Rounding Errors and Volatility Estimation

Transcription

1 Joural of Fiacial Ecoometrics Advace Access published February 27, 24 Joural of Fiacial Ecoometrics, 24, Vol., No., --27 Roudig Errors ad Volatility Estimatio YINGYING LI Departmet of Iformatio Systems, Busiess Statistics ad Operatios Maagemet, Hog Kog Uiversity of Sciece ad Techology PER A. MYKLAND Departmet of Statistics, Uiversity of Chicago ABSTRACT Fiacial prices are ofte discretized with smallest tick size of oe cet, for example. Thus prices ivolve roudig errors. Roudig errors affect the estimatio of volatility, ad uderstadig them is critical, particularly whe usig high frequecy data. We study the asymptotic behavior of realized volatility RV, which is commoly used as a estimator of itegrated volatility. We prove the covergece of the RV ad scaled RV uder varous coditios o the roudig level ad the umber of observatios. A bias-corrected volatility estimator is proposed ad a associated cetral limit theorem is show. The simulatio ad empirical results demostrate that the proposed method ca yield substatial statistical improvemet. JEL: C2, C3,C4 KEYWORDS: roudig errors, bias-correctio, diffusio process, market microstructure, realized volatility RV High frequecy data aalysis has received substatial attetio i recet years ad volatility estimatio is a cetral topic of iterest. The primary difficulty i estimatig daily volatilities by usig high frequecy data is the presece of market microstructure oise; otable developmets have bee made i this area. Zhag et al. 25; Zhag 26; Bardorff-Nielse et al. 28; ad Xiu 2 proposed ad evaluated various volatility estimators that exhibited favorable Fiacial support provided by the HKSAR RGC grats SBI9/.BM7 ad GRF-627, ad the Natioal Sciece Foudatio grats DMS , SES 6-365, ad SES is gratefully ackowledged. We thak Professor Michael J. Wichura for very helpful commets related to the Lemma 4 of this paper; ad we thak the Editor, Associate Editor, ad two aoymous referees for their costructive commets. Address correspodece to Yigyig Li, Departmet of Iformatio Systems, Busiess Statistics ad Operatios Maagemet, Hog Kog Uiversity of Sciece ad Techology, Clear Water Bay, Kowloo, Hog Kog, or yyli@ust.hk. Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24 doi:.93/jjfiec/bu5 The Author, 24. Published by Oxford Uiversity Press. All rights reserved. For Permissios, please jourals.permissios@oup.com

2 2 Joural of Fiacial Ecoometrics covergece properties, assumig that the microstructure oise was additive, ad idepedet ad idetically distributed i.i.d.. Li ad Myklad 27 ad Jacod et al. 29 studied cases i which the market microstructure oise was a combiatio of additive oise ad roudig error. Rosebaum 29 proposed a ovel volatility estimatio approach, usig absolute values of the icremets whe roudig is the oly source of the market microstructure oise. Roudig is a crucial source of market microstructure oise that should ot be igored. Because stocks are traded usig discrete price grids, their observatios are effectively rouded. I certai cases, particularly whe the stock prices are low, roudig ca be the mai source of market microstructure oise. Figure shows the secod-by-secod stock prices of Citigroup Ic o May, 27, idicatig that the log prices of the stock did ot exhibit the patter of a diffusio process or a diffusio process with additive oise. Rather, these prices seem like samples from a rouded diffusio. I this article, we focus o the extreme case where there is pure roudig. We explore the commoly used volatility estimator, the realized volatility RV ad how it ca be bias-corrected to yield cosistet volatility estimates. RV goes back to the path breakig work of Aderse ad Bollerslev 997, Aderse et al. 2, 23, Bardorff-Nielse ad Shephard 22, Jacod ad Protter 998, amog others. We cosider a security price process S, whose logarithm X =logs follows dx t =μ t dt+σ t dw t. I other words, S is the solutio to the followig stochastic differetial equatio: ds t =μ t + 2 σ 2 t S tdt+σ t S t dw t, t [,] 2 where W t is a stadard Browia motio. We assume that μ t ad σ t are cotiuous radom processes satisfyig the regularity coditios specified i Sectio. It is a commo practice i fiace to use the sum of frequetly sampled squared returs, the RV, to estimate the itegrated volatility σ t 2 dt. However, empirical studies have show that because of market microstructure oise, RV ca be severely biased whe prices are sampled at high frequecies, whereas samplig sparsely yields more reasoable estimates see, for example, the sigature plots itroduced by Aderse et al. 2. I this study, we ivestigate the case i which the cotamiatio caused by market microstructure results solely from roudig errors. Let α be a sequece of positive umbers represetig the accuracy of measuremet whe the price process is observed times durig a time period [,]. Suppose that at time i/ i =,, the value kα is observed whe the true value S i/ is i [kα,k+α with k Z. For every real s, we deote by s α =α s/α its rouded-off value at level α. Takig the Citigroup data as i Figure for example, the roudig level is α =.. O the day show, the k raged from 296 to 37. Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24

