ESTIMATING THE VOLATILITY OCCUPATION TIME VIA REGULARIZED LAPLACE INVERSION

Ecoometric Theory, 32, 216, 1253 1288. doi:1.117/s266466615171 ESTIMATING THE VOLATILITY OCCUPATION TIME VIA REGULARIZED LAPLACE INVERSION JIA LI Duke Uiversity VIKTOR TODOROV Northwester Uiversity GEORGE TAUCHEN Duke Uiversity We propose a cosistet fuctioal estimator for the occupatio time of the spot variace of a asset price observed at discrete times o a fiite iterval with the mesh of the observatio grid shrikig to zero. The asset price is modeled oparametrically as a cotiuous-time Itô semimartigale with ovaishig diffusio coefficiet. The estimatio procedure cotais two steps. I the first step we estimate the Laplace trasform of the volatility occupatio time ad, i the secod step, we coduct a regularized Laplace iversio. Mote Carlo evidece suggests that the proposed estimator has good small-sample performace ad i particular it is far better at estimatig lower volatility quatiles ad the volatility media tha a direct estimator formed from the empirical cumulative distributio fuctio of local spot volatility estimates. A empirical applicatio shows the use of the developed techiques for oparametric aalysis of variatio of volatility. 1. INTRODUCTION Cotiuous-time Itô semimartigales are widely used to model fiacial prices. I its geeral form, a Itô semimartigale ca be represeted as X t = X + t b s ds + t Vs dw s + J t, (1) where b t is the drift, V t is the spot variace, W t is a Browia motio, ad J t is a pure-jump process. Both the cotiuous ad the jump compoets are kow to be preset i fiacial time series. From a ecoomic poit of view, volatility ad We wish to thak a co-editor ad three aoymous referees for their detailed ad thoughtful commets, which helped greatly improve the paper. We would also like to thak Tim Bollerslev, Marie Carrasco, Peter Carr, Nathalie Eisebaum, Jea Jacod, Adrew Patto, Peter Phillips, Philip Protter, Markus Reiß, as well as semiar participats at various cofereces for may useful suggestios. Li s ad Todorov s work were partially supported by NSF grats SES-1326819 ad SES-95733 respectively. Address correspodece to Jia Li, Departmet of Ecoomics, Duke Uiversity, Durham, NC 2778; e-mail: jl41@duke.edu. c Cambridge Uiversity Press 215 1253 Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of

1254 JIA LI ET AL. jump risks are very differet ad this has spurred the recet iterest i separately idetifyig these risks from high-frequecy data o X; see, for example, Bardorff-Nielse ad Shephard (26) ad Macii (29). I this paper we focus attetio o the diffusive volatility part of X while recogizig the presece of jumps i X. Most of the existig literature has cocetrated o estimatig oparametrically volatility fuctioals of the form T g(v s)ds for some smooth fuctio g, typically three times cotiuously differetiable (see, e.g., Aderse et al. (213), Reault et al. (214), Jacod ad Protter (212), Jacod ad Rosebaum (213) ad may refereces therei). The most importat example is the itegrated variace T V sds, which is widely used i empirical work. These temporally itegrated volatility fuctioals ca be alteratively thought of as spatially itegrated momets with respect to the occupatio measure iduced by the volatility process (Gema ad Horowitz (198)). Motivated by this simple observatio, Li et al. (213) cosider the estimatio of the volatility occupatio time (VOT), defied by F T (x) = T 1 {Vs x}ds, x >, (2) which is the pathwise aalogue of the cumulative distributio fuctio (CDF). 1 Evidetly, the VOT also takes the form T g(v s)ds but with g discotiuous. The latecy of V t ad the osmoothess of g( ) tur out to cause substative complicatios i the estimatio of the VOT. To see the empirical relevace of the VOT, we ote that the widely used itegrated variace T V sds is othig but the mea of the occupatio measure xf T (dx), where the equivalece is by the occupatio formula 2. Therefore, the relatio betwee the VOT, the VOT quatiles, ad the itegrated variace is exactly aalogous to the relatio betwee the CDF, its quatiles ad the mea of a radom variable. Needless to say, i classical ecoometrics ad statistics, much ca be leared from the CDF ad quatiles beyod the mea. By the same logic, i the study of volatility risk, the VOT ad its quatiles provide additioal useful iformatio (such as dispersio) of the volatility risk which has bee well recogized as a importat risk factor i moder fiace. Li et al. (213) provide a two-step estimatio method for estimatig the VOT from a high-frequecy record of X by first oparametrically estimatig the spot variace process over [, T ] ad the costructig a direct plug-i estimator correspodig to (2). Their estimatio method is based o a thresholdig techique (Macii (21)) to separate volatility from jumps ad formig blocks of asymptotically decreasig legth to accout for the time variatio of volatility (Foster ad Nelso (1996), Comte ad Reault (1998)). I this paper we develop a alterative estimator for the VOT from a ew perspective. The idea is to recogize that the iformatioal cotet of the occupatio time is the same as its pathwise Laplace trasform, ad the latter Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of

ESTIMATING THE VOLATILITY OCCUPATION TIME 1255 ca be coveietly estimated as a sum of cosie-trasformed logarithmic returs (Todorov ad Tauche (212b)). Followig this idea, our proposal is to first estimate the Laplace trasform of the VOT ad the coduct the Laplace iversio. The iversio is otrivial because it is a ill-posed problem (Tikhoov ad Arsei (1977)). Ideed, Laplace iversio amouts to solvig a Fredholm itegral equatio of the first kid, ad the solutio is ot cotiuous i the Laplace trasform. I order to obtai stable solutios, we regularize the iversio step by usig the direct regularizatio method of Kryzhiy (23a,b). The fial estimator is kow i closed form, up to a oe-dimesioal umerical itegratio, ad ca be easily computed usig stadard software. The proposed iversio method ad the plug-i method of Li et al. (213) both ivolve some tuig parameters, but they play very differet roles ad reflect the differet tradeoffs uderlyig these two methods. For the iversio method, the first-step estimatio of the Laplace trasform does ot ivolve ay tuig. I fact, the first-step is automatically robust to the presece of price jumps ad achieves the parametric rate of covergece whe jumps are ot too active; see Todorov ad Tauche (212b). I the secod step, a tuig parameter is itroduced for stabilizig the Laplace iversio, at the cost of iducig a regularizatio bias. For the direct plug-i method of Li et al. (213), the key is to recover the spot variace process, for which two types of tuig are eeded. Oe is to select a threshold for elimiatig jumps, for which the trade-off is to balace the passthrough of small jumps ad the false elimiatio of large diffusive movemets. The other is to select the block size of the local widow by tradig-off the bias iduced by the time variatio of the volatility ad the samplig error iduced by Browia shocks. For both methods, the optimal choice of the tuig parameters remais a ope, ad likely very challegig, questio. We provide some simulatio results for assessig the fiite-sample impact of these tuig parameters. We ca further compare our aalysis here with Todorov ad Tauche (212a), where somewhat aalogous steps were followed to estimate the ivariat probability desity of the volatility process, but there are fudametal differeces betwee the curret paper ad Todorov ad Tauche (212a). First, ulike Todorov ad Tauche (212a), the time spa of the data is fixed ad hece we are iterested i pathwise properties of the latet volatility process over the fixed time iterval. This is further illustrated by our empirical applicatio which studies the radomess of the (occupatioal) iterquartile rage of various trasforms of volatility. Thus, i this paper we impose either the existece of ivariat distributio of the volatility process or mixig-type coditios. While such coditios may be reasoable for aalyzig data from a log sample period, they are ulikely to kick i sufficietly fast i short samples i view of the high persistece of the volatility process (Comte ad Reault (1998)). I our setup, we allow the volatility process to be ostatioary ad strogly serially depedet. This asymptotic settig provides justificatio for estimatig distributioal or, to be more precise, occupatioal properties of the volatility process usig data Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of

