Spot Volatility Estimation Using Delta Sequences

Transcription

1 Spot Volatility Estimatio Usig Delta Sequeces Cecilia Macii, Uiversità di Fireze, via delle Padette 9, 57, Fireze, Italy V. Mattiussi City Uiversity, Northampto Square, Lodo ECV HB, UK R. Reò Uiversità di Siea, Piazza S.Fracesco 7, 53 Siea, Italy July, Abstract. We itroduce a class of oparametric spot volatility estimators based o delta sequeces ad coceived to iclude may of the existig estimators i the field as special cases. The full limit theory is first derived whe uevely sampled observatios uder ifill asymptotics ad fixed time-horizo are cosidered, ad the state variable is assumed to follow a Browia semimartigale. We the exted our class of estimators to iclude Poisso jumps or fiacial microstructure oise i the observed price process. As a applicatio of our results, we relate the Fourier estimator to a specific delta sequece obtaied with the Fejér fuctio. The proposed estimators are applied to data from the S&P5 stock idex futures market. Itroductio I the last decade, the larger availability of high-frequecy fiacial data sets has spawed cosiderable ecoometric research o itegrated volatility, ad i particular o realized volatility reviews o the topic ca be foud i Bardorff-Nielse ad Shephard [], ad i Badi ad Russell [9]). More recetly, the iterest has moved to study the variability of the price dyamics at a particular poit i time, the so-called istataeous or spot volatility. The usage of spot volatility estimates is also icreasig i fiacial applicatios. For example, spot volatility estimates have bee show to be We are grateful to Simoa Safelici, Claudio Pacati, Jea Jacod, Kris Boudt, Fulvio Corsi, Mavira Macio, Imma Curato ad Davide Pirio for valuable commets ad helpful discussios. We would like to dedicate this work to the memory of Prof. Paul Malliavi Electroic copy available at:

2 beeficial, with respect to itegrated estimates, i estimatig ifiitesimal cross-momets [8] ad i testig for the presece of jumps []. Spot volatility is also the crucial igrediet i optio pricig with stochastic volatility, where the iitial volatility value, i additio to the iitial value of the uderlyig, is eeded to price the optio. With this paper, we try to wide ad sustai the class of the existig estimators i the field by proposig a method to estimate the spot volatility of a uivariate semimartigale which ca be adapted i the presece of Poisso jumps or microstructure oise. More importatly, we allow for jumps i the volatility process; i this case, our estimator will coverge to the average of spot volatilities observed immediately before ad after the evetual jump. Oe way of estimatig istataeous volatility cosists i assumig that the volatility process is a determiistic fuctio of the observable state variable, ad oparametric techiques ca be applied both i the absece see Flores-Zmirou [9], Badi ad Phillips [6], Reò [47] ad Hoffma [3]) ad i the presece of jumps i X see Johaes [3], Badi ad Nguye [5], ad Macii ad Reò [36]). Fully oparametric methods whe volatility is istead a càdlàg process have bee studied by Malliavi ad Macio [33, 34] ad Kristese [3] i the absece of jumps, ad by Zu ad Boswijk [53], Hoffma, Muk ad Schmidt-Hieber [] ad Ogawa ad Safelici [4] i the absece of jumps but with oisy observatios. Related studies iclude the idea of rollig sample volatility estimators i Foster ad Nelso [], see also Adreou ad Ghysels [4], the theory of spot volatility estimatio developed i Badi ad Reò [7], ad the kerel based methods of Fa ad Wag [7], ad Myklad ad Zhag [4]. I the presece of jumps but absece of oise), spot volatility have bee studied by Jacod ad Protter [8], Ngo ad Ogawa [4], Aït-Sahalia ad Jacod [] ad Dobrev, Aderse ad Schaumburg [5]. Alteratives are studied i Alvarez et al. [], Geo-Catalot et al [] ad Hoffma [4]. The purpose of our study is to defie a large class of o parametric estimators of istataeous volatility, which icludes may of the aforemetioed methods. Our ituitio suggests that a spot volatility estimator ca be writte as the covolutio of squared price returs with a sequece of fuctios, kow as delta sequece, which coverges to a Dirac delta fuctio cocetratig all the mass aroud oe poit for applicatios of delta sequeces i statistics see, for istace, Watso ad Leadbetter [5], ad Walter ad Blum [49]). I particular, we exted the kerel estimator of Kristese [3] by provig that a traditioal kerel fuctio ca be see as a delta sequece. Our class is show to be reasoably wide ad it icludes the Fejér sequece used i the work of Malliavi ad Macio [34] ad the idicator fuctio used i the work of Jacod ad Protter [8]. The study of the asymptotic theory see Sectio ) reveals that the estimators withi the class are, uder suitable coditios, ormally distributed, whe the umber of observatios diverges to ifiity i a fixed iterval [, T] ad the maximum iterval betwee the observatios ot ecessarily equally spaced) shriks to zero. Our fidigs are derived uder mild assumptios o the drivig coefficiets of the stochastic differetial equatio. I Sectio 3, we allow for microstructure oise i the data Electroic copy available at:

3 ad use a two-scale volatility techique, similar to the oe i Zhag et al [5], to make our estimator robust agaist the oise. I additio, we tackle the problem of discotiuities i the retur dyamics usig a threshold estimator as i Macii [35] to filter out jumps from the observed price process. I Sectio 4 we cosider the work by Malliavi ad Macio [34] ad further ivestigate the asymptotic behavior of their proposed Fourier estimator of spot σ. Sectio 5 presets a empirical aalysis usig high-frequecy stock idex futures, where the above estimators are applied to detect itraday volatility dyamics. Sectio 6 cocludes. Spot Volatility Estimatio i the Basic Settig I what follows, we will cosider a uivariate logarithmic price process X t defied o a filtered probability space Ω,F,F t ) t T,P) satisfyig the usual coditios, see e.g. Protter [46]. Our results are based o the set of assumptios outlied below. Assumptio. i) The logarithmic price X t is the solutio of the followig stochastic differetial equatio dx t = µ t dt+σ t dw t,.) where the iitial coditio X is measurable with respect to F, W t is a stadard Browia motio defied o the filtered probability space ad µ t, σ t are adapted processes with càdlàg paths. ii) Give a fixed poit t [,T], let B ε t) = [t ε,t + ε],with fixed ε >, ad assume that there exist Γ >, a sequece of stoppig times τ m ad costats C m) t ω,s) Ω B ε t) [,τ m ω)], a.s. such that for all m, for E u s [ σ u σ s ] C m) u s Γ,.) t where E t [ ] deotes E[ F t ]. The class of processes for σ t we wish to estimate poit-wise is larger tha the class of the processes with differetiable paths, ad it icludes the importat case where σ t is geerated itself by a Browia motio as i a stochastic volatility model. Ideed, every càdlàg process σ t is also locally bouded, ad for every possibly jumpig) Itô semimartigale σ t, the requiremet.) is satisfied with Γ = ) o every [,τ m ] where σ t is bouded, thaks to the Burkholder-Davis-Gudy BDG hereafter) iequality see [6], pag.5). I particular the estimator we are goig to propose below is robust to jumps i volatility. I order to work with irregular samplig, we adapt to our settigs the cocept of quadratic variatio of time defied i Myklad ad Zhag [39]. Assumptio. The process X t is observed + times at determiistic istats = t < t <... < t = T, ot ecessarily equally spaced ad with T fixed. We set i = t i ad 3