3 LI &MYKLAND Roudig Errors ad Volatility Estimatio 3 Secod by Secod Prices of Citigroup Ic o /May/27 Price Secods Figure Secod-by-secod stock prices of Citigroup Ic o May, 27. Roudig appears to be a mai feature of the data. We ivestigate the asymptotic behavior of the RV, as follows: V = i= Y i Y i 2, 3 where Y i =logs α i/, i =,, represets the observed log prices. Our mai results are preseted for the case i which the roudig is dow as previously described. This is primarily to facilitate presetig the proofs, where results of Delattre ad Jacod 997 for roud-off errors are applied. Same or similar results also apply to roudig up or roudig to the earest multiple of α see Remark 5 ad Sectio for additioal details. Jacod 996 ad Delattre ad Jacod 997 previously studied volatility iterferece based o a rouded Itô diffusio; although their work ispired the curret study, we seek i this article to spell out what esues whe roudig occurs o the origial e.g., the US dollar, euro, etc. scale rather tha the log scale. As we shall see later i this article, our fidigs idicate that this more realistic roudig yields a bias which requires a somewhat more complicated correctio. For example, i the simple case that the volatility is costat, the bias is o loger a fuctio of the volatility see Remark. We shall provide the limit of V, demostratig that the RV ca be problematic whe roudig errors are preset, ad elucidatig why sparse samplig could be a practical way to estimate volatility however, sparsely samplig does ot solve all the problems. We subsequetly propose a bias-corrected estimator, ad prove a associated cetral limit theorem. The simulatio results demostrate that the proposed bias-corrected estimator yields substatially ehaced statistical accuracy. Empirical studies show that the bias-correctio ca facilitate fiacial risk maagemet. Our mai bias correctio applies to the case of small roudig as i Delattre ad Jacod 997 ad Rosebaum 29. Such asymptotics are realistic i practice, cf. the fidigs for additive error i Zhag et al. 2. Small roudig asymptotics has also bee studied i Kolassa ad McCullagh 99, where it is Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24

4 4 Joural of Fiacial Ecoometrics show to be related to additive error. We also discuss the effects o RV whe the roudig is ot small. The theoretical results are preseted i Sectio ; the simulatio studies are preseted i Sectio 2, ad the empirical studies i Sectio 3; Sectio 4 cocludes. The proofs are show i the Appedix. ASYMPTOTIC RESULTS We assume that the latet security price process S t follows 2, where σ t is a oradom fuctio of S t, of class C 5 o [, I the Black Scholes model, σ t σ is a costat. Assume further that μ t is a cotiuous radom process i particular, it is locally bouded. Let β =α. Theorem : V P Whe α as such that β β [,, we have σ 2 t β2 dt+ 6 dt β2 π 2 k= k 2 exp 2π 2 k 2 σ 2 t S2 t β 2 Oe sees from this result that the bias is always positive whe β =, rapidly icreasig as β grows. Also, the bias is smaller whe the stock price S t, t [,] is larger. Figure 2 gives a visual represetatio of this. This result captures the empirical features that a subsamplig helps the same α value ad a smaller value yields a smaller β ad correspodigly smaller bias; ad b the roudig effect is less severe for more expesive stocks i.e., higher S t values correspod to smaller biases. Theorem shows that whe β, oe has the cosistecy of V.Ifβ decays polyomially i, we have the followig cetral limit theorem. Theorem 2: Whe β =O γ for some γ>, wehave V σt 2 dt β2 6 dt L stably where B is a stadard Browia motio idepedet of W. 2σ 2 t db t, dt. Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24 I this case, a fiite sample bias of β2 6 dt remais. The bias ca be estimated ad a bias-corrected estimator ca be determied as follows.

5 LI &MYKLAND Roudig Errors ad Volatility Estimatio Realized Volatility Versus Beta, S_= beta Realized Volatility Versus Beta, S_= beta Realized Volatility Versus Beta, S_= Figure 2 RV V versus β based o Theorem ad three simulated sample paths μ t, σ t. with startig prices S =, S = ad S =2. The dashed lie represets the true itegrated volatility, which is.; the solid curves represet the limits of the RV. The curve shapes idicate that the bias is icreasig i β, ad comparig the rages of the y axes of the plots demostrates that the bias is smaller whe S t, t [,] is larger. Theorem 3: The as, Assume that β =O γ for some γ>, ad let V V :=V α2 6 σ 2 t dt S α. i/ 2 i= L stably where B is a stadard Browia motio idepedet of W. 2σ 2 t db t, beta Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24 The simulatio results i the subsequet sectio demostrate that this biascorrectio yields substatially improved estimates. The empirical studies further show that the bias correctio ca be quite helpful i risk aalysis.

6 6 Joural of Fiacial Ecoometrics Remark : I the case where σ t σ, it is documeted i sectio 4 of Delattre ad Jacod 997 that whe roudig is implemeted o the log scale, the bias is a fuctio of σ 2 = σ t 2 dt. We see from the above results that i the presece of this more realistic type of roudig, the bias is o loger a fuctio of the targeted itegrated volatility eve whe σ t σ. Therefore this requires somewhat more complicated bias correctio. Remark 2: The coditio of small roudig is ecessary for the asymptotic results above. I practice, we apply these asymptotic results via expasio we observe oly oe α ad oe for a particular price process i a specified time period. Whe small roudig is relevat, we ca make a correctio as i Theorem 3, yieldig a superior estimator. I practice, we usually caot chage α.iftheβ is too big due to high samplig frequecy, we ca aalyze a subsample with a moderate frequecy to establish a situatio with small β. Ref. simulatio studies for additioal detail. Remark 3: The coditio that the radom process σ t is a oradom fuctio of S t is assumed so that the framework of Delattre ad Jacod 997 ca be applied. I Sectios 2 ad 3, we see i simulatio ad empirical studies that eve whe this coditio is ot ecessarily satisfied, the proposed bias correctio ca still be very helpful. We cojecture that similar results hold also i stochastic volatility settigs. 2 Whe the small roudig coditio is ot satisfied, the RV would blow up as the samplig frequecy becomes larger. Theorem 4 illustrates the asymptotic result of a simple case where σ t σ. I this case, simple bias correctio is isufficiet. A correctio after subsamplig may help. Theorem 4: Let the accuracy of measuremet α α be idepedet of the umber of observatios. Cosider the case whe σ t σ for t [,]. Redefie S α i/ =α if Sα i/ =.As, V 2 P σ π k= L logk+α log k+ 2, k where L a t is the local time at the level a of the cotiuous semimartigale X t =logs t see Revuz ad Yor 999, page 222. Remark 4: Redefiig S α i/ =α if Sα i/ = rules out the possibility of yieldig a logarithm of zero for log prices. I practice, this simply meas that the security price does ot go below the smallest roudig grid cet if α =. durig the specified time period. Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24 We emphasize that our derivatio builds o the geeral results of Delattre ad Jacod A formal extesio to this more geeral case ca use a simple parametric approximatio to the process, perhaps via the cotiguity argumets i Myklad ad Zhag 2.