1256 JIA LI ET AL. withi relatively short (sub)sample periods. Secod, ad quite importatly from a techical poit of view, the object of iterest here (i.e., the VOT) is a radom quatity with limited pathwise smoothess properties. It is well kow that smoothess coditios are importat i the aalysis of ill-posed problems (Carrasco et al. (27)). Ideed, our aalysis of the stochastic regularizatio bias demads techical argumets that are very differet from Todorov ad Tauche (212a), where the ivariat distributio is determiistic ad smoother. As a techical by-product of our aalysis, we provide primitive coditios for the smoothess of the volatility occupatio desity for a popular class of jump-diffusio stochastic volatility models. Overall, the curret paper ca be viewed as a extesio of the results i Todorov ad Tauche (212a) to the theoretically differet settig of fixed time spa ad provides the theoretical justificatio for applyig the method i Todorov ad Tauche (212a) over differet time horizos. Fially, the curret paper is also coected with the broad literature o ill-posed problems i ecoometrics; see Carrasco et al. (27) for a comprehesive review. I the curret paper, we adopt a direct regularizatio method for ivertig trasforms of the Melli covolutio type (Kryzhiy (23a,b)), which is very differet from spectral decompositio methods reviewed i Carrasco et al. (27). I particular, we do ot cosider the Laplace trasform as a compact operator for some properly desiged Hilbert spaces. We prove the fuctioal covergece for the VOT estimator uder the local uiform topology, istead of uder a (weighted) L 2 orm. The uiform covergece result is the used to prove cosistecy of estimators of the (radom) VOT quatiles. Our cotributio is twofold. First, the proposed estimator is theoretically ovel ad has fiite-sample performace that is geerally better tha the bechmark set by Li et al. (213) i the presece of active jumps. To be specific, we provide Mote Carlo evidece that the regularized Laplace iversio estimates are more accurate tha those of the direct plug-i method for estimatig lower volatility quatiles as well as the volatility media i jump-diffusio models. This patter appears i all Mote Carlo settigs ad is ideed quite ituitive: small jumps are u-trucated, ad they iduce a relatively large fiite-sample bias for volatility estimatio for lower quatiles. Moreover, this fidig exteds eve to the estimatio of higher volatility quatiles whe (asymptotically valid) oadaptive trucatio thresholds are used for the direct plug-i method. That oted, we do observe a partial reversal of this patter for estimatig higher volatility quatiles whe certai adaptive trucatio thresholds are used for the direct plug-i method, so the proposed method does ot always domiate that of Li et al. (213). We further illustrate the empirical use of the proposed estimator by studyig the depedece betwee the (occupatioal) iterquartile rage of various trasforms of the volatility ad the level of the volatility process. Such aalysis sheds light o the modelig of volatility of volatility. Secod, to the best of our kowledge, the ill-posed problem ad the associated regularizatio is the first ever explored i a settig with discretely sampled semimartigales withi a fixed time spa. Other ill-posed problems withi the high-frequecy settig will aturally arise, for example, i Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of

ESTIMATING THE VOLATILITY OCCUPATION TIME 1257 oparametric regressios ivolvig elemets of the diffusio matrix of a multivariate Itô semimartigale; see Härdle ad Lito (1994) for a review i the classical log-spa settig. This paper is orgaized as follows. I Sectio 2 we itroduce the formal setup ad state our assumptios. I Sectio 3 we develop our estimator of the VOT, derive its asymptotic properties, ad use it to estimate the associated volatility quatiles. Sectio 4 reports results from a Mote Carlo study of our estimatio techique, followed by a empirical illustratio i Sectio 5. Sectio 6 cocludes. The Appedix Sectio cotais all proofs. 2. SETUP 2.1. The uderlyig process We start with itroducig the formal setup. The process X i (1) is defied o a filtered space (,F,(F t ) t,p) with the jump compoet J t give by t ( ) J t = δ (s, z)1{ δ(s,z) 1} μ(ds,dz) R t ( ) + δ (s, z)1{ δ(s,z) >1} μ(ds,dz), (3) R where μ is a Poisso radom measure o R + R with compesator ν of the form ν (dt,dz) = dt λ(dz) for some σ -fiite measure λ o R, μ = μ ν ad δ : R + R R is a predictable fuctio. Regularity coditios o X t are collected below. Assumptio A. The followig coditios hold for some costat r (,2) ad a localizig sequece (T m ) m 1 of stoppig times. 3 A1. X is a Itô semimartigale give by (1) ad (3), where the process b t is locally bouded ad the process V t is strictly positive ad càdlàg. Moreover, δ (ω,t, z) r 1 Ɣ m (z) for all ω, t T m ad z R, where (Ɣ m ) m 1 is a seqeuece of λ-itegrable determiistic fuctios o R. A2. For a sequece (K m ) m 1 of real umbers, E V t V s 2 K m t s for all t,s i [, T m ] with t s 1. Assumptio A imposes very mild regularities o the process X ad is stadard i the literature o discretized processes; see Jacod ad Protter (212). The domiace coditio i Assumptio A is oly required to hold locally i time up to the stoppig time T m, which ofte take forms of hittig times of adapted processes; this requiremet is much weaker tha a global domiace coditio that correspods to T m +. This more geeral setup, however, does ot add ay techical complexity ito our proofs, thaks to the stadard localizatio procedure i stochastic calculus; see Sectio 4.4.1 i Jacod ad Protter (212) for a review o the localizatio procedure. Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of

