Estimating Volatility Using Intradaily Highs and Lows

Transcription

1 Estimatig Volatility Usig Itradaily Highs ad Lows Stefa Klößer Saarlad Uiversity October 8, 28 Abstract We ivestigate volatility estimators built by summig up quadratic fuctios of log-prices itradaily icremets, highs, ad lows, which we show to be cosistet for itegrated volatility whe log-prices are a Browia semimartigale. With respect to jumps, we sigle out four jump-robust estimators as well as estimators that accout oly for the sum of squared positive egative jumps. Next, we derive multivariate CLT s for the -scaled estimatio errors of some of these estimators i the presece of leverage effects. Fially we use these CLT s to develop highly efficiet jump-robust estimators of itegrated volatility ad a method for measurig the leverage effect. JEL-classificatios: C3, C4, G Keywords: Itegrated Volatility, Positive Negative Jumps-Iduced Variatio, High-Frequecy Data, High-Low-Prices S.Kloesser@mx.ui-saarlad.de

2 Itroductio Arguably, o cocept i fiacial mathematics is as loosely iterpreted ad as widely discussed as volatility. [...] volatility has may defiitios, ad is used to deote various measures of chageability. Volatility is a measure of price variability over some period of time. [...] Volatility ca be defied ad iterpreted i five differet ways. 2 Yet, the cocept of volatility is somewhat elusive, as may ways exist to measure it ad hece to model it. 3 Although the above quotatios make it clear that volatility is a rather vague cocept, volatility estimatio has bee a very promiet topic i fiacial ecoometrics for decades, sice may fiacial applicatios require kowledge or at least some estimate of price volatility, e.g. asset ad derivatives pricig, risk maagemet or portfolio selectio. The aim of this paper is to cotribute to the literature cocered with estimatig volatility i a cotiuous-time settig. Eve i this settig ad lettig aside implied volatility, volatility may refer to differet cocepts: we could be iterested i the quadratic variatio 4 of the log-price process o some day QV := p 2 2 p u dp u, where p t deotes the log-price at time t o that day ad the tradig iterval for ease of otatio is give by [, ]. QV coicides with IV := σ 2 udu, whe log-prices follow a Browia semimartigale, i.e. p t = p + µ u du + σ u dw u, Shiryaev 999, p Taylor 25, p Egle & Gallo 26, p. 4 4 The log-prices quadratic variatio is also called otioal volatility as well as ex-post variatio, e.g. Aderse et al. 22, Bardorff-Nielse et al. 28a.

3 where the drift process µ govers the mea, the process σ also called spot or istataeous volatility! accouts for the variability of the returs ad W deotes a stadard Browia motio. IV is called itegrated variace or itegrated volatility 5. I may cases, oe rather wats to estimate IV istead of QV, which, deotig the jumps by p t := p t p t, ad the sum of squared jumps by SSJ := p t 2, ca be decomposed ito IV ad SSJ: t [,] QV = IV + SSJ. As IV cotais iformatio about the cotributio of the cotiuous part of the log-prices to volatility, whereas SSJ ca tell us somethig about the variatio of returs due to jumps i prices, oe ofte is iterested i disetaglig QV ito IV ad SSJ, see e.g. Bardorff-Nielse & Shephard 24, Huag & Tauche 25, Bardorff-Nielse & Shephard 26, Bardorff- Nielse et al. 26b, Veraart 28. I may cotexts, for istace whe cosiderig the value at risk of some fiacial positio, it is ot oly importat to kow the risk due to jumps, but to have asymmetric measures of both risks of upward ad dowward jumps. Theoretically, this amouts to decomposig SSJ ito SSJ = SSpJ + SSJ, with the sum of squared positive jumps SSpJ := pt> p t 2 t [,] ad the sum of squared egative jumps SSJ := pt< p t 2, t [,] a topic to which the literature has tured oly recetly, see Bardorff-Nielse et al. 28b. To summarize, we re iterested i estimatig QV, IV, SSJ, SSpJ ad SSJ. The easiest task is to estimate quadratic variatio, because by defiitio it is 5 As is commo practice i ecoometrics, we call σ as well as σ 2 volatility. 2

