MICROSTRUCTURE NOISE, REALIZED VARIANCE, AND OPTIMAL SAMPLING

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1 ICROSTRUCTURE NOISE, REALIZED VARIANCE, AND OPTIAL SAPLING By Federico. Bandi y and Jeffrey R. Russell z First draft: November 23 Tis draft: July 27 Abstract A recent and extensive literature as pioneered te summing of squared observed intradaily returns, "realized variance," to estimate te daily integrated variance of nancial asset prices, a traditional object of economic interest. We sow tat, in te presence of market microstructure noise, realized variance does not identify te daily integrated variance of te frictionless equilibrium price. However, we demonstrate tat te noise-induced bias at very ig sampling frequencies can be appropriately traded o wit te variance reduction obtained by ig frequency sampling and derive an SE optimal sampling teory for te purpose of integrated variance estimation. We sow ow our teory naturally leads to an identi cation procedure wic allows us to recover te moments of te unobserved noise; tis procedure may be useful in oter applications. Finally, using te pro ts obtained by option traders on te basis of alternative variance forecasts as our economic metric, we nd tat explicit optimization of realized variance s nite sample SE properties results in accurate forecasts and considerable economic gains. Keywords: Realized variance, Quadratic variation, arket microstructure noise, Optimal sampling, Option pricing JEL Classi cation: G2, C4, C22 We are grateful to Tim Conley, Rob Engle, Neil Separd, two anonymous referees, and Bernard Selanié, te Editor, for valuable comments and suggestions. We tank te participants at te CIRANO conference Realized Volatility, ontreal, November 7-8, 23, te conference Econometric Forecasting and Hig-Frequency Data Analysis, Singapore, ay 7-8, 24, and seminar participants at various institutions for elpful discussions. We tank Jean Jacod for sending us a copy of is unpublised 994 manuscript. y Graduate Scool of Business, University of Cicago, 587 Sout Woodlawn Avenue, Cicago, IL federico.bandi@gsb.ucicago.edu. z Graduate Scool of Business, University of Cicago, 587 Sout Woodlawn Avenue, Cicago, IL je rey.russell@gsb.ucicago.edu.

2 . INTRODUCTION THE RECENT AVAILABILITY of quality ig-frequency nancial data as motivated a growing literature devoted to te model-free measurement of variance (see te review paper by Andersen et al. (22) and te references terein). Te main idea is to aggregate squared intra-daily returns to approximate te daily increments of te quadratic variation of te semimartingale tat drives te underlying logaritmic price process. Te consistency result justifying tis procedure is te convergence in probability of te sum of squared returns to te daily increment of te quadratic variation of te logaritmic price process as returns are computed over intervals tat are increasingly small asymptotically. Tis result is a cornerstone in semimartingale process teory (see, e.g., Cung and Williams (Teorem 4., page 76, 99)). ore recently, a nonparametric teory of inference for variance estimation as campioned empirical implementation of tese ideas and as termed tese estimates realized variance (Andersen et al. (23) and Barndor -Nielsen and Separd (22), BN-S ereafter). Te teoretical validity of te procedure inges on te observability of te true price process. However, te ne grain market dynamics (i.e., "te market microstructure") generate a divergence between te observed price process and te true or "frictionless equilibrium" price process, wose quadratic variation is te object of interest. Tis divergence could, for example, be induced by transaction price canges occurring as multiples of ticks (price discreteness) or by te existence of multiple prices for buyers and sellers (bid-ask spreads). It may be due to liquidity or information reasons (see, e.g., Stoll (2)). In agreement wit te extant literature, we denote te deviations of observed prices from frictionless equilibrium prices by "market microstructure noise" and te frictionless equilibrium price by "true price" or "equilibrium price." Te present paper provides a general treatment of market microstructure noise on realized variance estimates. Speci cally, we consider bot asymptotic and nite sample e ects of noise. Under our assumed price formation mecanism, we sow tat te realized variance estimates are asymptotically swamped by noise as te number of squared return data increases over a xed time period. Te teoretical manifestation of tis e ect is a realized variance estimator wic fails to converge to te increment of te quadratic variation (or "integrated variance") of te underlying logaritmic true price process but, instead, increases witout bound almost surely over any xed period of time, owever small. Hence, our results provide a teoretical justi cation for te diverging beavior at ig frequencies of te realized variance estimates as sown by Andersen et al. (999, 2) using "volatility signature plots," namely plots of realized volatility 2

3 versus alternative sampling frequencies. For illustration, Fig. contains te (averaged across days) variance signature plot for te IB data used in tis paper. As implied by our asymptotic teory, te plot is upward sloping at ig frequencies. Figure about ere Interestingly, despite te fact tat realized variance is not consistent for te conventional object of interest (i.e., increments of quadratic variation or integrated variance), we sow tat a standardized version of te realized variance estimator can be employed to identify te variance of te unobservable noise component. ore generally, we sow tat sample moments of te noisy return data can be put to work to identify moments of te underlying noise process. Tis contribution of te paper is important for te empirical analysis of te bias and variance of te realized variance estimator since, naturally, bot quantities depend on te distributional features of te unobserved noise component in te recorded returns. oreover, since te noise dictates deviations of observed prices from equilibrium prices, te noise moments may be of direct economic interest. Our nite sample results begin wit a caracterization of te bias/variance trade-o induced by te presence of microstructure noise. Wen te true price process is observable, iger sampling frequencies over a xed period of time result in more precise estimates of te integrated variance of te logaritmic price process. Wen te true price process is not observable, as is te case in te presence of microstructure frictions, frequency increases provide information about te underlying integrated variance but, necessarily, entail accumulation of noise a ecting bot te bias and te variance of te estimator. Te sampling frequency can be cosen optimally to balance tese two contrasting e ects. Speci cally, te bias/variance trade-o can serve as te basis for an optimal sampling teory for nonparametric variance estimates in te presence of microstructure noise. Under independence of te noise, we rst derive an expression for te conditional (on te underlying volatility pat) mean-squared-error (SE) of te contaminated realized variance estimator as a function of te sampling frequency. A robust metodology is ten provided to optimally coose te sampling frequency as te minimum of te conditional SE. Te metod relies on te computation of sample moments of te contaminated ig-frequency return data. As suc, it is simple to implement. Subsequently, we study several extensions of our proposed metodology. First, we consider a bias-corrected realized variance estimator and discuss optimal sampling for te purpose of nite sample variance minimization. Second, we evaluate optimal 3