7 LI &MYKLAND Roudig Errors ad Volatility Estimatio 7 Remark 5: Whe roudig is ot dow, but rather to the earest multiple of α,the results of Theorems 3 remai the same, but a small adjustmet must be made to the limit of Theorem 4: the local times will be at levels logk+ 2 α istead of logk+α. 2 SIMULATION STUDIES 2. Moderate Samplig Frequecies Cosider first the simplest case that σ t σ for t [,]. Deote by V _CI ad V _CI the omial 95% cofidece iterval CI based o V ad V, respectively. The aïve CI based o V relies o the classical theory for RV, which idicates the followig whe there is o observatio error: [V σ 2 ] L N,2σ 4. The resultig omial 95% CI is as follows: [ ] V _CI = V.96 2V 2 /, V V 2 /. Our fidigs i the previous sectio idicate that the RV is o loger reliable whe roudig errors are preset. We proposed the followig simple bias-corrected estimator that should fuctio whe α is reasoably small: By Theorem 3, V =V α2 6 S α. i/ 2 i= [V σ 2 ] L N,2σ 4. The adjusted omial 95% CI is as follows: [ ] V _CI = V.96 2V 2 /, V 2V /. To examie the performace of the volatility estimators V ad V, we perform the followig simulatio study. We simulate sample paths from 2 with μ=, σ =.. We examie two price levels, with startig prices of S =$ ad S =$5, respectively. At each price level,, simulatios were coducted for a oe-day period. We use a fixed roudig level of α α =., to mimic the fiacial market, i which stock prices are ofte rouded to the cet. Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24

8 8 Joural of Fiacial Ecoometrics Table Performace of the omial 95% CIs based o V ad V for stocks priced at approximately $ S =$ samp. freq. samp. itv mi mi mi.4 39 mi sec.279 β V _CI V _CI f: l: b: f: l: b: f: l: b: f: l: b: f: l: b: 94.29% % % f : actual coverage frequecy of the CIs; l : average CI legth; b : fiite sample bias % % % % % Tables ad 2 show the simulatio results. The first colums show the sample frequecies samp. freq., the secod colums show the correspodig sample itervals samp. itv., ad the third colums show the pre-limitig β =α, demostratig how the proposed small roudig asymptotic theory fuctios at a fiite sample size ad fixed roudig level. The fial two colums display three items each. The otatio f deotes the actual coverage frequecy, which is used to record the frequecy at which the true parameter is covered by the CIs based o the correspodig volatility estimators V ad V ; l deotes the average legth of the CI, which idicates the CI width; ad b deotes the fiite sample bias, which idicates how much ad i which directio the estimators are biased. Comparig V with V idicates that whe the sample frequecy is relatively low e.g., a 5-mi samplig iterval for stocks priced at approximately $, or 5 mi for stocks priced at approximately $5, both V ad V perform well because their actual coverage frequecy is ear the omial rate of 95%. This is cosistet with the empirical evidece that subsamplig is beeficial. But sice the covergece rate is the square root of, the CIs are wide whe is small. Whe the sample frequecy icreases slightly e.g., 3 mi - mi for stocks priced at approximately $ or 3 sec - 2 sec for stocks priced at approximately $5, the problems with the RV become apparet, the coverage frequecy decreases from approximately 95% to some much lower rates or eve zero i the former case, whereas the V _CI cotiues to exhibit a large coverage frequecy. The biases demostrate that the RV goes to some values much larger tha the true value, Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24

9 LI &MYKLAND Roudig Errors ad Volatility Estimatio 9 Table 2 Performace of the omial 95% CIs based o V ad V for stocks priced at approximately $5 S =$5 samp. freq. samp. itv. β α 78 5 mi mi mi.4 39 mi sec sec sec.484 f: l: b: f: l: b: f: l: b: f: l: b: f: l: b: f: l: b: f: l: b: V _CI 92.89% % % % % % % f : actual coverage frequecy of the CIs; l : average CI legth; b : fiite sample bias. V _CI 92.48% % % % % % % whereas the V remais close to the true parameter value. Hece, V substatially outperforms the ucorrected RV V. At extremely high frequecies less tha 3 sec for $ stocks or less tha sec for $5 stocks, the bias-corrected volatility estimator V demostrates decreased performace levels, although its bias remais substatially smaller compared with the RV. This is expected, because the bias-corrected estimator is built o asymptotic theory, which requires the coditio α, which is hypothetical, sice i practice oe is faced with a fixed data set ad a fixed tick size. If the sample frequecy were to cotiue to icrease at a fixed roudig level, the proposed bias correctio would evetually fail. The failure at extremely high frequecy is expected amog other RV-based volatility estimators as well ad is a direct cosequece of Theorem 4 Theorem 2 i Li ad Myklad 27 provides the result for the two scales realized volatility of Zhag et al. 25. The above simulatio shows that for a give price level ad roudig level, the proposed bias correctio method is effective whe the sample frequecy is ot excessively high. Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24

