we focus on he deal scenaro whou measuremen errors n he opon prces he crcal exenson o he case nvolvng such errors s provded n Secon 3 he heorecal value of he BSIV under he rsk-neural model s denoed κk, τ, S, θ 5 For each dae, we have a cross-secon of opon prces {O,kj,τ j } j=,,n for some neger N, where he ndex j runs across he full se of srkes and enors avalable on day he number of opons for he maury τ s denoed N τ he asympoc heory developed below reflecs he dsrbuon of he avalable opons n he sample across he days Henceforh, we rely on he followng noaon, N = τ N τ, N = N, N = mn =,, N 4 For each par,, τ, k, τ and k, τ denoe he mnmum and maxmum log-moneyness, respecvely he moneyness grd for he opons a me and enor τ s denoed, k, τ = k,τ < k,τ < k,τ N τ = k, τ he asympoc scheme sequenally adds new srkes o he exsng ones whn [k, τ, k, τ] he followng condon formally capures he noon of a large, ye heerogeneous, opon panel 2 Renaul 997 dscusses he homogeney of he opon prce wh respec o X, K more generally 3 For an alernave approach mposng mnmal assumpons on he objecve probably measure, whle employng a paramerc specfcaon for he sochasc dscoun facor, see Gaglardn e al 2 he nference procedures dffer very subsanally n oher mporan dmensons, as Gaglardn e al 2 nvoke large me span asympocs and deal wh a small and fxed cross-secon of opons for each day 4 hs assumpon s also almos unversally sasfed for he models used n praccal applcaons 5 Recall ha S s a Markov process If he dynamcs of S s non-saonary under Q, hen κ should also have a subscrp For noaonal smplcy, we mpose saonary, bu he analyss readly accommodaes non-saonary 6

8 Assumpon A For each =,, and each moneyness τ, he number of opons N τ wh N τ /N π τ and N /N ς, for some posve numbers π τ and ς Moreover, we have N τ,τ ψ,τ k unformly on he nerval k, τ, k, τ, where,τ = k,τ k,τ and ψ,τ akes on fne and srcly posve values Assumpon A allows for a grea deal of neremporal heerogeney n he observaon scheme For example, he enors need no be dencal across days and he assumpon of a fxed number of maures a each pon n me s mposed only o smplfy he exposon Imporanly, we allow for a dfferen number of opons n he panel across days, maures and moneyness Also, nuvely, he relave number of opons on a gven dae wll mpac he nference for he sae vecor on ha dae compared wh oher daes Lkewse, he relave number of opons across he varous maures and he local sparseness of he srkes should nfluence he qualy of nference for parameers and sae varables dfferenally dependng on her sensvy o enor and moneyness he quanes ς, π τ and ψ,τ k capure hese faces of he panel confguraon and hey do appear explcly n he asympoc dsrbuon heory esablshed laer Of course, alhough he rsk-neural measure s guaraneed o exs, s no unque because, n general, fnancal markes are ncomplee Gven our paramerc seng, our nex assumpon s exacly wha s requred o unquely denfy he parameerzed Q measure as well as he sae vecor gven he observaon scheme n assumpon A Assumpon A2 For every ɛ > and θ Θ, for some compac se Θ, we have, nf = { Z S ɛ} { θ θ ɛ} c k,τ = τ k,τ κk, τ, S, θ κk, τ, Z, θ 2 dk >, as We emphasze ha hs denfcaon condon vares across dsnc realzaons of he sae vecor Assumpon A and A2 mply ha, gven correc model specfcaon, we can recover he parameer vecor as well as he sae vecor realzaon whou error a any pon n me 6 Whle he sae varables change from perod o perod, he parameer vecor should reman nvaran Smlarly, he f o he opon prces provded by he model should be perfec hese resrcons may serve as he bass for specfcaon ess Moreover, he paramerc model has mplcaons for he pahwse behavor of X across all equvalen probably measures Mos noably, he dffuson coeffcen of X, ξ S, should be dencal for Q and P hs propery s also esable: he dffuson 6 In a seng wh an ncreasng me span, he me seres of he recovered sae vecor, S, may be furher used o esmae, paramercally or nonparamercally, he assocaed P law Hence, an opon panel wh ncreasng me span and wde cross-secon suffces for esmang boh he Q and P measures, and hus also he rsk premums assocaed wh he sae vecor dynamcs In prncple, here s no need for reurn daa on he underlyng asse 7