4 sampling for estimating nonlinear functions of increments of quadratic variation. Finally, we relax te assumption of independent noise made initially and study optimal sampling under more general noise dependence properties. We apply our metods to a sample of IB quote data and nd optimal sampling intervals tat are sorter tan te intervals typically implemented and/or conjectured in te literature. Our new intervals translate into considerable statistical gains. Te last part of te paper employes te pro ts obtained from option trading on te basis of alternative variance forecasts as te economic metric used to evaluate te relative bene ts of time-varying optimal (in an SE sense) sampling intervals versus xed, ad-oc intervals. We sow tat our proposed time-varying optimal intervals result in improved variance forecasts and pervasive economic gains. Despite general awareness of te necessity of accounting for price contaminations in igfrequency data for te purpose of integrated variance estimation (see, e.g., te discussions in Andersen et al. (2) and BN-S (22)), at te time of tis paper s original writing most existing treatments of te e ects of market microstructure noise on realized variance estimates were rater informal. Tere were important exceptions. Tese eiter imposed restrictions tat we wised to avoid or failed to be operational in practise. In some cases, te focus was di erent. Zou (996) proposes coosing te optimal sampling frequency of a bias-corrected realized variance measure by minimizing its variance expansion. Zou s framework inges on te i.i.d. nature of te noise components in te price process, on te Gaussianity of te noise, and on a constant spot price volatility over individual periods. Bai et al. (24) derive an SE expansion for variance estimates in te presence of dependent noise. However, tey take a less structural approac tan we do and are terefore unable to independently identify te two variance components of te observed returns, i.e., te equilibrium price variance and te variance of te microstructure noise. Consistent wit Zou (996) and under similar assumptions, Aït-Saalia et al. (25) derive a closed-form expression for te unconditional SE of a constant variance estimator. Tis paper s focus, owever, is on important e ciency gains from maximum likeliood estimation of parametric di usion models wit market microstructure noise. Parametric likeliood-based identi cation in te presence of noise is also studied in work by Gloter and Jacod (2a, 2b). Oomen (22) uses a structural model of price formation to provide simulated SE plots for noisy quadratic variation estimates as a function of te sampling interval. He allows for a stocastic variance but resorts to simulations in te absence of a closed-form speci cation for te relation between te relevant SE and te sampling frequency. Tere exists a related literature in probability teory wic studies te identi cation of functionals of nearly observed continuous-time 4

5 processes. Unobservability is generally eiter due to deterministic round-o errors or to Gaussian stocastic noise (see Delattre and Jacod (997) and Picard (993), for example). Contrary to te present paper and te papers cited previously, limiting results are obtained for contaminations tat are assumed to be small asymptotically. Wile interesting from a teoretical perspective, tese approaces are not appropriate for our observed price formation mecanism involving microstructure noise e ects. Not surprisingly, our price formation mecanism leads us to a di erent approac to identi cation. In contemporaneous and independent work, Zang et al. (25) ave also derived a conditional SE expansion for realized variance in te presence of i.i.d. noise, one of te teoretical contributions in te present paper. Under te same noise properties, tey propose a metodology to obtain consistent estimates of integrated variance. Teir suggested metod relies on averaging (bias-corrected) realized variance estimates sampled at a speci c frequency (i.e., it relies on "subsampling" in teir terminology). Wile our focus is on a general study of te asymptotic and nite sample properties of te classical realized variance estimator in te presence of noise, teir focus is on te asymptotic consistency properties of a novel approac to integrated variance estimation. A promising bias-corrected realized variance measure allowing for dependent noise and some form of dependence between noise and equilibrium price as been recently advocated by Hansen and Lunde (26). As in Zou (996), teir estimator is a kernel-based estimator. Te usefulness of a ric class of kernel estimates of integrated variance as been furter studied in recent work by Barndor -Nielsen et al. (26). Since tis paper s original draft, stimulating developments of optimal sampling metods ave been proposed for a variety of issues in nonparametric variance estimation in te presence of noise-contaminated ig-frequency data. We mention a few important contributions. Hansen and Lunde (26) discuss SE-based sampling of bias-corrected realized variance estimates. Oomen (26) study SE-based sampling for te purpose of evaluating te preferability of calendar time sampling versus business time sampling. Oomen (25) study SE-based optimal sampling in calendar time and business time for bias-corrected realized variance estimates. Bandi and Russell (26d) propose SE-based rules for selecting te optimal number of autocovariances (or subsamples) to be used in te de nition of kernel estimates of integrated variance as in Zang et al. (25) and Barndor -Nielsen et al. (26). ancino and Sanfelici (26) derive SE expansions for Fourier estimators of integrated variance. Andersen et al. (26) and Gysels and Sinko (26) study optimal sampling for te purpose of realized variance forecasting. Te researc on ig-frequency variance estimation and noise as been particularly vibrant in recent years. For 5