10 Joural of Fiacial Ecoometrics Estimated volatility e+ 5e 4 e Samplig iterval RV V V Figure 3 V RV ad V V at various samplig frequecies whe the roudig level is fixed at. fora sample path. The dotted horizotallie is the true itegrated volatility V. The samplig frequecies vary from observatio per secod left to observatio per 3 secods right. The price level is approximately $ i this simulatio. Estimated volatility e+ 5e 4 e 3 RV V V Samplig Frequecy Figure 4 V RV ad V V versus the square root of the samplig frequecies whe the roudig level is fixed at. for a sample path. The dotted horizotal lie is the true itegrated volatility V. The samplig frequecies vary from observatio per secod right to observatio per 3 secods left. The price level is approximately $ i this simulatio. 2.2 Large Samplig Frequecies at a Fixed Roudig Level At a fixed roudig level, whe is excessively large, the coditios of Theorems 3 are o loger met ad the feature described i Theorem 4 appears Figures 3 ad 4. Figures 3 ad 4 demostrate that as the samplig frequecy icreases samplig iterval decreases, both the RV ad the proposed bias corrected estimator V will Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24

11 LI &MYKLAND Roudig Errors ad Volatility Estimatio Table 3 Performace of V ad V whe volatility is ot a fuctio of price st Quartile Media 3rd Quartile Mea Root Mea Squares V V V V The estimatio errors are summarized by their st quartile, media, 3rd quartile, mea, ad root mea squares. rapidly icrease as described i Theorem 4. Figure 4 most clearly exhibits the rate of divergece. However, the V demostrates a clear advatage over the RV at a large rage of moderate-sized samplig itervals 2s - 2 mi i the case illustrated i Figure Whe Coditios Deviate From the Requiremets 2.3. Whe volatility is ot a fuctio of price. The theoretical results are established uder coditios specified i Sectio ; it is worth ivestigatig how the bias correctio performs if the required coditios are ot met. Therefore, we coduct simulatios based o a stochastic model i which the volatility process evolves by itself ad is ot a fuctio of the price process. The Hesto model Hesto 993 was adopted to determie the log price: dx t =μ ν t /2dt+σ t db t dν t =κη ν t dt+γν /2 t dw t where ν t =σt 2, B ad W are stadard Browia motios with EdB tdw t =ρdt, ad the parameters μ, η, κ, γ, ρ, ad the startig log-price X are set at.5/252,./252, 5/252,.5/252,.5, ad log9, respectively. Aït-Sahalia ad Kimmel 27 ad Aït-Sahalia et al. 23 were refereced whe selectig these parameter values. We used a moderate leverage effect parameter ρ =.5 to represet a idividual stock. We simulate, days ad obtaied pairs of the latet observatios X ti,σ ti for t =,t = 39,,t = for each day oe observatio per miute, =39. We compute the itegrated volatility V = i= σt 2 i ad use this value as the referece measure. The observed log prices are logexpx ti α where α =. rouded to cet. We compute the RV V ad the proposed bias-corrected estimator V ; ad summarize their performace i Table 3. The results idicate that although this model fails to meet the coditios for the theoretical results, the estimator V demostrates a clear advatage. Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, Whe roudig to the earest multiple of α. As metioed i Remark 5, roudig dow ad roudig to the earest multiple of α should yield

12 2 Joural of Fiacial Ecoometrics Table 4 Performace of V ad V whe roudig to the earest multiple of α st Quartile Media 3rd Quartile Mea Root Mea Squares V V V V The estimatio errors are summarized by their st quartile, media, 3rd quartile, mea, ad root mea squares. similar results. Table 4 presets the results, whe comparig the performace of V ad V based o the same simulated sample paths as the oe studied i Table 3. The oly differece is that the observatios become Y i/ =log[ expx t i/ α ] α with α =. i =,,, where [ ] deotes roudig to the earest iteger. The results show i Table 4 are similar to those i Table 3, as expected Whe jumps exist. We further ivestigate volatility estimatio i the presece of jumps. We simulated the followig model: dx t =μ ν t /2dt+ν /2 t db t +J t dn t 4 dν t =κη ν t dt+γν /2 t dw t, 5 where B t ad W t are Browia motios with correlatio ρ, N t is a Poisso process with itesity λ, ad J t deotes the jump size, which is assumed to follow a idepedet N,σJ 2. The prices are agai rouded to cets: Y i/ =logexpx i/ α i =,,. Oe way to remove the impact of jumps i volatility estimatio is usig the trucated RV which is defied as follows Aït-Sahalia ad Jacod, V,tr = Y i/ Y i / 2 { Yi/ Y i / a ϖ } 6 i= for some ϖ,/2 ad a>. We defie V,tr =V,tr α2 6 expy i/ 2 as the bias-corrected versio of the trucated RV. The parameters η =., γ =.5/252, κ =5/252, ρ =.5, μ=.5/252, λ=5, σ J =.5, ad i= Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24 3 See also Macii 2, Lee ad Myklad 28, ad Jig et al. 22. Bi- ad multipower methods Bardorff-Nielse ad Shephard, 24, 26 may also work, but we have ot ivestigated this.

