Estimating the Spot Covariation of Asset Prices. Statistical Theory and Empirical Evidence SFB E C O N O M I C R I S K B E R L I N

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1 SFB 649 Dscusso Paper Estmatg the Spot Covarato of Asset Prces Statstcal Theory ad Emprcal Evdece Markus Bbger* Markus Ress* Nkolaus Hautsch** Peter Malec*** *Humboldt-Uverstät zu Berl, Germay **Uversty of Vea, Austra, ad Ceter for Facal Studes, Germay ***Uversty of Cambrdge, Uted Kdom SFB E C O N O M I C R I S K B E R L I N Ths research was supported by the Deutsche Forschugsgemeschaft through the SFB 649 "Ecoomc Rsk". ISSN SFB 649, Humboldt-Uverstät zu Berl Spadauer Straße 1, D Berl

2 Estmatg the Spot Covarato of Asset Prces Statstcal Theory ad Emprcal Evdece Markus Bbger Nkolaus Hautsch Peter Malec Markus Ress October 014 Abstract We propose a ew estmator for the spot covarace matrx of a mult-dmesoal cotuous sem-martgale log asset prce process whch s subject to ose ad o-sychroous observatos. The estmator s costructed based o a local average of block-wse parametrc spectral covarace estmates. The latter orgate from a local method of momets LMM whch recetly has bee troduced by Bbger et al We exted the LMM estmator to allow for autocorrelated ose ad propose a method to adaptvely fer the autocorrelatos from the data. We prove the cosstecy ad asymptotc ormalty of the proposed spot covarace estmator. Based o extesve smulatos we provde emprcal gudace o the optmal mplemetato of the estmator ad apply t to hgh-frequecy data of a cross-secto of NASDAQ blue chp stocks. Employg the estmator to estmate spot covaraces, correlatos ad betas ormal but also extreme-evet perods yelds ovel sghts to traday covarace ad correlato dyamcs. We show that traday co-varatos follow uderlyg perodcty patters, reveal substatal traday varablty assocated wth co-varato rsk, are strogly serally correlated, ad v ca crease strogly ad early stataeously f ew formato arrves. Keywords: local method of momets, spot covarace, smoothg, traday co-varato rsk JEL classfcato: C58, C14, C3 Facal support from the Deutsche Forschugsgemeschaft va SFB 649 Ecoomc Rsk ad FOR 1735 Structural Iferece Statstcs: Adaptato ad Effcecy s gratefully ackowledged. Hautsch also ackowledges facal support from the Weer Wsseschafts-, Forschugs- ud Techologefods WWTF. Malec thaks the Cambrdge INET for facal support. Departmet of Mathematcs, Humboldt-Uverstät zu Berl. Emal: bbger@math.hu-berl.de. Address: Uter de Lde 6, D Berl, Germay. Faculty of Busess, Ecoomcs ad Statstcs, Uversty of Vea ad Ceter for Facal Studes, Frakfurt. Emal: kolaus.hautsch@uve.ac.at. Address: Oskar-Morgester-Platz 1, A-1090 Vea, Austra. Faculty of Ecoomcs, Uversty of Cambrdge. Emal: pm563@cam.ac.uk. Address: Sdgwck Aveue, Cambrdge CB3 9DD, Uted Kgdom. Departmet of Mathematcs, Humboldt-Uverstät zu Berl. Emal: mress@math.hu-berl.de. Address: Uter de Lde 6, D Berl, Germay. 1

3 1 Itroducto Recet lterature facal ecoometrcs ad emprcal face reports strog emprcal evdece for dstct tme varatos daly ad log-term correlatos betwee asset prces. Clearly rejectg the costacy of covaraces over tme uderles the mportace of sutable ecoometrc models to capture covarace dyamcs ad challeges rsk maagemet, portfolo maagemet ad asset prcg. Surprsgly lttle, however, s kow about traday varatos of asset retur covaraces. Whle the lterature proposes several approaches to estmate spot varaces 1, there s a lack of emprcal approaches ad correspodg statstcal theory to estmate spot covaraces usg hgh-frequecy data. I ths paper, we am at fllg ths gap the lterature ad propose a estmator for the spot covarace matrx of a mult-dmesoal cotuous sem-martgale log asset prce process whch s observed at o-sychroous tmes uder ose. The estmator s costructed based o local averages of block-wse parametrc spectral covarace estmates. The latter are estmated employg the local method of momets LMM estmator proposed by Bbger et al. 014, whch s show to be a rate-optmal ad asymptotcally effcet estmator for the tegrated covarato. As the LMM estmator bulds o locally costat approxmatos of the uderlyg covarace process ad estmates them block-wse, t provdes a atural settg to costruct a spot covarace estmator. Our methodologcal cotrbuto s as follows: Frst, we exted the LMM method to allow for autocorrelated market mcrostructure ose ad propose cosstet estmators of the autocorrelatos. Secod, we derve a stable cetral lmt theorem, showg the cosstecy ad asymptotc ormalty of the resultg spot covarace estmator. Apart from beg to our best kowledge the frst estmator for a spot covarace matrx, a mportat result s that the rate-optmalty of the uderlyg LMM estmator carres over to the spot estmator. It s show that t coverges cosderably faster tha exstg spot estmators. We provde smulato-based evdece o a optmal mplemetato of the estmator depedg o the choce of uderlyg smoothg parameters. Fally, a mportat objectve of ths paper s to provde frst emprcal evdece o the traday behavor of spot covaraces, correlatos ad asset prce betas. The quatfcato of spot covarace estmators s useful for traday rsk maagemet, market mcrostructure research, but also for market survellace ad motorg. For stace, wth the avalablty of estmates of co-volatltes the market, traday traders ca assess traday correlato rsks. Emprcal studes o the role of hgh-frequecy tradg, the mpact of market fragmetato ad the usefuless of volatlty crcut breakers mght heavly 1 See, e.g., Krstese 010, Myklad ad Zhag 008, Mac et al. 01, Bos et al. 01 or Zu ad Boswjk 014.