4 the limit of realized variace, at least whe there is o microstructure oise 6, where realized variace RV := p i p i equals the sum of squared icremets over small itervals. Much work has recetly bee devoted to the issue of estimatig IV i the presece of jumps, which for istace is possible by usig bipower or multipower variatio, which are cosistet estimators for IV i the presece of jumps, see e.g. Bardorff-Nielse & Shephard 24, Bardorff-Nielse & Shephard 26, Bardorff-Nielse et al. 26a. With these tools at had, estimatio of SSJ = QV IV is straightforward by subtractig a estimator of IV, e.g. bipower variatio, from a estimator of QV, e.g. realized variace. 7 Util ow, there is oly oe cotributio to the issue of estimatig SSpJ ad SSJ: the workig paper Bardorff-Nielse et al. 28b. There it is show that the realized semivariaces RS + := RS := p i p i >p i p i p i <p i p i 2, p i 2 coverge to IV + SSpJ ad IV + SSJ, resp., which easily leads the way 2 2 to cosistetly estimatig SSpJ ad SSJ by subtractig oe-half times bior multipower variatio. A differet strad of research is cocered with improvig volatility estimatio by cosiderig high-low-based estimators, where volatility ow meas the variace of daily returs. Startig with the semial paper Parkiso 98, umerous authors have cosidered differet estimators i order to make efficiet use of the iformatio cotaied i the widely available daily high ad low prices, see e.g. Garma & Klass 98, Beckers 983, Ball & Torous 984, Rogers & Satchell 99, Wiggis 99, Kuitomo 992, Yag & Zhag 2. As it has bee show theoretically, by simulatios 6 Recetly, a lot of work addresses the issue of microstructure oise, e.g. Zhag et al. 25, Badi & Russell 26, Zhag 26, Bardorff-Nielse et al. 28a. 7 Nevertheless, derivig CLT s or feasible tests is a demadig task, tackled e.g. by Bardorff-Nielse & Shephard 26, Bardorff-Nielse et al. 26b, Jacod 28, Veraart 28. 3

5 as well as empirically e.g., Taylor 987, Rogers et al. 994, Wiggis 99, Rogers & Satchell 99, Aderse & Bollerslev 998, Shu & Zhag 26, volatility estimatio ca be sigificatly improved by these methods. Because the daily rage cotais iformatio ot icorporated i realized volatility 8 Egle & Gallo 26, several authors have recetly proposed models that explicitly model the daily rage, e.g. Chou 25a, Bradt & Joes 26, Bruetti & Lildholdt 27, Che et al. 28. The use of the rage has also proved fruitful for the estimatio of cotiuoustime models: whereas Gallat et al. 999 ad Alizadeh et al. 22 use daily rages to estimate cotiuous-time stochastic volatility models, both Martes & va Dijk 27 ad Christese & Podolskij 27 estimate QV by the so-called realized rage-based variace 9 where p i, := p i, := RRV = 4 l2 s 2 i,, sup p t deotes the itradaily high o [ i, i ], i t i if p t deotes the itradaily low o [ i, i ], ad i t i s i, := p i, p i, deotes the itradaily rage o [ i, i ]. It has bee show both theoretically ad empirically that usig the itradaily rages sigificatly improves the estimatio of itegrated volatility whe there are o jumps. Alas, this improvemet i efficiecy comes at the cost of lackig robustess to jumps. I this paper, we combie the two strads of the literature discussed above. I particular, i sectio 2 we examie a whole buch of estimators, amely all estimators built by summig up quadratic fuctios of itradaily highs p i,, lows p i,, ad prices p i. We aalyze the correspodig six-dimesioal vector space of estimators, ad spa it by estimators desiged for special purposes: 8 Although Che et al. 26 call it couter-ituitive, it is ot too surprisig that measures desiged for two differet cocepts of volatility covey differet iformatio. 9 RRV is the realized versio of Parkiso s estimator. 4

6 first, we cosider four estimators for IV, called realized upside volatility RUV, realized dowside volatility RDV, realized geometric ragebased volatility RGRV ad realized time-reversed geometric ragebased volatility RTrGRV, ad show that these estimators are asymptotically robust ot oly to fiite-activity jumps, but eve to the additio of processes of fiite variatio, secod, we cosider estimators for IV + SSpJ ad IV + SSJ, called realized positive jumps icluded volatility RpJV ad realized egative jumps icluded volatility RJV, which are resp. asymptotically robust to egative ad positive jumps ad to cotiuous processes of fiite variatio as well. The remaider of the paper is orgaized as follows: I sectio 2, we itroduce the above-metioed estimators ad aalyze some of the properties they have i commo, amely their asymptotic robustess to cotiuous processes of fiite variatio ad their cosistecy for QV = IV whe log-prices follow a Browia semimartigale. I sectio 3 we ivestigate idividual properties of the estimators: the asymptotic robustess of RUV, RDV, RGRV ad RTrGRV with respect to jumps of fiite variatio, a cosistecy result for RUV, RDV, RpJV ad RJV whe log-prices are give by a cotiuous semimartigale, ad the asymptotic behaviour of RpJV ad RJV i the presece of jumps. Sectio 4 is devoted to multivariate CLT s for the -scaled estimatio errors of the estimators. After discussig some applicatios of the CLT s, for istace measurig the leverage effect or combiig the estimators to highly efficiet ew oes thereby recoverig the realized versios of some old oes, sectio 5 cocludes, while the appedix cotais most of the proofs. 2 Sums of Quadratic Fuctios of Itradaily Highs, Lows, ad Returs I the followig, we will be cocered with estimatig the volatility of some log-prices p, where we always at least assume that the log-price process p t t [,] is a semimartigale, i.e. it has càdlàg paths ad ca be writte as the sum of a local martigale ad a process of fiite variatio. For ease of otatio, we deote the time iterval by [, ] ad call it a day, although it does ot ecessarily have to correspod to a day. By volatility, we mea oe of the followig four cocepts: 5