6 extensive reviews, we refer te interested reader to Bandi and Russell (26c), Barndor -Nielsen and Separd (26), and caleer and edeiros (26). Te paper proceeds as follows. In Section 2 we lay out te model. Section 3 is about te limiting properties of te realized variance estimator wen microstructure noise a ects asset prices. In Section 4 we present an expansion of te conditional SE of te realized variance estimator wen i.i.d. noise plays a role and discuss optimal sampling by virtue of SE minimization. In Section 5 we study extensions of te metod. Speci cally, we discuss bias-correcting, optimal sampling for te purpose of estimating nonlinear functions of integrated variance, and optimal sampling in te presence of ricer noise dependence properties. Section 6 contains simulations. In Section 7 we use quote-to-quote IB price canges and apply our metods to te estimation of te integrated variance of te logaritmic price process and te second moment of te unobservable noise process. Section 8 discusses te economic bene ts of variance forecasting using optimal sampling. We do so in te context of option pricing and option trading. Section 9 concludes. Proofs, tecnical details, and a glossary of notation are contained in te Appendixes. 2. THE ODEL We introduce microstructure noise e ects in te context of a model tat is consistent wit previous teoretical approaces to model-free variance estimation (see, e.g., BN-S (22, 23, 24)). For convenience, we use similar notation as in BN-S (22, 23, 24). We consider a xed time period (a trading day, for instance) and write te observed price process at te end of te i-t period as ep i = p i # i i = ; 2; :::; n; (2.) were p i is te frictionless equilibrium price, i.e., te price tat would prevail in te absence of market microstructure frictions, and # i denotes microstructure noise. A simple logaritmic transformation gives us ln (ep i ) ln ep (i ) = ln(p i) ln(p (i )) + i (i ) i = ; 2; :::; n; (2.2) {z } {z } {z } er i r i " i were = ln(#): We can now divide eac period into sub-periods and de ne te observed ig-frequency continuously-compounded returns as er j;i = ln ep (i )+j ln ep (i )+(j ) j = ; 2; :::; ; (2.3) 6

7 were = is te time distance between adjacent logaritmic prices or, equivalently, te time orizon over wic te continuously-compounded returns are computed. Hence, er j;i is te j-t intra-period return over te i-t period. ore precisely, er j;i = r j;i + " j;i ; (2.4) were r j;i and " j;i (= (i )+j (i )+(j ) ) ave straigtforward interpretations in terms of equilibrium return and microstructure contamination in te return data, respectively. Bot te equilibrium return r j;i and te microstructure noise contamination " j;i are unobservable. Te econometrician only observes te noisy return data er j;i. To simplify te notation, we suppress te subscript i and deal wit te case i =. Tus, we write j in place of te double index j; i. Our discussion will ten apply to eac period from to n witout loss of generality. Our interest will be in caracterizing te asymptotic and nite sample properties of te classical realized variance estimator V b = P er2 j : As in Andersen et al. (23) and BN-S (22), among oters, b V will be used to estimate te integrated variance of te logaritmic price process over eac period. For clarity, we de ne tis quantity formally after a discussion of te assumptions tat we impose on te logaritmic price process (ln(p)) and te microstructure contaminations in te price process (te s), respectively. Assumption : (Te Price Process) Te logaritmic price process ln(p t ) is a continuous stocastic volatility Brownian semimartingale. Speci cally, ln(p t ) = ln(p ) + Z t s ds + Z t s dw s ; (2.5) were t is a continuous predictable drift process, t is a càdlàg spot volatility process, and W t is a standard Brownian motion. Assumption 2: (Te icrostructure Noise) Te microstructure frictions in te price process js are i.i.d. moment. mean zero wit a nite fourt Te object of econometric interest is te bounded integrated variance of te equilibrium price process over eac period, i.e., V = R 2 sds. 7

8 Assumption implies tat te equilibrium return process evolves in time as a stocastic volatility martingale di erence plus an adapted process of nite variation. Te stocastic spot volatility is allowed to display jumps, diurnal e ects, ig-persistence (possibly of te long-memory type), and nonstationarities. 2 Dependence between and te Brownian motion W (i.e., leverage e ects) is allowed. Assumption 2 implies tat te observed returns display an A() structure wit a negative rst-order autocorrelation. Te A() market microstructure model in returns (or te i.i.d. market microstructure model in prices) is typically justi ed by bid-ask bounce e ects (see, e.g., Roll (984)). It is, in general, an approximation. However, it is a credible approximation in decentralized markets were traders arrive in a random fasion wit idiosyncratic price setting beavior, te foreign excange market being an important example (see, e.g., Bai et al. (24)). It can also be a valid approximation in te case of equities wen considering transaction prices and/or quotes posted on multiple excanges (see Bandi and Russell (26b)). Awartani et al. (24) propose a formal test of te A() market microstructure model. In Section 5 we extend te model to a ricer noise dependence structure. Section 7 sows te empirical validity of te A() model for te sample of IB ig-frequency price data used in te present study. Importantly, wile te equilibrium return process is O p p over any intra-period time orizon of size =, te contaminations in te return process are O p(): Tis result, wic is a consequence of Assumptions and 2, implies tat longer period returns are less contaminated by noise tan sorter period returns. In oter words, te size of te contaminations does not decrease in probability wit te distance between subsequent time stamps. Te rounding of recorded prices to a grid, alone, makes tis feature of te model compelling. 3 One nal observation is in order before turning to our results. A large and successful literature in macroeconomics as focused on metods to study nonstationary time series (like our observed price process ln ep) expressed as te sum of a nonstationary component (like our unobserved equilibrium price process ln p wit ) and a residual stationary component (like our noise e ect Te study of jumps in te price process is beyond te scope of te present paper and is left for future researc. Te presence of jumps would require re-tinking about te object of interest since empasis could be placed on te di usive component of integrated variance over te period, on te total variance inclusive of squared jumps, or simply on te sum of te squared jumps. Importantly, te main limiting results in tis paper, namely te diverging beavior of realized variance at ig frequencies (Teorem, Point (ii)) and te consistency of te noise moment estimates (Teorem 2) would old even in te presence of infrequent Poisson jumps, for example. 2 For jumps in volatility, see Eraker et al. (23) and te references terein. For diurnal e ects, see Andersen and Bollerslev (998) and te references terein. For persistence in volatility, see Andersen et al. (23) and te references terein. For potential nonstationarities in volatility, see Bandi and Perron (26) and te references terein. 3 We are of course assuming tat sampling does not occur between ig-frequency price updates. 8