13 LI &MYKLAND Roudig Errors ad Volatility Estimatio 3 Table 5 Performace of V,tr ad V,tr whe jumps exist st Quartile Media 3rd Quartile Mea Root mea squares V,tr V V,tr V The estimatio errors are summarized by their st quartile, media, 3rd quartile, mea ad root mea squares. X =log9 were used, ad the trucatio level was set at a ϖ =4 α /2 ref. Aït-Sahalia ad Jacod 22, Aït-Sahalia et al. 23. The results are summarized i Table 5, which idicates that whe jumps exist, the proposed bias correctio method plays a sigificat role i reducig the bias of the trucated RV. Remark 6: We otice that there is some deterioratio i small sample behavior relative to the o jump-o trucatio case. Naturally, the trucatio leads to both higher bias ad higher variace i small samples. The issue may relate to whether jumps get over-detected i small samples Bajgrowicz et al., 23, but also to the fact that oe loses itervals that have cotiuous evolutio whether or ot they have jumps. This latter problem has bee discussed, with possible solutios, i Lee 25 ad Lee ad Haig 2, ad is beyod the scope of this article. 3 EMPIRICAL STUDY To further illustrate the effectiveess of the proposed bias correctio method, we coduct a empirical aalysis of Citigroup Ic. NYSE:C, CBS Corporatio NYSE:CBS, Dell Ic. NYSE:DELL, Host Hotels ad Resorts Ic. NYSE:HST, ad KeyCorp NYSE:KEY stock data from 29. We collected the stock prices every miute 39 observatios per day, computig the V ad V values for each day. Based o the estimated volatilities, ad the assumptio that the retur o each day is ormally distributed, exhibitig approximately zero mea ad variace as estimated as is commoly assumed i risk maagemet 4, we computed the 5% value at risk VaR for each day, coutig the total umber of days that the VaR was violated. Table 6 lists the VaR violatios that occurred amog the 252 days cosidered. Because we cosidered the 5% VaRs, the expected rate of violatio was 5%. The examied stocks based o the V demostrated rates closer to the expected rate tha did those based o V. The V teds to dramatically overestimate the daily volatilities, causig over-cautious VaRs. Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24 4 See, for example, Christofferse 2. We used the estimated volatility of the same day for computig VaR of that day as our mai purpose here is to examie the accuracy of the volatility estimators istead of to predict VaR.

14 4 Joural of Fiacial Ecoometrics Table 6 5% VaR violatio rate based o the miute-by-miute stock prices of C, CBS, DELL, HST, ad KEY stocks i 29 C CBS DELL HST KEY V.59% 2.78% 3.57% 3.57% 2.38% V 2.78% 3.7% 4.76% 3.97% 3.97% 4 CONCLUSIONS AND DISCUSSION We have explored the estimatio of the itegrated volatility whe roudig is the primary source of market microstructure oise. We established asymptotic results for the RV based o small roudig coditios. We proposed a bias-corrected estimator for which cosistecy ad cetral limit theorems were established. Results were also preseted for the case whe small roudig coditios were ot satisfied. The effectiveess ad practicailty of the proposed bias correctio method was demostrated i both simulatio ad empirical studies. Note that while we work with observatios o a time iterval [,], results for the more geeral case of time iterval [,T] are obtaied by rescalig. The case of uequal observatio times ca be studied based o the methods of Jacod ad Protter 998 ad Myklad ad Zhag 26. APPENDIX A. Preparatio We assume without loss of geerality see Sectio A.4 for further justificatio that μ t =, i which case dlogs t =σ t dw t ; A. ad that there exist oradom costats L σ, U σ,, such that More Notatio: L σ σ t U σ for t [,]. A m := {ω :S t ω t [,] [ m },m] ; { S i/ B := ω : max i S }; 2log i / Y i, := S α i/ Sα i / ; Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24 U,φ:= i= φ S α i /,Y i, for φ :R 2 R; A.2

15 LI &MYKLAND Roudig Errors ad Volatility Estimatio 5 h : desity of the stadard ormal law; h s : desity of the ormal law N,s 2. Lemma : PB as. Proof. ByA., Note that for ay i =,2,, i/ S i/ /S i / =exp σ s dw s. i / i/ E exp σ s dw s i / i/ E exp =exp 2 U2 σ. Hece for ay a> Therefore, σ s dw s i/ i / 2 σs 2 ds+ i / 2 U2 σ i/ P σ s dw s >a i / i/ =P exp σ s dw s >exp a i / exp 2 Uσ 2 exp a. =P =P S i/ P max i max i max i Si/ S i / S i / > 2log + i/ σ s dw s >log i / >2log 2log + Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24

16 6 Joural of Fiacial Ecoometrics exp 2 Uσ 2 log 2log exp + as. A parallel argumet gives hece the coclusio. P max i S i/ S i / >2log as, Lemma 2: If α β [,, the for ay m, there exist N large ad c m, m ] such that for all N, i =,,2,,, Proof. ad hece the coclusio. Lemma 3: Proof. OA m B, S α i/ c m o A m. i =,,2,,, S α i/ S i/ α ; S i/ m o A m, ad α as, Suppose that β = α β [,, the for ay fixed m>, sup ω A m B Y i, S α i / log =O. Y i, = S α i/ Sα i / S i/ S i / +2α 2mlog+2β. By Lemma 2, oe ca fid a c m, m ] such that for large, oa m B, Y i, 2mlog+2β. α S cm i / Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24 Sice β β<, the above iequality implies that for ay fixed m, sup Y i, log ω Am B α is O S. i /

17 LI &MYKLAND Roudig Errors ad Volatility Estimatio 7 Lemma 4: Let β>, the for all σ,x >, Proof. β u+yσ x/β hy x 2 dydu=σ 2 + x 2 β 2 6 β2 π 2 k 2 exp 2π 2 k 2 σ 2 x 2 β 2. k= β u+yσ x/β 2 hy dydu x β U +Yσ x/β 2 =E, U uif [,], Y N, x = β2 E U +Yσ x/β 2 x2 = β2 x 2 E U +Z 2, Z N k=, σ 2 x 2 β 2 = β2 x 2 E E U +Z 2 Z = β2 x 2 E Z {Z} 2 {Z}+Z+ {Z} 2 {Z} = β2 x 2 EZ 2 +E{Z} {Z} =σ 2 + β 2 x 2 6 β2 π 2 k 2 exp 2π 2 k 2 σ 2 x 2 β 2, where {z}=z z is the fractioal part of z. The last equality above is proved by usig the Fourier expasio for fuctio f z={z} {z} 2. A.2 Proof of Theorem Recall that V is defied i 3. For large, = = V I Am B logs α i/ logsα i / 2 I Am B i= log i= Sα i/ Sα i / S α i / + 2 I Am B A.3 Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24