7 quadratic variatio QV = p 2 2 itegrated volatility IV = p t = p + µ u du + p dp, σ 2 udu, whe we additioally assume that σ u dw u is a Browia semimartigale, itegrated volatility IV plus the sum SSpJ = squared positive jumps: IV + SSpJ. t [,] itegrated volatility IV plus the sum SSJ = squared egative jumps: IV + SSJ. t [,] pt> p t 2 of the pt< p t 2 of the I order to estimate volatility, we assume kowledge of some itradaily data, specifically we assume that we are give p ad, for some iteger ad all i =,...,, itradaily highs p i, = itradaily lows p i, = itradaily log-prices p i. sup t [ i, i ] p t, if t [ i, i ] p t, I practical applicatio, these data may either stem from ultra-high-frequet data such as order book data, or from aggregatig some very high-frequet data, e.g. of samplig frequecy sec, to some lower frequecy, e.g. 5 mi, which is the most ofte used procedure i practice to avoid udesired effects caused by microstructure oise, although this is typically ot the best way of dealig with microstructure oise e.g., Zhag et al. 25. I the followig, we therefore wo t elaborate o the issue of microstructure oise, but rather assume that we are give the above-described data, may they come from order book data or from aggregatig higher-frequet data. Give itradialy highs ad lows, we cosider estimators of the form f p i, p i, p i, p i, p i p i, where f is some quadratic fuctio of its three argumets. Clearly the resultig vector space is of dimesio 6, ad our first goal is to fid six special 6

8 estimators spaig it. Buildig o the estimators V max = 2p p p p, V mi = 2p p p p recetly itroduced by Becker et al. 27b, we start by itroducig realized up- ad dowside volatility, which are the realized versios of V max, V mi ad correspod to f RUV a, b, c := 2aa c ad f RDV a, b, c := 2bb c: RUV := RDV := 2 2 p i, p i p i, p i p i, p i, p i, p i. Usig the geometric rage itroduced by Gumbel & Keeey 95, we further defie realized geometric rage-based volatility RGRV := 2 l2 p i, p i p i p i, ad its time-reversed pedat, realized time-reversed geometric rage-based volatility RTrGRV := p i, p i p i p i,, 2 l2 which correspod to f RGRV := a b ad f 2l 2 RTrGRV := a cc 2l2 b. Fially, we itroduce realized positive jumps icluded volatility RpJV ad realized egative jumps icluded volatility RJV : RpJV := RJV := 2 p i, p i p i p i,, 2 2 p i, p i + p i p i,, 2 which correspod to f RpJV a, b, c := a2 +b c 2 ad f 2 RJV a, b, c := a c2 +b 2. 2 We have the followig lemma, whose simple proof we omit: For the amig of RUV ad RDV see Chou 25b, Becker et al. 27a. The reaso for baptizig the estimators i this way will become clear later. 7

9 Lemma 2. The six-dimesioal vector space give by is spaed by RUV, RDV, RGRV, RTrGRV, RpJV ad RJV. The lemma i particular etails that RV ad RRV ca be writte as liear combiatios of these six estimators: RV = 2 RUV RDV + RpJV + RJV, 2 RRV = 2 l2 RGRV + 2 l2 RTrGRV + 4 l 2 4 l2 4 l 2 RpJV + 4 l2 RJV. We ow give a result o the cosistecy of these estimators for IV, whe p follows a Browia semimartigale. Theorem If p t = p + µ u du + σ u dw u with a Browia motio W, locally bouded ad predictable drift µ ad càdlàg spot volatility σ, the RUV, RDV, RGRV, RTrGRV, RpJV ad RJV ad all covex combiatios thereof coverge i probability to IV = Proof: See appedix. σ 2 u du. The assumptios of the previous theorem seem somewhat restrictive. Nevertheless, much work is cast withi this settig, see e.g. Aderse et al. 22, Christese & Podolskij 27 ad the refereces give there. By Theorem, we have a six-dimesioal coe of cosistet estimators for IV. We ow show robustess of these estimators with respect to perturbatios i form of cotiuous processes of fiite variatio. It holds i a much more geeral settig tha Theorem. Theorem 2 Suppose that y is a cotiuous process of fiite variatio ad RpJVp + RJVp is bouded i almost surely. The the differeces RUVp + y RUVp, RDVp + y RDVp, RGRVp + y RGRVp, RTrGRVp + y RTrGRVp, RpJVp + y RpJVp, RJVp + y RJVp coverge i probability to. 8