4 = T ad assume max,...,{ i } = O ). The quadratic variatio of time up to a give t T is defied as Ht) = lim H t), where H t) = i )..3) Assumig that the above limit exists, we require that H is Lebesgue-almost surely differetiable i [, T], with H such that for some K ot depedig o i) H t i ) i K i..4) t i t I the special case of equally spaced observatios, i =, H t) = ad.4) is satisfied with K =. Whe the observatios are more less) cocetrated aroud t, the we have H t) < H t) > ). The assumptio max,..., { i } = O ) is techical, ad meas that the partitio should ot vary asymptotically too wildly with respect to the equally spaced partitio. Coditio i.4) for the partitio {t i } i is differet from coditio v) i assumptio A of [39]. For istace, cosider the sequece of partitios where the amplitude of the first [/] itervals ],t i ] is adoftheremaiig [/]is.the = /3adHt) = 4t/3I {t T}+4T/9+t/3)I {t>t} with T = T/3. This fuctio H is ot differetiable i T, however, we have that H t i ) i = for all other poits. It follows that our Assumptio is satisfied. O the cotrary, if we oly cosider the time poits t where H is differetiable, Myklad ad Zhag assumptio is ot fulfilled, sice H t) H t ) sup H t) +. t>t Deote the log-price) retur by X i = X ti X ti. Our proposed estimator takes the form of a discrete covolutio σ,ft) = f t) X i ),.5) where f ) is a give sequece of real fuctios belogig to the class specified below. Defiitio. A sequece F. = {f, IN} of fuctios f : D R, with D R beig a give set ad D is said to be a delta sequece if, for all processes σ t satisfyig Assumptio, as here ad throughout all the paper it is iteded that the itegrals are defied over the itersectio with s D), where R σ ) t) = o p ) ad σ ) t = f ) f ) T T T f s t)σ sds = σ ) t +Rσ ) t),.6) f s t)σ sds = c f σ ) t +o p),.7) f 4 s t)σ sds = O p f )),.8) ) ψ + f σ t +ψ f σ t I {t ],T[} +ψ f σ T I {t=t} +ψ + f σ I {t=} where x< f x)dx ψ f ad ψ+ f = ψ f ψ f = ψ+ f = for symmetric delta sequeces). 4

5 Note that if σ is cotiuous i t ],T[, the σ ) t = σ. If we istead estimate at boudaries t t = or t = T) we have to weight for the exact mass of the delta sequece respectively at the right ad at the left of t. Coditio.6) resembles the typical defiitio of delta sequece i aalysis. The techical coditios.7) ad.8) are required to guaratee the existece of a cetral limit theorem. Delta sequeces have bee itroduced i statistics to estimate the desity of a radom variable, see e.g. Watso ad Leadbetter [5]. The mai result of this sectio is stated uder a set of additioal coditios that we collect i the followig Assumptio. Assumptio 3. We assume that F = {f, IN} is a delta sequece with f ) + ad D f x)dx, ad that further the fuctios f satisfy: i) sup x D f x) Cf ) for a suitable costat C ii) f is Lipschitz i a eighborhood of with Lipschitz costat L such that L /f ) ; further, either f or Γ/ i f t) i. iii) there exists a costat M ε > ot depedig o for which sup f x) M ε..9) x Bε c) Theorem. below ca be show to hold also whe the coditio L /f ) i Assumptio 3 ii) is replaced with the coditio, less striget but less direct to be verified, f s t) f t) ds for the cosistecy part, ad ti f s t) f t) ds/ f ) for the CLT part. Assumptio 3 is ot straightforward to verify for a give sequece f. For this reaso, we specify the followig propositio, which ivolves a set of sufficiet coditios usig oly the features of f istead of the features of the process σ t. Propositio.. Cosider a sequece of oegative fuctios f : D R, with D R ad D, such that, as, coditios i) iii) i Assumptio 3 are fulfilled, ad further: iv) D f x)dx.) v) there exists a sequece ε such that ε ε f x)dx.) vi) where c f is a real costat D f x) f ) dx c f.) 5

6 The f is a delta sequece. I coditio iii) of Assumptio 3 we chose to ormalize fx) by f ), but alteratively f ) ca be replaced by ay sequece a able to deliver similar results, such as a = fx)dx. Some relevat examples of sequeces f x) satisfyig Assumptio 3 are listed below. Other examples ca be derived from Walter ad Blum [49]. Example : Kerels Kerel estimators, used by [3] to estimate spot volatility, ca ideed be used to geerate a class of delta sequeces. Cosider a fuctio K : R R ad a positive sequece h, ad defie: f x) = ) x K..3) h The sequece h is typically called badwidth, ad sice f ) = h K), we ca iterpret f ) as the iverse of the badwidth. I the case i which we write the delta sequece as.3), Assumptio 3 ca be reformulated as follows: Assumptio 3 for kerels):. + Kx)dx = ad K x)dx = c c f = c K) ). sup x R Kx) CK) 3. Kx) is almost everywhere differetiable ad K is bouded. 4. h is such that sup x R K ) xh h. 3 ) 5. sup x B c ε t) x h K h Mε. For example, the Gaussia kerel: h Kx) = e x π has c = π ad c f =, ad Assumptio 3 is readily verified, while the Epaechikov kerel Kx) = 3 4 x )I { x } has c = 3 5 ad c f = 4 5 ad also verifies 3. The idicator kerel: Kx) = I { x } also verifies Assumptio 3 ad has c = ad c f =. Example : Trigoometric fuctios 6

7 Trigoometric fuctios used i Fourier aalysis are traditioal approximats of the Dirac delta, ad aturally appear i the costructio of the Fourier estimator of Malliavi ad Macio [33]. The first example is the Dirichlet sequece give by g x) = π D N x), with domai [ π,π], where D N x) := e ihx = si[ ) ] N + x si x,.4) h N ad N is a divergig sequece. The Dirichlet sequece ca be egative at some poits. A positive trigoometric example, which will become crucial i Sectio 4, is give by f x) = π F N x) with domai [ π,π], where F N x) is the Fejér sequece F N x) := s N s ) e isx = si N+ N + N + si x ad N is aother divergig sequece. The followig properties hold N: x ),.5) i) π π π F N x)dx = iii) D Nx) = N +)F N x) ii) iv) π π π F N ) D N x)dx = π π F Nx)dx = 4π 3, provig that f ad g itegrate to ad that c f = 3 ad c g =. Now, otice that / six/) /siε/) if ε x π with < ε < π. This easily proves coditios iv) ad vi) i Propositio. for f. Moreover, with ε = ε we have F N x)dx N +si ε /) π ε ) ε x π which coverges to zero if ǫ N. This proves, together with the remaiig trivial coditios i Propositio., that F N is a delta sequece. The followig theorem derives the asymptotic distributio of the proposed volatility estimator.5). We will use MN,V) to deote a mixed ormal distributio with stochastic variace V. Theorem.. Let Assumptios,, 3 hold. If,f ) i such a way that f ), the for ay t [,T] we have σ,f t) p σ ) t. If furthermore, ) Rσ t) = o p f ) ), the [ σ,f t) σ ) L s) t] MN,c f H t)σ 4 ) t), f ) where the above covergece is stable i law. A similar result is obtaied i Kristese [3] whe f x) is of the form.3). O the otio of stable covergece, see Jacod [5]. Remark. O the validity of a CLT) The crucial coditio for the validity of a CLT is R σ ) t) = o p f ) ). This coditio is, however, typically satisfied with suitable choices of the sequece f ) or, i the kerel case, of the 7

