Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
The University of Chicago Department of Statistics TECHNICAL REPORT SERIES DETERMINING THE VOLATILITY OF A PRICE PROCESS IN THE PRESENCE OF ROUNDING ERRORS ∗ Yingying Li and Per A. Mykland TECHNICAL REPORT NO. 573 Departments of Statistics The University of Chicago Chicago, Illinois 60637 September 2006 ∗ Yingying Li is Doctoral Student, Department of Statistics, The University of Chicago, Chicago, IL 60637. E-mail: [email protected]. Per A. Mykland is Professor, Department of Statistics, The University of Chicago, Chicago, IL 60637. E-mail: [email protected]. We gratefully acknowledge the support of the National Science Foundation under grant DMS-0204639 and DMS0604758. We thank Professor Michael J. Wichura for very helpful comments related to the Lemma 4 of this paper. Determining the Volatility of a Price Process in The Presence of Rounding Errors ∗ Yingying Li and Per A. Mykland September 2006. This version: September 22, 2006 Abstract Let S denote the price process of a security, and suppose that S follow a geometric Brownian motion with volatility σ 2 . We consider the case when the observations at the discrete time (α ) points 0, 1/n, 2/n, · · · , 1 are the rounded-off values Si/nn = αn bSi/n /αn c (i = 0, · · · , n), where αn > 0 is the round-off level corresponding to the sample frequency n. We investigate Pn (α ) (αn ) ))2 , the asymptotic behavior of the “Realized Volatility” V n = i=1 (log(Si/nn ) − log(S(i−1)/n which is commonly used as an estimator of the volatility σ 2 . We prove the convergence of V n or scaled V n under different conditions on αn . A bias corrected estimator of the volatility is proposed and an associated central limit theorem is shown. Simulation results show that improvement in statistical properties can be substantial. KEY WORDS: Bias-correction; Diffusion Process; Market Microstructure; Realized Volatility; Rounding Errors. ∗ Yingying Li is Doctoral Student, Department of Statistics, The University of Chicago, Chicago, IL 60637. Email: [email protected]. Per A. Mykland is Professor, Department of Statistics, The University of Chicago, Chicago, IL 60637. E-mail: [email protected]. We gratefully acknowledge the support of the National Science Foundation under grant DMS-0204639 and DMS-0604758. We thank Professor Michael J. Wichura for very helpful comments related to the Lemma 4 of this paper. 1 Determining the Volatility of a Price Process in The Presence of Rounding Errors 1 2 INTRODUCTION We consider a security price process S, which is the solution to the following stochastic differential equation dSt = µSt dt + σSt dBt , t ∈ [0, 1] (1) where Bt is a standard Brownian motion, and µ ∈ R and σ ∈ (0, ∞) are constants. It is a common practice in finance to use the sum of frequently sampled squared returns (the R1 “Realized Volatility” (RV)) to estimate the integrated volatility 0 σ 2 dt (for process (1), this is simply the volatility σ 2 ). However, recent empirical studies in finance showed evidence that market microstructure makes this estimator fail when the prices are sampled at high frequencies, and sampling sparsely gives more reasonable estimates. We investigate the case when the contamination due to market microstructure is simply round-off errors, and we are interested in the limiting behavior of the RV. More specifically, let αn be a sequence of positive numbers which represents the accuracy of measurement when one observes the process n times during the time period [0,1]. Suppose at time i/n (i = 0, · · · n), one observe the value kα when the true value Si/n is in [kα, (k + 1)α) with k ∈ Z. For every real s we denote by s(αn ) = αn bs/αn c its rounded-off value at level αn . We investigate the asymptotic behavior of the estimator (the RV) V n n X (α ) (αn ) = (log(Si/nn ) − log(S(i−1)/n ))2 . (2) i=1 Jacod (1996) and Delattre and Jacod (1997) have previously studied the problem of inference for volatility based on a rounded Brownian motion. While this work is the inspiration for the current paper, these earlier results are not quite as relevant to securities prices, as rounding happens on the original (dollar, euro, etc) scale and not on the log scale. As we shall see in this paper, the more realistic type of rounding leads to a bias which is no longer nonrandom (as in section 4 of Delattre Determining the Volatility of a Price Process in The Presence of Rounding Errors 3 and Jacod (1997)), but instead requires a somewhat more complicated correction. We emphasize, however, that we owe a lot to the earlier developments by Delattre and Jacod. Rounding has also been studied by Zeng (2003), who has developed a Bayesian inference algorithm for this problem. We prove the convergence of V n or scaled V n , under different assumptions on αn . The theorems show the problems of using realized volatility as an estimator of the volatility in the presence of rounding errors, and explain why “sampling sparsely ” could be a practically helpful way to estimate the volatility (but “sampling sparsely ” doesn’t solve all the problems). We then propose a bias corrected estimator, and prove an associated central limit theorem. Simulation results show that substantial improvement in statistical properties can be achieved. These main results are presented in section 2. Section 3 is devoted to prove the theorems. And Section 4 for conclusions and discussion. Our main bias correction applies to the case of “small rounding”, as in Delattre and Jacod (1997). This kind of asymptotics is quite realistic in practice, cf. the findings for additive error in Zhang, Mykland, and Aı̈t-Sahalia (2005a). Small rounding asymptotics has also been studied in Kolassa and McCullagh (1990), where it is shown to be related to additive error. 2 2.1 THE MAIN RESULTS The Theorems Theorem 1. Let the accuracy of measurement αn ≡ α be independent on the number of observa(α) (α) tions n. Redefine Si/n = α if Si/n = 0. Then, 1 1 √ V n →P σ n r ∞ 2 X log(kα) 1 L1 (log(1 + ))2 , π k k=1 Determining the Volatility of a Price Process in The Presence of Rounding Errors 4 where LaT is the local time of the continuous semimartingale Xt = log St , t ∈ [0, T ]; defined as in Revuz and Yor (1999), page 222. One sees that the realized volatility V n blows up as the sample frequency n becomes higher, just as in Jacod (1996), though the form of the limit is different. √ Now let βn = αn n. Theorem 2. If αn → 0 as n → ∞ in such a way that βn → β ∈ [0, ∞), then V n β2 →P σ + 6 Z 2 0 1 β2 1 dt − π2 St2 Z 0 1 ∞ 2 2 1 X 1 2 2 σ St exp{−2π k }dt. k2 β2 St2 k=1 The above limit is a quantity increasing in β, and always bigger than σ 2 when β 6= 0. It blows up to ∞ as β approaches ∞. In the case that βn → 0, V n converges to σ 2 at the limit; but before the limit, V n is always biased as an estimator of σ 2 . If βn → 0 fast enough, one has the following central limit theorems: Theorem 3. If √ 2 nβn → γ < ∞, then √ β2 n[V n − σ 2 − n 6 Z 1 0 1 dt] →L N (0, 2σ 4 ). St2 In this case, one can estimate the bias and find a bias-corrected estimator of σ 2 : Theorem 4. If √ 2 nβn → γ < ∞, let V0n := V n − α2n 6 Pn 1 i=1 (S (αn ) )2 , i/n √ n[V0n − σ 2 ] →L N (0, 2σ 4 ). one has Determining the Volatility of a Price Process in The Presence of Rounding Errors 5 The above theorems clearly show the problems of using realized volatility as an estimator of the volatility in the presence of rounding errors. And one can see that in all the cases, when one uses a sub-sampled frequency nsparse < n to do the estimation, the ill-posedness of the estimation problems could be weaken. This is consistent with what people have recently found in the context of additive error (see, for example, Zhang, Mykland, and Aı̈t-Sahalia (2005b)). When αn → 0 fast enough, the realized volatility converges to σ 2 , but one needs to do a bias correction to solve the finite-sample estimation problem. This can give rise to substantial improved confidence intervals, as we shall now see. 2.2 The Simulation Results Denote by V n CI the nominal 95% confidence interval (CI) based on V n and V0n CI the nominal 95% confidence interval based on V0n , as follows. The naive CI based on V n relies on the classical theory of the realized volatility, which says that when there is no observation error, √ n[V n − σ 2 ] →L N (0, 2σ 4 ). The