9 ) (see, e.g., Stock and Watson (988), for a review). Wile tese metods ave also proved very useful in empirical work on market microstructure (see, e.g. Hasbrouck (993)), our identi cation procedures are novel. As we will sow below, we rely on te di erent stocastic orders of te unobserved components of te observed returns as described in te previous paragrap. In oter words, we rely on te identi cation potential of ig-frequency return data sampled at di erent frequencies. 3. ASYPTOTIC THEORY FOR REALIZED VARIANCE Under te set-up in Eq. (2.4), we can rewrite te realized variance estimator as te sum of tree components, namely bv = r 2 j + {z } A {z } B {z C } If te true price process were observable, only te term A would of course drive te limiting " 2 j + 2 r j " j : (3.) properties of b V. Te presence of microstructure noise introduces two additional terms. We will sow tat it is mainly term B tat makes standard consistency arguments fail. Intuitively, B diverges to in nity almost surely as te number of observations increases asymptotically (or, equivalently, as te observation frequency increases in te limit) since more and more noise is being accumulated for a xed period of time. Teorem below caracterizes our asymptotic ndings regarding realized variance. Te term Q in te teorem denotes te bounded integrated quarticity discussed, e.g., by BN-S (22), i.e., Q = R 4 sds. We use tis notation trougout te paper. ten ten Teorem : (i) Assume absence of market microstructure noise, i.e., j 8j. If Assumption is satis ed, r bv V ) N (; 2Q) as! : (3.2) (ii) Assume presence of market microstructure noise. If Assumption and 2 are satis ed, Proof: See Appendix A. bv a:s:! as! : (3.3) 9

10 Remark : (Absence of icrostructure Noise) In te absence of market microstructure contaminations, te estimation error between te realized variance estimator and integrated variance over te period converges weakly to a mean-zero mixed Gaussian distribution at speed p. Result (i) replicates a nding obtained by Jacod (994, Proposition 9.) and Jacod and Protter (998). Under te additional restriction of independence between and W (absence of leverage e ects), te result was also obtained by BN-S (22). Here, we obtain it in te context of a functional central limit teory. Our derivation (in Appendix A) is of independent econometric interest. It is novel and simpler tan te original derivation in Jacod (994) and Jacod and Protter (998). Remark 2: (Presence of icrostructure Noise) Wen microstructure e ects play a role, te realized variance estimator does not consistently estimate te integrated variance over a period. Intuitively, te summing of an increasing number of contaminated return data entails in nite accumulation of noise as te frequency increases asymptotically. Speci cally, wile te term A in Eq. (3.) above converges to te integrated variance over te period (from Eq. (3.2) in (i)), te term B diverges to in nity almost surely. 4 Te term C is stocastically dominated by B. Te limiting result in (ii) is an asymptotic approximation suggesting tat for large, as is te case for ig-frequency data, te researcer must be wary of microstructure contaminations as te e ect of te noise can be substantial. Hence, any statement about te informational content of te conventional realized variance estimator as a measurement of te integrated variance of te underlying logaritmic price process ougt to be a nite sample statement. Interestingly, owever, sample moments of te observed return series can be used to learn about population moments of te unobserved noise returns at ig frequencies. Tis result is formalized in te following teorem. Teorem 2: If Assumptions and 2 are satis ed and E 8 <, ten as!. Proof: See Appendix A. er q j p! E (" q ) q = 2; 3; 4 (3.4) For large, Teorem 2 implies tat (over any xed period ) one can consistently estimate 4 Tis result does not inge on te dependence properties of te noise. It simply relies on te di erent stocastic orders of te noise returns (O p()) and equilibrium returns (O p( p )). In oter words, te result will also old under te more general noise structure in Subsection 5.3 below.

11 moments of te unobserved noise process at all frequencies by using data sampled at te igest frequency. We now move from asymptotic arguments to a caracterization of te nite sample bias/variance trade-o induced by te accumulation of noise. 4. FINITE SAPLE THEORY FOR REALIZED VARIANCE Tis section derives te SE of te realized variance estimator conditional on te volatility pat. Our strategy is to learn about te underlying integrated variance of te logaritmic price process by sampling at a rate wic minimizes te conditional expected squared loss of te realized variance estimator. We sow tat te minimum SE is acieved for a nite number of return observations. We also sow tat depends on moments of te microstructure noise distribution as well as on te integrated variance and te integrated quarticity of te underlying logaritmic price process. For empirical tractability, te results in tis section are stated under tree additional conditions wic we jointly collect in Assumption 3. We empasize tat tese conditions are not required to establis te asymptotic results in te previous section. Assumption 3: (). (2)? W. (3) Te frictions js are independent of te price process. Assumption 3() implies unpredictability of te equilibrium returns. Wile te presence of time-varying risk premia would invalidate tis assumption, te drift component is known to be negligible at te sampling frequencies considered in te realized variance literature. 5 Coerently, classical market microstructure teory predicts tat te unobservable equilibrium price sould evolve as a martingale at ig frequencies (see, e.g., O Hara (995)). 5 Consider a stock like IB, for example. Assume a realistic annual constant drift of.8. Te magnitude of te drift over a minute interval would ten be :8=( ) = :52 7. Computing te standard deviation of IB return data over te same orizon by using data discussed in Section 7 yields a value equal to 9:5 4. Hence, at te one minute interval te drift component is :6 4 ; or nearly =;, te magnitude of te return standard deviation. Similarly, te daily value of te drift based on a 6-our trading period would be 5:47 5. Te average daily standard deviation is :5 2. Terefore, te daily drift is 3:6 3 te magnitude of te daily standard deviation.