8 badwidth). I the Appedix, we explicitely prove that, for the Gaussia, Epaechikov ad idicator kerel, this coditio is fulfilled whe h Γ+, ad for other kerels the coditio ca be verified i a similar way. The Fejér sequece is explicitely treated i Propositio 4.. Remark. Small sample correctio) I small samples, it is advisable to use the estimator σ,ft) = f t) X i ) f,.6) t) i from which it is immediate to derive the same asymptotic results as i Theorem. give that f t) i, as. Remark 3. Choice of the optimal f ) The choice of the optimal sequece f relies o usual bias-variace trade-off cosideratios see, for example, Fa ad Yao [8], or the discussio i Kristese [3]). From the proof of theorem., we ca see that the bias depeds both o the choice of the kerel ad of the regularity of σ. For example, i the case i which f is the idicator fuctio, we get see the proof of Remark ) that the bias is O f ) Γ/) ad, give that the variace is Of ) ), we get that the optimal choice of f ) is proportioal to +Γ, ad the speed of covergece of the spot volatility estimator is /4. About the choice of the optimal delta sequece, there are almost o results for Γ <, with the exceptio of [6], who suggests the usage of a double expoetial kerel, that is f x) = f )e x f). To get some isight o this problem, we simulate the model: dx t = µ+σ t dw ) t dlogσ t = ηdw ) t +dj t.7) where corrdw ),dw ) ) = ρ, the jump occurs, exactly, i t =.5, with size ormally distributed with mea.44 ad stadard deviatio.. We set η =.6, ρ =.5 ad µ =.6/5 these parameters are based o estimates i [8]). We also set =, ad f ) = ad cosider six delta sequeces: idicator, triagular, Gaussia, Epaechikov, Fejér ad double expoetial. We adopt the correctio.6). The relative) root mea square error i estimatig σ ) is show i Figure t=.5., showig that there is o substatial differece betwee the five delta sequeces, eve if the double expoetial kerel seems to preset some advatages. 3 Estimatio i the presece of microstructure oise/jumps This sectio shows that, with proper adjustmets, the estimator σ,f t) ca be employed to the aalysis of a more geeral data geeratig process where prices are affected by microstructure oise or ca display a fiite umber of jumps, two importat aspects that play a relevat role i the study of fiacial time series. 8

9 3.5 Epaechikov Idicator Gaussia Fejer Double Expoetial Triagular Frequecy Relative RMSE Figure.: Relative RMSE distributio for the estimatio of σ ) t differet delta sequeces listed i the leged. o, replicatios of model.7), for the five 3. Robustess to microstructure effects The followig results emphasize the suitability of our theoretical framework to deal with microstructure oise effects i the observed data. For semplicity, cosider logarithmic asset prices X ti which are observed at equispaced discrete times t,...,t ad are subject to a observatio error due to microstructure oise. Assumptio 4. Assume that observatios are equally spaced i = ). Let X ti = Y ti +ε i, 3.) where Yt i ) is the uobservable efficiet price satisfyig Assumptio, ad ε i deotes the oise compoet. The oise process {ε i } i is i.i.d. ad idepedet of Y with E[ǫ i ] = ad E[ǫ 8 i ] < +. I what follows, we deote by V ε = E[ǫ i ] ad κ ε = E[ǫ 4 i ]. ) Lemma 3.. Let Assumptios 3 ad 4 hold. If R σ ) t) = o p f ), the f ) where the above covergece is i distributio. ) σ,ft) V ε N, c f ad f ) as κε +Vε ) ), 3.) It is immediate to see that the market microstructure-iduced bias is give by E[ σ,ft) σ t)] = V ) ε +o, 3.3) 9

10 whichdivergesatrate. However, wheappropriatelycorrectedbyafactor, acosistetestimate of the oise variace ca be obtaied ad this is of the form V ε = σ,ft). To obtai a cosistet ad asymptotically ormally distributed estimator of the spot variace, we follow the two-scale approach i Zhag et al [5] ad propose a estimator with overlappig prices at the lower frequecies. The idea is to remove the market microstructure oise by subtractig volatility estimated at two differet frequecies, leavig the latet volatility uaffected. The approach we are proposig here is ot efficiet. Efficiet estimatio could be achieved, for example, usig multiscales [5], by smoothig the observed time series via pre-averagig as i Jacod et al [7], or by usig autocovariaces ad a flat-top kerel as i Bardorff-Nielse et al []. Defie a iteger < ad set σ,,ts t) = + f t) [ X ti+ X ti ) X ti X ti ) ]. 3.4) The followig Theorem shows that σ,ts, t) is a cosistet ad ormally distributed estimator i the presece of microstructure oise. Theorem 3.. Let Assumptios 3 ad 4 hold. If,f ), i such a way that f ), R σ ) t) = o p ) ad L /f ), we have σ,,ts p t) σ ) t. Furthermore, if ) Rσ t) = ) o p f ) ad = c ) 3 with c R, the f ) ) 3 where the above covergece is stable i law. [ ] σ,,ts t) σ ) L s) t MN [,c f V ε +cσ 4 ) t]), 3.5) Notice that the speed of covergece of the estimator i 3.5) is i lie with that obtaied i [5] i the case of itegrated volatility estimatio. Ideed, whe estimatig itegrated volatility i the presece of oise, you get a speed of covergece of /6 istead of the / that you would get i the absece of oise. With spot volatility, we get /6 f ) / istead of / f ) /. 3. Robustess to jumps We ow cosider the case where a fiite umber of Poisso jumps is added to the stochastic itegral drivig the state variable dyamics. Assumptio 5. The adapted process X t defied o [,T] satisfies X t = Y t +J t 3.6) with dj t = c J t)dn t, where Y t fulfills Assumptio, J t is a doubly stochastic Poisso process, ad N t is a o-explosive Poisso coutig process whose itesity is a adapted stochastic process λ t. The

11 size of the jumps occurrig at times τ,...,τ Nt) is give by i.i.d. radom variables c J τ j ) such that P{c J τ j ) = }) = t [,T]. Followig the approach i Macii [35], we defie our estimator to be /σ,f t) = f t) X i ) I { Xi) ϑ, 3.7) } where I { } deotes the idicator fuctio ad ϑ is a suitable sequece. The aim of the threshold ϑ is to disetagle the discotiuous variatio iduced by the Poisso jumps from the cotiuous variatio iduced by the Browia motio. Asymptotically, this happes whe ϑ coverges to zero slower tha the modulus of cotiuity of the Browia motio, as specified i the ext Theorem. Note that ϑ ca also be either a fuctio of time or a stochastic process see Macii ad Reò [36]). Alterative optios to 3.7) are the flat kerel estimator i Aït-Sahalia ad Jacod [], or the locally averaged bipower variatio proposed by Veraart [48]. Both approaches admit ifiite jump activity i the data. Theorem 3.3. Let Assumptios, 3 ad 5 hold. If,f ) ad ϑ i such a way that ) f ), ϑ / log ad R σ ) t) = o p ) we have /σ,f t) p σ t). Furthermore, if R σ ) t) = o p f ) ), the ] [ /σ,f t) σ ) L s) t MN,c f H t)σ 4 ) t), f ) where the above covergece is stable i law. 4 Relatio to the Fourier estimator I this sectio, we aalyze the Fourier estimator first itroduced i Malliavi ad Macio [33]. I particular, we show that the Fourier estimator ca be writte as the sum of a delta sequece estimator, whe the fuctio f ) is set to be equal to the Fejér sequece see Example ), ad a zero-mea oisy term. I the Fourier method, the classical harmoic aalysis is combied with stochastic calculus to coect the Fourier trasform of the log-price process X t to the Fourier trasform of the volatility fuctio σt. Specifically, the spot volatility estimator is defied to be σ,f,,n t) = k ) H, k)e ikτ, 4.) N k N where ad H, k) := T + F dx)s) := T s F dx)s)f dx)k s), 4.) e isτj X j 4.3) j=