10 Proof: See appedix. Theorem 2 tells us that all the estimators give by are asymptotically robust to cotiuous processes of fiite variatio, wheever boudedess of RpJVp+RJVp holds, which i particular holds i the settig of Theorem. However, Thereom 2 oly presumes boudedess of RpJVp + RJVp, which allows it to be applied i much more geeral situatios, for istace whe prices are allowed to exhibit jumps. 3 Idividual Properties of RUV, RDV, RGRV, RTrGRV, RpJV, ad RJV Further elaboratig o robustess, we ow show that RUV, RDV, RGRV, RTrGRV are ot oly asymptotically robust to cotiuous processes of fiite variatio, but to ay process of fiite variatio. Theorem 3 If y is a process of fiite variatio ad RpJVp + RJVp is bouded i almost surely, the differeces RUVp + y RUVp, RDVp + y RDVp, RGRVp + y RGRVp, RTrGRVp + y RTrGRVp coverge i probability to. Proof: Here we give the argumets for robustess to fiitely may jumps, relegatig the more tedious details to the appedix. Thus suppose that y exhibits oly oe jump, occurig at some stoppig time T. This jump affects oly the subiterval [ i, i ] with i < T i, i.e. i = T. As p i, maxp T, p T, p i, mip T, p T, p i p T, p i p T, the differeces will vaish asymptotically because of p i, p i p i, p i p T + p T =, p i, p i p i, p i p T p T + =, p i, p i p i, p i p T + p T =, 9

11 p i, p i p i, p i p T p T + =. By Theorem 3, we have robustess of RUV, RDV, RGRV, RTrGRV to all processes of fiite variatio, i particular to fiite-activity jump processes as well as iifiite-activity jump processes of fiite variatio. We ow assume that p is a semimartigale with cotiuous paths, ad show that RUV RDV, as soo as it coverges, is cosistet for QV. 2 Theorem 4 Assume that p is a cotiuous semimartigale. The the followig coditios are equivalet: RUV coverges, 2 p i, p i coverges, 2 p i, p i coverges, all the above estimators coverge to QV. 2, 2, For RDV, p i, p i ad p i, p i a aalogous assertio holds. Proof: See appedix. We ed this sectio by turig our attetio to RpJV, RJV ad explaiig their amig. Theorem 5 RpJV is robust to egative jumps of fiite variatio ad accouts for the squares of positive jumps, while RJV is robust to positive jumps of fiite variatio ad accouts for the squares of egative jumps. Proof: As i the proof of Theorem 3, we have 2 p i, p i pt +2, 2 It may well be doubted whether there is a aalogo to Theorem 4 for RGRV or RTrGRV, as, due to their ormig costat 2 l2, these estimators seem to be somewhat less atural tha RUV ad RDV, or put differetly, it looks as if RGRV ad RTrGRV exploit the properties of Browia motio to a greater extet tha RUV ad RDV do.

12 2 p i, p i pt +2, 2 p i, p i pt 2, 2 p i, p i pt 2, ad the proof ow follows alog the same lies as the proof of Theorem 2. Combiig Theorems 4 ad 5, we fid that for sums of Browia semimartigales ad processes of fiite variatio, RpJV RJV is a cosistet estimator for IV + SSpJ IV + SSJ, which explais the ame realized positive egative jumps icluded volatility. Combiig the results of this sectio, we ca estimate SSpJ SSJ by subtractig ay covex combiatio of the jump-robust estimators RUV, RDV, RGRV, RTrGRV or ay jump-robust cosistet estimator of IV from RpJV RJV. 4 Cetral Limit Theorems for the Estimatio Errors To be able to costruct cofidece regios or statistical tests for IV based o the ewly itroduced estimators, we eed a cetral limit theorem for the properly scaled estimatio errors. The aim of this sectio is to derive such a CLT whe there are o jumps i the price process. 3 To achieve this, we have to itroduce some stroger assumptios o the spot volatility σ t. I the sequel, we will assume that p t = p + µ u du + σ u dw u 2 is a Browia semimartigale, that σ ever vaishes ad that it is give by σ t = σ + µ u du + σ u dw u + ṽ u db u, 3 where µ, σ, ṽ are càdlàg, µ is locally bouded as well as predictable, ad B is a Browia motio idepedet of W. 4 3 The more geeral case with jumps is delayed for further research. 4 These are exactly the same assumptios as i Christese & Podolskij 27.

13 We start with a CLT for six related, yet ifeasible estimators, which are built usig ot oly the drivig Browia motio W s itradaily icremets, but also its highs ad lows W i, := sup W t, W i, := if W t. t [ i, i ] t [ i, i ] The result is stated usig stable covergece i law, which is slightly stroger tha mere covergece i law 5. Theorem 6 Let p be give by 2, 3 ad defie for the auxiliary estimators V := V {RUV, RDV, RGRV, RTrGRV, RpJV, RJV} σ 2 i f V W i, W i, W i, W i, W i W i. 4 The we have RUV IV RDV IV RGRV IV RTrGRV IV RpJV IV RJV IV d S 8 3 2π u σ 2 σs 2 dw s + R σ 2 s db6 s for a six-dimesioal Browia motio B 6 idepedet of the filtratio F to which p is adapted, where u σ 2 =,,,,, ad R = For details o stable covergec i law, itroduced by Réyi 963, see e.g. Aldous & Eagleso 978, Jacod 997, Bardorff-Nielse et al. 28a, Appedix A. 2