4 beeft from the avalablty of hgh-frequecy covarace estmators whch are applcable hgher dmesos. Moreover, aalyzg the behavor of spot co-volatltes o days wth dstct formato arrval or flash crashes provdes mportat sghts to hgh-frequecy depedece structures ad ther reacto to commo shocks. For example, t s a emprcally well-documeted fact that correlatos are hgher ad dversfcato opportutes are smaller durg bear markets tha durg bull markets at a lower frequecy level see, e.g., De Sats ad Gerard, 1997; Log ad Solk, 001. Ths rases the questo f such chagg correlato structures mght be observed also o a traday level, e.g., durg flash crash perods. Fally, spot covarace estmates are a ecessary buldg block for co-jump tests see Bbger ad Wkelma, 013. Our study s maly related to two felds of lterature. Frst, there s a vast body of papers o the estmato of tegrated covarace matrces, whle accoutg for market mcrostructure ose ad the asycrocty of observatos. Startg from the semal realzed covarace estmator by Bardorff-Nelse ad Shephard 004 whch eglects both types of frctos, Hayash ad Yoshda 011 propose a cosstet ad effcet estmator uder asychrocty, but the absece of mcrostructure ose. Estmators accoutg for both types of frctos are, amog others, the quas maxmum lkelhood estmator by At-Sahala et al. 010, the multvarate realzed kerel estmator by Bardorff-Nelse et al. 011, the multvarate preaveragg estmator by Chrstese et al. 013, the two-scale estmator by Zhag 011, ad the LMM estmator by Bbger et al Secod, there s cosderable lterature o spot volatlty estmato. A oparametrc kerel-type estmator the absece of mcrostructure ose s put forward by Foster ad Nelso 1996, Fa ad Wag 008 ad Krstese 010. To accout for ose, the predomat approach s to compute a dfferece quotet based o a ose-robust tegrated volatlty estmator, e.g., the uvarate realzed kerel, the preaveragg estmator or the two-scale estmator. Here, examples clude Myklad ad Zhag 008, Mac et al. 01, Bos et al. 01 ad Zu ad Boswjk 014. Fally, estmators that are robust to jumps, but eglect mcrostructure ose are put forward, e.g., by At-Sahala ad Jacod 009, Aderse et al. 009 ad Bad ad Reo 009. Iterestgly, the problem of estmatg the spot covarace matrx the presece of mcrostructure ose ad asychrocty effects has ot yet bee addressed a study o ts ow. Our paper thus brdges the gap betwee the two felds of lterature outled above. Compared to tegrated co-varace estmators, spot estmators heretly feature slower covergece rates due to the addtoal smoothg volved. Hece, to maxmze precso, as may observatos as possble should be used. The latter ca be acheved by employg data sampled at the hghest frequecy possble,.e., at tck-by-tck level. Hase ad Lude 006 ad, more recetly, At-Sahala et al. 011, however, show that at ths frequecy, mcrostructure ose appears 3

5 to volate the tradtoal..d. assumpto, exhbtg more complex depedece structures. Extedg the LMM estmator to allow for serally depedet ose ad proposg a datadrve procedure to select the order of seral depedece s therefore a crucal step to make the estmator applcable to spot covarace estmato ad to beeft from ts optmalty propertes. I partcular, we show that t satsfes a cetral lmt theorem at almost optmal rate. The approach preseted here does ot accout for jumps the log-prce process. From a methodologcal pot of vew, a exteso to dsetagle jumps ad cotuous compoets utlzg a trucato techque as Bbger ad Wkelma 013 appears feasble. I the gve framework, however, due to addtoal tug parameters volved ths would requre a comprehesve exteso, whch would dlute the ma ew estmato deas. Cosequetly, our proposed spot covarace estmator does ot separate betwee a dffusve ad jump compoet. For our emprcal results ad correspodg coclusos, ths s ot a lmtato sce ay case, potetal jumps are cosstetly captured by the spot estmator. Moreover, Chrstese et al. 014 show that, whe cosderg data sampled at the tck-by-tck level, jumps are detected far less frequetly tha based o a coarser samplg grd. Based o extesve smulato studes, we vestgate the mpact of dfferet choces of the relevat put parameters o the estmator s fte sample precso ad provde gudace of how to choose these parameters applcatos. Moreover, we demostrate that the proposed procedure for estmatg the order of seral depedece the mcrostructure ose process performs well uder realstc codtos. Applyg the spot covarace estmator to four years of trade ad quote data for 30 of the most lqud costtuets of the NASDAQ100 ad a ETF trackg the latter, provde ovel emprcal evdece o the traday behavor of covaraces ad correlatos. Frst, there s a dstct traday seasoalty patter wth covaraces declg ad correlatos creasg throughout the day, whle betas rema relatvely stable. Secod, spot co-varato reveals substatal traday varablty ad thus reflect co-varato rsk. Thrd, spot co-varato s strogly serally correlated wth a day ad across days. Fally, spot covaraces ad correlatos substatally chage durg flash crashes or the arrval of fudametal formato. We show that these extreme scearos, spot covaraces ad varaces of the 30 blue chps strogly ad early stataeously shoot up, causg the resultg correlatos ether to substatally crease or decrease. These ovel sghts show that depedece structures betwee assets ca be very varable ad sestve to ews. It turs out that our estmator s able to capture extreme co-varace movemets o a hgh tme resoluto. The remader of the paper s structured as follows. Secto troduces the proposed spot estmator, gves ts asymptotc propertes ad shows how to estmate uderlyg ose autocovaraces. I Secto 3, we preset a smulato study aalyzg the estmator s sestvty 4

6 to the choce of put parameters ad examg the performace of the procedure for estmatg the autocovarace structure of the ose process. Secto 4 provdes emprcal evdece o spot co-varaces, correlatos ad betas based o NASDAQ data. Fally, Secto 5 cocludes. Estmato of Spot Covaraces.1 Theoretcal Setup ad Assumptos Let X t t 0 deote the d-dmesoal effcet log-prce process. I le wth the lterature ad motvated by well-kow o-arbtrage argumets, we assume that X t follows a cotuous Itô sem-martgale X t = X 0 + t 0 b s ds + t 0 σ s db s, t [0, 1], 1 defed o a fltered probablty space Ω, F, F t 0, P wth drft b s, d-dmesoal stadard Browa moto B s ad stataeous volatlty matrx σ s. The latter yelds the d d- dmesoal spot covarace matrx Σ s = σ s σs, whch s our object of terest. We cosder a settg whch dscrete ad o-sychroous observatos of the process 1 are dluted by market mcrostructure ose,.e., Y p = X p + ɛ p t p, = 0,..., p, p = 1,..., d, wth observato tmes t p, ad observato errors ɛ p. Observed returs for compoet p {1,..., d} are gve by Y p = Y p = X p t p Y p 1 = X p + ɛ p 3 X p + ɛ p t p ɛ p 1, = 1,..., p. 1 Let = m 1 p d p deote the umber of observatos of the slowest asset. I Secto.3, we cosder hgh-frequecy asymptotcs wth / p ν p for costats 0 < ν p 1, such that the asymptotc varace-covarace matrces for estmators of Σ s are regular. Feasble ferece o Σ based o the cosdered methods, however, s tractable eve a broader framework wth dfferet speeds creasg sample szes. For ths techcal geeralzato, we refer to Bbger et al Below we summarze the assumptos o the stataeous volatlty matrx ad drft process, ose propertes ad observato tmes. Covergece rates of spot estmators crucally deped o the smoothess of the uderlyg fuctos. We therefore cosder balls Hölder 5

