Estimating spot volatility with high-frequency financial data

1、Estimating spot volatility with high-frequency financial dataliJEL classification:C58C13C14Keywords:Spot volatilityMarket microstructure noiseSubsamplingScale selectionBandwidth selection? 2014 Elsevier B.V. All rights reserved.1. IntroductionSpot volatility, also known as instantaneous volatility,

2、measuresthe strength of return variations at a certain time point, ex-pressed per unit of time (Andersen et al. (2010). Spot volatilityhas important applications in studying the intraday patterns of thevolatility process, testing price jumps (Lee and Mykland (2007),Veraart (2010), and estimating par

3、ametric stochastic volatilitymodels (Bandi and Reno (2009), Kanaya and Kristensen (2010).In this paper, we are interested in the nonparametric estimation ofspot volatilitywith high-frequency financial data.Spot volatility estimation in the literature dates back to Mer-ton (1980), who considered a co

4、nstant volatility model. Later on,researchers tended to estimate volatility in the context of theARCHmodel (Engle (1982), the GARCHmodel (Bollerslev (1986),and their numerous variations. Nonparametric estimation of spotvolatility in the context of diffusion models was firstly consideredby Foster and

5、 Nelson (1996). Andreou and Ghysels (2002) con-ducted simulation studies using Foster and Nelsons estimator and? Corresponding author. Tel.: +44 0 20 7040 8619; fax: +44 0 20 7040 8580.E-mail addresses: Yang.Zucity.ac.uk (Y. Zu), H.P.Boswijkuva.nl(H. Peter Boswijk).1 Tel.: +31 0 20 525 4316.some rel

6、ated estimators. Recent contributions include Myklandand Zhang (2008), Fan and Wang (2008) and Kristensen (2010).High-frequency financial data have becomemore accessible foracademic research in recent years. In contrast to low frequency(daily, weekly or longer sampling frequency) financial datasets,

7、high-frequency datasets are characterized by the large number ofobservations they contain and the existence of so-called marketmicrostructure noise. OHara (1998) made theoretical studies ofmarket microstructure noise; Andersen et al. (2000) and Hansenand Lunde (2006) analyzed the empirical character

8、istics of thenoise.Existing research on volatilitymeasurement for high-frequencydata focuses mainly on the ex post nonparametric measurementof the integrated volatility of the underlying efficient price process.Andersen et al. (2001) and Barndorff-Nielsen and Shephard(2002) made important early cont

9、ributions to the use of realizedvariance to estimate the integrated volatility. However, they didnot consider the effects of market microstructure noise, andthe realized variance estimator can only be applied to sparselysampled data, where the effects of noise are small. The problemof estimating int

10、egrated volatility under noise was first studied byZhou (1996), who gave an unbiased but inconsistent estimator forintegrated volatility. A?t-Sahalia et al. (2005) considered a constantvariance model and gave a Maximum Likelihood Estimator for theconstant variance. Later, four types of estimators we

11、re proposed//10.1016/j.jeconom.2014.04.001Journal of EconometricContents lists availaJournal of Ecjournal homepage: .eEstimating spot volatility with high-frequYang Zu a,?, H. Peter Boswijk b,1a Department of Economics, City University London, Northampton Square, EC1V 0HB, Londb Amsterdam

12、School of Economics, University of Amsterdam, Valckenierstraat 65-67, 1018a r t i c l e i n f oArticle history:Received 14 January 2014Received in revised form14 January 2014Accepted 19 January 2014Available online 19 April 2014a b s t r a c tWe construct a spot volatilcrostructure noise. We provedr

13、ivenmethod is proposed toMonte Carlo simulations, weestimators. Empirical examp0304-4076/? 2014 Elsevier B.V. All rights reserved.s 181 (2014) 117135ble at ScienceDirectonometricssevier4/locate/jeconomency financial dataon, United KingdomXE Amsterdam, The Netherlandsty estimator for high-frequency f

14、inancial data which contain market mi-consistency and derive the asymptotic distribution of the estimator. A data-select the scale parameter and the bandwidth parameter in the estimator. Incompare the finite sample performance of our estimator with some existingles are given to illustrate the potent