1258 JIA LI ET AL. We ote that Assumptio A imposes o parametric structure o the uderlyig process, allowig for jumps i X t ad V t, ad depedece betwee various compoets i a arbitrary maer. I particular, we allow the stochastic volatility process V t to be depedet o the Browia motio W t, so as to accommodate the leverage effect (Black (1976)). The costat r i Assumptio A1 cotrols the activity of small jumps, as it provides a boud for the geeralized Blumethal Getoor idex. The assumptio is stroger whe r is smaller. Assumptio A2 requires the spot variace process V t to be (locally) 1/2-Hölder cotiuous uder the L 2 orm. This assumptio holds i the well-kow case i which V t is also a Itô semimartigale with locally bouded characteristics. It also holds for log-memory specificatios that are drive by fractioal Browia motio; see Comte ad Reault (1996). Assumptio A2 coicides, albeit with a differet orm, to the oe maitaied by Reault et al. (214). 2.2. Occupatio times We ext collect some assumptios o the VOT ad the associated occupatio desity. I what follows we defie F t ( ) as (2) with T replaced by t. Assumptio B. The followig coditios hold for some localizig sequece (T m ) m 1 of stoppig times ad a costat sequece (C m ) m 1. B1. Almost surely, the fuctio x F t (x) is piecewise differetiable with derivative f t (x) for all t [, T ]. For all x, y (, ), P({the iterval (x, y) cotais some odifferetiable poit of F T ( )} {T T m }) C m x y. B2. For ay compact K (, ), sup x K E [ f T Tm (x) ] <. Assumptio B is used i our aalysis o the estimatio of F T (x) for fixed x. As i Assumptio A, we oly eed the domiace coditios to hold locally up to the localizig sequece T m. Assumptio B1 holds if the occupatio desity of V t exists, which is the case for geeral semimartigale processes with odegeerate diffusive compoet ad large classes of Gaussia processes; see, for example, Gema ad Horowitz (198), Protter (24), Marcus ad Rose (26), Eisebaum ad Kaspi (27) ad refereces therei. Assumptio B1 holds more geerally uder settigs where F t ( ) ca be odifferetiable (ad eve discotiuous) at radom poits, as log as these irregular poits are located diffusively o the lie, as formulated by the secod part of Assumptio B1. This geerality accommodates certai pure-jump stochastic volatility processes, such as a compoud Poisso process with bouded margial probability desity. 4 Assumptio B2 imposes some mild itegrability o the occupatio desity ad is satisfied as soo as the probability desity of V t is uiformly bouded i the spatial variable ad over t [, T ], which is the case for typical stochastic volatility models. To derive uiform covergece results, we eed to stregthe Assumptio B as follows. Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of

ESTIMATING THE VOLATILITY OCCUPATION TIME 1259 Assumptio C. The followig coditios hold for some localizig sequece (T m ) m 1 of stoppig times ad costats γ >ε>. C1. Almost surely, the fuctio x F t (x) is differetiable with derivative f t (x) for all t [, T ]. C2. For ay compact K (, ), sup x K E [ f T Tm (x) 1+ε] <. C3. For ay compact K (, ), there [ exist costats (C m ) m 1 such that for all x, y K, wehavee sup t T f t Tm (x) f t Tm (y) 1+ε] C m x y γ. Assumptios C1 ad C2 are stroger tha Assumptio B. I additio, the Hölder-cotiuity coditio i Assumptio C3 is otrivial to verify. We hece devote Sectio 2.3 to discussig primitive coditios for Assumptio C that cover may volatility models used i fiacial applicatios, although this set of coditios is far from exhaustive. Fially, we ote that C3 ivolves expectatios ad for establishig pathwise Hölder cotiuity i the spatial argumet of the occupatio desity (via Kolmogorov s cotiuity theorem or some metric etropy coditio, see, e.g., Ledoux ad Talagrad (1991)), oe typically eeds a stroger coditio tha that i C3. 2.3. Some primitive coditios for Assumptio C We cosider the followig geeral class of jump-diffusio volatility models: dv t = a t dt + s (V t )db t + dj V,t, (4) where a t is a locally bouded predictable process, B t is a stadard Browia motio, s( ) is a determiistic fuctio, ad J V,t is a pure-jump process. This example icludes may volatility models ecoutered i applicatios. It is helpful to cosider the Lamperti trasform of V t. More precisely, we set Ṽ t = g (V t ), where g ( ) is ay primitive of the fuctio 1/s( ), that is, g(x) = x du/s (u) ad the costat of itegratio is irrelevat. By Itô s formula, the cotiuous martigale part of Ṽ t is B t. Lemma 2.1(a) shows that uder some regularity coditios, the trasformed process Ṽ t satisfies Assumptio C. To prove Lemma 2.1(a) we compute the occupatio desity of Ṽ t explicitly i terms of stochastic itegrals via the Meyer Taaka formula 5 ad the we boud the correspodig spatial icremets. The Lemma 2.1(b) shows that V t iherits the same property, that is, it satisfies Assumptio C, provided that the trasformatio g( ) is smooth eough. LEMMA 2.1. (a) Let k > 1. Cosider a process Ṽ t with the followig form t t Ṽ t = Ṽ + ã s ds + B t + δ (s, z)μ(ds,dz), (5) R Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of

126 JIA LI ET AL. where ã t is a locally bouded predictable process, B t is a Browia motio ad δ( ) is a predictable fuctio. Suppose the followig coditios hold for some costat C >. (i) δ (ω,t, z) Ɣ m (z) for all (ω,t, z) with t S m, where (S m ) m 1 is a localizig sequece of stoppig times ad each Ɣ m is a oegative determiistic fuctio satisfyig ( Ɣ m (z) β + Ɣ m (z) k) λ(dz) <, for some β (,1). R (ii) The probability desity fuctio of Ṽ t is bouded o compact subsets of R uiformly i t [, T ]. (iii) The process Ṽ t is locally bouded. The the occupatio desity of Ṽ t, deoted by f t ( ), exists. Moreover, for ay compact K R, there exists a localizig sequece of stoppig times (T m ) m 1, such that for some K > ad for ay x, y K, we have E[ f T Tm (x) k ] K ad E[sup t T Tm f t (x) f t (y) k ] K x y (1 β)k (1/2). (b) Suppose, i additio, that Ṽ t = g (V t ) for some cotiuously differetiable strictly icreasig fuctio g : R + R. Also suppose that for some γ (,1] ad ay compact K (, ), there exists some costat C >, such that g (x) g (y) C x y γ for all x, y K. The V t satisfies Assumptio C. 3. ESTIMATING VOLATILITY OCCUPATION TIMES We ow preset our estimator for the VOT ad its asymptotic properties. We suppose that the process X t is observed at discrete times i, i =,1,..., o [, T ] for fixed T >, with the time lag asymptotically whe. Our strategy for estimatig the VOT is to first estimate its Laplace trasform ad the to ivert the latter. We defie the volatility Laplace trasform over the iterval [, T ]as T L T (u) e uv s ds, u >. By the occupatio desity formula (see, e.g., (6.5) i Gema ad Horowitz (198)), the temporal itegral above ca be rewritte as a spatial itegral uder the occupatio measure, that is, L T (u) = e ux f T (x)dx = e ux F T (dx), u >. Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of