7 spaces of order α 0, 1] ad wth radus R > 0: C α,r fx fy [0, 1], E = {f : [0, 1] E f α R}, f α := f + sup x y x y α, where deotes the usual spectral orm ad f := sup t [0,1] ft for fuctos o [0, 1]. I our setup, we have E = R d d for matrx-valued fuctos, E = R d for vectors or E = [0, 1] for dstrbuto fuctos. Frst, for the drft process 1, we oly assume a very mld regularty: Assumpto 1. b s s [0,1] s a F s -adapted process wth b s C ν,r [0, 1], R d for some R < ad some ν > 0. Furthermore, the assumptos regardg the stataeous volatlty matrx process 1 ca be summarzed as: Assumpto. σ s s [0,1] follows a F s -adapted process satsfyg Σ s = σ s σs Σ uformly for some strctly postve defte matrx Σ. σ s s [0,1] satsfes σ s = f σ s 1, σ s wth f : R d d R d d beg a cotuously dfferetable fucto all coordates, where For α 0, 1/], σ 1 s s a cotuous Itô sem-martgale of the form 1 wth càdlàg adapted volatlty of volatlty σ s satsfyg Σ s = σ s σ s Σ uformly for some strctly postve defte matrx Σ. The drft of σ 1 s a adapted càdlàg process. For α 1/, 1], σ 1 vashes. σ s C α,r [0, 1], R d d wth some R <. Hece, σ s s a fucto of a cotuous Itô sem-martgale σ s 1 ad a addtoal compoet σ s. The latter ca capture traday perodcty effects see, e.g., Aderse ad Bollerslev, The larger α, the more restrctve becomes Assumpto. If α > 1/, the semmartgale compoet vashes ad σ s s exclusvely drve by the compoet σ s. Hece, the more terestg case s α 1/. Importatly, the above assumptos also allow for leverage effects,.e., a o-zero correlato betwee σ s ad the Browa moto B s 1. It s atural to develop results uder ths geeral smoothess assumpto as t s commoly kow that oparametrc estmato problems, the uderlyg regularty determes the sze of smoothg wdows ad a fortor the resultg optmal covergece rates. Our assumptos o the mcrostructure ose process are stated observato tme, whch s le wth, e.g., Hase ad Lude 006 ad Bardorff-Nelse et al. 011: 6

8 Assumpto 3. ɛ = {ɛ p, = 0,..., p, p = 1,..., d} s depedet of X ad has depedet compoets,.e., ɛ p At least the frst eght momets of ɛ p ɛ p Cov ɛ p s depedet of ɛ q j for all, j ad p q., = 0,..., p, exst for each p = 1,..., d., = 0,..., p, follows a R-depedet process for some R <, mplyg that, ɛ p +u = 0 for u > R ad each p = 1,..., d. Defe by η p = η p 0 + R u=1 η u p, wth η u p := Cov ɛ p, ɛ p +u, u R, 4 the compoet-wse log-ru ose varaces, where the η u p, 0 u R, are costat for all 0 u. We mpose that η p > 0 for all p. The depedece betwee ose ad the effcet prce, as stated part of Assumpto 3, s stadard the lterature see, e.g., Zhag et al., 005. Cosderg serally depedet ose s o-stadard ad motvated by emprcal results, e.g., Hase ad Lude 006. The movg-average-type depedece structure the ose process part of Assumpto 3 follows, e.g., Hautsch ad Podolskj 013, mplyg the log-ru varace 4. Fally, we assume that the tmg of observatos s drve by c.d.f. s F p goverg the trasformatos of observato tmes to equdstat samplg schemes by meas of sutable quatle trasformatos: Assumpto 4. There exst dfferetable cumulatve dstrbuto fuctos F p, p = 1,..., d, such that the observato regmes satsfy t p = Fp 1 / p, 0 p, p {1,..., d}, where F p C α,r [0, 1], [0, 1], p = 1,..., d, wth α beg the smoothess expoet Assumpto for some R <. Assumpto 4 mples that observato tmes are ether o-stochastc or radom, but depedet from the log-prce process. A treatmet of edogeous tmes the gve theoretcal framework s beyod the scope of ths paper. See Koke 014 for a recet study of edogeous tmes ad L et al. 014 for a study a settg eglectg mcrostructure ose. Combg tme-varat log-ru ose varaces η p ad locally dfferet observato frequeces from Assumptos 3 ad 4 mples locally varyg ose levels: Defto 1. I the asymptotc framework wth / p ν p, where 0 < ν p <, p = 1,..., d, for, we defe the cotuous-tme ose level matrx H s = dag η p ν p F 1 p s 1 / 1 p d. 5 For the case of edogeous ose, see, e.g., Kala ad Lto

9 Note that for equally-spaced observatos, we have F p s = s, such that F 1 p s = 1. The, the p-specfc asymptotc ose level s η p ν p 1/ wth ν p expressg the verse of the sample sze of the p-th process relatve to the slowest process. Hece, havg less frequet observatos o a sub-terval s equvalet to havg hgher ose dluto by mcrostructure effects o ths sub-terval. Ths terplay betwee ose ad lqudty has bee dscussed by Bbger et al Local Method of Momets Estmato of the Spot Covarace Matrx Our approach for estmatg the stataeous covarace matrx rests upo the cocept of the local method of momets LMM troduced Bbger et al We partto the terval [0, 1] to equdstat blocks [kh, k + 1h ], k = 0,..., h 1 1, wth the block legth h asymptotcally shrkg to zero, h 0 as. The key dea s to approxmate the uderlyg process 1 model by a process wth block-wse costat covarace matrces ad ose levels. I the more smplfed settg of Bbger et al. 014, t s show that such a locally costat approxmato duces a estmato error for the tegrated covarato, whch, however, ca be asymptotcally eglected for suffcet smoothess of Σ t ad F p f the block szes h shrk suffcetly fast wth creasg. Ths opes the path to costruct a asymptotcally effcet estmator of the tegrated covarato matrx based o optmal block-wse estmates. I the preset settg, we buld o the dea of block-wse costat approxmatos of the uderlyg covarace ad ose process ad show that t allows costructg a cosstet spot covarace estmator, whch ca atta a optmal rate. A major buldg block s the costructo of a ubased estmator of the block-wse covarace matrx Σ kh = σ kh σ kh based o the local spectral statstcs S jk = πjh 1 p =1 Y p p t Y p 1 Φ jk 1 + tp 1 p d, 6 where Φ jk deote orthogoal se fuctos wth spectral frequecy j, whose dervatves Φ jk form aother orthogoal system correspodg to the egefuctos of the covarace operator of a Browa moto, ad are gve by Φ jk t = h jπ s jπh 1 t kh 1 [kh,k+1h t, j