18 8 Joural of Fiacial Ecoometrics log Y i, α i= S i / Y i, α i= S 2 i / Y i, for θ,. α S i / = = By Lemma 2, oe ca fid + 2 I Am B Y i, S α i / θ 3 I Am B, c m, m ] such that for large, Sα i/ c m for all i =,,2,,. A.4 Defie y 2, whe x cm ; x φ cm x,y= 3 c 4 x2 8 m c 3 x+ 6 A.5 y 2, whe x <c m c 2 m. m Note i particular that φ cm is a fuctio satisfyig Hypothesis L r i Delattre ad Jacod 997 with r =2. For large eough, by Lemmas 2 ad 3, A.3 ca be rewritte as V I Am B i= where U, is defied i A.2. Furthermore, φ cm S α i /,Y i,i Am B +O log 3 /2 log 3 =U,φ cm I Am B +O I Am B, /2 V I Am =V I Am B +V I Am B c U,φ cm I Am +V U,φ cm I Am B c +O log 3 =U,φ cm I Am +o p by Lemma. /2 I Am B I Am B By Theorem 3. of Delattre ad Jacod 997, hyφ cm S t,β u+yσ t S t /β dydudt, if β>; U,φ cm P hyφ cm S t,yσ t S t dydt, if β =. Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24

19 LI &MYKLAND Roudig Errors ad Volatility Estimatio 9 Note that sice c m /m, wehave φ cm S α i /,Y= Y S α i / 2 I Am +φ cm S α i /,YI A c m. Lemma 4 gives, whe β>, U,φ cm I Am P σt 2 S2 t + β2 6 β2 π 2 k 2 exp 2π 2 k 2 σ t 2S2 t β 2 dti Am. It is easy to check that the above covergece is also true whe β =. Therefore, for β [,, V I Am =U,φ cm I Am +o p P σt 2 S2 t + β2 6 β2 π 2 k 2 exp 2π 2 k 2 σ t 2S2 t β 2 dti Am. k= That is to say, for ay δ>, ɛ>, there exists N, such that for all >N, P V I Am k= σt 2 S2 t + β2 6 β2 π 2 k 2 exp 2π 2 k 2 σ t 2 S2 t β 2 dti Am >δ <ɛ. k= O the other had, sice A m as m, there exists M large, such that Therefore, for >N, P V PA c M + PA c M <ɛ. σt 2 S2 t + β2 6 β2 π 2 k 2 exp 2π 2 k 2 σ t 2S2 t β 2 dt >δ P V I AM <2ɛ. k= σt 2 S2 t + β2 6 β2 π 2 k 2 exp 2π 2 k 2 σ t 2 S2 t β 2 dti AM >δ k= Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24 This proves Theorem.

20 2 Joural of Fiacial Ecoometrics A.3 Proof of Theorem 2 ad Theorem 3 By A.3, for large, ad V I Am B = Y i, α i= S 2 i / Y i, for θ,. α S i / Y i, S α i / θ 3 I Am B, Usig the c m, m ] as i A.4, we defie y 3, whe x cm ; x ψ cm x,y= 4 c 3 3x y 3, whe x <c m c 4 m. m A.6 ca be further writte as V I Am B log 4 U,φ cm I Am B U,ψ cm I Am B +O I Am B ; V I Am = V I Am B + V I Am B c /2 U,φ cm U,ψ cm I Am + V U,φ cm +U,ψ cm I Am B c +O log 4 = U,φ cm I Am U,ψ cm I Am +o p, /2 I Am B A.6 A.7 where φ cm is defied i A.5, ψ cm i A.7 ad U, ia.2, ad we have used Lemma 3 i the above. Note that ψ cm S t,σ t S t y is a odd fuctio of y, ad β =; by Theorem 3. of Delattre ad Jacod 997, U,ψ cm P hyψ cm S t,σ t S t ydydt=. Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24

21 LI &MYKLAND Roudig Errors ad Volatility Estimatio 2 Therefore, As a cosequece, U,ψ cm I Am P. V I Am = U,φ cm I Am +o p. A.8 ad Also by Corollary 3.3 of Delattre ad Jacod 997, sice φ cm x,y iseveiy, where B W, ad [U,φcm Ɣφ cm S t,β dt] stablyilaw φ cm,φ cm S t, /2 db s, Ɣφ cm S t,β = hyφ cm S t,β u+yσ t S t /β dydu = = β u+yσ t S t /β 2 hy I +φ Am cm St,β u+yσ t S t /β I A dydu cm σ 2 t + β2 6 β2 π 2 S t k 2 exp 2π 2 k 2 σ t 2S2 t β 2 I Am + k= hyφ cm S t,β u+yσ t S t /β dydui A c m by Lemma 4 ; φ cm,φ cm S t, = h σt S t yφc 2 m S t,ydy = = h σt S t y [ y S t h σtst y [ y 2 h σt S t yφ cm S t,ydy 4 I Am +φ2 c m S t,yi A c m ]dy S t 2 I Am +φ cm S t,yi A c m ] dy 2 y 4 y 2 2 h σt S t y dy h σtst S y dy I Am t S t +[ h σt S yφ t c S m t,y 2 dy ] 2 h σt S t yφ cm S t,ydy I A c m ; A.9 A. Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24