ESTIMATING THE VOLATILITY OCCUPATION TIME 1261 The Laplace trasform of the VOT ca the be obtaied by usig Fubii s theorem ad is give by L T (u) = e ux F T (x)dx. (6) u Followig Todorov ad Tauche (212b), we estimate the volatility Laplace trasform L T (u) usig the realized volatility Laplace trasform defied as [T/ ] L T, (u) = i=1 cos ( ) 2u i X/ 1/2, u >, (7) where [ ] deotes the largest smaller iteger fuctio ad i X X i P X (i 1). Todorov ad Tauche (212b) show that L T, ( ) L T ( ) locally uiformly with a associated cetral limit theorem. These covergece results are robust to the presece of price jumps without appealig to the thresholdig techique as i Macii (21) ad Li et al. (213). Cosequetly, u 1 P L T, (u) u 1 L T (u) for each u (, ). Oce the Laplace trasform of the VOT is estimated from the data, i the ext step we ivert it i order to estimate F T (x). Ivertig a Laplace trasform, however, is a ill-posed problem ad hece requires a regularizatio (Tikhoov ad Arsei (1977)). Here, we adopt a approach proposed by Kryzhiy (23a,b) ad implemet the followig regularized iversio of u 1 L T (u): F T,R (x) = L T (u) (R,ux) du, x >, (8) u where R > is a regularizatio parameter ad the iversio kerel (R, x) is defied as 6 (R, x) = 4 ( s cos(r l(s)) sih(π R/2) 2π 2 s 2 si(xs)ds + 1 ) s si(r l(s)) + cosh(π R/2) s 2 si(xs)ds. + 1 It ca be show that the regularized iversio F T,R (x) ca be also writte as (see (A.14)) F T,R (x) = 2 F T (xu) si(r lu) u π u 2 du. 1 That is, F T,R (x) is geerated by smoothig the VOT via the kerel 2u 1/2 si(r lu)/π(u 2 1), which approaches the Dirac mass at u = 1asR. 7 Our estimator for the VOT is costructed by simply replacig L T (u) i (8) with L T, (u), that is, it is give by F T,,R (x) = L T, (u) (R,ux) du u = L T, (e z ) (R, xe z )dz. (9) Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of

1262 JIA LI ET AL. Todorov ad Tauche (212a) use a similar strategy to estimate the (determiistic) ivariat probability desity of the spot volatility process uder a settig with T. However, the problem here is more complicated, sice the estimad F T ( ) itself is a radom fuctio, which i particular reders the regularizatio bias radom, whereas i Todorov ad Tauche (212a) ad Kryzhiy (23a,b), the object of iterest is determiistic. We ow tur to the asymptotic properties of the estimator F T,,R (x), where R is a sequece of (strictly positive) regularizatio parameters that grows to + asymptotically. Here, we allow R to be radom so that it ca be data-depedet, while the rate at which it grows is give by a determiistic sequece ρ.itis coceptually useful to decompose the estimatio error F T,,R (x) F T (x) ito two compoets: the regularizatio bias F T,R (x) F T (x) ad the samplig error F T,,R (x) F T,R (x). Lemmas 3.1 ad 3.2 characterize the order of magitude of each compoet. LEMMA ( ) 3.1. Let x > be a costat. Suppose that R = O p (ρ ) ad R 1 = O p ρ 1 for some determiistic sequece ρ with ρ. Uder Assumptios A ad B, F T,R (x) F T (x) = O p ( ρ 1 l(ρ ) ). LEMMA 3.2. Let η (,1/2) be a costat ad K (, ) be compact. Suppose that R ρ for some determiistic sequece ρ with ρ. Uder Assumptio A, sup x K F T,,R (x) F T,R (x) ( πρ )( )) = O p (exp ρ (r 1)/2 (r 1)(1/r 1/2) + ρ l (ρ ) 1/2 + ρ 2 2 (1+η)/2. Lemma 3.1 describes the order of magitude of the regularizatio bias. Lemma 3.2 describes the order of magitude of the samplig error uiformly over x K, where the set K is assumed bouded both above ad away from zero. Lemma 3.2 holds for ay costat η (,1/2). This costat arises as a techical device from the proof ad should be take close to 1/2 so that the boud i Lemma 3.2 is sharper. Combiig Lemmas 3.1 ad 3.2 ad choosig the regularizatio parameter properly, we obtai the poitwise cosistecy of the VOT estimator. THEOREM 3.1. Suppose (i) Assumptios A ad B; (ii) ρ = δ l ( 1 ) for some δ (,2 δ/π ), where δ mi{(r 1)(1/r 1/2),1/2}; (iii) R ρ ad R 1 = O p ( ρ 1 ). The for each x >, F T,,R (x) F T (x). P Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of