12 Assumption 3(2) implies absence of leverage e ects. Wile assuming absence of leverage e ects is known to be empirically reasonable in te case of excange rate data, te same condition appears restrictive wen examining ig-frequency stock returns. Noneteless, recent work uses tractable parametric models to sow tat te e ect of leverage on te unconditional SE of te realized variance estimator in te absence of market microstructure noise is asymptotically negligible (eddai (22) and Andersen et al. (25)). Tis work provides a justi cation for te standard assumption of no-leverage in te literature (see te review paper by Andersen et al. (22) for furter discussions). Assumption 3(3) implies independence between te equilibrium returns and te noise components at all frequencies. At low frequencies tis assumption provides a reasonable approximation. One way of assessing te empirical plausibility of te assumption at ig frequencies is by using volatility signature plots (see Andersen et al. (999, 2)), as advocated by Hansen and Lunde (26). Hansen and Lunde (26) sow tat te presence of a negative correlation between noise and equilibrium returns migt determine realized variance estimates tat do not diverge as te sampling frequency increases. Suc estimates migt even underestimate te true integrated variance over te period. Bandi and Russell (26b) point out tat te diverging beavior of realized variance at ig frequencies is rater pervasive across markets, sampling metods (trade time sampling versus calendar time sampling), and observed prices (mid-quotes versus transaction prices). In agreement wit our limiting results (Teorem, Point (ii)), Fig. in te Introduction sows te diverging beavior of te variance signature plot of te stock analyzed in Section 7 (IB). In our data we do not nd obvious evidence of a signi cant, negative correlation. Teorem 3 lays out te conditional (on te volatility pat) SE of realized variance. In wat follows, te symbol E will denote expectation conditional on te pat of volatility over te period. We continue to use te symbol E wen te conditioning statement is moot. Te symbol " j will denote te j-t lag of te variable ". Teorem 3: If Assumptions, 2, and 3 are satis ed, ten E bv were te parameters a; b; and c are de ned as: V 2 = 2 (Q + o()) + b + 2 a + c; (4.) a = E(" 2 ) 2 ; (4.2) b = E " 4 + 2E(" 2 " 2 ) 3 E(" 2 ) 2 ; (4.3) 2

13 and c = 4E(" 2 )V 2E(" 2 " 2 ) + 2 E(" 2 ) 2 : (4.4) Proof: See Appendix A. Sould te observed return series not be a ected by microstructure noise, ten te SE would decrease to zero as te number of observations diverges to in nity. In fact, it would simply reduce to te conditional variance of te sum of squared equilibrium returns, i.e., 2 (Q + o()) (see BN-S (22) and Appendix A for a derivation). Wen microstructure noise is present, te SE does not vanis as te number of observations diverges to in nity asymptotically (or, equivalently, as te sampling frequency increases over time). Summing up contaminated squared returns induces bot a bias term, i.e., E bv V = E " 2 ; (4.5) and an additional variance term (jointly captured by ). Tus, te term induces a trade-o wic can be optimized by coosing te number of observations so tat arg min 2 (Q + o()) + : (4.6) Notice tat E(" 2 " 2 ) = 2 E "4 under Assumption 2. Hence, te minimum of te conditional SE expansion of te realized variance estimator only depends on te second and fourt moment of te noise component in te return data as well as on te integrated quarticity Q and te integrated variance V. Te second and fourt moment of te noise can be estimated consistently by virtue of Teorem 2. 6 BN-S (22) ave provided an estimator of te integrated quarticity wic is consistent in te absence of market microstructure noise, i.e., b Q = 3 P er4 j : Inevitably, bq, like b V, loses its consistency properties wen microstructure e ects play a role. In ligt of tis observation, to evaluate te SE empirically, we suggest using relatively low frequencies (i.e., 5- or 2-minute frequencies) to compute preliminary (rougly unbiased) estimates of b V and b Q. 7 Te simulations in Section 6 provide evidence for te empirical usefulness of tis straigtforward estimation procedure. 8 6 Te estimators nite sample biases and bias-corrections are discussed in Subsection Te design of e cient quarticity estimators is an important topic for future researc. 8 We refer te reader to te simulations in Bandi and Russell (26a) and te unpublised Appendix to tis paper (posted on te journal s web site) for furter evidence. 3