12 is the discrete Fourier trasform of dx t. Here τ = πt/t ad τ i = πt i /T are rescaled times. Malliavi ad Macio [34] prove that, whe N =, σ,f, =N,N t) p σ t, whe,n. They also provide a weak covergece result for a Lebesgue average of σ,f, =N,N t) σ t o [,T], but do ot provide a cetral limit theorem for the estimatio error of the spot variace see also Clemet ad Gloter, [3] for a discussio ad a geeralizatio of their results). I order to apply the Fourier estimator, it is ecessary to set the umber of coefficiets of the price process used i the computatio of the volatility coefficiets, ad the umber of volatility coefficiets N used i the recostructio of the volatility trajectory. Both ad N are sequeces depedig o. Importatly, here we do ot restrict to the choice = N, suggestig that a higher is beeficial. A referece value for equally spaced data is = /, also kow as Nyquist frequecy see Priestley [45]). I what follows, we show that the Fourier estimator does ot belog directly to our class but it ca be rearraged ito the sum of two terms: the volatility estimator σ,f t), where f ) is a rescaled Fejér sequece, ad a cross-product term with zero mea. Propositio 4.. Defie σ,f t) as i.5) with f x) = π F Nx), where F N is defied i Eq..5), with x ] π,π[. Assume that Assumptios, ad 3 hold, with the exceptio of property ii) i Assumptio 3. Assume ow that, as,n, N/ i such a way that there exists a sequece ε such that N 3 ε 4 / ad N/ε Γ ). The, if t ],T[, f ) [ σ,f t) σ ) t] L s) MN Propositio 4.. The Fourier estimator give i 4.) is such that where ad φ,f,g t) = g ) σ,f,,n t) = σ,ft)+φ,f,g t) σ,ft) = i j= f t) X i ), 4 3 H σ 4 ) t ). f t j t)g t j ) X i X j. with f x) = T F N πx/t) ad g x) = T D πx/t). It holds that E[φ,f,gt)] = ad, if σ is idepedet from W t, the covariace betwee σ,f t) ad φ,f,gt) is zero. Propositio 4. shows that, uder our set of Assumptios, the Fourier estimator ca be redefied usig delta sequeces with a improvemet i terms of the variace. The cross-term typically adds oise ad computatioal burde. Kaatai [3] made a similar remark i the case of the itegrated volatility estimatio. Note that, i geeral, the cross-terms might be beeficial to the reductio of the

13 3.5 Geerated variace Fourier prime = sqrt)) Fourier prime = /) Delta sequece Fejer) Spot Volatility Time Figure 4.: Fourier estimates of the variace of a sigle simulated path the geerated variace is the thick solid lie) with = 5 ad N = 8, i the case = / thi solid lie) ad = dashed lie). It is clear that a lower leads to a higher variace. We also report the estimator.5) with the Fejér delta sequece: it is almost idetical to the Fourier estimator with = / but computatioally faster. mea square error i the presece of market microstructure oise, see Macio ad Safelici [37, 38], ad Bardorff-Nielse et al []. The above fidigs are clearly illustrated i Figure 4. where it is apparet as the trajectory estimated with the Fourier method without cross products ca either have a larger oise dashed lie), or perfectly overlap circles) to the delta sequece estimator i the case i which we choose =. Further simulatio evidece suggests that, i the uequally spaced case, the optimal choice of is H t). 5 Empirical applicatio I this fial Sectio, we apply the proposed estimators 3.4) ad 3.7) to a set of market data cosistig of high-frequecy trasactios of the S&P 5 stock idex futures. We restrict our attetio to year 999 ad to cotracts closer to maturity. Trasactios are recorded over 5 tradig days betwee 8.3 a.m. to 3.5 p.m. ad iterpolated to a 5 secods grid. Every day, we the have a total of 4,86 price returs. For both estimators we use the Epaechikov kerel with h = 5 miutes. To calculate the low frequecy estimator σ, t) o the right-had side of 3.4), we apply a subsamplig techique similar to that described i Zhag et al [5] with =, which correspods to oe-miute returs. I order to avoid the effect of jump dyamics i the observed data, we first remove from the sample all the days characterized by sigificat price chages usig the procedure described 3

14 x 4 x 8 Microstructure.6.5 Spot Volatility Time Figure 5.: Itraday spot volatility for the S&P5 stock idex futures over oe year of data calculated usig the two scale estimator 3.4). Days with relevat jump activity are previously removed from the sample. The iset shows the average estimate of the microstructure oise variace V ǫ. below. Figure 5. plots the estimated itraday spot volatility averaged across days ad calculated i daily time uits. The well kow U shape is clearly detected, as it was already observed i previous studies, see, for istace, Aderse ad Bollerslev [3]. The estimate of the microstructure variace V ε is also provided. We ow tur to the jump-robust estimator, ad we use 5-miute returs for computatio of spot volatility estimators, to softe the impact of microstructure oise. To show that our threshold estimator /σ,f t) is robust to price jumps, we compare it with the origial spot volatility estimator.5) usig a data-set created by removig all days with relevat jump activity. The resultig itraday volatility curves the should be almost idetical. To idetify the jumps, we employ the C Tz statistics i Corsi et al [4], After settig the daily sigificace level of a jump to 99%, a total of 8 days are detected ad the excluded from the sample. The top pael i Figure 5. shows that the volatility curves obtaied with the two aforemetioed estimators match almost everywhere, meaig that that /σ,f t) is ot affected by large price movemets ad is able to provide robust estimates of the itraday volatility dyamics. We the apply the same estimatio procedure to a sample made of the 8 days iitially removed; the result i plotted i Figure 5., bottom pael. As expected, ow the two curves behave quite differetly, especially aroud the market opeig time. 4

15 x 4 volatility estimator threshold volatility estimator.6 Spot Volatility Time 6 x 4 5 volatility estimator threshold volatility estimator 4 Spot Volatility Time Figure 5.: Itraday spot volatility for the S&P5 stock idex futures averaged over oe year of data calculated usig the origial volatility estimator.5) ad the threshold estimator 3.7) respectively. Top pael: origial data-set without relevat jump activity. Bottom pael: sample made of 8 days characterized by large price movemets. The sigificace level of jump detectio is set to 99%. The volatility is measured i daily uits. 6 Coclusios We elarged the class of spot volatility estimators usig localizig sequeces of fuctios which coverge to a Dirac delta. Uder mild hypotheses o the data geeratig process, we provide a asymptotic theory for the estimators withi the class ad we propose suitable modificatios to assess the effect of microstructure oise or price discotiuities. As a special case, we related the Fourier estimator with the delta sequece obtaied with the Fejér sequece, showig that the latter is more efficiet i 5