ESTIMATING THE VOLATILITY OCCUPATION TIME 1263 I Theorem 3.1, we set the regularizatio parameter R to grow slowly to ifiity so that both the regularizatio bias ad the samplig error vaish asymptotically. Coditio (ii) specifies the admissible rage of the tuig parameter, which depeds o r whe r > 1 ad shriks as r approaches 2 (the theoretical upper boud for jump activity of semimartigales). This pheomeo reflects the wellkow difficulty of disetaglig active jumps from the diffusive compoet. Estimators for jump activity (see, e.g., Aït-Sahalia ad Jacod (29)) may be used to assess the restrictiveess of this coditio i a give sample. More geerally, the iversio method is ot limited to the realized Laplace trasform estimator L T, ( ). With a geeric estimator L T, ( ) for L T ( ), weca associate a iversio estimator for the VOT as F T,,R (x) = L T, (u) (R,ux) du, x >. u Theorem 3.2 shows that F T,,R (x) is a cosistet estimator for F T (x) uder a high-level coditio cocerig the estimatio error of L T, ( ) uder the L 1 orm. THEOREM 3.2. Suppose (i) there exist a localizig sequece (T m ) m 1 of stoppig times ad a sequece (C m ) m 1 of positive costats such that for some c (,1/2), δ > ad all u >, E L Tm, (u) L Tm (u) C m (u c + u 1+ c) δ ; (1) (ii) ρ = δ l ( 1 ) ( ) for some δ,2 δ/π ; (iii) R ρ ad R 1 ( ) = O p ρ 1. The for each x >, F T,,R (x) F T (x). P Theorem 3.1 ca also be proved by usig Theorem 3.2. Ideed, it ca be see from the proof of Lemma 3.2 that the estimator L T, ( ) verifies (1) for ay c (, 1/2) with δ as give i Theorem 3.1. I other settigs, alterative estimators might be required to verify these coditios. The key to the proof of Theorem 3.2 is a extesio of Lemma 3.2 uder coditio (1), but with a coarser boud. A pessimistic theoretical boud o the rate of covergece for Theorem 3.1 is essetially l ( 1 ), which is drive by the regularizatio bias. The plug-i estimator of Li et al. (213), i cotrast, ca formally be bouded by a polyomial rate of covergece. However, the bouds might ot be sharp. Efficiecy issues i the estimatio of itegrated volatility fuctioals of the form T g(v s)ds has recetly bee tackled by Jacod ad Reiß (214), Jacod ad Rosebaum (213) ad Reault et al. (214) for smooth g( ). The VOT, o the other had, correspods to a discotiuous trasform g( ) = 1 { x}. Assessig the efficiecy of the VOT estimators remais to be a ope questio that is likely very challegig. That beig said, at least ituitively, more efficiet estimators of the itegrated Laplace trasform of volatility tha the oe i (7), like the oes cosidered i Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of

1264 JIA LI ET AL. Reault et al. (214), ca help improve the efficiecy of the VOT estimators based o regularized iversio. The theoretical results i Theorem 3.2 provide the foudatios for doig this. Aother ope questio is how to optimally choose tuig parameters i order to miimize some loss criterio, such as the mea square error. Such aalysis remais to be a techical challege, ot oly for the curret paper, but also for the i-fill aalysis of high-frequecy semimartigale data i geeral. 8 I Sectio 4, we provide simulatio results i a realistically calibrated Mote Carlo settig for comparig the fiite-sample performace of the two methods ad for assessig the robustess of the proposed estimator with respect to the tuig parameter. The poitwise covergece i Theorems 3.1 ad 3.2 ca be further stregtheed to be uiform i the spatial variable, as show below. THEOREM 3.3. Suppose Assumptio C. The the followig statemets hold for ay compact K (, ). (a) Uder the coditios of Theorem 3.1, sup F T,,R (x) F T (x) P. (11) x K (b) Uder the coditios of Theorem 3.2, sup F T,,R (x) F T (x) P. (12) x K Next, we provide a refiemet to the fuctioal estimator F T,,R ( ). The discussio below oly requires the uiform covergece (11) to hold, so it also applies to the geeric estimator F T,,R ( ) uder (12). While the occupatio time x F T (x) is a pathwise icreasig fuctio by desig, the proposed estimator F T,,R ( ) is ot guarateed to be mootoe. We propose a mootoizatio of F T,,R ( ) via rearragemet, ad, as a by-product, cosistet estimators of the quatiles of the occupatio time. To be precise, for τ (, T ), we defie the τ- quatile of the occupatio time as its pathwise left-cotiuous iverse: Q T (τ) = if{x R + : F T (x) τ}. For ay compact iterval K (, ), we defie the K-costraied τ-quatile of F T ( ) as Q K T (τ) = if{x K : F T (x) τ}, where the ifimum over a empty set is give by supk. While Q T (τ) is of atural iterest, we are oly able to cosistetly estimate Q K T (τ), although K (, ) ca be arbitrarily large. This is due to the techical reaso that the uiform covergece i Theorem 3.3 is oly available over a oradom idex set K, which is bouded above ad away from zero, but every quatile Q T (τ) is itself a radom Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of

ESTIMATING THE VOLATILITY OCCUPATION TIME 1265 variable ad thus may take values outside K o some sample paths. Such a complicatio would ot exist if F T ( ), ad hece Q T (τ), were determiistic the stadard case i ecoometrics ad statistics. Of course, if the process V t is kow a priori to take values i some set K (, ), the Q T ( ) ad Q K T ( ) coicide. I practice, the K-costrait is typically ubidig as log as we do ot attempt to estimate extreme (pathwise) quatiles of the process V t. We propose a estimator for Q K T (τ) ad a K-costraied mootoized versio F T,,R K ( ) of the occupatio time as follows: supk 1 { F T,,R (y)<τ} ifk Q K T,,R (τ) = if K+ { F T,,R K (x) = if τ (, T ) : Q K T,,R (τ) > x dy, τ (, T ), }, x R, where o the secod lie, the ifimum over a empty set is give by T. By costructio, Q K T,,R : (, T ) K is icreasig ad left cotiuous ad F T,,R K : R [, T ] is icreasig ad right cotiuous. Moreover, Q K T,,R is the quatile fuctio of F T,,R K, i.e., for τ (, T ), Q K T,,R (τ) = if{x : F T,,R K (x) τ}. Asymptotic properties of F T,,R K ( ) ad Q K T,,R (τ) are give i Theorem 3.4. THEOREM 3.4. Let K (, ) be a compact iterval. If F T ( ) is cotiuous ad sup F T,,R (x) F T (x) P, x K the we have the followig. (a) sup F T,,R K P (x) F T (x). x K (b) For every τ {τ (, T ) : Q T ( ) is cotiuous at τ almost surely}, Q K ( T,,R τ ) P Q K ( T τ ). We ote that the mootoizatio procedure here is similar to that i Cherozhukov et al. (21), which i tur has a deep root i fuctioal aalysis (Hardy et al. (1952)). Cherozhukov et al. (21) shows that rearragemet leads to fiite-sample improvemet uder very geeral settigs; see Propositio 4 there. 9 Our asymptotic results are distict from those of Cherozhukov et al. (21) i two aspects. First, the estimad cosidered here, i.e. the occupatio time, is a radom fuctio. Secod, as we are iterested i the covergece i probability, we oly eed to assume that sup x K F T,,R (x) F T (x) P ad, of course, our argumet does ot rely o the fuctioal delta method. Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of