22 22 Joural of Fiacial Ecoometrics hece φ cm,φ cm S t, /2 = y 4 y 2 2 h σt S t y dy h σtst S y dy t S t +[ h σt S yφ t c S m t,y 2 dy =2σt 4 /2 I Am +[ h σt S yφ t c S m t,y 2 dy /2 I Am h σt S t yφ cm S t,ydy 2 ] /2 I A c m h σt S t yφ cm S t,ydy 2 ] /2 I A c m. A. Plug A. ad A. ito A.9, ad ote that by the assumptio that β = O γ, Oe has, β 2 π 2 [ + k= U,φ cm [ stably ilaw ZI Am + U,φ cm k 2 exp 2π 2 k 2 σ 2 t S2 t β 2 σt 2 dt+ β2 6 Ɣφ cm S t,β dt ] dt ] I A c m [ h σtst yφ cm S t,y 2 dy where Z 2σ 4 t /2 db s, B W. dt a.s. o A m as. I Am h σt S t yφ cm S t,ydy 2 ] /2 db s I A c m, For ay cotiuous fuctio g that vaishes outside a compact set, the above stable covergece implies that E F, E E [ [ g g [ U,φ cm σt 2 dt+ β2 6 2σt 4 /2 db s I Am I Am I E ]. ] dt ]I Am I Am I E A.2 Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24

23 LI &MYKLAND Roudig Errors ad Volatility Estimatio 23 oe has, ad Ad by defiig η cm, tobe 2, whe x c m ; x η cm x,y= 3 c 4 x2 8 m c 3 x+ 6, whe x <c m c 2 m, m V I A m =V I Am β2 6 U,η c m I Am. A.3 Agai, by Theorem 3. of Delattre ad Jacod 997, U,η cm I Am = i= β 2 6 U,η c m β2 6 By A.8, A.3, ad A.4, S α i/ 2 I A m P dti Am. dt I Am =O P β 2 =o P. V I Am = U,φ cm β2 6 Also sice that g is uiformly cotiuous, E F, lim E = lim E[g =E [ g [ V g [ U,φ cm dti Am +o p. σ 2 t dti A m I Am I E ] σt 2 dt+ β2 6 2σ 4 t /2 db t I Am I Am I E ] by A.2, which implies, for ay ɛ>, there exists N, such that N, ] dt I Am I Am I E ] [ E g [ ] [V σ t 2 ]I A M I AM I E ] E g 2σt 4 /2 db t I AM I AM I E <ɛ. A.4 Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24 Note also that g is bouded, suppose g M g. Recall that PA c M, oe ca choose M such that PA c M <ɛ/m g.

24 24 Joural of Fiacial Ecoometrics So for N, E[g [V E[g [V +2M g PA c M 3ɛ σ 2 t dt]i E] E [ g σ 2 t dt]i A M I AM I E ] E 2σ 4 t /2 db t I E ] [ g 2σ 4 t /2 db t I AM I AM I E ] Hece we ve proved that for all cotiuous fuctio g that vaishes outside a compact set, E F, i.e., [ lim E[g [V σt 2 dt]i E]=E g [V σ 2 t dt] L stably 2σ 4 t /2 db t I E ], 2σ 4 t /2 db t. This fiishes the proof of Theorem 3. The proof of Theorem 2 is basically cotaied i the proof above. A.4 The Case of Geeral μ t ad σ t Step : For geeral cases whe μ t =, if there exists L σ, U σ, C μ,, such that L σ σ t U σ ad μ t C μ for t [,], the previous results all hold. For the simplicity of otatio, we cosider the log scale. Let P be the probability measure correspodig to the system dx t =σ t dw t ad Q the probability measure correspodig to the system dx t =μ t dt+σ t dw Q t, where W t ad W Q t are stadard Browia motios uder P ad Q respectively. Note that by the Girsaov Theorem see, for example, page 64 of Øksedal 23, for bouded σ t ad μ t as stated i the coditios of Step, P ad Q are mutually absolutely cotiuous. The followig propositio justifies the coclusio of Step. Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24

25 LI &MYKLAND Roudig Errors ad Volatility Estimatio 25 Propositio Myklad ad Zhag 29 Suppose that ζ is a sequece of radom variables which coverges stably to Nb,a 2 uder P meaig that Nb,a 2 =b+an,, where N, is a stadard ormal variable idepedet of F, also a ad b are F measurable. The ζ coverges stably i law to b+an, uder Q, where N, remais idepedet of F uder Q. Step 2: for locally bouded σ t ad μ t, the stable covergece ad the covergece i probability stay valid. This ca be proved by a localizatio argumet which uses essetially the same techiques as i the derivatio i the last part of Sectio A.3. For example, to uboud σ t, oe cosiders a sequece of stoppig times τ m correspodig to a sequece of positive costats σ m which icreases to ifiity as m : τ m =mi{t: σt 2 σ m 2 }, ad ote the fact that the sets {τ m >T}. I particular, the locally bouded assumptio is automatically satisfied whe σ t ad μ t are cotiuous. A.5 Proof of Theorem 4 Similar argumet as the Proof of Theorem 3 i Li ad Myklad 27 gives the result. Received April 5, 2; revised December 4, 23; accepted Jauary 8, 24. REFERENCES Aït-Sahalia, Y., J. Fa, ad Y. Li. 23. The Leverage Effect Puzzle: Disetaglig Sources of Bias at High Frequecy. Joural of Fiacial Ecoomics 9: Aït-Sahalia, Y., ad J. Jacod. 22. Aalyzig the Spectrum of Asset Returs: Jump ad Volatility Compoets i High Frequecy Data. Joural of Ecoomic Literature 5: 7 5. Aït-Sahalia, Y., ad R. Kimmel. 27. Maximum Likelihood Estimatio of Stochastic Volatility Models. Joural of Fiacial Ecoomics 83: Aderse, T. G., ad T. Bollerslev Itraday Periodicity ad Volatility Persistece i Fiacial Markets. Joural of Empirical Fiace 4: Aderse, T. G., T. Bollerslev, F. X. Diebold, ad P. Labys. 2. Great Realizatios. Risk 3: 5 8. Aderse, T. G., T. Bollerslev, F. X. Diebold, ad P. Labys. 2. The Distributio of Exchage Rate Realized Volatility. Joural of the America Statistical Associatio 96: Aderse, T. G., T. Bollerslev, F. X. Diebold, ad P. Labys. 23. Modelig ad Forecastig Realized Volatility. Ecoometrica 7: Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24