9 coeffcen may be recovered nonparamercally from a connuous record of X and conrased wh he model-mpled ξ S We develop formal ess for such pahwse resrcons of he rsk-neural model n Secon 4, coverng he relevan case of nosy opon and asse prce observaons here are marked dfferences n he nformaon conen of he opon panel wh fxed me grd versus he prce pah of he underlyng asse hs s mos readly llusraed n he deal, and nfeasble, seng of frconless radng and error-free prcng A connuous record for X allows us o oban he dffusve volaly, whou error, from a local neghborhood of he curren me, and o denfy he mng and sze of any prce jump In conras, error-free opon daa enable us o drecly observe he sae vecor, S If he sae vecor consss of a sngle volaly facor, V, as s ofen assumed, he wo approaches provde equvalen, and perfec, nference abou he sae of he sysem If he model allows for prce jumps, he opon panel les us nfer he, possbly me-varyng, rsk-neural jump nenses and jump dsrbuons, bu does no reveal he acual jump realzaons In conras, he prce pah for X denfes he jumps, bu does no pn down he jump dsrbuon Fnally, f we move o a mul-facor volaly seng, as mpled by much recen research, he opons daa are even more pvoal for nference For example, f here are wo volaly facors, e, V = V, + V 2,, he hgh-frequency daa for X drecly nforms us abou he aggregae value, V, only, whle he opon daa le us denfy V, and V 2, separaely Whle hese conclusons only apply for an deal seng, clarfes wha ype of nformaon one may aspre o oban from eher source, even f wll nvolve esmaon and nferenal errors n pracce 3 Inference for Opon Panels wh a Fxed me Span We now urn o he emprcally relevan case of nosy observaons Fgure depcs a nonparamerc kernel regresson esmae of he relave bd-ask spread n he quoes for S&P 5 ndex opons, n uns of BSIV, as a funcon of moneyness, and normalzed by volaly he spread s non-rval and ncreases que sharply for deep ou-of-he-money OM calls Clearly, he nose n any ndvdual opon prce s que sgnfcan hs fac movaes our use of an exensve cross-secon of opon prces o mgae and dversfy he mpac of measuremen error 7 In he remander of hs secon, we develop nference procedures for he parameer vecor, θ, governng he rsk-neural dsrbuon and he realzed rajecory of he sae vecor {S } =,, based on an opon panel, observed wh error We frs nroduce our assumpons regardng opon errors, hen defne our esmaor and, n urn, esablsh conssency and asympoc normaly 7 A smlar perspecve underles he Chcago Board Opons Exchange CBOE compuaon of he volaly VIX ndex I ncludes all relevan shor maury S&P 5 ndex opons whn he prescrbed srke range, wh he mplc premse ha he observaon errors largely wash ou n he negraon 8

10 3 IV error kernel regresson 2 IVa IVb IVa+IVb/ Log Moneyness: logk/f σ τ Fgure : Kernel regresson esmae of he bd-ask spread of opon mpled volaly as a funcon of moneyness he esmaes are based on he bes bd and ask quoes for he S&P 5 opons on he CBOE a he end-of-radng for each Wednesday durng January, 996 July 2, 2 We use all avalable opons wh maures up o a year F and σ denoe, respecvely, he fuures prce and he Black-Scholes a-he-money mpled volaly a he end of he radng day 3 he Opon Error Process We spulae ha opon prces, quoed n erms of BSIV, are observed wh error, e, we observe κ,k,τ = κ,k,τ + ɛ,k,τ, where he errors, ɛ,k,τ, are defned on a space Ω = A,k,τ for A,k,τ = R, wh Γ N,k R,τ Γ denong he se of all possble enors Ω s equpped wh he produc Borel σ-feld F, wh ranson probably P ω, dω from he orgnal probably space Ω on whch X s defned o Ω We defne he flraon on Ω va F probably space Ω, F, F, P s gven as follows, = σ ɛ s,k,τ : s hen he flered Ω = Ω Ω, F = F F, F = s> F s F s, Pdω, dω = P dω P ω, dω Processes defned on Ω or Ω may rvally be vewed as processes on Ω as well, eg, W connues o be a Brownan moon on Ω We henceforh adop hs perspecve whou furher menon Inuvely, hs formal represenaon may be movaed as follows he opon errors are defned on he space Ω We equp hs space wh he smple produc opology as, a any pon n me, only a counable number of hem appear n our esmaon Snce he opon errors can be assocaed wh any srke, pon n me and maury, we need a large space o suppor hem 9