1266 JIA LI ET AL. 4. MONTE CARLO We ow examie the fiite-sample performace of our estimator ad compare it with the direct plug-i method proposed by Li et al. (213). We cosider the followig jump-diffusio volatility model i which the log-volatility is a Lévydrive Orstei Uhlebeck (OU) process, that is, dx t = e V t 1 dw t + dy t, dv t =.3V t dt + dl t, (13) where L t isalévy martigale uiquely defied by the margial law of V t which i tur has a self-decomposable distributio (see Theorem 17.4 of Sato (1999)) with characteristic triplet (Defiitio 8.2 of Sato (1999)) of (, 1,ν) for ν(dx) = 2.33e 2. x 1 x 1+.5 {x>} dx with respect to the idetity trucatio fuctio. Our volatility specificatio is quite geeral as it allows for both diffusive ad jump shocks i volatility, with the latter beig of ifiite activity. The mea ad the persistece of the volatility process are calibrated realistically to observed fiacial data. I particular, we set E[e V t 1 ] = 1 (our uit of time is a tradig day ad we measure returs i percetage) ad the persistece of a shock i V t has a half-life of approximately 23 days. Fially, Y t i (13) is a tempered stable Lévy process, i.e., a pure-jump Lévy process with Lévy measure c e λ x, which is idepedet from L t ad W t. The tempered stable process is a flexible jump speci- x β+1 ficatio with separate parameters cotrollig small ad big jumps: λ cotrols the jump tails ad β coicides with the Blumethal Getoor idex of Y t (ad hece cotrols the small jumps). We cosider three cases i the Mote Carlo: (a) o price jumps, which correspods to c =, (b) low-activity price jumps, with parameters c = 6.298, λ = 7, ad β =.1, ad (c) high-activity price jumps, with parameters c = 1.348, λ = 7, ad β =.9. The value of λ i each case is set to produce jump tail behavior cosistet with oparametric evidece reported i Bollerslev ad Todorov (211). Further, i all cosidered cases for Y t, we set the parameter c so that the secod momet of the icremet of Y o uit iterval is equal to.3 which produces jump cotributio i total quadratic variatio of X similar to earlier oparametric empirical evidece from high-frequecy fiacial data. 1 I the Mote Carlo we fix the time spa to be T = 22 days, equivalet to oe caledar moth, ad we cosider = 8 which correspods to 5-miute samplig of itraday observatios of X ia6.5-hour tradig day. For each realizatio we compute the 25-th, 5-th, ad 75-th volatility quatiles over the iterval [, T ] ad assess the accuracy i measurig these radom quatities by reportig bias ad mea absolute deviatio (MAD) aroud the true values for the cosidered estimators. We first aalyze the effect of the regularizatio parameter R o the volatility quatile estimatio. For brevity, we coduct the aalysis i the case whe X t does ot cotai price jumps, while otig that similar results hold i the other cases. I Table 1 we report results from the Mote Carlo for regularized Laplace iversio Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of

Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of TABLE 1. Mote Carlo Results: Effect of R Q T, (.25) Q T, (.5) Q T, (.75) Start Value True Bias MAD True Bias MAD True Bias MAD Pael A: Regularized Laplace Iversio with R = 2.5 V = Q V (.25).1737.21.282.286.178.322.4519.8.43 V = Q V (.5).3293.392.527.5243.324.583.869.78.746 V = Q V (.75).6337.716.975.9945.6.114 1.5162.178.1383 Pael B: Regularized Laplace Iversio with R = 3. V = Q V (.25).1737.128.244.286.82.31.4519.1.448 V = Q V (.5).3293.251.465.5243.16.558.869.38.745 V = Q V (.75).6337.453.86.9945.34.188 1.5162.94.1394 Pael C: Regularized Laplace Iversio with R = 3.5 V = Q V (.25).1737.93.262.286.42.335.4519.114.556 V = Q V (.5).3293.18.5.5243.93.632.869.8.947 V = Q V (.75).6337.298.981.9945.211.121 1.5162.154.1736 Note: I each of the cases, the volatility is started from a fixed poit beig the 25-th, 5-th ad 75-th quatile of the ivariat distributio of the volatility process, deoted correspodigly as Q V (.25), Q V (.5) ad Q V (.75). The colums True report the average value (across the Mote Carlo simulatios) of the true variace quatile that is estimated; MAD stads for mea absolute deviatio aroud the true value. The Mote Carlo replica is 1. ESTIMATING THE VOLATILITY OCCUPATION TIME 1267

1268 JIA LI ET AL. with values of the regularizatio parameter of R = 2.5, R = 3., ad R = 3.5. Overall, the performace of our volatility quatile estimator is satisfactory with biases beig small i relative terms. I geeral, the differece across the differet values of the regularizatio parameter are relatively small. From Table 1 we ca see the typical bias-variace tradeoff that arises i oparametric estimatio: for lower value of R (more smoothig) the biases are larger but the samplig variability is smaller, while for higher value of R (less smoothig) the opposite is true. The value of R that leads to the smallest MAD is R = 3. ad heceforth we keep the regularizatio parameter at this value. We ext compare the performace of the regularized Laplace iversio approach for volatility quatile estimatio with the direct plug-i method of Li et al. (213). The latter is based o local estimators of the volatility process over blocks give by V i = 1 u k j=1 ( ) 2 i+ j X 1 { }, i =,...,[T/ i+ j X ] k, v,i where u = k ad k deotes the umber of high-frequecy elemets withi a block (k satisfies k ad k ); v,t is the threshold which takes the form v,t = α,t ϖ for some strictly positive process α,t ad ϖ (,1/2). These local estimators are the used to approximate the volatility trajectory via V t = V iu, t [ iu,(i + 1)u ), ad V t = V ([T/u ] 1)u, [T/u ]u t T, ad from here the direct estimator of the volatility occupatio time is give by F d T, (x) = T 1 { V s x} ds, x R. The direct estimator F T, d (x) has two tuig parameters. The first is the block size k which plays a similar role as the regularizatio parameter R i the regularized Laplace iversio method. We follow Li et al. (213) ad set k = 4 throughout. The secod tuig parameter is the choice of the threshold v,t. There are various ways of settig this threshold which all lead to asymptotically valid results. Oe simple choice is a time-ivariat threshold of the form v,t = 3σ.49, where σ is a estimator of E(V t ). Aother is a time-varyig threshold that takes ito accout the stochastic volatility. Here we follow Li et al. (213) (ad earlier work o threshold estimatio) ad experimet with v,t = 3 BV j.49 ad v,t = 4 BV j.49 for t [ j 1, j), where BV j = π [ j/ ] 2 i=[( j 1)/ ]+2 i 1 X i X is the Bipower Variatio estimator of Bardorff-Nielse ad Shephard (24). 11 I Tables 2 ad 3 we compare the precisio of estimatig the mothly volatility quatiles via regularized Laplace iversio (with R = 3.) ad via the Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of

Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of TABLE 2. Mote Carlo Results I Presece of Low Activity Jump Compoet Q T, (.25) Q T, (.5) Q T, (.75) Start Value True Bias MAD True Bias MAD True Bias MAD Pael A: Regularized Laplace Iversio with R = 3. V = Q V (.25).173.195.318.286.75.754.4564.1693.1858 V = Q V (.5).3234.241.55.5122.961.198.8139.262.248 V = Q V (.75).6235.271.854.9686.138.1691 1.518.2753.3216 Pael B: Direct Method with Costat Threshold 3.49 V = Q V (.25).173.128.117.286.1456.188.4564.1839.2912 V = Q V (.5).3234.95.1215.5122.999.2142.8139.68.356 V = Q V (.75).6235.272.1523.9686.466.2958 1.518.2538.5589 Pael C: Direct Method with Adaptive Threshold 3 BV j.49 V = Q V (.25).173.576.625.286.878.96.4564.1294.1443 V = Q V (.5).3234.595.716.5122.947.1148.8139.1421.1767 V = Q V (.75).6235.486.875.9686.896.1414 1.518.1554.2174 Pael D: Direct Method with Adaptive Threshold 4 BV j.49 V = Q V (.25).173.852.892.286.1345.1394.4564.218.218 V = Q V (.5).3234.935.124.5122.1488.1622.8139.2226.2443 V = Q V (.75).6235.893.1179.9686.1542.187 1.518.2441.2876 Note: Descriptio as for Table 1. ESTIMATING THE VOLATILITY OCCUPATION TIME 1269

Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of TABLE 3. Mote Carlo Results I Presece of High Activity Jump Compoet Q T, (.25) Q T, (.5) Q T, (.75) Start Value True Bias MAD True Bias MAD True Bias MAD Pael A: Regularized Laplace Iversio with R = 3. V = Q V (.25).1619.726.833.276.1526.1626.4579.2833.2992 V = Q V (.5).384.879.997.4936.178.1849.8169.34.3371 V = Q V (.75).5985.933.1278.9326.1888.2117 1.5245.3127.3944 Pael B: Direct Method with Costat Threshold 3.49 V = Q V (.25).1619.1666.1678.276.217.241.4579.2317.3411 V = Q V (.5).384.1556.1675.4936.17.2562.8169.162.414 V = Q V (.75).5985.162.1694.9326.361.348 1.5245.2139.612 Pael C: Direct Method with Adaptive Threshold 3 BV j.49 V = Q V (.25).1619.1338.1347.276.176.182.4579.2133.232 V = Q V (.5).384.1331.1375.4936.1726.1899.8169.213.2541 V = Q V (.75).5985.1191.1361.9326.1621.21 1.5245.2.2951 Pael D: Direct Method with Adaptive Threshold 4 BV j.49 V = Q V (.25).1619.1586.159.276.2123.221.4579.2758.2897 V = Q V (.5).384.169.1642.4936.2173.2315.8169.279.316 V = Q V (.75).5985.1527.1643.9326.2137.2452 1.5245.2745.358 Note: Descriptio as for Table 1. 127 JIA LI ET AL.

ESTIMATING THE VOLATILITY OCCUPATION TIME 1271 direct method for the above discussed ways of settig the threshold parameter. 12 We cosider oly the empirically realistic scearios i which X cotais jumps i the compariso. A immediate observatio from the tables is that the threshold parameter i F T, d plays a crucial role. Ideed, a costat threshold does a very poor job: it yields huge biases ad also results i very oisy estimates. The volatility quatile estimators based o F T, d work well oly whe a time-varyig adaptive (to the curret level of volatility) threshold is selected. Comparig these estimators with the oe based o the regularized Laplace iversio, we see a iterestig patter. The estimatio of the lower volatility quatiles is doe sigificatly more precisely via the iversio method. For the lower volatility quatiles, the estimates based o F T, d cotai otrivial bias. This is due to small u-trucated jumps which play a relatively bigger role whe estimatig the lower volatility quatiles. The above observatio cotiues to hold, albeit to a far less extet, for the volatility media. For the highest volatility quatile, we see a partial reverse. This volatility quatile is estimated more precisely via F T, d but maily whe the lower time-varyig threshold v,t = 3 BV j.49 is used. Overall, we fid mixed results i this comparative aalysis, but the evidece i Tables 2 ad 3 clearly illustrates that the proposed volatility quatile estimator based o the regularized Laplace iversio provides a importat alterative to the direct plug-i method. 5. EMPIRICAL APPLICATION We illustrate the oparametric quatile recostructio techique with a empirical applicatio to two data sets: Euro/$ exchage rate futures (for the period 1/1/1999 12/31/21) ad S&P 5 idex futures (for the period 4/22/1982 12/3/21). Both series are sampled every 5 miutes durig the tradig hours. The time spas of the two data sets differ because of data availability but both data sets iclude some of the most quiescet ad also the most volatile periods i moder fiacial history. These data sets thereby preset a serious challege for our method. I the calculatios of the volatility quatiles we use a time spa of T = 1 moth ad as i the Mote Carlo we fix the regularizatio parameter at R = 3. Figure 1 shows the results for the Euro/$ rate ad Figure 2 shows those for the S&P 5 idex. The left paels show the time series of the 25-th ad 75-th mothly quatiles of the spot variace V t, the spot volatility V t ad the logarithm of the spot variace l(v t ).The estimated quatiles appear to track quite sesibly the behavior of volatility durig times of either ecoomic moderatio or distress. The right paels show the associated iterquartile rage (IQR) versus the media of the logarithm of the spot variace; we use the IQR to measure the variatio of the (trasformed) volatility process. The aim of these plots is to discover how the dispersio of volatility relates to the volatility level. We see that for both data sets, the IQRs of the spot variace ad the spot volatility exhibit a clearly positive, ad geerally covex, relatioship with the media log-variace. I cotrast, Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of

1272 JIA LI ET AL. FIGURE 1. Estimated Quatiles of the Mothly Occupatio Measure of the Spot Volatility of the Euro/$ retur, 1999 21. The three left-had paels show the 25 ad 75 percet quatiles of the mothly occupatio measure of volatility expressed i terms of the local variace (left-top), the local stadard deviatio (left-middle), ad the local log-variace (left-bottom). Each right-side pael is a scatter plot of the iterquartile rages of the associated mothly left-side distributios versus the medias of the distributios (i logvariace). Volatility is quoted aualized ad i percetage terms. the IQR of the log-variace process shows o such patter, suggestig that the log volatility process is homoscedastic, or at least idepedet from the level of volatility, iovatios. To guide ituitio about our empirical fidigs, suppose we have f (V t ) = f (V ) + L t o [, T ], for L t alévy process ad f ( ) some mootoe fuctio (this is approximately true for the typical volatility models like the oes i the Mote Carlo whe T is relatively short ad the volatility is very persistet as i the data). 13 I this case, the iterquartile rage of the volatility occupatio time of f (V t ) o [, T ] will be idepedet of the level V. O the other had, for other fuctios h(v t ) the dispersio will deped i geeral o the level V. The IQR of the volatility occupatio measure ca be used, therefore, to study the importat questio of modelig the variatio of volatility. The evidece here poits away from affie volatility models towards those models i which the log volatility has iovatios that are idepedet from the level of volatility like the expoetial OU model i (13). This is cosistet with earlier parametric evidece for superior performace of log-volatility models over affie models. 14 Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of