14 with c c2 c2 c3 c3 c c2 c2 c3 c3 RR = Σ := c2 c2 c4 c5 c6 c6 c2 c2 c5 c4 c6 c6 c3 c3 c6 c6 c7 c8 c3 c3 c6 c6 c8 c = with costats c = 8 l ζ3, c 2 = ζ3 2l 2, c 3 = 2 2 l ζ3, c 4 = 2 2l22, c 2l = + 2 l 2 4l ζ3 8, 2l2 2 c 6 = 3 + ζ l2, c 7 = ζ3 8 9π, c 8 = l π ad Riema s Zeta fuctio ζ. 6 Proof: See appedix. As V is ifeasible i practice, the ext theorem cosiders the -scaled differece betwee the actual volatility estimator V ad its auxiliary versio V. 6 ζ

15 Theorem 7 Let p be give by 2, 3. The we have RUV RUV RDV RDV RGRV RGRV P RTrGRV RTrGRV 8 3 2π RpJV RpJV RJV RJV Proof: See appedix. σ s σ s ds σ s σ s ds ds ds σ s µs + σs 2 σ s µs + σs 2 Combiig Theorems 6, 7, we have the followig result: Theorem 8 Let p be give by 2, 3. The we have RUV IV RDV IV RGRV IV d S RTrGRV IV C RpJV IV RJV IV with C = 8 3 2π σ s µs + σs 2 ds + σs 2dW s σ s µs + σs 2 Theorem 8 ca i particular be used i two ways: σ s σ s ds σ s σ s ds +R ds σsdw 2 s σ 2 s db6 s we ca estimate the leverage effect by usig RUV RDV as a asymptotically ubiased estimator for 2C σ s σ s ds. From Theorem 8, it ca easily be derived that the asymptotic distributio of RUV RDV σ s σ s ds 2C 4

16 is mixed ormal MN, c L IQ, with c L = 9π 6 l2 7ζ3.59 ad itegrated quarticity IQ := σ 4 s ds. we ca look for optimal covex combiatios of the estimators desiged for special purposes, where optimal loosely speakig meas optimal variace, e.g. we could look for the best jump-robust estimator of IV that is ot affected by the presece of leverage as give by σ. Defiig u µ,σ :=,,,,,, u σ, σ :=,,,,, 2 2, we ca rewrite the limit i Theorem 8 as 8 3 2π u µ,σ σ s µ s ds + u σ, σ σ s σ s ds + u σ 2 28 σsdw 2 s + R σ 2 sdb 6 s, from which it is easily see that the estimatio error of a covex combiatio w RUV, RDV, RGRV, RTrGRV, RpJV, RJV will coverge stably i law to C w µ,σ σ s µ s ds + w σ, σ σ s σ s ds + w σ 2 σsdw 2 s + w RR w σ 2 sd B s, with w µ,σ := w u µ,σ, w σ, σ := w u σ, σ, w σ 2 := w u σ 2 ad B a Browia motio idepedet of F. I particular, i case of vaishig w µ,σ, w σ, σ, ad w σ 2, we will have a mixed ormal limit MN, c w IQ with c w := w RR w. We obtai RRV by w RRV =,, 2 l2 4 l2, 2 l2 4 l2, 4 l2,, 4 l2 with vaishig w µ,σ, w σ, σ, w σ 2, ad c wrrv = , thereby regaiig the CLT of Christese & Podolskij 27. I order for the limitig distributio to have zero mea, the weights of RpJV ad RJV as well as those of RUV ad RDV must be the same. Optimizig c w uder these restrictios leads to w =.298,.298,.93,.93,.8799,.8799, which happes to be the realized versio of the Garma & Klass 98 estimator, c w the equals Alas, as RRV, this estimator will lose cosistecy for IV i the presece of jumps because it ivolves RpJV ad RJV. 5

17 RUV + RDV Averagig RUV ad RDV, we have RUDV := with c 2 w =.339 ad the additioal advatage of RUDV beig a jump-robust estimator. RUDV turs out to be the realized versio of the estimator of Rogers & Satchell 99. Combiig RUV, RDV, RGRV, ad RTrGRV, we are able to fid a eve more efficiet jump-robust estimator of IV. The optimal covex combiatio is give by w = , , , ,, with c w = , which is a lot better tha the correspodig costat 2.6, whe usig bipower variatio as a jump-robust estimator of IV 7. 5 Coclusio I this paper, we have itroduced ad aalyzed the class of volatility estimators that are built by summig up quadratic fuctios of itradaily returs, highs, ad lows. I particular, we have foud several ew estimators, four of which are robust to jumps of fiite variatio, while two others eable us to separately estimate the positive ad egative jumps cotributio to quadratic variatio. After deliverig cosistecy results uder varyig assumptios, we proved CLT s for the properly scaled estimatio errors of these estimators. These theorems the lead the way to the costructio of highly efficiet estimators of IV that are robust to jumps. As a further by-product, we foud a possibility for estimatig the leverage effect. We leave for further research the issues of extedig the CLT s by allowig for jumps i the price process, costructig tests for jumps, fidig aalogous estimators of itegrated quarticity IQ as well as providig small-sample corrected versios of the estimators for fiite. 6 Appedix 6. Proof of Theorem First of all, we ca without loss of geerality restrict µ ad σ to be bouded e.g., Bardorff-Nielse et al. 26a. The proof proceeds i two steps: i the first step, we show V P IV for all V 7 Of course, bipower variatio does ot eed itradaily highs ad lows. 6