14 Next, we o er a simple approximation to coose te optimal frequency in te presence of A() noise. Remark 3: For a ig optimal number of observations, we can write Wen te quadratic term in Eq. Q =3 (E(" 2 )) 2 : (4.7) (4.) dominates te linear term (i.e., for values of su ciently large), te approximation in Eq. (4.7) provides a good representation of te optimal number of observations. In Section 7 we sow tat tis property is valid for a liquid stock like IB. Bandi and Russell (26a) con rm te validity of tis approximation for a large number of S&P stocks. Remark 3 is important for two reasons. First, it provides a straigtforward rule-of-tumb to coose witout aving to go troug te oterwise simple minimization of Eq. (4.) above. Second, it clearly illustrates wat te main determinants of te optimal frequency are, namely te integrated quarticity of te logaritmic price process and te squared variance of te microstructure noise component in te return process. Naturally, can be regarded as a signal-to-noise ratio: te stronger te signal, te iger te optimal frequency. 5. ETENSIONS We consider tree extensions of te previous framework: ( ) bias-corrected realized variance, (2 ) non-linear functions of integrated variance, and (3 ) dependent noise. 5.. Bias-corrected realized variance Eq. (4.5) sows tat te bias of V b is equal to E(" 2 ). Te squared bias term can be a substantial component of te SE and bias-correcting te realized variance estimator migt be very bene- cial in practise (see Zou (996), Hansen and Lunde (26), and Oomen (25) for important, alternative approaces to bias-correcting in te context of integrated variance estimation). Given an estimate of E(" 2 ) (obtained as in Teorem 2), a bias-corrected realized variance estimator can be derived. Te following teorem presents te conditional SE in tis case. Te SE provides a means of selecting an optimal number of observations bc (bc) b V. to be used to optimally bias-correct Teorem 4: De ne b V bc = b V E(" 2 ). If Assumptions, 2, and 3 are satis ed, ten 4

15 E bv bc 2 V = 2 (Q + o()) + b + c; (5.) were te parameters b and c are de ned in Eq. (4.3) and Eq. (4.4). Remark 4: closed-form as were In tis case, te minimum of te SE (variance) is conveniently de ned in bc = 2Q =2 2E (" 4 ) 3 (E(" 2 )) 2 ; (5.2) 2E " 4 3 E(" 2 ) 2 = 4E 4 > : (5.3) Te optimal number of observations bc of te bias-corrected estimator is in general larger tan. Tis is intuitive in tat bias reduction permits sampling at iger frequency. Section 7 provides an empirical application. It sould be noted tat we assume absence of estimation uncertainty in te identi cation of te bias term E " 2. Tis assumption is justi ed by te fact tat te second moment of te noise as te potential to be estimated accurately in te presence of data sampled at ig frequency. We expect te empirical relevance (for te purpose of evaluating bc ) of incorporating estimation uncertainty regarding E " 2 to be small. Work on tis subject is beyond te scope of te present paper and is left for future researc Nonlinear functions of integrated variance Interest could be placed in nonlinear functions of integrated variance. Typical examples in nance are volatility ( p V ) and Sarpe ratios. Neglecting scale factors (i.e., expected excess returns), te latter require computation of p V. As is te case for Remark 3 and Remark 4 above, Remark 5 provides a simple SE-based rule to coose te optimal frequency in tis case. Remark 5: Consider a twice continuously di erentiable function f(:). For a ig optimal number of observations f, and a second-order Taylor expansion of f(:), we can write f! 2(f =5 (V )) 2 Q (f (V )) 2 (E(" 2 )) 4 ; (5.4) provided f (:) 6= and f (:) 6= : Proof: See Appendix A. 5

16 As earlier, f depends on a signal-to-noise ratio. Importantly, as expected, "te signal" (i.e., te variance term) in te numerator depends on te rst derivative of te function, wereas "te noise" term in te denominator depends on te function s curvature. 9 For te types of nonlinearities routinely encountered in nance, te non-zero condition on te function s derivatives is of course easily satis ed in general. In Section 7 we employ Eq. (5.4) to compute te optimal frequency of te realized volatility estimator for IB and a typical day in our sample Noise dependence Tis subsection generalizes te persistence properties of te noise. Assumption 2. To tis extent, we revise Assumption 2b : (Te icrostructure Noise) () Te microstructure frictions in te price process js ave mean zero and are strictly stationary wit joint density f (:). (2) Te variance of " j = j j is K + o() wit K >. Assumption 2b permits general dependence features for te microstructure noise components in te recorded prices. Te correlation structure of te microstructure noise contaminations can capture, as earlier, rst-order negative autocorrelation in te recorded ig-frequency returns, as determined by bid-ask bounce e ects, as well as iger-order dependence as induced by clustering in order ows. In tis case, owever, te caracteristics of te noise returns " s may, in general, depend on te sampling frequency =. Te joint density of te s as a subscript to make tis dependence explicit. Consistently, te symbol E will be used in wat follows to denote expectation given (and te measure f (:) in te case of te noise returns). As before, te symbol E ; will denote expectation conditional on te volatility pat. Assumption 2b(2) implies tat te variance of te noise returns does not converge to zero as te observation frequency increases. Tis is e ectively an identi cation condition justi ed by te economics of ig-frequency price formation provided sampling does not occur witin price updates. Tis assumption yields, as earlier, te asymptotic diverging beavior of realized variance 9 Naturally, te accuracy of te approximation depends on te function f(:) as well as on te moments of te unobserved price components. Given a Taylor expansion of te relevant function, one could, in principle, minimize te full-blown SE. Tis said, even in te second-order expansion case, te form of te SE is very involved and of little practical applicability. Tis SE can be provided by te autors upon request. 6