16 the case i which the price follows a Browia semimartigale ad there is either leverage effect or microstructure oise. We fially applied the resultig estimators to a data-set of high-frequecy stock idex futures ad successfully recovered the traditioal U-shaped itraday volatility patter. The paper leaves ope the possibility of further developmets. For example, we would like to study the joit cotributio of microstructure oise ad jumps, possibly usig the techiques i Jacod et al [7] ad Podolskij ad Vetter [43, 44]. Also, the challegig problem of the optimal choice of the delta sequece is to be addressed. Fially, the asymptotic distributio of the spot volatility obtaied with the Fourier estimator is ukow. We leave all these iterestig issues for future research. 6

17 A Proofs I what follows, we will use...)dx to deote a itegral over R. C or K idicate a costat which does ot deped o i, or o the sequece F = {f, IN}, but ca deped o t ad the localizig sequece τ m, ad which keeps the same ame eve whe chagig from lie to lie or from oe side to aother of the iequality. Without loss of geerality, we assume σ. Recall that for ay Lebesgue-itegrable fuctio a, for all l we have, by Jese iequality, i a s ds i a s l ds ) l A.) We remark that assumig µ,σ,h càdlàg etails that they are locally bouded. By a localizatio procedure similar to that i [9] sectio 5.4, p.549), we ca assume without loss of geerality that they are bouded as ω,t) vary withi Ω [,T]). I order to prove Theorem. we use Lemma A. below several times with A beig equal to σ k for some powers k {,,,3,4}. Note that, by property.) ad the boudedess of σ, we have, for k, ad, similarly, we also have, for k, E u s [ σ u σ s k ] = E u s [ σ u σ s σ u σ s k ] C u s Γ. A.) E u s [ σ u σ s k ] C u s Γ. A.3) For k = we istead have, by Jese iequality, E u s [ σ u σ s ] C u s Γ/ ad E u s [ σ u σ s ] C u s Γ/. Propositio.. Assume first that < t < T. Usig the boudedess of σ ad the property.), we ca write: T ) T R σ ) t) = f s t)σsds f s t)ds+o) σ ) t = T ad, usig.) ad property.), E[ T f s t) σ s σ ) t) ds+op ), f s t) ] σs σ ) ) T t ds f s t)e [ σ s σ ) ] t ds = C f s t)e [ σ s σ ) ] t ds s t <ε +C f s t)e [ σ s σ ) ] t ds s t ε Cε Γ/ +C f x)dx. x ε If istead t = T, we repeat the same reasoig above usig: ψ = s T< f s T)ds+o), ad we proceed i a similar way if t =. This proves.6). To prove.7) for f, it is thus eough to prove 7

18 that g x) = f x) c f f ) satisfies Eqs..), which is straightforward from property.), ad.), which is obtaied, usig sup x f x) Cf ) as: g x)dx = f x) f x) x ε c f x ε f ) dx C f x)dx. x ε To prove.8), use the boudedess of σ ad.) ad write: T ft t) 4 f) σsdt C T ft t) f )dt Cf ) f ) Lemma A.. i) For a sequece of processes A ) bouded by the same costat K, if f are Lipschitz fuctios ad max i i = O ) the T f s t)a ) s ds f t) A ) s ds = O a.s. L ). A.4) As a corollary, uder.6) ad if L we have f t) i, as. ii) Cosider a bouded càdlàg process A. If either f or Γ/ T f s t) ds ad.9) holds, if both f ad g = f/c f f )) satisfy.6), uder max i i = O ),.4) ad.) we have f t)a ) p i H A) t A.5) ad f ) ft i t)a ) i p c fh A) t. A.6) iii) Uder.) ad the boudedess of σ, for ay p =,,,3 there exists α > such that for all i =,..., ad for a suitable costat C p, E i σ ti W i p ) 4 p σs σ ti dws have α >, C p iσ p α i I {ti B εt)} +I { B εt)} ). iv) Uder the same assumptios as for ii) above, for ay bouded càdlàg process M ad α > we f t) f ) M ti i α i I {ti B εt)} +I { B εt)} ) p v) Uder.) ad the boudedess of σ, for ay p =, we have for all i =,..., ad for some E i σ ti W i p ) p σs σ ti dws C p i σ p t i α i I + {ti B εt)} +I {ti B εt)} ). vi) Uder.) ad the boudedess of σ, for p [,8], [ ) p ] ti E i σs σt i ds K p i Γ/ i I +I {ti B εt)} {t ). i B εt)} 8

19 Proof. i) Notig that max i i = O ) implies i i K, we have K i T f s t)a s ds f t) A s ds f s t) f t) A s ds L s ds KL i KL. As for the corollary, by.6) with σ we have T f s t)ds, as. Thus it is sufficiet to show that f t) i T f s t)ds, which is guarateed by A.4) with A. ii) It is eough to show Eq. A.5), sice the assumptios o f imply that also g = f /c f f )) is Lipschitz with Lipschitz costat G KL ad thus g is also satisfyig Γ G ad.9). By applyig.6) we obtai H A) t = T f s t)h A) s ds+o p ). We ow show that this last term has the same asymptotic behavior as f t)a ti i /. I fact, T f s t)h A) s ds i i i f t)a ti i/ = [f s t) f t)]h A) s ds+ f t)[h A) s A ti i / ]ds. Give the assumed boudedess of H A, the first term i the rhs of the display above is a.s. bouded by KL i i KL. As for the secod sum above, we ca write it as i i f t)h sa s A ti )ds+ f t)h s i / )A ti ds. Usig A.4) ad.4), the last term has the same limit as i which is bouded i absolute value by K i f s t)h s i / )A ti ds, T f t) i K f s t) ds, A.7) which i tur coverges to zero by coditio f ) ad the Lipschitz property of f. Now we deal with the term i A.7) by splittig it ito the sum over the idexes i s.t. B ε t) ad the sum of the other terms. Sice the càdlàg process A has at most coutably may jumps withi [,T] ad each jump time has Lebesgue measure, we have {i: B εt)} f t)h sa s A ti )ds = A.8) 9

20 {i: B εt)} f t)h sa s A ti )ds. UsigtheA.4)adtheboudedessofH adaweobtaithatthelastsumhasthesameasymptotic behavior as Bε ct)ψ) s ds, where ψ ) s. = f s t)h s A s j ) A tj I ]tj,t j]s). Note that, for Lebesgue-almost all s, ψ ) s, because ay fixed s [,T] belogs to oly oe iterval, say ]t j,t j] ad t j is always o the left had side of s, so that A t j A s, as, ad thus j A t j I ]tj,t j]s) A s. Moreover.9) ad the boudedess of A ad H imply that ψ ) s) KM ε which belogs to L B c εt)). By the domiated covergece theorem we coclude that T ψ) s ds ad A.8) is asymptotically egligible. Fially we show that {i: B εt)} f t)h sa s A ti )ds P. I fact usig.) its L Ω) orm is domiated by [ t i ] E E i f t) H s A s A ti ds K {i: B εt)} {i: B εt)} f t) ds Γ/ i K Γ/ T f s t) ds. iii) Usig Hölder iequality, the cosidered coditioal expectatio is domiated by [ ] ti E / i [ σ W i p ]E / i σ s σ ti dw s 8 p usig the the Burkolder-Davis-Gudy iequality this is less tha [ ] ti ) 4 p C p σ p p i Ei σ s σ ti ) ds. A.9) Sice p 3 the by A.) with l = 4 p, i σ s σ ti ) ds i we obtai that [ ] [ ti ) 4 p E i σ s σ ti ) ds E i σ s σ ti ) 4 p) ds 3 p i 3 p i E i [ σ s σ ti 8 p ]ds. ) /4 p), σ s σ ti ) 8 p ds Usig ow assumptio.) whe B ε t) ad the boudedess of σ otherwise, the last term is domiated by ) 3 p i I {ti B εt)} K Γ/ i +I {ti B εt)} i ]