10 The statstcs 6 de-correlate the osy observatos 3 ad ca be thought of as represetg ther block-wse prcpal compoets. 3 They bear some resemblace to the pre-averaged returs as employed Jacod et al Whle pre-averagg estmators, however, utlze rollg local wdows aroud each observato, our approach reles o fxed blocks ad optmal combatos the spectral frequecy doma. It ca be show that CovS jk = Σ kh + π j h H k 1 + O1, 8 where H k deotes the block-wse costat dagoal ose level matrx wth etres H pp k = 1 p η p Fp 1 kh, 9 ad Fp 1 kh deotes the verse of the block-wse costatly approxmated observato frequecy evaluated at kh. The relato 8 suggests estmatg Σ kh based o the emprcal covarace S jk S jk, whch s bas-corrected by the ose-duced term π j h H k. A tal pre- estmator of the spot covarace matrx at tme s [0, 1], Σ s, s the costructed based o bas-corrected block-wse emprcal covaraces S jk Sjk, whch are averaged across spectral frequeces j = 1,..., J, p ad a set of adjacet blocks, vec ˆΣpre kh = Us, L s, U s, k=l s, J p 1 J p j=1 vec S jk Sjk π j h Ĥ k, 10 wth the vec operator stackg the elemets of a d d-matrx to a d 1 -vector columwse, whle L s, = max{ sh 1 K, 0} ad U s, = m{ sh 1 +K, h 1 1}, such that the legth of the smoothg wdow obeys U s, L s, + 1 K + 1. Ĥ k s a -cosstet estmator of H k wth p-th dagoal elemet Ĥ k pp = ˆη p h kh t p k+1h t p t p Detals o the costructo of the estmator of the compoet-wse log-ru ose varaces, ˆη p, are provded Secto.5 below. For each spectral frequecy j, the statstc S jk Sjk π j h Ĥ k s a asymptotcally ubased though effcet estmator of Σ kh. Averagg across dfferet frequeces therefore creases the estmator s effcecy. Equally weghtg as 10, however, s ot ecessarly optmal. A more effcet estmator ca be devsed by cosderg 10 as the pre-estmated spot 3 Somewhat related approaches for a uvarate framework ca be foud Hase et al. 008 ad Curc ad Cors 01. 9

11 covarace matrx ad the, derve estmated optmal weght matrces Ŵj, yeldg the fal LMM spot covarace matrx estmator as vec ˆΣs = Us, L s, U s, J k=l s, j=1 Ŵ j Ĥ k, vec ˆΣ pre kh S jk Sjk π j h Ĥ k. 1 As outled detal Bbger et al. 014, the true optmal weghts are gve proportoally to the local Fsher formato matrces accordg to W j H k, Σ kh = J u=1 = I 1 k I jk, Σkh + π u h H 1Σkh k + π j h H k 13 wth I jk beg the Fsher formato matrx assocated wth block k ad spectral frequecy j, gve by I jk = Σ kh + π j h H k, 14 ad I k = J j=1 I jk deotg the k-specfc Fsher formato explotg the depedece across frequeces j. Here, A = A A deotes the Kroecker product of a matrx wth tself ad A = A 1 A 1 = A A 1. We show Secto.3 that the estmator 1, whch bulds o the dealzed model cosdered Bbger et al. 014, s cosstet ad satsfes a stable CLT uder the more realstc ad geeral assumptos of Secto.1. For practcal purposes, ote that, whle both the plot estmator 10 ad the LMM estmator 1 are symmetrc, oe s guarateed to yeld postve sem-defte estmates. Cofdece s based o estmated Fsher formato matrces, whch are by costructo postve-defte. For the estmates themselves, we ca set egatve egevalues equal to zero, whch s tatamout to a projecto o the space of postve sem-defte matrces. Ths adjustmet does ot affect the asymptotc propertes of the estmator as outled below, but oly mproves the fte sample performace..3 Asymptotc Propertes As a prerequste for the dscusso of the cetral lmt theorem for the estmator 1, some cosderatos regardg K, whch determes the legth of the smoothg wdow, are eeded. For ths purpose, suppose that a certa smoothess α 0, 1] of the stata- 10

12 eous volatlty matrx s grated accordg to Assumpto. The, a smple computato yelds COV ˆΣs = O K 1, mplyg a bas-varace trade-off the mea square error MSE ˆΣs [ := E ˆΣ s Σ s ]. More precsely, for a specfc α > 0, we have MSE ˆΣs = OP K 1 + OP K α h α, 15 where the frst term orgates from the varace ad the secod term s duced by the squared bas. Cosequetly, for gve h log 1/, whch optmally balaces ose ad dscretzato error as derved Bbger et al. 014, choosg K α/α+1 mmzes the MSE ad facltates a estmator wth K covergece rate. Fally, the desred cetral lmt theorem for the estmator 1 requres a slght udersmoothg, resultg a smaller choce of K : Theorem 1. We assume a setup wth observatos of the type, a sgal 1 ad the valdty of Assumptos 1-4. The, for h = κ 1 log 1/, K = κ β log 1 wth costats κ 1, κ ad 0 < β < αα+1 1, for J ad / p ν p wth 0 < ν p <, p = 1,..., d, as, the spot covarace matrx estmator 1 satsfes the potwse stable cetral lmt theorem: β/ vec ˆΣs d st Σ s N 0, Σ Σ 1/ H + Σ1/ H, Σ s Z s [0, 1], 16 where Σ H = H H 1 ΣH 1 1/ H, wth ose level H from 5 ad Z = COVvecZZ for Z N0, E d beg a stadard ormally dstrbuted radom vector. Theorem 1 s proved Appedx A below. The covergece 16 s stable, whch s equvalet to jot weak covergece wth ay measurable bouded radom varable defed o the same probablty space as X. Ths allows for a feasble verso of the lmt theorem f we re-scale the estmator by the mplctly obtaed estmated varace: Corollary 1. Uder the assumptos of Theorem 1, the spot covarace matrx estmator 1 satsfes the feasble cetral lmt theorem gve by U s, L s, + 1 1/ 1/ d ˆV s vec ˆΣs Σ s N 0, Z, s [0, 1], 17a U s, where ˆV s = U s, L s, k=l s, J 1 Î jk, j=1 17b 11