15、ial applications of the estimator.E118 Y. Zu, H. Peter Boswijk / Journal offor estimating integrated volatility in the presence of noise. Theseare the subsampling-based Two Scale Realized Variance (TSRV)estimator by Zhang et al. (2005) and the Multiscale RealizedVariance (MSRV) estimator by Zhang (2

16、006); the Realized Kernel(RK) estimator by Barndorff-Nielsen et al. (2008), which is basedon Zhou (1996)s first order moving average correction; the pre-averaging method by Podolskij and Vetter (2009), and Jacod et al.(2009); and the Quasi-Maximum Likelihood Estimator (QMLE) byXiu (2010), which is b

17、ased on the estimator in A?t-Sahalia et al.(2005).In this paper, we study the problem of estimating spot volatilitywith high-frequency data and we explicitly consider the effectsof market microstructure noise. Our approach is closely related tothe literature on integrated volatilitymeasurementwith n

18、oise.Weconstruct our estimator based on the Two Scale Realized Varianceestimator by Zhang et al. (2005) our estimator calculates theincrement of the Two Scale Realized Variance estimator over asmall interval and applies an appropriate normalization. Underappropriate conditions, we prove consistency

19、and derive theasymptotic distribution of our estimator and propose a data-drivenprocedure to select tuning parameters. In practically meaningfulMonte Carlo simulations, we compare our estimator with existingmethods in terms of several error measures and we demonstratethe improved accuracy in using o

20、ur estimator.Some recent research is closely related to this paper. Myklandand Zhang (2008) independently proposed the same estimator asin our paper, but did not provide a complete asymptotic theory.Bandi and Reno (2009), Ogawa and Sanfelici (2011), Bos et al.(2012), among others, have considered sp

21、ot volatility estimatorsbased on the Realized Kernel estimator and the Pre-Averagingestimator. In a concurrent paper, Mancini et al. (2012) (Section 3.1)have proposed a two-scale estimator for spot volatility weightedby the so-called delta sequence and have provided theoreticalanalysis; our estimato

22、r is a special case of their estimator withequal weights. We provide more comprehensive asymptotic andfinite sample studies for our estimator; we also study the problemof bandwidth and scale parameters selection, which is importantfor practical implementation. In the presence of jumps butwithoutnois

23、e, spot volatility has been studied by A?t-Sahalia and Jacod(2009), Ngo and Ogawa (2009) and Andersen et al. (2009).Munk and Schmidt-Hieber (2010b,a) studied the best possibleconvergence rate of any spot volatility estimator in a volatilitymodel observedwith noise,where the volatility process is ass

24、umedto be a deterministic function. Hoffmann et al. (2010) deriveda minimax bound for the same problem in a genuine stochasticvolatility model observed with noise, they showed that this boundis nearly optimal in their definition and they proposed a waveletestimator that achieves this rate. Their rat

25、e is n?1/8 if translatedto the present context, up to some logarithmic corrections. Ourestimator does not have the best rate of convergence in their sense,we discuss possible extensions to improve the convergence rate inSection 6.The structure of this paper is as follows. Section 2 introducesthe set

26、up of the problem. Section 3 defines the estimator, studiesits asymptotic properties and the problem of bandwidth and scaleselection. Section 4 conducts Monte Carlo studies on the finite-sample properties of the estimator and Section 5 contains twoempirical applications to Euro FX futures data. Sect

27、ion 6 discussespossible extensions to our model. Section 7 concludes the paper.Proofs are collected in Appendix A and technical lemmas arecollected in Appendix B.Throughout the paper, ?X, Y ? denotes the quadratic covariationof two processes X and Y ;d? denotes converge in distribution; st?pdenotes

28、stable convergence in distribution; ? denotes convergein probability; for a real number x, ?x? denotes its integer part. Weconometrics 181 (2014) 117135call 2t the spot variance at time t , andwe call t the spot volatility attime t . However, as in the financial econometrics literature, whenwe use t