11 Fnally, snce we wan o allow he opon prces and he underlyng process X o be dependen, we defne he probably measure va a ranson probably dsrbuon from Ω o Ω For he formal analyss of Ω, see chaper I of Dellachere and Meyer 978 Gven he presence of observaon error, we canno denfy he parameers and sae vecor smply by nverng he opon prcng formula We mus explcly accommodae he mpac of nose on he nference In parcular, f a lmed se of opons s ncluded, hen nference s only feasble under src paramerc assumpons regardng he error dsrbuon hs s problemac, as we have lle evdence peranng o he naure of hese prce errors In conras, a large cross-secon allows us o average ou he errors and reman fully nonparamerc regardng her dsrbuon However, hs error dversfcaon only works f we can ensure ha he effec of he opon prce errors vanshes n a suable manner he followng condon suffces for esablshng conssency of our esmaor recall he noaon N = mn =,, N Assumpon A3 For every ɛ >, =,,, and any fne posve-valued F -adaped process ζ k, τ on R Γ connuous n s frs argumen, we have, for N and θ Θ, sup { Z S >ɛ} { θ θ >ɛ} N j= ζ k j, τ j [κk j, τ j, S, θ κk j, τ j, Z, θ] ɛ,kj,τ j N j= [κk j, τ j, S, θ κk j, τ j, Z, θ] 2 P If he sae vecor S has bounded suppor, assumpon A3 follows from a unform Law of Large Numbers on compac ses for whch prmve condons are well known, see, eg, Newey 99 Of course, for ypcal asse prcng models he sochasc volaly process, and hus S, has unbounded suppor Assumpon A3 may hen be shown o follow from unform convergence on a space of funcons vanshng a nfny; see, eg, heorem 2 n Ibragmov and Has mnsk 98 Assumpon A3 provdes suffcen condons for Assumpon A3 o hold Assumpon A3 For every =,,, we assume: κk, τ, S, θ s connuously dfferenable n S, θ, κ k, τ, S, θ = Olog S as S and θ Θ for some regon n k, τ, k, τ wh posve Lebesgue measure, ɛ,k,τ and ɛ,k,τ are ndependen condonal on F, whenever k, τ k, τ, E ɛ,k,τ F =, E ɛ,k,τ max{p,q}+ι F = ζ,k,τ, beng a connuous funcon n s second argumen and ι > arbrary small for ζ,k,τ Assumpon A3 provdes condons drecly on he opon error and opon prce In parcular, condon requres he opon prce o dverge n sep wh he sae vecor hs s prey nuve Indeed, he BSIV for shor maury AM opons approxmaely equals spo volaly and hence ncreases ndefnely as volaly does so he smoohness of he opon prces n as

12 well as he ndependence and unbasedness of he measuremen errors n are mpled by he condons for he asympoc lm heory o hold, and hey are dscussed furher below hs leaves he weak momen resrcon on he measuremen nose n as he only separae assumpon on he error process In he Appendx we verfy ha Assumpon A3 mples Assumpon A3 We requre addonal regulary for our lmng dsrbuonal resuls o hold Assumpon A4 For he error process, ɛ,k,τ, we have, E ɛ,k,τ F =, E ɛ 2,k,τ F = φ,k,τ, for φ,k,τ beng a connuous funcon n s second argumen, ɛ,k,τ and ɛ,k,τ are ndependen condonal on F, whenever, k, τ, k, τ, v E ɛ,k,τ 4 F <, almos surely Assumpon A4 mples ha he observaon errors, condonal on he flraon F, are ndependen Noneheless, he error process may dsplay a sochascally evolvng volaly whch can depend on opon moneyness and enor as well as any oher process defned on he orgnal probably space such as he enre hsory of X and S Relave o he earler leraure, we avod paramerc modelng of he error and allow for sgnfcan flexbly for s condonal dsrbuon, ncludng he varance and hgher order momens Assumpon A4 does, however, rule ou correlaed opon errors, alhough hs requremen may also be weakened Assumpon A4 s analogous o he condons mposed on he mcrosrucure nose process for hgh-frequency asse prces n Jacod e al 29 and subsequen papers We sress ha par s crcal for our resuls, alhough may be weakened by allowng for a bas ha vanshes asympocally Par excludes correlaon n he error across srkes, bu we can accommodae weak spaal dependence, a he cos of more complex noaon and proof On he oher hand, f he opon errors conan a common componen across all srkes, hs error, obvously, canno be averaged ou by spaal negraon n he moneyness dmenson For example, Baes 2 assumes ha opon prces on a gven day, for moneyness whn ceran ranges, may conan a common error componen He nerpres hs as a model specfcaon error In our seng, such feaures mus be ncluded n he heorecal value κk, τ, S, θ raher han beng reaed as errors Fnally, f s more approprae o assume unbased errors for he opon prce raher han he BSIV whch consues a nonlnear ransformaon of he prce one should nsead mnmze he dsance beween observed and model-mpled opon prces In our emprcal applcaon, we fnd he BSIVs o be approxmaely lnear n prces across he relevan srke range, so he dsncon beween unbasedness of mpled volales or prces s no a praccal concern; see, eg, Chrsoffersen and Jacobs 24 for a dscusson of he mpac of he error specfcaon