13 wth U s, ad L s, defed as 10 ad 1. Îjk s defed accordg to 14 wth H k ad Σ kh, k = 0,..., h 1 1, replaced by the estmators 10 ad 11, respectvely. Ulke 16, whch we obta a mxed ormal lmtg dstrbuto, the matrx Z s completely kow. It s gve by twce the symmetrzer matrx troduced by Abadr ad Magus 005, ch. 11 ad correspods to the covarace structure of the emprcal covarace of a d-dmesoal stadard Gaussa vector. The asymptotc varace-covarace matrx 16 s the same stataeous process that appears tegrated over [0, 1] as varace-covarace matrx of the tegrated covarace matrx estmator Bbger et al Accordgly, Theorem 1 s le wth the results o classcal realzed volatlty the absece of ose for d = 1 ad the oparametrc Nadaraya-Watsotype kerel estmator by Krstese 010 wth asymptotc varace σs 4 R k z dz, where k deotes the used kerel. I our case, the estmator s of hstogram-type ad the rectagle kerel does ot appear the asymptotc varace. Let us pot out that estmator 1, buldg o optmal combatos over spectral frequeces, s more advaced tha a usual hstogramestmator. Whe comparg our oparametrc estmator 1, e.g., to the aforemetoed oe by Krstese 010, our case, the actual badwdth s K + 1h or smaller, sce we smooth over up to K + 1 adjacet blocks of legth h. I ths cotext, oe ca as well thk of employg h 1 de-osed block statstcs as uderlyg observatos. Regardg the covergece rate 16, we may focus o the case α 1/, whch s tatamout to the spot volatlty matrx process σ s s [0,1] beg at least as smooth as a cotuous sem-martgale. Ths assumpto yelds the rate 1/8 ε, for ay ε > 0, such that we almost atta the optmal rate 1/8, whch s obvously lower tha the correspodg rate for tegrated co-varace estmators the settg wth ose, 1/4. Notably, our spot covarace matrx estmator 1 coverges cosderably faster tha exstg ose-robust spot volatlty estmators based o the dfferece quotet of, e.g., pre-averagg estmates Zu ad Boswjk, 014. Though the udersmoothed estmator satsfyg 16 has a slghtly slower covergece rate tha the optmal oe, the two-step approach 1 wth combatos over dfferet frequeces strogly reduces the estmator s varace compared to smpler methods. Ths s well cofrmed our fte-sample smulatos Secto 3. Theorem 1 ad Corollary 1 hold for estmato pots s [0, 1], both the teror ad the boudary rego of the ut terval. Ths result s a cosequece of the estmators 10 ad 1 beg of hstogram-type, mplyg that smoothg s coducted by averagg over a set of adjacet blocks, whch merely eeds to cota tme t, ad does ot have to be cetered aroud the pot of estmato. The above property s a major dfferece compared to kerel-based spot volatlty estmators, as e.g., the oe proposed by Krstese 010, whch requre sutable 1

14 correcto methods to elmate the so-called boudary bas, thus makg mplemetato more volved. Fally, Theorem 1 may be employed to deduce asymptotc results for the estmators of spot correlatos ad spot betas. These ca be cosdered as the stataeous couterparts to the tegrated quattes studed, e.g., Aderse et al. 003 ad Bardorff-Nelse ad Shephard 004. I ths cotext, focus o those elemets of the spot covarace matrx Σ t, t [0, 1], volvg oly the dces p, q {1,..., d}. Further, deote the spot correlato ad beta estmators based o 1 by ˆρ pq pq s = ˆΣ s / ˆΣ s pp ˆΣqq pq pq s ad ˆβ s = ˆΣ s /ˆΣ pp s. The, Theorem 1 mples by smple applcato of the Delta-method that wth β/ pq vec ˆρ s β/ vec ˆβpq s ρ pq d st s β s pq d st N N 0, AV ρ,s, s [0, 1], 18a 0, AV β,s, s [0, 1], 18b AV ρ,s = Σ pp s + Σ qq s AV p 1d+q,p 1d+q s + Σ pq s 4 Σ qq s Σpq s Σ qq s AV β,s = Σ pp s Σ p,q s 3Σ pp s Σ pq s 4 Σ pp s AV q 1d+q,q 1d+q s AV p 1d+q,q 1d+q s + Σ pq s Σ pp s 3Σ qq s Σpq s Σ pp s Σ s qq AV p 1d+p,p 1d+p s AV p 1d+q,p 1d+p s AV p 1d+p,q 1d+q s, AV p 1d+q,p 1d+q s + Σ pq s Σ pp 4AV p 1d+p,p 1d+p s s Σ p,p 3AV p 1d+q,p 1d+p s s, 18c 18d where AV s deotes the asymptotc varace-covarace matrx 16. Feasble versos of the cetral lmt theorems 18a ad 18b ca be readly obtaed aalogously to Corollary 1..4 Choce of Iputs The proposed spot covarace matrx estmator 1 depeds o four put parameters to be chose: the block legth h, the maxmum spectral frequecy J, the maxmum frequecy for the pre-estmator 10, J, p as well as v the legth of the smoothg wdow, K. For, Theorem 1 requres that h = O log 1/. s gve by m p p h, but a spectral cut-off J = O log ca be chose, sce the weghts decay fast wth creasg frequecy j, makg hgher frequeces asymptotcally eglgble. The effect of 13

15 quckly dmshg weghts mples that should be fxed at a value ot too large, e.g., J p = 5. The reaso s that the cut-off drectly determes the uform weghts the preestmator 10. For v, we geerally set K = O α/α+1. The latter choce mples udersmoothg, thereby forfetg rate-optmalty of the estmator, but provdes us a cetral lmt theorem. Uder the cotuous sem-martgale or smoother assumpto α 1/ for the spot volatlty matrx process, whch seems admssble most facal applcatos, we set K = O 1/4 ε for ay ε > 0. I practce, we troduce proportoalty parameters for, ad v,.e. h = θ h log 1/, J = θ J log ad K = θ K 1/4 δ, where θ h, θ J, θ K > 0 ad δ deotes a small postve umber. Ths approach s le wth the selecto of the wdow legth the cotext of pre-averagg estmators Jacod et al., 009. We dscuss the specfc choce of the above put parameters more detal Sectos 3 ad Estmatg Nose Autocovaraces Accordg to part of Assumpto 3, the estmato of the log-ru ose varace η p, p = 1,..., d, defed 4, oly requres estmates of compoet-wse auto-covaraces, but o covaraces across processes. Therefore, for ease of exposto, we restrct the aalyss to d = 1, focusg o a oe-dmesoal model wth + 1 observatos of Y = X t + ɛ, = 0,...,. Further, we set η u = η u 1 = Covɛ, ɛ +u for the u-th order autocovarace, whle the log-ru varace s ow smply deoted by η. Followg part of Assumpto 3, for some R 0, we may eglect all depedeces η u, u > R. Hece, ɛ, = 0,...,, s a R-depedet process ad the returs Y have a MAR-structure. Fx R 0 as the order of seral depedece. We shall dscuss below how to choose R from the data practce. We successvely estmate the autocovaraces by ˆη R = 1 ˆη r ˆη r+1 = 1 Y R r + 1 Y +r Y, =1 Y + 1 r=1 =1 u =1 u=1 =1 r Y +u Y, 0 r R 1. 19a 19b I partcular, ths cludes ˆη 0 ˆη 1 = 1 Y, 19c whch s the classcal estmator of η 0 a..d. settg as Zhag et al The estmators are -cosstet ad satsfy cetral lmt theorems. To costruct a estmator for the varace =1 14