18 {RUV, RDV, RGRV, RTrGRV, RpJV, RJV} ad the auxiliary estimators V give by 4. The proof will the be completed by showig that the differece betwee the ifeasible estimator V ad the feasible volatility estimator V vaishes asymptotically. To accomplish the first step, we write V = ξ V, with i, ξ V i, := σ2 i f V W i, W i, W i, W i, W i W i. 5 We have, due to the scalig property of Browia motio ad because f V is quadratic: E ξ V F i = i, σ2 i E f V W, W, W, with W := sup W t ad W := if W t. As it is easy to see 8 that t [,] t [,] E f V W, W, W = for all V, we fid E ξ V i, F i P IV. 6 Usig essetially the same argumets, we have for the obviously ucorrelated, zero-mea variables η V which implies η V 2 E F i i, i, := ξ V = 2σ4 i i, E ξ V η V i, F i Varf V W, W, W, i, : 7 P. 8 Combiig 5-8, the first step is complete. I the secod step, we will ow prove that f V σ i { V V = f V W i, W i p i, p i, σ i, p i, p i, p i p i W i, W i, σ i W i 8 By, e.g., usig the geeratig fuctio give i Garma & Klass 98. W i } 7

19 vaishes asymptotically. To simplify otatio, we defie a i, := σ i W i, W i, ã i, := p i, p i, b i, := σ i c i, := σ i W i, W i W i W i, b i, := p i, p i,, c i, := p i p i. It is the straightforward to show that a i, ã i,, b i, b i,, c i, c i, are bouded by κ i, := sup µ τ dτ + σ τ σ i dw τ. s,t [ i, i ] Because f V is quadratic, the differece s f Vã i,, b i,, c i, f Va i,, b i,, c i, ca up to some costats be writte as sums of terms of the form d i, ẽ i, e i,, d i, d i, ẽ i, e i,, with d i,, e i, either a i,, b i, or c i, ad d i,, ẽ i, either ã i,, b i, or c i,. Usig the rage s i, := W i, W i,, it is clear that a i,, b i, ad c i, are bouded by σ i s i,, while differeces d i, d i,, ẽ i, e i, are bouded by κ i,. I order to complete the proof, we therefore oly have to show that the sums σ i s i, κ i,, κ 2 i, coverge to i probability, which ca be easily see by applyig the Cauchy- Schwarz iequality to the first sum ad the usig exactly the same argumets as i Christese & Podolskij 27, p. 342f. 6.2 Proof of Theorem 2 First of all, because y is of fiite variatio, it ca be writte as the differece of two mootouos cotiuous processes. We oly give the proof for robustess to o-decreasig cotiuous processes y, as the proof for oicreasig cotiuous processes ca be doe i a exactly aalogous way. By straightforward argumets, p + y i, p + y i 8 s p i, p i,

20 are all bouded by y i p + y i, p + y i p + y i p + y i p i, p i p i p i y i. Usig the otatios ã i, := p + y i, p + y i bi, := p + y i, p + y i c i, := p + y i p + y i, c i, := p i,, a i, := p i, p i,, b i, := p i, p i, p i, the differeces RUVp + y RUVp etc. ca be writte as fã i,, b i,, c i, fa i,, b i,, c i,. As the fuctios f are quadratic, usig the iequalities established above, up to some costats we ed up with sums of the form d i, y i y i, y i y i 2, with d i, either a i,, b i,, c i,, a i, c i,, or b i, c i,. Usig Cauchy-Schwarz iequality as well as the boudedess of RpJVp + RJVp, the result ow follows from the fact that y i y i 2 P, because y is cotiuous ad of fiite variatio. 6.3 Proof of Theorem 3 Due to Theorem 2, we have robustess with respect to cotiuous processes of fiite variatio, whereas the part of the proof give i the mai text gives us robustess to fiitely may jumps. So we re left with provig robustess to pure-jump processes of fiite variatio, possibly havig ifiitely may jumps. As i the proof of Theorem 2, we may ad will restrict the proof to o-decreasig processes y. Now, for every fixed ε >, we ca decompose y ito its large jumps part y l,ε, with y y l,ε ε ad oly fiitely may jumps, ad 9