13 32 Conssency In order o formally defne our nference procedure, we frs nroduce an arbrary conssen nonparamerc esmaor for he spo varance, V, obaned from hgh-frequency daa on he underlyng asse We denoe hs esmaor V n, where n sgnfes he number of hgh-frequency observaons of X ha are avalable whn a un nerval of me an explc example of V n s provded n Secon 33 Our esmaes for he sae vecor and he rsk-neural parameers based on he opon panel and he hgh-frequency daa are hen obaned as follows, {Ŝn } =,,, θ n N = argmn κ,kj,τ {Z } =,,, θ Θ N j κk j, τ j, Z, θ 2 + λn V n 2 ξ Z, 5 = j= for a deermnsc sequence of nonnegave numbers {λ n } he esmaon s based on mnmzng he mean squared error n fng he panel of observed opon mpled volales, wh a penalzaon erm ha reflecs how much he mpled spo volaly devaes from a model-free volaly esmae he presence of V n n he objecve funcon serves o help denfy he parameer vecor by penalzng values ha mply unreasonable volaly levels he presence of he penalzaon erm n 5 s remnscen of he ncluson of nformaon regardng he P dynamcs n opon-based esmaon, eg, Baes 2 and Pan 22 here s, however, a fundamenal dfference We do no model he P dynamcs and he penalzaon n 5 concerns he pahwse behavor of he opon surface, no s P law hs s herefore, a more robus parameer-free and sronger pahwse resrcon on he opon dynamcs he conssency of Ŝn, θ n follows from he nex heorem heorem Suppose assumpons A-A3 hold for some N fxed and ha { V n } =,, s conssen for {V } =,,, as n hen, f N and λ n λ for some fne λ as n, we have ha Ŝn, θ n exss wh probably approachng and furher ha, Ŝn S P, θ n θ P, =,, 6 hus, n he presence of observaon errors sasfyng assumpon A3, we can sll recover he sae vecor as well as he rsk-neural parameers conssenly from he opon panel he key dfference beween he parameers and he sae vecor s ha he laer changes from day o day, whle he former mus reman nvaran across he sample he longer he me span covered by he sample, he more resrcve s hs nvarance condon for he rsk-neural measure Anoher major dsncon sems from he penalzaon erm consruced from hgh-frequency daa as hs erm nvolves only he sae vecor and no drecly he rsk-neural parameers 2

14 33 he Lmng Dsrbuon of he Esmaor In analogy o he hgh-frequency based realzed volaly esmaors, whch also rely on n-fll asympocs, our lmng dsrbuon resuls nvolve sable convergence We use he symbol L s o ndcae hs form of convergence I s an exenson of he sandard noon of convergence n law o he case where he lmng sequence converges jonly wh any bounded varable defned on he orgnal probably space I s parcularly useful when he lmng dsrbuon depends on F, as n our seng For formal analyss of hs concep, see, eg, secon VIII5c n Jacod and Shryaev 23 he sable convergence resul n he followng heorem s crcal for enablng our feasble nference as well as he developmen of our dagnosc ess n Secon 4 heorem 2 Assume A-A4 hold and κ, τ, Z, θ s wce connuously-dfferenable n s argumens hen, f N and λ 2 n N, for n, we have, N Ŝn S N Ŝn S N/ θn θ L s H Ω /2 E E E, 7 where E,,E are p vecors and E s q vecor, all defned on an exenson of he orgnal probably space beng d wh sandard normal dsrbuon, and we defne Φ, p p Φ Φ = p p Φ, Φ Φ +, Φ +, Φ, +, + +, +, Φ = H, Ω, 8 wh he blocks of H and Ω for =,, gven by H, = τ πτ k,τ k,τ ψ,τ k Sκk, τ, S, θ S κk, τ, S, θ dk, +, + H = k,τ = τ πτ k,τ ψ,τ k θκk, τ, S, θ θ κk, τ, S, θ dk, =, + H = H +, τ πτ k,τ k,τ ψ,τ k Sκk, τ, S, θ θ κk, τ, S, θ dk, 3

15 Ω, = τ πτ k,τ k,τ ψ,τ k φ,k,τ S κk, τ, S, θ S κk, τ, S, θ dk, +, + Ω = = k,τ ς τ πτ k,τ ψ,τ k φ,k,τ θ κk, τ, S, θ θ κk, τ, S, θ dk, =, + Ω = Ω +, ς τ πτ k,τ k,τ ψ,τ k φ,k,τ S κk, τ, S, θ θ κk, τ, S, θ dk Several commens are n order Frs, we reerae ha he lm resul n 7 holds sably condonal on he flraon of he orgnal probably space he lm s mxed-gaussan, wh a mxng varable, H Ω /2, ha s adaped o F 8 he random asympoc varance of he esmaor sgnfes ha he precson n recoverng he sae vecor vares from perod o perod, and ha he qualy of nference n general depends on he values of he sae vecor and asse prces as well as he number and characerscs of he opons, e, enor and moneyness hs provdes mporan flexbly as he feaures of he opon daa change from day o day I also allows us o formally compare esmaes across dfferen me perods and we make frequen use of hs fac n he nex secon We sress ha heorem 2 does no requre any form of saonary or ergodcy of he sae vecor, respecvely volaly, under he sascal dsrbuon As noed prevously, many aspecs of he lmng dsrbuonal heory esablshed above resemble he correspondng heory for volaly esmaors based on hgh-frequency daa, see, eg, Barndorff-Nelsen e al 26 Our seup may be conrased o he cross-seconal regressons wh common shocks analyzed by Andrews 25; see also Kuersener and Prucha 2 for exensons Andrews 25 analyzes cross-seconal leas squares esmaors where boh he errors and regressors, condonal on an F -adaped random vecor, are d In our seng, he role of he regressors s aken on by he sae vecor, S, bu s no drecly observable and, crcally, exhbs srong emporal dependence Mos mporanly, he sable convergence resuls of heorem 2 are vald for a much wder σ-feld han a subfeld of F as n Andrews 25 enablng feasble nference from opon prces ha are hghly correlaed wh he evolvng reurn and volaly nnovaons Furher, n heorem 5 below, we show ha he sable convergence of heorem 2 holds jonly wh ha of a hgh-frequency esmaor for spo volaly, anoher resul for whch our general sable convergence resul s ndspensable o mplemen feasble nference, we need o oban he requse conssen esmae of he condonal asympoc varance of } =,,, θ {Ŝn n, whch n urn, can be done usng a conssen esmaor of he opon error, ɛ,k,τ he formal resul s saed n he followng heorem 8 More formally, he marces should be denoed H ω and Ω ω o hghlgh he fac ha hey depend on he parcular realzaon on he orgnal probably space 4