16 of ˆη r, deote for q, r, r {0,..., R} Γ rr q r q+r = 1 Y +r Y, +q Y +q+r Y [ˆη r ˆη r+1 ˆη r 1 ˆη r ] 0 =1 [ˆη r ˆη r +1 ˆη r 1 ˆη r ], where ˆη r ˆη r+1, ˆη r 1 ˆη r, ˆη r ˆη r +1 ad ˆη r 1 ˆη r are computed accordg to 19b. The, the varace of ˆη r, 0 r R, s cosstetly estmated by Varˆη r = 1 Vr+1 + Vr + Cr,r+1, 1a wth Cr,r+1 = ˆΓ r r r+1 ˆΓ 00 ˆΓuu R u ˆΓ uu q, 1b u=1 u=0 u =1 ad Vr = Cr,r. Partcularly, for r = R, we have Varˆη R = 1 VR. Below, we gve a feasble cetral lmt theorem, whch etals a asymptotc dstrbuto-free test of the hypotheses H Q 0 : η u = 0 for all u Q, Q = R + 1. Theorem. Uder Assumpto 3, the followg cetral lmt theorem apples to the estmators defed by 19a ad 19b: V r q=1 + V r+1 + C r,r+1 1/ˆη r η r d N0, 1. Cosequetly, uder H Q 0 : T QY = /VQ ˆη Q d N0, 1. 3 Theorem s proved Appedx A. The statstc TQ Y serves as a test statstc for the sgfcace of o-zero autocovaraces for certa lags. A accurate strategy to select the order of seral depedece R thus requres computg the test statstcs T Q Y for Q Q = R + 1 large eough ad corporatg all autocovaraces utl the frst hypothess of a zero autocovarace caot be rejected for a gve sgfcace level. The, deotg the determed order by ˆR, a estmate of the log-ru ose varace, ˆη, s obtaed accordg to 4 based o the dvdual estmates ˆη 0,..., ˆη ˆR. 3 Smulato Study We coduct a smulato study to exame two ssues. Frst, we aalyze the mpact of dfferet choces of the put parameters θ h, θ J ad θ K troduced Secto.4 o the estmator s 15

17 fte sample performace. Secod, we study the precso of the procedure for estmatg the log-ru ose varace η as outled Secto.5. As these ssues are ot related to the dmesoalty of the process, we set d = 1 for reasos of parsmoy. We assume that the effcet log-prce process corporates both a stochastc ad a ostochastc seasoal volatlty compoet, whch s modeled by a Flexble Fourer Form as troduced by Gallat Correspodgly, we assume the uderlyg process as gve by dx t = µdt + φ t σ t db t, l φ t = αφ t + β φ t + 4a Q [γ φ,q cosπqt + δ φ,q sπqt], 4b q=1 where B t s a stadard Browa moto ad σ t the stochastc volatlty compoet. The drft term s set to µ = The seasoalty compoet s ormalzed such that 1 0 φ t dt = 1 wth parameter values gve by meda estmates for md-quotes of 30 hghly lqud costtuets of the NASDAQ See Secto 4.1 for a summary of the dataset. I ths cotext, we employ the estmato procedure by Aderse ad Bollerslev The stochastc volatlty compoet σ t s assumed to follow oe- ad two-factor models accordg to Huag ad Tauche 005. Detals are provded Appedx B. I all settgs, we smulate the process 4a ad 4b by a Euler dscretzato scheme based o a oe-secod grd assumg 3, 400 secods per tradg day, whle settg = 3, 400. We dlute the observatos of the effcet log-prce process by serally depedet mcrostructure ose wth R = 1,.e., ɛ = θ ɛ ɛ 1 + u, u N 0, η/ 1 + θ ɛ, = 1,...,. To esure that the absolute ose level s le wth the varato the volatlty process, we determe η by choosg the ose-to-sgal rato per trade ξ 1 := η/ dt see Oome, φ4 t σ4 t We cosder low-ose ad hgh-ose scearos, settg ξ = ad ξ = 1.941, respectvely. These umbers correspod to the frst ad thrd quartle of respectve estmates based o the NASDAQ data employed our emprcal applcatos. Here, η s estmated followg the procedure from Secto.5, whle the tegrated quartcty s approxmated by the squared sub-sampled fve-mute realzed varace. Fally, we choose θ ɛ = 0.441, yeldg a frst-order autocorrelato of η 1 = 0.6, whch s the meda estmate for the uderlyg NASDAQ data. To vestgate the mpact of chose put parameters, we compute the LMM estmator 1 over a grd of values for θ h, θ J ad θ K. For each combato, we evaluate a ormalzed root mea tegrated squared error, RMISE := M 1 M m=1 [ˆσ t,m / σ t,mφ ] t 1 dt, where 4 We estmate the daly volatlty compoet based o sub-sampled fve-mute realzed varaces stead of a parametrc GARCH approach. The umber of susods Q s chose based o the BIC

18 Table 1: RMISE-optmal values of θ h, θ J ad θ K based o grd search. Normalzed root mea tegrated squared errors M 1 M 1 m=1 [ˆσ 0 t,m / σ ] t,mφ t 1 dt computed for M = 3000 Mote Carlo replcatos. 1F ad F represet a oe- ad two-factor model for the stochastc volatlty compoet of the effcet log-prce, respectvely. ξ deotes the square root of the ose-to-sgal rato per trade,.e., ξ 1 := η/ 0 φ4 t σ t 4 dt. Normalzed RMISEs are reported percetage pots. ξ Vol. Spec. θ h θ J θ K RMISE F F F F ˆσ t,m s the spot varace estmate for tme t replcato m, σ t,m s the correspodg true stochastc volatlty compoet ad M s the umber of replcatos. Table 1 reports the values of the put parameters yeldg mmal RMISEs for M = For the settg cludg the oe-factor stochastc volatlty model 1F ad a low ose level, the RMISE-optmal values of the put parameters yeld a cofgurato wth h 1 = 77 blocks spag about 5 mutes each, a spectral cut-off J = 10 ad a smoothg wdow of K = 4 blocks. A hgher ose level mples a decrease the cut-off to J = 80 ad a legtheg of the smoothg wdow to K = 5 blocks. I the two-factor specfcato F, the rougher volatlty paths result a smaller optmal value of θ h, traslatg to h 1 = 305 blocks wth legth of oly about 1 mute ad 15 secods each. For the hgh ose level, the cut-off reduces to J = 15, whle the smoothg wdow legthes to K = 15. I geeral, we see that the two-step method 1 clearly outperforms a smple hstogram estmator whch reles oly o the frst frequecy J = 1. The effect of a devato from the RMISE-optmal values of θ h, θ J ad θ K s llustrated Fgure 1 for the oe-factor model assumg a hgh ose level. The plots depct RMISEs for a grd of two of the three put parameters, whle the thrd oe s fxed at ts optmal value. Evdetly, the performace of the estmator s ot overly sestve to the choce of θ h. Uless the latter s set to a extremely low value or θ K s extremely small, resultg excessve udersmoothg, the RMISE s farly stable. Regardg θ J, oly very small choces mply a cosderable reducto the estmator s precso, as the spectral frequeces are cut off too early. Fally, although θ K appears to be the most fluetal parameter, devatos from ts optmal value mply rather mld creases the RMISE both drectos. We ca coclude that the performace of the spot covarace matrx estmator 1 s qute robust for a rage of sesble put choces. To evaluate the proposed estmator of the log-ru ose varace, we cosder mcrostructure ose processes based o dfferet orders of seral depedece R. Hece, we assume 17