29、he term spot volatility in general discussions, it could referto either 2t or t , depending on the context.2. The modelLet Xt be a univariate log price process, assumed to be aBrownian semimartingale, satisfyingdXt = ?tdt + tdWt , t 0, 1,where Wt is a standard Brownian motion; ?t is the spot driftpr

30、ocess, and t is the spot volatility process; both are predictable.We further assume:A1 the processes ?t and t have continuous sample paths.A2 the process t is positive.Since X is a Brownian semimartingale, it has continuous samplepaths, and its quadratic variation process satisfies?X, X?t =? t0 2s d

31、s, t 0, 1,such that the spot volatility satisfies 2t =d ?X, X?tdt. (1)Taking into account the market microstructure noise existing inhigh-frequency financial data, we further assume8A3 X is not observable, butYt = Xt + tis observed over the interval 0, 1 in discrete time over a gridti = i/n for i =

32、0, 1, . . . , nwith equal distance?n = 1/n.A4 tini=1 are independent and identically distributed (i.i.d.)with mean 0, variance 2, and with finite fourth moment.Furthermore, tini=1 are independent of the Xt process.The continuity assumption on the volatility sample paths ac-commodates a large class o

33、f spot variance processes such as dif-fusion processes, long memory, deterministic patterns as well asnonstationarity. The model allows for possible dependence be-tween Wt and t, so leverage effects are allowed in this model.The model specification and Assumption A1 exclude the possi-bility of jumps

34、 in both the price process and the volatility process.Assumption A4 excludes the possibility that the noise is dependentover time (so called dependent noise) and that the noise is depen-dent of the efficient price process (so-called endogenousnoise).Wediscuss possible extensions to these cases in Se

35、ction 6.3. The estimator and its properties3.1. The estimatorWe are interested in estimating the realization of the spotvariance process? 2t?at any time t (0, 1). Our estimator is basedon the TwoScale RealizedVariance estimator (TSRV) by Zhang et al.(2005).The TSRV estimator uses a subsampled and av

36、eraged RealizedVariance (RV) estimator over a scale K , together with a usualRealized Variance estimator to correct the effects of noise. It isdefined asTSRV = Y , Y K ? n?nY , Y ,EY. Zu, H. Peter Boswijk / Journal ofwhereY , Y K := 1Kn?i=K(Yti ? Yti?K )2,Y , Y :=n?i=1(Yti ? Yti?1)2,n? := n? K + 1K.

37、The definition of Y , Y K appears to be different from the originalformulation in Zhang et al. (2005), but is an equivalent reformula-tion of their definition, which was also used by Zhang (2006).Our estimator is constructed using Eq. (1)a theoretical timederivative is approximated by a numerical de

38、rivative over a smallinterval. When the interval is t ? h, t, a filtering version of theestimator for the spot variance 2t is defined as:? 2t = TSRVt ? TSRVt?hh . (2)We call it a filtering version of the estimator because it only usesdata up to time t . Similarly, a smoothing version of the estimato

39、rwhichuses both lead and laggeddata at time t could be constructedas follows:? 2t = TSRVt+h/2 ? TSRVt?h/2h .Here we split the bandwidth evenly between leads and lags. In thefollowing, all the theoretical results will be stated for the filteringversion of the estimator. Analogous results apply to the

40、 smoothingversion. In the Monte Carlo experiment, we study both versions ofthe estimator.We call our estimator the Two Scale Realized Spot Variance(TSRSV) estimator. To have a direct expression for the estimator,we first introduce some notation. For a sequence Zti , i = 0,1, . . . , n, defineZ, ZK ,

41、ht := 1h?t?htit(?KZti)2K= 1h?t?htit(Zti ? Zti?K )2K,where we use the notation ?K for the K th difference operator;and we use a to signify this is a numerical time derivative; whenK = 1, we write Z, Zht := Z, Z1,ht . Denote V?K ,ht as the filteringversion of the TSRSV estimator at time t as in (2), i