16 heorem 3 Under he condons of heorem 2, conssen esmaes for H and Ω are gven by Ĥ and Ω, where for he same paron of he marxes as n 8, we se Ω, = N N +, + Ω =, + Ω = Ĥ, = N N j= Sκk j, τ j, Ŝn, θ S κk j, τ j, Ŝn, θ, +, + Ĥ, + Ĥ = j= = N Ω +, = Ĥ +, N 3 = N N j= θκk j, τ j, Ŝn, θ θ κk j, τ j, Ŝn, θ, = N N N j= j= Sκk j, τ j, Ŝn, θ θ κk j, τ j, Ŝn, θ, θ 2 κ j κk j, τ j, Ŝ, S κk j, τ j, Ŝn, θ S κk j, τ j, Ŝn, θ, = θ 2 N N 2 j= κ j κk j, τ j, Ŝn, θ κk j, τ j, Ŝn, θ θ κk j, τ j, Ŝn, θ, θ 2 = = N κ j κk j, τ j, Ŝn, S κk j, τ j, Ŝn, θ θ κk j, τ j, Ŝn, θ Based on equaons 9 and, as well as he lm resul n 7, pvoal ess such as -ess for he parameers are readly consruced hs s a by-produc of he sable convergence n equaon 7, whch ensures ha he resul holds jonly wh he convergence n probably of Ĥ and Ω o her random asympoc lms We also noe ha heorem 2 allows for condonal heeroskedascy n he opon prce error When he laer s presen, he esmaor n 5 s neffcen n he sense ha he observaons are no weghed opmally he nex heorem accommodaes he use of weghed leas squares WLS heorem 4 Under he seng of heorem 2, suppose here exs 9 φ,k,τ P φ,k,τ >, unformly on [k, τ, k, τ], for τ Γ and =,,, as N and n Defne {S n } =,,, θ n N κ,k,τ κk j, τ j, Z, θ 2 = argmn {Z } =,,, θ Θ wh λ 2 n N where E,,E hen = j= φ,k,τ {S n } =,,, θ n s conssen and furher, S n S N S n S θ n θ L s Λ /2 E E E + λ n V n ξ Z 2, 2, 3 are p vecors and E s q vecor, all defned on an exenson of he orgnal probably space and jonly consung an d sandard normal vecor, and for he same 5

17 paronng of Λ as n equaon 8, he Λ marx s defned by, Λ, = ς k,τ τ πτ φ,k,τ k,τ ψ,τ k Sκk, τ, S, θ S κk, τ, S, θ dk, +, + Λ = = ς k,τ τ πτ φ,k,τ k,τ ψ,τ k θκk, τ, S, θ θ κk, τ, S, θ dk, = ς, + Λ = Λ +, τ πτ k,τ k,τ φ,k,τ ψ,τ k Sκk, τ, S, θ θ κk, τ, S, θ dk A conssen esmae for Λ s gven by Λ where, for he dencal paron of he marces as used n equaon 8, we se, Λ, = N = N j= φ,k,τ Sκk j, τ j, Ŝn, θ S κk j, τ j, Ŝn, θ, +, + Λ = N = N j= φ,k,τ θκk j, τ j, Ŝn, θ θ κk j, τ j, Ŝn, θ,, + Λ = N = N j= φ,k,τ Sκk j, τ j, Ŝn, θ θ κk j, τ j, Ŝn, θ he weghng of he observaons n 2 renders he condonal expeced Hessan of he objecve funcon equal o he lmng condonal covarance marx of he graden or score of he objecve funcon, evaluaed a {S } =,,, θ hs mples ha he asympoc condonal covarance marx of {S n } =,,, θ n s Λ he dervaon of he WLS esmaor n heorem 4 depends crcally on he sable convergence As n he classcal heory for M-esmaors, he proof explos Slusky s heorem whch mples X n, Y n L X, Y for wo sequences X n L X and Y n P Y, bu, mporanly, only when Y s non-random hs resul may be exended o he case where he lm Y s random only under he sable form of convergence, see, eg, equaon 225 n Jacod and Proer 22 For our WLS esmaor, he lms of he weghs and he elemens of he Hessan marx are random, unlke he classcal WLS esmaor for whch hey are non-random consans heorem 4 references a generc conssen esmaor of he condonal asympoc varance of he opon error, φ,k,τ A conssen esmae for he condonal varance of ɛ,k,τ, as a funcon of k paramerc or nonparamerc, for each par, τ, may be consruced n a manner smlar o he sandard WLS esmaors, see, eg, Robnson 987 and Newey and McFadden 994 he assumpon, λ 2 n N, n heorem 2 and heorem 4 ensures ha he penaly erm n 5 and 2 has no frs-order asympoc effec n he esmaon hs s convenen from an emprcal pon of vew as only boundedness n probably, and no conssency, s requred of V n Hence, we can apply he esmaon procedure even n sengs where V n o mcrosrucure nose whou havng o perform nose-robus correcons 9 may be mldly based due 9 We are graeful o a referee for ponng hs ou Despe hs convenen propery, we explo suffcenly sparse hgh-frequency reurn observaons n our emprcal applcaon n Secon 6, ha hs bas wll be neglgble 6