21 sice Γ i Γ/ i ) ad combiig this with A.9) the thesis follows. iv) The cosidered sum is domiated by K α i: B εt) f t) f ) im ti +K i: B εt) f t) f ) im ti. By A.6) with A t = M t I Bεt) t) we obtai that the first sum teds to zero i probability. The same result holds for the secod sum, with A t = M t I B c ε t), as At) = At ) =. v) This proceeds exactly as i iii) by substitutig 4 p each time it appears with p. vi) Usig A.) with l = p we obtai E i [ σ s σ ds ) p ] E i [ p i σ s σ p ds Sice σs σt i = σ s σ ti σ s +σ ti K σ s σ ti, the last term above goes as follows [ ] ti ) K p i E i σ s σ ti p ds I +I {ti Bεt)} {ti Bεt)} K p i Γ/ i I {ti B εt)} +I { B εt)} ). ]. Theorem.. It is ot restrictive to set µ t =. We start by provig the stated covergece i law. Usig Eq..6) ad the A.4) ad that µ, we have = = = = [ ] ˆσ,Ft) σ ) t f ) [ ] f t) Xi σ ) t f ) [ ] T f t) Xi f s t)σ s)ds+r σ ) t) f ) ) f ) [ f t) U i +O a.s. L f ) where for i =.. Sice we assumed L U i := f t) f ) f ad ) ) Rσ X i ) + Rσ σ s)ds +O a.s. L )+R σ ) t) t), A.) f ) σ s dw s ) σsds. A.) t) = o p f ) ), the last two terms above ted to zero i probability, ad thus it is sufficiet we derive a cetral limit theorem stable i law for ]

22 U i For that we refer to Theorem IX.7.8 i Jacod ad Shiryaev [9] esurig that the followig are sufficiet coditios i) E i [U i ] p ii) E i [Ui ] p V t iii) iv) E i [Ui 4 ] p E i [U i H i ] p, where E i [ ] abbreviates E[ F ti ] ad iv) has to hold i both the cases where H = W or H = B, with B ay bouded martigale orthogoali the martigale sese) to W. Coditio i) is immediately proved usig the Itô isometry E i [U i ] = ) f t) ti E i σ s dw s σsds f ) t i =. As for coditio ii), cosider [ ] E i U i = f t) f ) E i ) ti ti σ s dw s σ sds = { [ ) ] [ f ti t) ti 4 ) ] ti E f i ) σ s dw s +E i σsds A.) E i [ ti σ s dw s ) σ sds) ]}. All the three coditioal expectatios cotai some leadig terms, which we eed to compute exactly. Basically, for s ],t i ] we write σ s = σ ti +σ s σ ti ), we fid exact equalities for the expressios cotaiig σ ti ad by usig assumptio.) we show that the other terms are asymptotically egligible. Now write ) 4 ti ) 4 ti [ ] E i σ s dw s = E i σti +σ s σ ti dws = 3σt 4 i i + ) 4 p ti c p E i σ ti W i ) p [ ] σs σ ti dws, p=,,,3 with suitable costats c p. Usig Lemma A. iii) ad iv), the first term withi brackets i A.) cotributes by ad thaks to A.6) this i tur has Plim equal to f t) f ) 3σ 4 i +o p ), 3c f H σ 4 ) t. As for the secod term withi brackets i A.) we similarly decompose it as ) E i σ i + σ s σ ds

23 = σt 4 i i + ) q ti c p E i σt i i ) q σs σt i ds. q=, Usig Lemma A. vi) with p = q, both terms with q =, are bouded by σ q q i q i α i I {ti B εt)} +I { B εt)} ), for some α >, which by Lemma A. iv) give asymptotically egligible cotributio to A.), so that the secod term withi brackets i A.) cotributes by by A.6). f t) f ) σ 4 i +o p ) c f H σ 4 ) t As for the third term withi brackets i A.) we still decompose it as ) ti ) ti ) E i σ ti W i + σs σ ti dws σt i i + σ ds) s σ ti = σ 4 i + +σ E i [ + q=, p=, c p σ +p i E i [ W i ) p t i W i ) ] ) σs σ ti ds ] ) ) p σs σ ti dws [ c q σ q E i W i ) q t i ) ) q σs σ ti dws ] ) σs σ ti ds. By Lemma A. v) ad iv) the terms with p =, give asymptotically egligible cotributio to A.). Notig that t i σ s σ ds K i, the terms with q =, are reduced to terms of exactly the same type as the oes with p =, above ad thus are asymptotically egligible. Now we deal with the term σ E i [ W i ) which, by the Hölder iequality is domiated by [ σt i Ei [ W i ) 4 ] Ei ] ) σs σ ti ds ) σs σ ti ds ]. Usig Lemma A. vi) with p =, we obtai that i tur this is less tha K i α i I {ti B εt)} +I { B εt)} ), with a suitable α > ad by A. iv) also the cotributio of this term is asymptotically egligible. Therefore the third term withi brackets i A.) has the same limit i probability as f t) f ) σ 4 i P c f H σ 4 ) t. A.3) 3

24 by lemma A. ii). Summig up, the probability limit i coditio ii) is V t = c f H σ 4 ) t. We ow deal with the above coditio iii), where we oly have to check the egligibility of the fourth coditioal momets, ad eve some rough estimates are sufficiet. [ ] E i U 4 f 4 i = t) ) ti 4 ti f) E i σ s dw s σ sds ft 4 i t) f ) E i ) 8 ti ) 4 ti σ s dw s + σsds. As σ s is assumed wlog to be bouded, by the BDG iequality we have that E i [ ti σ s dw s ) 8 ] K 4 i ad the last sum above is domiated by K f 4 t) i f ) 3 i K havig used that, by Assumptio, max i i K. However f 4 T t) i f) f4 s t)ds P f), ft 4 i t) i f), A.4) as f 4 = gf ), with gx) = x 4, is also Lipschitz o D with Lipschitz costat KK 3 L ad thus the sum withi the last display is bouded by f ) ft 4 i t) fs t) ds 4 K f) K3 L KL f ) L 3 f ). Cosequetly, usig Eq..8) with σ, the probability limit of A.4) is the same as by assumptio. plimk T f4 s t)ds f ) plimk f ) =, We fially cosider coditio iv), startig from the case H = B. Deote R t = t σ sdw s ad M t = R t t σ sds. Sice B is orthogoal to W we have that d[m,b]. I fact dr ) t = R t dr t + d[r,r] t = R t dr t + σ tdt, thus d[m,b] t = d[r,b] t = R t d[r,b] t = R t σ t d[w,b] t. Therefore also M ad W are orthogoal, meaig that E ti [ M i B i ] = for all i =.., ad coditio iv) is verified. Whe istead H = W, E i [U i W i ] 4