42、t is defined as:V? K ,ht = Y , Y K ,ht ? n?n Y , Y ht ,wheren? = nh? K + 1Kh.The estimator depends on a subsampling size parameter K , whichwe call the scale parameter, and an interval length parameter h,which we call the bandwidth parameter.3.2. DecompositionTo study the statistical properties of V

43、? K ,ht as an estimator of 2t , we first make a biasvariance-like decomposition for thedifference of the two:V? K ,ht ? 2t =?V? K ,ht ? 1h? tt?h 2s ds?+?1h? tt?h 2s ds? 2t?= : P0 + P1,where P0 and P1 are defined implicitly and can be viewed as thevariance and the bias part, respectively. However, no

44、te that thisconometrics 181 (2014) 117135 119is not the biasvariance decomposition defined for usual nonpara-metric estimators, because under our assumption of possible exis-tence of leverage effects, EV? K ,ht =? tt?h 2s ds/h. The variance partP0 is closely related to a similar quantity studied for

45、 the TSRV es-timator, and can be analogously decomposed into two parts as inZhang et al. (2005):V? K ,ht ? 1h? tt?h 2s ds =?Y , Y K ,ht ? X, XK ,ht ? n?n Y , Y ht?+?X, XK ,ht ? 1h? tt?h 2s ds?:= P2 + P3,where P2 and P3 are defined implicitly. Using the terminologies inthe above mentioned paper, thes

46、e are the error due to noise partand error due to discretization part, respectively.We study the bias part P1 first, for which we need makemore specific assumptions on the path properties of the volatilityprocess. We assume:A5 The spot variance process? 2t?is an It? process, satisfy-ingd 2t = tdt +t

47、dBt ,where t and t are stochastic processes with con-tinuous sample paths. Bt is a standard Brownian Mo-tion, possibly correlated with Wt.This was the assumption used in Mykland and Zhang (2008)about the volatility process. Under this assumption, the limitingdistribution of P1 can be derived:Proposi

48、tion 1. Under Assumptions A1, A2 and A5, as n ,h 0, nh ,h?1/2P1st?132t ZP1 ,where ZP1 is a standard normal variable independent of the -algebragenerated by the X process. The stable convergence is with respect tothe -algebra generated by the X process.Here P1 converges to amixed normal distribution

49、stably, wheret is the diffusion parameter of the spot variance process, which israndom. The rate of convergenceh?1/2 is implied by the continuoussemimartingale assumptionwemake for the spot variance process.The diffusion assumption on the spot variance process is ratherrestrictive. To accommodate mo

50、re classes of models for the spotvariance process, we could relax Assumption A5 to H?lders typecontinuity assumption:A5-1 For almost all paths of the spot variance process,? 2t ?(m) ? ? 2s ?(m)? 6 C |t ? s| , for all t, s 0, 1 ,wherem > 0 is an integer, the superscript (m) denotes themth derivati

51、ve with respect to time, and 0 < < 1.This assumption was used in Kristensen (2010). It is very gen-eral and accommodates diffusion process, long memory stochasticdifferential equations (as in Comte and Renault (1998), determin-istic patterns as well as nonstationarity in the spot variance pro-

52、cess. In particular, continuous semimartingales are allowed for inthe assumption with m = 0 and < 1/2 (see for example, Ch. V,Exercise 1.20 of Revuz and Yor (1998). Lvys modulus of conti-nuity type assumption sup|t?s|6h? 2t ? 2s ? = Op ?h1/2 |log h|1/2?(e.g. in Fan andWang (2008) is also included

53、 in this assumption, aswell as a deterministic spot variance function. Under A5-1, a cen-tral limit theorem for P1 is not available, but the following upperbound can be easily derived:E120 Y. Zu, H. Peter Boswijk / Journal ofProposition 2. Under Assumptions A1, A2 and A5-1,P1 = Op?hm+?.We then study

54、 the due to noise part P2 and the due todiscretization part P3. We show that P2 converges to a normaldistribution conditional on the X process, while P3 converges toa mixture of normal distributions stably. The results for thesequantities will be extensions of Zhang et al. (2005)s correspondingresults for a fixed interval to our context of shrinking intervals.Proposition 3. Under Assumptions