18 We can exend he above analyss o cover scenaros n whch he penaly erm s refleced n he lmng dsrbuon he requremen s ha we can esablsh he jon lmng dsrbuon of he nonparamerc esmaor V n and he emprcal processes arsng from he opon prcng error deermnng he lm n 7 hs s feasble for he wo nonparamerc jump-robus realzed volaly esmaors defned below, V ±,n = n,n X 2,n X αn ϖ,,n k n I ±,n X = log X + log n X + n where α >, ϖ, /2, k n denoes a deermnsc sequence wh k n /n and, V,n and V +,n I,n = { k n +,, } and I +,n = {,, k n }, 4 are esmaors for he spo varance from he lef and rgh, respecvely, and may be vewed as localzed versons of he runcaed varaon esmaor proposed orgnally by Mancn 2 If we denoe he se of jump mes for he varance process by J = {s : V s > }, hen, under weaker regulary condons han n Assumpon A, V +,n and V,n are boh conssen for V, provded / J We only need o esmae he spo volaly for a fne number of pons n me Snce he jump compensaor, conrollng he dsconnues n V, s absoluely connuous n, he probably of havng jumps a any dscree me pon s zero, snce, almos surely, / J he heorem below provdes he jon lm dsrbuon of V ±,n and he opon-based Ŝn Defnng he coeffcen, γ = 2 βϖ [ β /2 + [ βϖ] ] /4, we have, heorem 5 Under assumpon A, provded k n wh k n / n γ, and wh β defned as n A, we have for N, kn V +,n V V +,n V L s 2V 2V Ẽ Ẽ, 5 where Ẽ,, Ẽ s a sandard normal vecor ndependen of he orgnal flraon F and defned on an exenson of he orgnal probably space If he condons of heorem 2 hold, he vecor Ẽ,, Ẽ s ndependen from he vecor E,, E, E deermnng he lm dsrbuon of Ŝn,, Ŝn n equaon 7,n +,n Moreover, If,, J =, he resuls reman vald for V replacng V, =,, We sress ha, for praccal mplemenaon, s mporan o le he runcaon level reflec he pronounced nraday paern n volaly and we do so 7

19 I s opmal o choose ϖ close o /2 and, nex, k n close o n /2, provded he jumps are no oo acve, e, her acvy ndex sasfes β < 4/3 a farly mld resrcon recall Assumpon A he heorem reveals ha he convergence of V ±,n holds jonly wh ha of Ŝn and hey are asympocally ndependen condonal on he flraon of he orgnal probably space Usng heorem 5, we can exend heorem 2 o he case where he penalzaon has an asympoc effec on he esmaon an analogous exenson holds for he WLS esmaor n heorem 4 heorem 6 Assume A-A4 are sasfed and κ, τ, Z, θ s wce connuously-dfferenable n s argumens Le V n = V,n for V,n defned n 4 and he condons of heorem 5 hold hen, f N and n, such ha λ n λ and λ n N/ kn λ, we have: N Ŝn S N Ŝn S N/ θn θ L s H + D Ω + Σ /2 E E E, 6 where E,,E are p vecors and E s a q vecor, all defned on an exenson of he orgnal probably space and all muually ndependen sandard normal vecors Furhermore, wh he same paronng of D and Σ as n equaon 8, we defne, D, = λ Sξ S S ξ S, Σ, = λ2 ς 2V 2 S ξ S S ξ S, =,,, where λ = lm n λ n and he remander of he elemens n D and Σ are zero Conssen esmaes for D and Σ are gven by D and Σ where, for he same paron of he marxes as n 8, D, = λ n S ξ Ŝ S ξ Ŝ, Σ, and he res of he elemens of D and Σ are zero = λ2 nn 2 V n 2 S ξ Ŝ S ξ Ŝ, =,,, k n he condons on he sequence λ n n heorem 6 are weaker han hose n heorem 2, as k n recall heorem 5 he penalzaon erm affecs boh he condonal expeced Hessan of he objecve funcon and he condonal covarance marx of he score of he objecve funcon evaluaed a {S } =,,, θ heorem 6 allows us o sudy he effec of he penalzaon n he objecve funcon more formally We dsngush wo cases Frs, N k n hen we have λ = and hus D = he relave speed condon beween k n and n n heorem 5 can be weakened slghly, f β, a he cos of more lenghy dervaons 8