19 5 30 RMISE RMISE θ h θ J θ h θ K a θ K = 0.4 b θ J = RMISE θ J c θ h = θ K Fgure 1: RMISE depedg o put parameters oe-factor model ad hgh ose level. Normalzed root mea tegrated squared errors M 1 M 1 m=1 [ˆσ 0 t,m / σ ] t,mφ t 1 dt computed for M = 3000 Mote Carlo replcatos ad reported percetage pots. I each subplot, the remag put parameter s fxed at ts optmal value accordg to Table 1. 18

20 Table : Descrptve statstcs of the estmated order of seral depedece R ad estmated log-ru varace η of the ose process. R deotes the true order of depedece of the ose process. The settgs are: R = 0, R = 1 wth θ ɛ,1 = 0.5, ad R = wth θ ɛ,1 = 0.5 ad θ ɛ, = F ad F represet a oe- ad two-factor model for the stochastc volatlty compoet of the effcet log-prce, respectvely. ˆR ad ˆη are computed followg Secto.5 usg α = 0.05 ad Q = 15. BIASˆη ad STDˆη are re-scaled by Results based o M = 5000 Mote Carlo replcatos. R Vol. Spec. ˆR STD ˆR BIASˆη STDˆη 0 1F F F F F F ɛ = Θ R L u, Θ R L := R r=0 θ ɛ,rl r, θ ɛ,0 = 1, u N 0, η/θ R 1, = 1,...,. We employ the followg settgs: R = 0, R = 1 wth θ ɛ,1 = 0.5, ad R = wth θ ɛ,1 = 0.5 ad θ ɛ, = 0.3. Moreover, we select a hgh ose level by settg ξ = 3, whch ca be cosdered as a stress test for the proposed procedure. For M = 5000 replcatos, Table shows meas ad stadard devatos for the estmates of R, as well as bases ad stadard devatos for the estmates of the log-ru ose varace η based o α = 0.05 ad Q = 15. We observe that the procedure proposed Secto.5 slghtly over-estmates R, resultg more coservatve estmates of the order of seral depedece the ose process. Moreover, whe comparg the results for the oe- ad two-factor stochastc volatlty model, the precso of the estmate of R s smlar. The two-factor model, however, cosstetly mples a hgher volatlty the estmates of η. Geerally, we ca coclude that the proposed approach provdes a satsfactory precso a realstc scearo. 4 Emprcal Study 4.1 Data We cosder md-quote ad trade data at the hghest possble frequecy for 30 of the most lqud costtuets of the NASDAQ100 dex as well as the PowerShares QQQ Trust, a ETF trackg the NASDAQ100. The sample perod s from May 010 to Aprl 014. Md-quotes are computed from frst-level lmt order book data provded by the LOBSTER database. 5 The latter recostructs the order book from a message stream, whch s part of NASDAQ s hstorcal TotalVew-ITCH data ad cotas all lmt order submssos, cacellatos ad executos o each tradg day see Huag ad Polak, 011. Accordgly, the correspodg trasacto 5 See 19

21 data ca be read out from the above message fles drectly. A crucal advatage of the resultg datasets, e.g., compared to smlar oes sampled from the Trade ad Quote TAQ database, s the fact that all recorded evets are tme stamped wth at least mllsecod precso, whch allows for a ecoometrc aalyss at the hghest resoluto possble. Despte the clea ature of the data, resultg from the latter beg drectly take from NASDAQ s message stream, we hadle remag errors the trade ad md-quote samples usg the cleag procedures proposed by Bardorff-Nelse et al Table 3 gves summary statstcs of ask ad bd quotes recorded wheever the uderlyg lmt order book chages,.e., duced by a submsso of a lmt or market order or a cacellato of a exstg lmt order. 6 The umber of lmt order book updates s eormous, amoutg to several hudred thousads for some stocks. It turs out, however, that a cosderable amout of best ask/bd quote revsos equals zero. Ths s most remarkable for Mcrosoft, where ths umber amouts to more tha 99%. Sce ths sheer amout of data makes the computato of the estmators challegg ad cumbersome, we costruct the estmators employg quote revsos. Table 3 also reports the log-ru ose varace estmates computed accordg to Secto.5 ad the correspodg ose-to-sgal ratos per observato. It s show that there s a cosderable varato of the ose-to-sgal rato across the dfferet assets. Fally, we fd strog emprcal support for sgfcat autocorrelatos the ose process, clearly volatg the tradtoal..d. assumpto. We use the md-quote revsos for the NASDAQ100 costtuets to estmate spot covarace matrces accordg to 1, yeldg par-wse spot covaraces ad correlatos, as well as dvdual volatltes. Further, we also clude the QQQ ETF ad estmate spot covarace matrces to obta estmates of spot betas wth QQQ as market proxy. I both cases, we select the relevat puts as dscussed Secto.4. The correspodg proportoalty parameters are set to the values foud to be optmal the smulato study of Secto 3. Takg a more coservatve stace, we rely o the 1F -settg assumg a hgh ose level. 7 Hece, we set θ h = 0., θ J = 8, θ K = 0.4 ad J p = 5. Table 4 reports summary statstcs for the umber of blocks, the spectral cut-off ad the legth of the smoothg wdow as duced by the uderlyg data for both the etre sample ad each year. O average, we use approxmately 3 blocks per day, resultg a average block legth of 17 mutes. Spectral frequeces are cut off at early 54, whle the average legth of the smoothg wdow s about blocks, traslatg to roughly half a hour. Regardg the evoluto over tme, the average umber of blocks creases from the frst to secod year, 6 All tables ad fgures uderlyg the emprcal study are gve the Appedx C. 7 We alteratvely compute spot covaraces usg the cofgurato that s optmal the F -scearo ad employg trasacto prces ad fd qualtatvely smlar results. For sake of brevty, we do ot report them here. 0