26 26 Joural of Fiacial Ecoometrics Bajgrowicz, P., O. Scaillet, ad A. Treccai. 23. Jumps i High-Frequecy Data: Spurious Detectios, Dyamics, ad News. Workig paper. Bardorff-Nielse, O. E., P. R. Hase, A. Lude, ad N. Shephard. 28. Desigig Realized Kerels to Measure Ex-post Variatio of Equity Prices i the Presece of Noise. Ecoometrica 76: Bardorff-Nielse, O. E. ad N. Shephard. 22. Ecoometric Aalysis of Realized Volatility ad Its Use i Estimatig Stochastic Volatility Models. Joural of the Royal Statistical Society, B 64: Bardorff-Nielse, O. E. ad N. Shephard. 24. Power ad Bipower Variatio with Stochastic Volatility ad Jumps with Discussio. Joural of Fiacial Ecoometrics 2: 48. Bardorff-Nielse, O. E. ad N. Shephard. 26. Ecoometrics of Testig for Jumps i Fiacial Ecoomics Usig Bipower Variatio. Joural of Fiacial Ecoometrics 4: 3. Christofferse, P. 2.Elemets of Fiacial Risk Maagemet, 2d ed. Oxford, UK: Academic Press. Delattre, S., ad J. Jacod A Cetral Limit Theorem for Normalized Fuctios of the Icremets of a Diffusio Process, i the Presece of Roud-Off Errors. Beroulli 3: 28. Hesto, S A Closed-Form Solutio for Optios with Stochastic Volatility with Applicatios to Bods ad Currecy Optios. Review of Fiacial Studies 6: Jacod, J La Variatio Quadratique du Browie e Présece d Erreurs d Arrodi. Astérisque 236: Jacod, J., Y. Li, P. A. Myklad, M. Podolskij, ad M. Vetter. 29. Microstructure Noise i the Cotiuous Case: The Pre-Averagig Approach. Stochastic Processes ad their Applicatios 9: Jacod, J., ad P. Protter Asymptotic Error Distributios for the Euler Method for Stochastic Differetial Equatios. Aals of Probability 26: Jig, B.-Y., X.-B. Kog, Z. Liu, ad P. A. Myklad. 22. O the Jump Activity Idex for Semimartigales. Joural of Ecoometrics 66: Kolassa, J., ad P. McCullagh. 99. Edgeworth Series for Lattice Distributios. Aals of Statistics 8: Lee, S. S. 25. Essays i Fiacial Ecoometrics. Ph.D. thesis, The Uiversity of Chicago, Booth School of Busiess. Lee, S. S., ad J. Haig. 2. Detectig Jumps from L evy Jump Diffusio Processes. Joural of Fiacial Ecoomics 96: Lee, S. S., ad P. A. Myklad. 28. Jumps i Fiacial Markets: A New Noparametric Test ad Jump Dyamics. Review of Fiacial Studies 2: Li, Y., ad P. A. Myklad. 27. Are Volatility Estimators Robust with Respect to Modelig Assumptios? Beroulli 3: Macii, C. 2. Disetaglig the Jumps of the Diffusio i a Geometric Jumpig Browia Motio. Giorale dell Istituto Italiao degli Attuari LXIV: Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24

27 LI &MYKLAND Roudig Errors ad Volatility Estimatio 27 Myklad, P. A., ad L. Zhag. 26. ANOVA for Diffusios. The Aals of Statistics 34:#4. Myklad, P. A., ad L. Zhag. 29. Iferece for Cotiuous Semimartigales Observed at High Frequecy: A Geeral Approach. Ecoometrica 77: Myklad, P. A., ad L. Zhag. 2. The Double Gaussia Approximatio for High Frequecy Data. Scadiavia Joural of Statistics 38: Øksedal, B. 23. Stochastic Differetial Equatios, 6th ed. Uiversitext, Berli: Spriger-Verlag. Revuz, D., ad M. Yor Cotiuous Martigales ad Browia Motio, 3rd ed. Berli, Germay: Spriger-Verlag. Rosebaum, M. 29. Itegrated Volatility ad Roud-off Errors. Beroulli, 5: Xiu, D. 2. Quasi-Maximum Likelihood Estimatio of Volatility with High Frequecy Data. Joural of Ecoometrics 59: Zhag, L. 26. Efficiet Estimatio of Stochastic Volatility Usig Noisy Observatios: A Multi-Scale Approach. Beroulli 2: Zhag, L., P. A. Myklad, ad Y. Aït-Sahalia. 25. A Tale of Two Time Scales: Determiig Itegrated Volatility with Noisy High-Frequecy Data. Joural of the America Statistical Associatio 472: Zhag, L., P. A. Myklad, ad Y. Aït-Sahalia. 2. Edgeworth Expasios for Realized Rolatility ad Related Estimators. Joural of Ecoometrics 6: Dowloaded from at Hog Kog Uiversity of Sciece ad Techology o February 28, 24