20 Consequenly, he penaly erm n he objecve funcon ncreases he covarance marx of he esmaor Hence, s opmal o pck λ n o ensure λ =, e, choose λ n such ha he penalzaon has no frs-order asympoc effec on he esmaor hs s nuve Snce N k n, he recovery of he volaly sae s done more effcenly va he opons daa Comparavely speakng, he hgh-frequency daa only nfuse nose no he esmaon procedure For N k n or N k n, may be preferable o have λ > he logc s ransparen n he nfeasble case of k n = and N k n, e, a connuous record of X, from whch V may be recovered whou error hen opon prces observed wh error are subopmal for recoverng V Of course, he dffusve volaly s only a mnor par of he full sae and parameer vecor, {S n } =,,, θ, ha we seek o esmae, so opons reman crcal for he nference 4 Pahwse Rsk-Neural Model ess he hypohess ha our model for he rsk-neural dynamcs s well-specfed has numerous mplcaons he prevous secon develops he lm heory necessary o devse formal ess for hs hypohess We propose a baery of dagnoscs, fallng no hree caegores: he frs concerns he f o he opon surface, he second checks for sably of he rsk-neural parameers, and he hrd assesses he equaly beween he opon-mpled volaly and a nonparamerc volaly esmae based on hgh-frequency daa he ess are all pahwse as hey nvolve resrcons on he observed pah of he opon surface and he underlyng asse prce Imporanly, hey do no resrc he sascal law for X, beyond wha s mpled by he rsk-neural law As such, hey do no rely on a jon hypohess ha he model s correcly specfed under boh he P and Q measures For smplcy n hs secon, we derve all ess for he orgnal seup n heorem 2, e, he case of ordnary leas squares n whch he penalzaon has no frs-order asympoc effec 4 Opon Prce F We frs develop a es based on he f afforded by he paramerc model he prevous secon supples us wh ools o formally separae observaon errors from model msspecfcaon errors n fng he opon prces he corollary below provdes a -es ha capures he qualy of he model f o he opon surface a a specfc pon n me for a gven enor Corollary Le K k, τ k, τ be a se wh posve Lebesgue measure and denoe by N K he number of opons on day wh enor τ and log-moneyness belongng o he se K hen, under he assumpons of heorem 2, we have, 9

21 j:k j K κ,kj,τ κk j, τ, Ŝn, θ n Π Ξ Π L s N,, Ξ = Ĥ Ω Ĥ Ĥ Υ, Υ, Ĥ Υ2,, 7 Υ, = NN K p j:k j K κ,kj,τ κk j, τ, Ŝn, θ 2 n S κk j, τ, Ŝn, θ n +p = N N K N j:k j K κ,kj,τ κk j, τ, Ŝn, θ 2 n θ κk j, τ, Ŝn, θ n, Π = p N j:k j K Υ 2, = N K j:k j K κ,kj,τ κk j, τ, Ŝn, θ n 2, S κk j, τ, Ŝn, θ n +p = N j:k j K θ κk j, τ, Ŝn, θ n N K he logc behnd he es n Corollary s sraghforward By aggregang he model-mpled opon f spaally, we average ou, and hus allevae, he effec due o he observaon error n he opons bu we rean he error due o nadequae model f Hence, for he resul n equaon 7 o apply, s necessary ha K has posve Lebesgue measure and ha κk, τ, Z, θ s a smooh funcon of log-moneyness he -sascs mpled by he asympoc lm resul n equaon 7 resemble he condonal momen ess proposed by Newey 985 and auchen 985 he asympoc varance of he opon f j:k j K κ,kj,τ κk j, τ, Ŝn, θ n, s esmaed feasbly by Π Ξ Π I accouns for he effec of he esmaon error of Ŝn, θ n I s crcal for he dervaon of Corollary ha he convergence n equaon 7 holds sably so ha he sandardzaon of he model f n equaon 7 yelds a varable wh a lmng sandard normal dsrbuon he es n equaon 7 can, of course, be exended o pool ogeher he esmaed errors across opons wh dfferen enors as well as for opons observed on dfferen days he es wll be powerful agans alernaves for whch he errors n fng he opons n he regon K end o be hghly correlaed, as hs blows up he numeraor whou affecng he denomnaor of he rao n 7 hs wll ypcally be he case, as sandard models mply smoohness n opon prces as a funcon of moneyness ha s, f he f s poor for a gven srke, due o model msspecfcaon, he model-mpled opon prces wll end o devae n he same drecon for nearby srkes Furhermore, he es of Corollary allows us o check he model f over shorer perods of me hs s more nformave abou poenal sources of model falure han assessng he me-averaged opon prce f, as s common pracce For example, we may be able 2