25 f t) f ) E i σ s dw s ) Usig the computatios doe i order to evaluate E i [Ui ] above, we have ) ti ti E i σ s dw s σ sds σ 4 i +K i σsds E i [ Wi )]. ) α i I +I {ti B εt)} { B εt)}, for a suitable α >, therefore the last term withi the above display is domiated by f t) σ f 4 ti +K αi I {ti Bεt)} +I {ti Bεt)} ) K ) 3 i f t) i f ) By A.5) i) ad the assumptios L ad f ), this last has the same probability limit as T fs t)ds f ), which is zero, as T f s t)ds by.6), ad coditio iv) is verified also whe H = W ad this completes the proof of the stable covergece of σ,f t). ) ForthecovergeceiprobabilitythecoditioR σ ) t) = o p f ) is ot required. Ideed by multiplyig both sides of Eq. A.) by f ) we fid ˆσ,Ft) σ ) t = ) ti f t) σ s dw s σsds +O a.s. f ) )+R σ ) t). Last two terms are o p ) by the assumptio f ) ad.6), while we check the egligibility of the first term by usig the law of large umbers for the sum of martigale differeces see e.g. Lemma 4. i Jacod [6]). It is sufficiet to show that ) ] E i [ f ) U i = f ) E i [Ui ], i which is esured by i E i [U i ] c fh σ 4 ) t f ). i obtaied with the computatios for ii) above ad by We ow check the egligibility of the drift i order to reach the cosistecy ad the CLT for ˆσ,F t). If µν, by followig A.) ad substitutig X i we have ˆσ,F t) σ ) t = f ) U i + i + i f t) f ) σ s dw s f t) ti µ s ds) f +O a.s. L ) t i µ s ds f ) ) Rσ )+ t), f ) 5

26 ) ad we see firstly that agai the assumptio Rσ t) p is oly eeded for the CLT ad secodly f ) that showig the egligibility of the secod ad third terms above is sufficiet also to state the cosistecyof ˆσ,F t).todealwithsuchterms, whicharebothofthetype i ξ i,weapplylemma4.i[6], ad i both cases we check coditio 4.4). Sice i E i [ ξ i ] i Ei [ ξ i ] K E i [ ξ i ], it is sufficiet to check that i E i [ξi ]. By the boudedess of µ ad BDG iequality ad the A.6) we have ) ft i t) ti ) ti E i σ s dw s µ s ds K f i ) i K f t) f ) 3 i i f t) f ) i p ad as desired. i ft i t) [ ] E i µ s ds) 4 K ft i t) i p f ) f i ) Remark. Assume t ],T[. For the Gaussia kerel, with ε as i.) we have t) = A + f ) R σ ) f ) [ h s [,T]: s t ε ) ] s t K σsds σ ) t, A.5) h with By chagig variable via x = s t)/h, the secod term i A.5) is writte as B +C +D +E B = ε/h Kx)σ t h σ x t f )dx ε/h Kx)σ t+h, C = σ x t )dx, ) f ) D = ε/h Kx)σ t dx, E = f ) + ε/h Kx)σ tdx f ). Usig Eq..), we have: E[ B ] f ) A similar result holds for C. Moreover, ε/h Kx)xh ) Γ/ dx C h +Γ. ) s t A = K σ h f ) s [,T]: s t >ε h sds = h ε/h T t)/h Kx)σ K) h dx+ x+t Kx)σ h dx x+t t/h ε/h ε/h h Kx)dx+ h t/h T t)/h ε/h Kx)dx. 6

27 For the first term, o [ t/h, ε/h ], for small h we have Kx) K ε/h ) = e ε h ) = h 4, so that h ε/h t/h Kx)dx C h 7. The secod term ca be dealt with i a similar way. o ε h ) 4 ) Fially for D, the well kow iequality ad the boudedess of σ imply D = f ) E is dealt with similarly. ε/h for y > : + y Kx)dx C y Ky) ε/h Kx)σ t dx C h Kx)dx h h C h 4. ε e ε h For the Epaechikov kerel, A = for large eough, ad the rest is similar. For the idicator kerel, for large eough, [ ] ) σ E Rσ t) x /f ) f )E t+x/f σ ) ) t dx = C f) Γ/, f ) f ) f ) providig the same result. Lemma 3.. The proof is based o that of Theorem.. I what follows, we compute coditioal expectatios with respect to a ew augmeted filtratio F ε t obtaied by icludig the observed oise ε i ) ti t for each t [,T]. As show i the proof of Theorem., it is harmless to set µ t = so we do it i what follows. Write, where σ,ft) V ε = A +B +C ) V ε, A = B = C = f t) Y i ) f t) Y i ε i ε i ) f t)ε i ε i ). For the first term A, we ca simply apply Theorem. ad obtai that A = O p ). Next, write B = f ) U B,i where U B,i := f ) f t) Y i ε i ε i ). 7

28 We have E i [U B,i ] = ad, usig the idepedece of the oise,: [ ] E i U B,i = 4 = 4 ft i t) [ E i Yi ) ] E i [ε i ε i ) ] f ) ft i t) [ E i Yi ) ] V ǫ +ǫ f ) i ) Now otice that, usig the boudedess o σ s ad Eq. A.), E i [ Y i ) ] = E i [σs]ds = σt i + ) = σt i +O p +Γ/ E i [σ s σ ]ds A.6) so that we ca write: [ ] E i U B,i = 4 = 8 f t) f ) σt i +O p +Γ/ ))V ε +ε i ) ft i t) σt f ) i V ε +o p )+R B, where the o p ) terms is the term multiplyig O p R B, = 4 +Γ/ ), while ft i t) σt f ) i ε p i V ε ) by a geeralized versio of the law of large umbers, sice E[R B, ] = ad 6 ft 4 i t) f) E[σt 4 i ] E[ε i V ε )] = O p f ) ) by the boudedess of σ s, the fiiteess of the momets of the ǫ ad by Eq..8). This proves that E [ ] i U p B,i 8Vε c f σ ) t). Similarly, usig also the BDG iequality, [ ] E i U 4 B,i = 6 6K = 6K ft 4 i t) [ f) E i Yi ) 4] E i [ε i ε i ) 4 ] = O p f ) ) ft 4 i t) [ f) E i Yi ) ] Ei [ε i ε i ) 4 ] ft 4 i t) f) σ 4 ) ie i [ε i ε i ) 4 ]+o p ) The B f ) MN,8V ε c f σ ) t) ), that is, B = O p f )). Now cosider the term D := C V ε. Write: U C,i = f [ t) εi ε i ) ] V ε f ) 8

29 ad otice that, sice f t ) = OL ) + o f ) ), we have D = f ) ) U C,i +o p f ). We immediately have E[U C,i ] =, ad: while: E[UC,i] = 4 E[UC,i] 4 = 6 p c f = O p f ) ) f ) f t) κε +Vε ) κε +Vε ), [ f ) f4 t) εi E ε i ) ) 4 ] V ε which proves that D = O p f ) ), which is the leadig order, ad also proves the statemet i Eq. 3.). Theorem 3.. Write Y i := Y ti ad X i := X ti. We have: σ,,ts t) = + f t)y i+ Y i ) f t)ǫ i+ ǫ i ) f t)ǫ i+ ǫ i )Y i+ Y i ) f t)y i Y i ) f t)ǫ i ǫ i ) f t)ǫ i ǫ i )Y i Y i ) := A +A +A 3 +B +B +B 3 ad defie C = A +B ad C 3 = A 3 +B 3. Start with A ad write: A = = + f t) Y i +j Y i +j ) j= i Y i Y i ) f t i j t) := a+b + j= f t) j>k Y i +j Y i +j )Y i +k Y i +k ) 9