22 subsequetly drops, oly to crease aga the last year. Each tme, chages the block umber are accompaed by moves of the spectral cut-off ad the legth of the smoothg wdow the same drecto. Notably, the varato all three puts s cosderably hgher the frst two years tha the last two years of the sample. 4. Itraday Behavor of Spot Co-Varaces Fgure shows the cross-sectoal decles of across-day averages of spot covaraces ad correlatos for each asset par. Fgure 3 depcts the uderlyg volatltes ad betas wth respect to the ETF trackg the NASDAQ100 dex. We observe dstct traday seasoalty patters. Covaraces clearly decle at the begg of the tradg day, stablze aroud oo o a wdely costat level ad slghtly crease before market closure. Iterestgly, the resultg correlatos show a reverse patter ad sgfcatly crease durg the frst tradg hour. The latter s caused by spot volatltes, that decay faster tha the correspodg covaraces at the begg of the tradg day. Hece, the co-varablty betwee assets s hghest after start of tradg whch mght be caused by the processg of commo formato. The assets dosycratc rsk, however, as reflected by spot volatltes, s eve hgher, overcompesatg the effect of hgh covaraces ad leadg to lower correlatos at the begg of the tradg day. Iterestgly, spot volatltes drop sgfcatly faster tha uderlyg covaraces durg the frst tradg hour. Shortly after opeg, spot volatltes are approxmately twce as hgh as the average daly based o the ope-to-close tegrated varace estmate volatlty, but strogly decle thereafter. Ths makes correlatos sharply creasg betwee 10:00 ad 11:00 am. Accordgly, we observe that meda spot correlatos rage betwee approxmately 0. ad 0.4 across a day. 8 Ths s cotrast to a daly correlato computed from the ope-to-close tegrated covarace estmate of approxmately 0.3 ad shows that eve o average, traday varablty of correlatos ad covaraces s substatal. Moreover, daly covaraces ted to vary wdely locksteps wth the volatlty of the uderlyg ETF, makg the correspodg betas close to be costat across the day. Hece, systematc rsk wth respect to the NASDAQ100 vares much less tha dosycratc rsk. I Fgure 4, we compute, for each asset par ad each pot durg the day, the stadard devato of spot covaraces ad correlatos across days. We observe that the across-day varablty covaraces s hghest after market opeg ad shortly before closure. A smlar pcture s also observed for spot volatltes Fgure 5. We assocate these patters wth effects arsg from overght formato processg the morg ad creased tradg actvtes the afteroo, where traders ted to re-balace or close postos before the ed of tradg. 8 Note that we obta very smlar pctures for absolute correlatos ad covaraces. Hece, durg the aalyzed perod, correlatos betwee NASDAQ100 assets are wdely postve. 1

23 Hece, dosycratc effects seem to become stroger durg these perods, creasg the varablty of co-varaces. Iterestgly, the across-day stadard devatos correlatos show a reverse patter. Thus, across-day varablty traday correlatos s lowest at the begg of tradg, creases utl md-day ad s wdely costat durg the afteroo hours. Here, creased across-day covarace ad volatlty rsk seem to compesate each other. Lkewse, the daly varablty of spot betas s wdely costat through dfferet tra-day tme pots except for the hghest decle. To evaluate the traday varablty of all spot quattes, we compute a proxy for the total traday varato ormalzed by the L 1 -orm,.e., Ṽ orm f = g f t f t 1 =1 [ g ] 1 f t t, where f stads for the respectve spot quatty of terest ad g deotes the umber of uderlyg grd pots per day. Fgures 6 ad 7 show the tme seres of cross-sectoal medas of the correspodg traday varato measures. The measures ormalzato makes them comparable across days ad quattes, whle provdg sghts to traday co-varato rsks. We observe that the latter are strogly tme-varyg ad are clustered over tme. Hece, traday co-varato rsks seem to be relatvely persstet, followg some log-term movemets. O dvdual days, we observe, evertheless, exceptoally hgh traday fluctuatos of these quattes. These are lkely days, where the market faces fudametal ews arrval or uusual actvtes, e.g., duced by flash crashes. Ideed, Secto 4.3, we wll focus o three selected days where markets are cofroted wth extreme evets. These three days, correspodg to the dates 05/06/10, 1/7/1 ad 04/3/13 more detals wll follow Secto 4.3, are dcated by the vertcal les ad deed reflect salet traday co-varato rsks. Iterestgly, ormalzed varatos are hghest for covaraces ad correlatos, whch mght be also partly duced by hgher estmato errors for covaraces compared to varaces. Accordgly, the varablty of spot volatltes s geerally lower, but ca stll be substatal o dvdual days. Cofrmg the fdgs above, the varablty betas s geerally lowest. However, eve betas reflectg =1 systematc rsk are far from beg costat o selectve days. Fgure 8 reports the averaged autocorrelato fuctos ACFs of all four quattes wth the fgures beg costructed such that oe lag correspods to approxmately fve mutes. We observe that all co-varablty measures are strogly serally correlated across short tme tervals wth frst-order autocorrelatos beg aroud Nevertheless, the ACFs decay relatvely fast wth a day. Ths s most extreme for spot volatltes, where the ACF decles from 0.95 at the 5 mute lag to below 0.1 after approxmately 3.5 hours. The otceable seasoalty patter the ACFs for covaraces ad volatltes uderles dstct daly autocorrelatos, whch

24 cosderably exceed the tradaly autocorrelatos at slghtly smaller lags. For correlatos ad betas, ths patter s less proouced. Here, log-term autocorrelatos stablze aroud 0.5 ad decay very slowly. 4.3 Evet Studes The prevous secto shows that spot correlatos ad covaraces ca substatally vary durg a day, eve f these patters are averaged across tme ad assets. Here, we am at aalyzg the behavor of spot co-varablty extreme market perods. The frst study aalyzes the flash crash o 05/06/10. Fgure 9 shows the traday movemets of the QQQ ETF over the etre tradg day ad a tme wdow startg at 1:30 pm. We assocate dvdual tme pots wth the followg evets: 9 1 At :00 pm, protests Athes related to the Euro crss trgger a sharp dow movemet of the Euro especally vs. the Ye. I the U.S., fud maagers start large-scale short-sellg of futures cotracts o the S&P500 E-M, leadg to a tradg volume whch s sx tmes hgher tha usual. At :35 pm, the E-m market makers cut back tradg. 3 At :37 pm, NASDAQ stops routg orders to ARCA, the electroc tradg platform of NYSE, due to huge lags order ackowledgemet. 4 At :45 pm, rumors spread suggestg that the decle occurred due to a fat-fger error of a Ctgroup trader, ad ot because of a adverse ews shock. Ths helps stablzg markets ad lqudty futures tradg rebouds. 5 At 3:01 pm, NASDAQ resumes routg to ARCA. Utl market closure, tradg remas erratc. Fgure 10 shows the cross-sectoal decles of resultg spot covaraces ad correlatos o ths day. We observe that covaraces are vrtually costat durg the morg, but early stataeously crease shortly after :00 pm,.e., after large-scale short sellg of E-M futures started. The decles show that the cross-sectoal dstrbuto of covaraces across all asset pars s very skewed, revealg huge upward shfts some covaraces, but oly very moderate reactos others. Fgure 11, however, shows that the correspodg reactos spot volatltes have bee much stroger. Ths eve leads to declg correlatos wth meda correlatos droppg from approx. 0.5 before :00 pm to approx. 0.3 aroud the peak of the crash. Hece, ths evet s characterzed by a exploso of dosycratc rsk, whch overcompesates creases covaraces ad ultmately decreases correlatos. Lkewse, systematc rsk, as reflected by the correspodg spot betas sgfcatly decles durg ths perod. Ths s due to the fact that the spot volatlty of the NASDAQ100 creases much stroger tha the uderlyg covaraces. Hece, we ca summarze that the May 010 flash 9 See For a detaled overvew, we refer to CFTC ad SEC