22 o assocae specfc ypes of model falure wh broader economc developmens ha pon owards omed sae varables n he model or a fundamenal lack of sably n he rsk-neural measure 42 me-varaon n Parameer Esmaes Our second es s based on he varaon of he rsk-neural parameers over me Under sandard regulary condons, model msspecfcaon wll mply ha he esmaor converges o a pseudorue parameer vecor, see, eg, Whe 982 and Goureroux e al 984 However, n our seng he sae varables change from perod o perod hs should nduce a correspondng movemen n he pseudo-rue parameer vecor for he msspecfed model ha s, under msspecfcaon, he me-varaon n he opon prces canno be raonalzed by shfs n he sae varables, so we should expec spll-over n erms of neremporal varaon n he pseudo-rue rsk-neural parameer esmaes over dsnc me perods Desgnng a es for parameer varaon s sraghforward usng heorem 2, as he esmaes obaned from opon panels spannng dsjon me perods should be asympocally ndependen when condoned on he flraon of he orgnal probably space 2 Corollary 2 In he seng of heorem 2, denoe he rsk-neural parameer esmaes from wo opon panels coverng dsjon me perods by θ n and θ 2 n If he rsk-neural model s vald for boh of hese dsnc me perods, we have, θn θ n 2 Âvar θn + Âvar θ 2 n θn θ 2 n L s χ 2 q, 8 where Âvar θ n and Âvar θ n 2 denoe conssen esmaes of he asympoc varances of θ n θ n 2 based on equaons 9- n heorem 3, and q denoes he dmenson of he parameer vecor he analogous resul apples for a subse of he parameer vecor of dmenson r < q, bu wh r replacng q n equaon 8 43 Dsance beween Model-Free and Opon-Impled Volaly Our fnal dagnosc ess wheher he spo volaly esmaed nonparamercally from hghfrequency daa on he underlyng asse s equal o he spo volaly, V, mpled by he opon daa gven he model for he rsk-neural dsrbuon of X hs resrcon follows from he fac ha he dffuson coeffcen of X should be nvaran under an equvalen measure change recall P and Q are locally equvalen Hence, f he opon prce dynamcs s successfully capured by he sae 2 Of course, hs can be generalzed o he case of overlappng esmaon perods by appropraely accounng for he condonal covarance of he wo parameer esmaes and 2

23 vecor S, he wo esmaes should no be sascally dsnc hs s, of course, he dencal consran ha we explo n our penalzaon erm durng esmaon Noneheless, he condon may be formally esed f we accoun suably for he specfcaon of he objecve funcon n 5 V ±,n o render he es feasble, we use he wo nonparamerc jump-robus volaly esmaors defned n 4 n he prevous secon Usng he jon asympoc convergence resul for V ±,n, Ŝn n heorem 5, we can derve he asympoc behavor of he dfference We sae hs mporan resul n he followng corollary V ±,n ξ Ŝn Corollary 3 Under he condons of heorems 2 and 5, we have for k n, N and λ 2 n N, V,n ξ Ŝn S ξ Ŝn χ S ξ Ŝn N + 2 V,n 2 k n =,, L s Ĕ Ĕ, 9 where χ s he par of Ĥ Ω Ĥ correspondng o he varance-covarance of Ŝn and where Ĕ,, Ĕ s a vecor of sandard normals ndependen of each oher and of F Ye agan, we sress ha we do no need a paramerc model for V under he sascal measure, P, o es he equaly of he spo volaly mpled by he underlyng asse dynamcs and he modeldependen opon-mpled dynamcs However, he es does hnge crcally on he characerzaon of he jon sable asympoc law n heorem 5 Consequenly, hs pahwse resrcon on he spo volaly canno be formally esed under he usual approach o opon-based paramerc nference whch precludes he applcaon of hs ype of lm heory he es n Corollary 3 compares wo alernave esmaors of he spo dffusve volaly: a paramerc one based on he opon daa and a nonparamerc one based on he hgh-frequency record for X As dscussed afer heorem 6, dependng on he relave growh of he opon daa ±,n and he hgh-frequency ncremens used n esmaon of V, we can have eher he opon-based or he hgh-frequency based esmaor be more effcen for recovery of V In ypcal applcaons wh a rch se of dervaves daa as s he case n our emprcal applcaon laer, however, he opon-based esmaor ends o domnae and hence he second erm n he denomnaor on he lef sde of 9 s usually subsanally larger han he frs one In hs case, e, for N k n, our es n Corollary 3 s remnscen of he Hausman 978 specfcaon es 3 Under he null of correc rsk-neural model specfcaon, ξ Ŝn s more effcen han 3 We hank a referee for ponng ou hs connecon V ±,n whle V ±,n s a robus 22