30 Usig the Lipschitz property of f : f t i j t) = f t)+ol j ) ) A.7) so that, usig Theorem.,: a = f t)y i Y i ) i + i OL j ) ) = = j= f t)y i Y i ) + i ) +OL ) )) σ ) t)+o p f ) +ed effect)+o p L ) The explicit evaluatio of the ed effect, usig the properties of f i Assumptios 3 ad Eq. A.), gives: f t)y i Y i ) i = f t)y i Y i ) i = O p ) A.8) Thus we have proved that a σ ) t) = o p f ) ). For the term b, write, usig agai the Lipschitz propertya.7), rearragig the terms i the sum ad evaluatig ed effects as i Eq. A.8), b = = = = := Y i Y i ) f t i j t)y i Y i j )+ed effects) i= i= j= Y i Y i ) f t)+ol ) ) ) Y i Y i j )+o p f ) j= j ) f t)y i Y i ) Y i j+k Y i j+k )+o p f ) i= f t)y i Y i ) i= i= b,i +o p f ) ) j= k= j= j )Y i j Y i j )+o p f ) ) Now, E i [b,i ] = ad, usig Eq. A.6): E i [b,i] = 4 ft i t)e i [Y i Y i ) ] j ) Y i j Y i j ) i= = 4 i= i= ft i t) σt i +O p +Γ/ ) i= ) j= j= j ) Y i j Y i j ) ) = +O p Γ/ ) 4 ft i t)σt i j ) Y i j Y i j ) j= 3

31 +8 ft i t)σt i i= ) := +O p Γ/ ) c+d) j>k j ) k ) Y i j Y i j )Y i k Y i k ) Now write Y i j Y i j ) = = = j t i j σ s dw s j ) = t i j σs σ ti j ) dws +σ ti j i j W s j t i j σs σ ti j ) dws +σ ti j i j W s j ) ) t i j σs σ ti j ) dws +σ t i j i j W s ), ad, accordigly, c = c +c +c 3. Usig the boudedess of σ s, BDG iequality ad property.) it is straightforward to prove that c is o p f )), ad further usig the Holder iequality also c is proved to be o p f )). We the deal with the term c 3. Start by oticig that we ca rearrage the sums ad eglect ed-effects which are give by O ) terms of the same kid of the O p ) terms that we are goig to retai, which are the followig: c 3 =4 ft i t)σt i j ) σt i j i j W s ) i= + =4 i= j= σt i i+ W s ) j= ) j σt i+ j ft i+ j t) Now we use the properties of f, the boudedess of σ ad Lemma A. to show that: + c 3 =4 i= + =4 i= σt i i+ W s ) j= σt i i+ W s ) +o p f )) :=c 4 +c 5 +o p f )) j= ) j σ t i+ j f t)+o p L ) ) ) [ j ] σt i +σt i+ j σt i ) ft i t) The term c 4 = 4 + i= σ t i i+ W s ) j σ j= ) ti ft i t) = i= c 4,,i +ed effects is the leadig term sice: Ei [c 4,,i ] = 4 + f ) i= ft i t) σ 4 t f ) i j= ) j p 4 3 c fσ 4 t ) 3

32 ad, usig property.9 ad the Lipschitz property of f 4 : which shows that c 4 / f )) + f ) E[ c 5 ] K Ei [c p 4,,i] f )) p 4 3 c fσ 4 t ). The term c 5 is istead egligible, sice: i= K ) Γ j= ) j E [ ] σ f σ ti+ j ti t) f ) Now cosider the d term. Write d = i= δ,i ad, usig Cauchy-Schwartz, BDG ad the boudedess of σ t, E i [δ,i] =64 ft 4 i t)σt 4 i i= i= E i i= j>k j ) k ) Y i j Y i j )Y i k Y i k ) K ft 4 i t) E i j ) Y i j Y i j ) E i 64K k j k ) Y i k Y i k ) ft 4 i t) j ) i= k ) 64K k j ft 4 i t) 4 = of) 3 ) i= sice j/). This proves that d is egligible with respect to c. Fially, usig the same techique: E i [c 4,i] = 6 ft 4 i t)e i [Y i Y i ) 4 ] j ) Y i j Y i j ) i= i= i= j= j= 6K ft 4 i t) j ) Y i j Y i j ) ad the expected value of the last term is limited, usig agai BDG, by 6K ft 4 i t) 4 = of ) 3 ), i= 4 4 3

33 implyig that also the limit i probability is. Fially cosider f ) i= E i [b,i H i ] where H = W or H = B as i the proof of Theorem.. Whe H = B, : E i [b,i B i ] = f t)e i [Y i Y i ) B i ] j ) Y i j Y i j ) i= i= ad this coverges to zero sice, usig the same otatios as i the proof of Theorem. Y i Y i ) = M i ad E i [ M i B i ] =. Whe H = W, usig the same techique as above,: E i [b,i W i ] E i [b,i f ) f i= ) ]E i [ W i ) ] i= = ft i t)e i [Y i Y i ) ] j ) Y i j Y i j ) f ) i= j= = f f ) t i t)o p 4 ) i= Altogether, this proves that: A σ ) t) f ) MN j=, 4 ) 3 c fσ 4 ) t stably i law. Next, cosider B. By Theorem., B = O p f ) /) ad ca be eglected with respect to A. Cosider ow C ad C 3. Start with C := + α,i where α,i = f t ) ǫ i+ ǫ i ) ǫ i ǫ i ) ) Usig the fact the the ǫs are iid, we have E[α,i ] = ad: + E[α,i] = f + ) f t ) f ) 6V ǫ +κ ǫ ) which implies, provided + E[α,i 4 ] p which is readily proved, that f ) C N )),c f 6V ǫ +κ ǫ. We ext have C 3 := + β,i where β,i = f t )ǫ i+ ǫ i )Y i+ Y i ) ǫ i ǫ i )Y i Y i )) Agai, E i [β,i ] = ad, usig the same techiques above ad the law of large umbers i the last step: + E i [β,i] = 4 + f t ) [ V ǫ +ǫ i )E i [Y i+ Y i ) ] 33

34 + V ǫ +ǫ i )E i [Y i Y i ) ]+ǫ i E i [Y i+ Y i )Y i Y i )] ] = 4f ) + f t ) f ) [ V ǫ +ǫ i )σ +V ǫ +ǫ i )σ +ǫ i σ ]+o p ) =O p f )/) which implies that C is the leadig term. Thus, the domiatig terms are A σ ) t ) = O pf ) ) / ) ad C = O p f ) / ) / ) ). The two asymptotic rates coicide whe /3 the desired result. as stated, leadig to Theorem 3.3. Deote by X = Y +J where Y is a cotiuous semimartigale. By virtue of Theorem i Macii [35], for large eough, we ca write, almost surely, /σ,f t) = f t) Y i ) f t) Y i ) I { Ni }. Theorem. ca be applied to the first term, while the secod term is O p N T f )), or equivaletly, ) o p f ), where N T is the Poisso coutig process ad is vaishig i the limit. Lemma A.. Result A.6) cotiues to hold also whe f x) is give by the Fejér sequece. Proof. Firstly, remark that for ay bouded process A we have C j= tj t j C k N j= j= tj t j tj t j f s t) f t j t) As ds k N k ) e iks t) ik j N k N ) e iks t) e iktj t)) ds ds = C j= tj t j f s t)nds CN, where deotes asymptotic equivalece. It follows from sup x f x) f ) that tj T f ) ft j t) A s ds fs t)a s ds t j = f ) j= j= tj f ) ft j t) fs t))a s ds j= t j = tj f t j t) f s t))f t j t)+f s t))a s ds t j C j= tj t j f t j t) f s t) A s ds CN. A.9) A.) 34