17 over every xed time period (as in Teorem above). If te variance of te noise returns converged to zero at te same speed as te variance of te equilibrium returns (as implied by a di usion model for te price contaminations, for example), ten realized variance would converge to te sum of (increments of) equilibrium price and noise price quadratic variation. However, given te discreetness in te observed prices, it is unlikely tat tis modelling alternative would provide a meaningful approximation. Teorem 4 presents te conditional SE of te realized variance estimator under general noise dependence. Teorem 5: If Assumptions, 2b, and 3 are satis ed, ten were E ; bv = E " V 2 = 2 (Q + o()) + ; (5.5) ( j)e " 2 " 2 j + 4E (" 2 )V: (5.6) Empirical evaluation of te SE (and coice of ) is arder in tis case tan in te A() case. On te one and, te SE is not a closed-form function of since te noise moments cange wit te sampling frequency. On te oter and, sample moments of te observed returns (over eac period) cannot deliver consistent estimates of te noise moments for eac. However, for eac, we can derive (rougly) unbiased estimates of te noise moments. Tese can be used, in conjunction wit low-frequency estimates of V b and Q; b as earlier, to evaluate te SE for every frequency before nding its minimum. To tis extent, we notice tat E (" 2 ) = E E (" 4 ) = E er 2 j er 4 j A V ; (5.7) A 6E (" 2 )V + O 2 ; (5.8) and E (" 2 " 2 s) = E s j=s+ er 2 j er 2 j s A 2E (" 2 )V s + O ( s) for all xed s <, given te volatility pat. Tese expressions can be used to bias-correct te sample moments of te observed returns for eac frequency (5.9) witin eac period of interest. 7

18 Wen ignoring te order terms in Eqs. (5.8) and (5.9), te procedure results in a rst-order bias-correction. However, if one is worried about accuracy, te full bias can be caracterized. Speci cally, te order term in Eq. (5.8) is equal to is equal to 3Q 2 wereas te order term in Eq. (5.9) s + E (" 2 ) s (j ) j= s+ 2 udu (j )! Z (j s) 2 udu (j s )! + 2 udu s! (j ) 2 udu! A : (5.) Bot te rst-order bias-correction and te full bias-correction can be applied to straigt sample moments of te observed return data. As in Teorem 2, tese moments can of course be computed over a xed time span. Hence, we do not need to use information across multiple periods to evaluate te single-period SEs. However, under te assumption tat te properties of te noise do not cange from period to period, we can average te sample moments of te observed returns across periods and employ conventional limiting arguments as te number of periods n diverges to in nity for a xed. Given te volatility pat, tis procedure would result in consistent estimation of te relevant moments wit te quantities tat depend on volatility being replaced by teir averages across periods. Te single-period object V, for example, would ave to be replaced by its average across periods, i.e., P n i= V i=n. Section 7 applies tese metods. One can of course bias-correct realized variance even in te presence of noise dependence and ten optimize te estimator s SE (i.e., variance). In tis case, te SE of te bias-corrected estimator is te same as tat in Teorem 5 witout a term equal to 2 E (" 2 ) 2. Wile, as earlier, te SE can be evaluated for any xed time period, meaningful bias-correcting requires computation of E (" 2 ) by using information across periods: bv bc = b V bc 2 4 n bc bc i= er 2 j;i V i bc 3 A5 : (5.) Of course, te averaging in te bias-correction can be local (and rely on a moving window of periods, rater tan on te full sample n) if one wises to allow for properties of te noise tat cange over time. 8

19 6. SIULATIONS Our teoretical results deliver an optimal sampling frequency wic is determined by moments of te market microstructure noise and moments of te equilibrium price process. In practice tese moments must be estimated from te data. In tis section we provide onte Carlo evidence about te empirical performance of our optimal sampling teory wen te moments are estimated from data. In our onte Carlo setting we can evaluate te true SE (given moments of te data generating process) for any coice of sampling interval. We can terefore compare te relative performance (in an SE sense) of realized variance estimators constructed using any sampling interval of interest. Speci cally, we compare te SEs of tree estimators. Te rst estimator is te true minimum SE estimator wic cooses an interval wic minimizes te true SE (i.e., te estimator wic uses as de ned in Eq. (4.6)). Tis estimator is intended to represent te best realized variance estimator one can obtain in an SE sense. Obviously, tis estimator is infeasible in practice since it requires knowledge of te true noise moments, integrated quarticity, and integrated variance. Te second estimator relies on xed 5-minute intervals. Te SE for tis estimator represents a bencmark re ecting practical applications of realized variance estimates (see, e.g., Andersen et al. (999, 2)). Finally, we consider te estimator wic uses sample estimates of te noise moments, integrated quarticity, and integrated variance. Tis estimator represents empirical application of our proposed optimal sampling teory. As suc, we refer to it as te feasible optimal estimator. Since te observed price is te equilibrium price plus market microstructure noise, our onte Carlo experiment requires a model for te noise and for te equilibrium price process. Recent researc suggests te importance of a two-factor volatility model for te dynamics of te equilibrium price. We follow Cernov et al. (23) and write: d ln(p t ) = dt + s- exp ( + v t + 2 v 2t ) dw pt ; (6.) dv t = v t dt + dw v t; (6.2) dv 2t = 2 v 2t dt + ( + 3 v 2t ) dw v2 t; (6.3) were dw v t and dw v2 t are independent Brownian motions and s-exp denotes te exponential function splined wit appropriate growt conditions ensuring te existence of a unique, stationary solution (see Appendix A of Cernov et al. (23)). Te leverage correlations are 9