21 the small jumps part y s,ε = y y l,ε, which may have ifiitely may jumps whose sum is bouded by ε. As we have robustess to fiitely may jumps, we oly have to cosider the small jumps part y s,ε. Obviously, we have p + y s,ε p + y s,ε i p i, p i < y s,ε i y s,ε i < ε, i, ad aaloguous formulæ for the ifimum ad fial value i [ i, i ], which etail, as i the proof of the previous theorem, that we re left, up to some costats, with sums that cotai the factor y s,ε i y s,ε i. Usig Cauchy Schwarz iequality, we get uiformly i RUVp + y s,ε RUVp Cε etc., which completes the proof. 6.4 Proof of Theorem 4 We first prove the followig lemma, which itself is of some iterest. Lemma 6. For ay cotiuous semimartigale X t t with X =, ruig maximum X t = sup X s, ad ruig miimum X t = if X s, we have: s t s t 2X X X = X 2 2 X dx, 2X X X = X 2 2 X dx. Proof: We oly have to prove the first assertio, as the secod oe follows by applyig the first oe to X. As X is o-decreasig, it is of fiite variatio, hece X 2 = 2 X dx. Because for cotiuous processes, X X vaishes o the support of dx, we further have X XdX =, which implies X 2 = 2 XdX. By itegratio by parts, we fid due to [X, X ] = : XdX = X X X dx, which etails X 2 = 2X X X dx, from which the assertio follows easily. We ow proceed to the proof of Theorem 4. Agai we oly have to prove the assertio for RUV, as the proof for RDV follows i a completely aalogous 2

22 way. Applyig Lemma 6. to X t := p t+ i have RUV = where p i, := t This gives with p t := p i, p i 2 2 i i p i p i, p t i for every i =,...,, we dp t p i, sup p s deotes the ruig maximum of p o [ i, i ]. i s t RUV = p i, p i 2 2 p dp, p + t, p t. As the itegrad coverges to, the itegral t p dp vaishes asymptotically, showig the equivalece of the covergece of RUV ad 2 p i, p i ad their covergece to the same limit i case of covergece. Applyig this to p t := p t, we get the same result for RUV ad 2. p i, p i The result ow follows from the decompositio RV = RUV + p i, p i 6.5 Proof of Theorem p i, p i. First of all, we ca, as i the proof of Theorem, without loss of geerality restrict µ ad σ to be bouded. From Bardorff-Nielse et al. 26a, 2.7 we have σ2 P i IV, 2

23 which, due to the properties of stable covergece i law, allows us to replace IV by, i.e. we have to show the stable covergece i law σ2 i RUV RDV RGRV RTrGRV RpJV RJV σ2 i σ2 i σ2 i σ2 i σ2 i σ2 i d S 8 3 2π u σ 2 σsdw 2 s + R σ 2 sdb 6 s. Usig for V {RUV, RDV, RGRV, RTrGRV, RpJV, RJV} the F i -measu- rable r.v. s η V give by 7 ad defiig χ V i, := i, η V = σ 2 i f V i, we have RUV RDV RGRV RTrGRV RpJV RJV W i, W i σ2 i σ2 i σ2 i σ2 i σ2 i σ2 i =, W i, W i, W i W i χ RUV i, χ RDV i, χ RGRV i, =: χ RTrGRV i, χ RpJV i, χ RJV i, χ i,. The proof ow follows by applyig Theorem 3-2 of Jacod 997, for which we have to show: t sup Eχ i, F i P, 9 t 22,

24 t Eχ i, χ i, t E t E F i χ i, W i Eχ i, F i Eχ i, W i for F t := F i F i P F t t 32 9π u σ 2u σ + 2 RR P 8 3 2π u σ 2 σs 4 ds, σs 2 ds, t E χ i, 2 P { χi, >ε} F i ε >, 2 χ i, N i N i F i P N M b with [W, N] = holds due to E χ i, F i = E η i, F i =, as see i the proof of Theorem. t 2. To prove, we oly have to cosider E χ i, χ i i, F, because E χ i, F i =. As f V is quadratic, we fid, by usig the scalig property of Browia motio, that the compoets of this matrix are give by t E i, = σ2 i η V i, η V 2 i, F i t = E η V η V2 σ4 i with V, V 2 {RUV, RDV, RGRV, RTrGRV, RpJV, RJV}, η V f V W i, W i, W i, W i, W i W i Obviously, we have η V := f V W, W, W. E η V t η V 2 σ4 i 23 P E η V t η V 2 σ 4 sds,,

25 so that i order to prove, we oly have to show that the compoets of 32 u 9π σ 2u σ + RR are give by E η V η V2. This ca be see by 2 simple, but tedious calculatios usig the fourth momets of W, W ad W, 9 which we omit. 3. I order to prove, we agai make use of the scalig property of Browia motio to compute E χ V i, W i W i F i = σ 2 i E W f V W, W, W, whose sum coverges to E W f V W, W, W σs 2 ds. We have E W f RUV W, W, W = E W 2 W W W =, because W ad W W W are idepedet e.g., Seshadri 988. The same reasoig applies to RDV. Cocerig RGRV, RTrGRV, we also get the value, because these estimators are symmetric i W. For RpJV, RJV, we have to compute E W W W W 2 = 8 3 2π, E W W W W 2 = 8 3 2π, which ca be see e.g. by usig the geeratig fuctio give i Garma & Klass 98 or by usig the joit distributios of W, W ad W, W, which ca be foud i Borodi & Salmie Due to the boudedess of σ, it suffices to prove 2 for σ, givig V f V W i, W i, W i, W i, W i W i 2 for χ i, 2, where the sum rus over V {RUV, RDV, RGRV, RTrGRV, RpJV, RJV}. 9 These momets ca be foud i Garma & Klass 98, p