ESTIMATING THE VOLATILITY OCCUPATION TIME 1273 FIGURE 2. Estimated Quatiles of the Mothly Occupatio Measure of the Spot Volatility of the S&P5 idex futures retur, 1982 21. The orgaizatio is the same as Figure 1. 6. CONCLUSION I this paper we use iverse Laplace trasforms to geerate a quick ad easy oparametric estimator of the volatility occupatio time (VOT). The estimatio is coducted based o discretely sampled Itô semimartigale icremets over a fixed time iterval with asymptotically shrikig mesh of the observatio grid. We derive the asymptotic properties of the VOT estimator locally uiformly i the spatial argumet ad further ivert it to estimate the correspodig quatiles of volatility over the time iterval. Mote Carlo evidece shows good fiite-sample performace that is sigificatly better tha that of the bechmark estimator of Li et al. (213) for estimatig lower volatility quatiles. A empirical applicatio illustrates the use of the estimator for studyig the variatio of volatility. NOTES 1. To make the aalogy exact, oe may ormalize the expressio i (2) by T 1. Here, we follow the covetio i the literature (see, e.g., Gema ad Horowitz (198)) without usig this ormalizatio. 2. See, for example, (6.4) i Gema ad Horowitz (198). 3. A localizig sequece of stoppig times is a sequece of stoppig times which icreases to +. 4. Whe V t is a compoud Poisso process, each odifferetiable poit of F T ( ) is a realized level of V t. Therefore, the probability i Assumptio B1 is bouded by P(V t (x, y) for some t [, T ]). Sice the expected umber of jumps is fiite ad V t has a bouded desity, this probability is further bouded by x y up to a multiplicative costat. 5. This is possible because the cotiuous martigale part of Ṽ t is a Browia motio. Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of

1274 JIA LI ET AL. 6. This seemigly complicated iversio kerel correspods to a simple stabilizig procedure for the Melli trasform of the itegral equatio (6); see Sectio 2 i Kryzhiy (23a) for techical details. 7. The fact that the regularized quatity ca be cosidered as a type of covolutio betwee the object of iterest ad a smoothig kerel that asymptotically collapses to a Dirac mass also arises i oparametric kerel desity estimatio; see, for example, (9) i Härdle ad Lito (1994). 8. The key difficulty lies i the characterizatio ad estimatio of asymptotic bias terms uder a settig with asymptotically varyig tuig parameters. I a recet paper, Kristese (21) cosiders the challegig questio of the optimal choice of a badwidth parameter i the estimatio of spot volatility. I the settig without leverage effect ad jumps, Kristese (21) remarks (see p. 77) that the optimal choice of tuig parameter is still difficult whe the sample path of the volatility process is ot differetiable with respect to time. Nodifferetiable paths, however, are commo for the typical stochastic volatility models as the oes cosidered here. 9. Mootoizatio methods may also improve the rate of covergece; see Carrasco ad Flores (211) for such a example i the study of decovolutio problems. It may be iterestig to explore this theoretical possibility i future research. 1. The price jumps specificatios cosidered here are both of ifiite activity, hece there are ifiite umbers of jumps withi a fiite iterval. However, big jumps are always of fiite umber. For example jumps of size bigger tha.34%, which correspods to a average three stadard deviatio move of the cotiuous price price icremet at the 5-miute iterval, occur o average 9.17 (case b) ad 3.87 (case c) times o a iterval of legth 22 days. The low-activity jump specificatio geerates more big jumps tha the high-activity oe, with the role reversed for the small jump sizes (recall that the quadratic variatio of both jump specificatios is costraied to be the same). 11. I priciple, the direct plug-i method of Li et al. (213) ca be applied to other jump-robust spot volatility estimators ad may achieve better fiite-sample performace. Improvig the direct plug-i method i this directio is beyod the scope of the curret paper. 12. Comparig the results for F T, i Tables 2 ad 3 with those i Table 1, we otice that the egative biases for the first two quatiles i the case of o price jumps tur ito positive biases i the two cases of price jumps. I the simulatio scearios with price jumps, the estimator F T, cotais biases both due to the regularizatio error ad due to the separatio of volatility from jumps. The bias due to the presece of price jumps is positive ad domiates the bias due to the regularizatio error. 13. This also holds approximately true for two-factor models i which oe of the factors is fast mea revertig ad the other is very persistet (which is the case for most of the estimates of such models reported i empirical work). I such a settig, the fast mea revertig factor plays miimal role i the depedece of the iterquatile rage of various trasforms of the spot variace over the iterval o the level of volatility. 14. Regardig log volatility, preset evidece from time series data while Cot ad da Foseca (22) preset evidece from the optios-implied volatility surface. REFERENCES Aït-Sahalia, Y. & J. Jacod (29) Estimatig the degree of activity of jumps i high frequecy fiacial data. Aals of Statistics 37, 222 2244. Aderse, T. G., T. Bollerslev, P. F. Christofferse, & F. X. Diebold (213) Fiacial risk measuremet for fiacial risk maagemet. I G. Costaides, M. Harris, ad R. Stulz (Eds.), Hadbook of the Ecoomics of Fiace, Vol.II. Elsevier Sciece B.V. Bardorff-Nielse, O. & N. Shephard (24) Power ad Bipower Variatio with Stochastic Volatility ad Jumps. Joural of Fiacial Ecoometrics 2, 1 37. Bardorff-Nielse, O. & N. Shephard (26) Ecoometrics of Testig for Jumps i Fiacial Ecoomics usig Bipower Variatio. Joural of Fiacial Ecoometrics 4, 1 3. Black, F. (1976) Studies of stock price volatility chages. Proceedigs of the Busiess ad Ecoomics Sectio of the America Statistical Associatio, 177 181. Bollerslev, T. & V. Todorov (211) Estimatio of Jump Tails. Ecoomtrica 79, 1727 17783. Dowloaded from http:/www.cambridge.org/core. Duke Uiversity Libraries, o 15 Dec 216 at 17:46:9, subject to the Cambridge Core terms of