20 corr(dw p ; dw v ) = and corr(dw p ; dw v2 ) = 2 : We use te same parameter values as tose used in Huang and Taucen (26). We simulate 5, days, eac of 6.5 ours, using -second discretized increments. Te initial value of te volatility process is set equal to its unconditional mean of 4% annualized (obtained from a preliminary set of simulations). Te variance of te simulated volatilities is extremely ig. Te smallest and largest daily volatilities in te sample correspond to annualized return volatilities of :8% and over %, respectively. Tis is re ected in a very large value of te return kurtosis of 23. Wile it is not clear tat any single stock would ave suc a broad volatility range, tis process is useful to empirically establis te performance of our realized variance estimates for a broad range of states of te underlying volatility process. We specify te market microstructure noise as a mean zero, i.i.d. Gaussian process. We set te variance of te market microstructure noise equal to :2 2, were 2 is te unconditional variance of te daily true return process. Te ratio between noise variance and integrated variance of te underlying true price is fairly typical (see Bandi and Russell, 26a). Te daily unconditional true return variance is :76. Our noise variance is terefore equal to :92. Te observed return data is generated by summing te equilibrium returns and te noise returns over -second intervals. For eac day we estimate te moments of te noise using -second sampling intervals. Preliminary estimates of Q and V are obtained using 5-minute return data. For eac day, we construct te true SE function using te true noise moments, te true daily integrated variance, and te true daily integrated quarticity. We ten evaluate te SE at tree points, te infeasible minimum SE sampling interval, te 5-minute sampling interval, and te feasible minimum SE sampling interval. We rst discuss te performance of our feasible minimum SE estimator relative to te infeasible minimum SE estimator. Fig. 2 presents te empirical cumulative distribution function (CDF) for te ratio of te root SE (RSE) of te infeasible optimal estimator to te RSE of te feasible optimal estimator. Figure 2 about ere Tis plot summarizes te e ciency loss wic results from using estimated moments to construct te optimal sampling intervals rater tan true moments. Te feasible estimator performs very well. For alf of te days in te simulated sample te RSE of te infeasible estimator 2

21 is at least 95% tat of te feasible estimator. For tree quarters of te days te RSE of te infeasible estimator is at least 83% of te RSE of te feasible estimator. Te largest di erences result in a RSE of te infeasible estimator wic is about 35% of te RSE of te feasible estimator. Tus, te imprecision induced by moment estimation error appears small from an SE perspective. We now turn to te comparison between te feasible optimal sampling interval and te xed 5-minute interval. Fig. 3 presents te empirical CDF for te ratio of te RSE of te feasible optimal estimator to te RSE of te 5-minute estimator. Figure 3 about ere Tis plot compares two feasible estimators wic migt be used in practice. Te feasible optimal sampling interval outperforms te 5-minute estimator 88% of te time. Additionally, te improvement in RSE can be substantial - as muc as 5 times more accurate. Not surprisingly, te days in wic te 5-minute interval outperforms te optimal interval always correspond to optimal intervals near 5 minutes. Even on te rare occasions wen te 5-minute interval outperforms te optimal interval, te loss in e ciency is generally extremely small wit te RSE for te feasible optimal interval exceeding te 5-minute RSE by % or more on fewer tan 2.5% of te days. Because te relevant comparison is between te feasible optimal estimator and xed-interval estimators used and/or suggested in te extant literature (suc as te 5-minute estimator), tese results are important. Te results suggest tat, wile estimation error does not play an important role in practise, te upside to optimal sampling can be large and te downside is very small. 7. A SPECIFIC STOCK: IB 7.. Te data We use quote-to-quote mid-point prices. Te quotes were obtained from te TAQ data set for te mont of February 22. We restrict our attention to NYSE and IDWEST quote updates. Quotes prior to 9:3am are removed to ensure our sample contains no opening quotes. We ave a total of 4; 64 quote arrivals over te mont. On average a new quote arrives every 2:92 Ideally, one would want to use all available quotes from te consolidated market to construct te mid-quote return series. However, quotes from te satellite markets tended to eiter be far noisier tan tose generated by te NYSE specialist or ave a suspicious correlation structure. Tis observation is particularly true in te case of te NASDAQ quotes. A notable exception was te IDWEST excange. We terefore constructed mid-quote return series for IB by using quotes obtained bot from te NYSE and te IDWEST excange. 2

22 seconds. We construct -second continuously-compounded returns. Te smallest return in our sample is :92% and te largest is %. Te rst autocorrelation is signi cantly negative and equal to :2. Te second autocorrelation is :45 and te tird is :2. Tus, te A() approximation appears to capture te main economic e ects in our data. We con rm tis nding in te next subsection How big is te unobserved noise component of te observed IB price process? If te second moment of te noise return does not vary across frequencies, as in te A() model, ten Teorem 2 implies tat a re-scaled version of te realized variance estimator computed at ig frequencies consistently estimates te variance of te unobservable noise process over eac day. Here we also average te single period estimates across days. If te second moment of te noise component canges from day to day, as typically te case in practise, ten te resulting value can be interpreted as an average daily estimate. If te second moment does not cange from day to day, ten averaging te single period estimates across days would result in e ciency gains in te estimation of te constant noise second moment. We nd tat te square root of te re-scaled realized variance (computed over a 6:5-our trading period) is :38%. Notice tat tis estimate is essentially te sample standard deviation imposing a mean return of zero. To put tis in dollar context, consider tat te average price for IB over te mont of February in our sample was around dollars. Also, recall tat, under te A() model, te variance of te noise term in Eq. (2.2) is one alf te variance of te return contamination ". Tus, te standard deviation of te logaritmic noise price is given by = :38%= p 2 = :26%. Since, were = exp(), ten te standard deviation of te average IB price over te period is about 2:6 cents. For added perspective, te average spread for IB in our sample is :6 cents. Hence, te standard deviation is small relative to te spread wit a += 2 standard deviation interval just about equal to te average spread. Since most trades take place inside te bid/ask spread, te estimated magnitude of te noise variance appears very plausible Computing te optimal frequency for IB In Fig. 4 we plot te estimated conditional SE of te realized variance estimator under te A() assumption as well as te (nonparametric) SE for te case of dependent noise as described in Subsection 5.3. Te corresponding numbers for te rst tree autocorrelations of te 5 second returns are :5, :3, and :22. 22