26 We the have t E χ i, 2 { χi, >ε} F i = t E A {A>ε}, with A = V f V W, W, W 2. Now 2 follows easily from domiated covergece, as EA < ca be prove by usig the predictable represetatio property of Browia motio, the argumet beig exactly the same as give by Christese & Podolskij 27, p Proof of Theorem 7 As i the proofs of Theorems ad 6, we ca, without loss of geerality, assume that µ, σ, µ, σ, ad ṽ are bouded. Defiig θ V := i, f V p i, p i, p i, p i, p i p i, the proof proceeds i two steps: usig 5, we first show θ V i, ξ V i, E θ V i, ξ V i, F i P, while the proof will we completed by showig that E θ V i, ξ V i, F i 4 coverges to the limit give i the theorem. To prove the first step, we obviously have that θ V i, ξ V i, E θ V i, ξ V i, F i is a zero-mea, ucorrelated sequece, so that it is eough to show that E θ V i, ξ V E i, θ V i, ξ V i, F i 2 P, 25

27 which ca be doe with the same methods as i the proof of Theorem. For the secod step, we decompose σ t, p t for t [ i, i ] as follows: with R σ,t := σ t = σ i i µ s ds + + σ i i i dw s + ṽ i i σ s σ i dw s + i db s + R σ,t ṽ s ṽ i db s, p t = p i + µ i i ds + σ i i dw s s s with R p,t := + σ i i i i dw τ dw s + ṽ i µ s µ i ds + i i i R σ,s dw s. db τ dw s + R p,t Agai usig Cauchy-Schwarz, Burkholder-Davis-Gudy iequalities, ad the decompositio itroduced i the proof of Theorem, we fid that i 4, we ca replace θ V by i, θ V := f V p i,, p i,, p i, 5 where for t [ i, i ] i, p t := σ i i dw s + µ i i ds + σ i i s i dw τ dw s + ṽ i i s i db τ dw s. Obviously, for fixed i,, p equals o [ i, i ] σ i t d W s + t t s t s µ i ds + σ i d W τ d W s + ṽ i 26 d B τ d W s.

28 with t := t i W := [, ] ad the idepedet Browia motios 2 W i + W i, B := B i + B i We ow follow the lies of Christese & Podolskij 27, p. 346 f by applyig their Lemma Christese & Podolskij 27, p. 343 to ε = ad the fuctios g i, := µ i ds + σ i f i, := σ i s d W s, d W τ d W s + ṽ i defied o [, ], which gives us the decompositios p i, σ i W i, W i p i, σ i W i, W i p i σ i W t W i with E R 2 i, = o P = E s. d B τ d W s, = g i, t W + R i,, = g i, t W + R i,, = g i,, R 2 i, uiformly i i ad t W := arg sup t [,] Wt, t W := arg if t [,] Wt. Combiig this with 4, 5, we have to fid the limit of E where f V σ i W + a i,, σ i W + b i,, σ i f V σ i W, σ i W + b i, W, σ i W F i a i, := g i, t W + R i,, b i, := g i, t W + R i,, c i, := g i,, W := W, W := W. 2 For otatioal coveiece, we suppress the depedece o i,., 27

29 As f V is a quadratic polyomial, decomposig it as i Theorem results up to costats i sums of products of σ i W, g i,, ad R i, possibly with or. Usig Cauchy-Schwarz ad Burkholder-Davis-Gudy iequalities oce agai, we see that all the sums of products icludig R i, will vaish, as well as those where both factors iclude g i,, while the products of σ i W cacel out. I the ed, we therefore have to cosider oly sums of products of σ i W ad g i,, with the actual form depedig o V: for RUV: 2 E for RDV: 2 E for RGRV: σ i σ i 2 l 2 for RTrGRV: 2 l 2 for RpJV: E W g i, t W g i, +σ i W g i, t W g i, E E σ i +σ i W W g i, t W W W g i, t W F i F i W g i, + σ t W W i g i, t W F i σ i W W g i, g i, t W W W g i, t W g i, +σ i σ i W g i, t W +σ i W W g i, g i, t W 28 F i F i

30 for RJV: E σ i W g i, t W +σ i W W g i, t W g i, F i Usig s d W τ d W s = 2 W 2 t t, we ow decompose the process g i,: g i, t = µ i 2 σ σ i i t + 2 W 2 t + ṽ i g 3 t with g 3 t := B s d W s. Replacig B by B shows that g 3 plays o role i the sums above. Replacig W by W W shows that for RUV, RDV the drift term of g i, ca be eglected. Replacig W by W shows that for RGRV, RTrGRV the drift term as well as the leverage term ca be eglected. The theorem ow follows easily by computig the correspodig joit momets of W, W, W, t W, ad t W. 2 2 The desity of W, t W ca be foud i Borodi & Salmie 22, p

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