resulting nominal 95% CI is h i p p V n CI = V n − 1.96 ∗ 2(V n )2 /n, V n + 1.96 ∗ 2(V n )2 /n . Since our findings above indicate that there are some problems with using the classical theory of the realized volatility when the rounding errors are present. We propose a new estimator V0n = V n − n 1 αn2 X . (α n) 2 6 i=1 (Si/n ) Determining the Volatility of a Price Process in The Presence of Rounding Errors 6 By Theorem 4, √ n[V0n − σ 2 ] →L N (0, 2σ 4 ). Our adjusted nominal 95% CI is then · V0n CI = V0n ¸ q q n n n 2 2 − 1.96 ∗ 2(V0 ) /n, V0 + 1.96 ∗ 2(V0 ) /n . To examine the performance of the new estimator V0n , we did the following simulation: We simulated sample paths from (1) with µ = 0.02 and σ = 0.2. Assume the observations at the (α ) time points 0, 1/n, 2/n, · · · , 1 are the rounded-off values Si/nn = αn bSi/n /αn c (i = 0, · · · , n) with round-off level αn satisfying αn2 n3/2 = 1. For each sample frequency n = 1000, 2000, · · · , 10000, 1000 sample paths were simulated. And correspondingly, 1000 V n CI’s and 1000 V0n CI’s were worked out. The (actual) coverage probability of V n CI is estimated by the ratio of the total number of times that the true parameter σ 2 lies inside the V n CI’s to the total number of sample paths 1000. And by the same way the coverage probability of V0n CI is estimated. Figure 1 records the estimated coverage probabilities of V n CI and V0n CI. One can read from Figure 1 that for each n, the nominal 95% CI based on V n only has an estimated coverage probability of 20% − 30%. While the nominal 95% CI based on V0n seems to be worthy of the name. 1.0 Determining the Volatility of a Price Process in The Presence of Rounding Errors o o o o o o o o o 0.8 o 7 0.4 0.6 o: estimated coverage probability of V0^n_CI *: estimated coverage probability of V^n_CI * * * * * * * * * 0.0 0.2 * 0 2000 4000 6000 8000 10000 n Figure 1: The Estimated Coverage Probabilities, V0n CI versus V n CI. The nominal coverage probability is 95%. Determining the Volatility of a Price Process in The Presence of Rounding Errors 3 3.1 8 PROOFS OF THE MAIN RESULTS Proof of Theorem 1 The proof of this theorem is very similar to the proof of Theorem 3 in Li and Mykland (2006) with T = 1 (except that here we consider the rounding to be “rounding down”; in Li and Mykland (2006), the rounding is “rounding to the nearest multiple of α” ). The basic framework of the proofs relies on Jacod (1996). Please refer to the above papers for the details. 3.2 Preparations for proofs of Theorem 2-4: Notations: Am := {ω ∈ Ω : St (ω)t∈[0,1] ∈ [ Bm,n := {ω ∈ Ω : max 1≤i≤n Yi,n := 1 , m]}; m √ 1 n|Si/n − S(i−1)/n | ≤ 2mσ(2 log n) 2 }; √ (α ) (αn ) n(Si/nn − S(i−1)/n ); n U (n, φ) = 1X (αn ) φ(S(i−1)/n , Yi,n ) for function φ on R2 ; n i=1 h(·) : density of the standard normal law ; hs (·) : density of the normal law N (0, s2 ). Lemma 1. ∀m, P (Am T c ) → 0 as n → ∞. Bm,n (3) Determining the Volatility of a Price Process in The Presence of Rounding Errors 9 Proof: dSt = µSt dt + σSt dBt 1 ⇒St = S0 exp{(µ − σ 2 )t + σBt } 2 √ √ σ 1 1 ⇒ n|Si/n − S(i−1)/n | = S(i−1)/n | n(exp{ √ Zi + (µ − σ 2 ) } − 1)|, 2 n n where Zi ∼ N (0, 1), i.i.d., i = 1, 2, · · · , n. For any kn , P ( max 1≤i≤n √ n|Si/n − S(i−1)/n | > kn , St {t∈[0,1]} ∈ [1/m, m]) 1 kn σ 1 ≤P ( max | exp{ √ Zi + (µ − σ 2 ) } − 1| > √ , St {t∈[0,1]} ∈ [1/m, m]) 1≤i≤n 2 n n m n σ 1 kn 1 ≤P ( max | exp{ √ Zi + (µ − σ 2 ) } − 1| > √ ) 1≤i≤n 2 n n m n kn 1 2 1 σ ≤P ( max (1 − exp{ √ Zi + (µ − σ ) }) > √ )+ 1≤i≤n 2 n n m n 1 kn σ 1 P ( max (exp{ √ Zi + (µ − σ 2 ) } − 1) > √ ). 1≤i≤n 2 n n m n 1 1 1 Let Mn = max1≤i≤n Zi , cn = (2 log n) 2 − 12 (2 log n)− 2 (log(4π) + log log n), sn = (2 log n)− 2 . The extreme value theorem says (c.f. Aldous (1989) page 46 or Ferguson (1996) page 99) Mn − cn →D ξ, sn where ξ has support (−∞, ∞) and P (ξ ≤ x) = e−e −x . It follows that Determining the Volatility of a Price Process in The Presence of Rounding Errors 10 σ 1 1 kn P ( max (1 − exp{ √ Zi + (µ − σ 2 ) }) > √ ) 1≤i≤n 2 n n m n σ 1 1 kn =P ( min exp{ √ Zi + (µ − σ 2 ) } < 1 − √ ) 1≤i≤n 2 n n m n √ 1 2 1 kn n √ =P ( min Zi < [( σ − µ) + log(1 − )] ) 1≤i≤n 2 n m n σ √ n 1 1 kn =P (Mn > [(µ − σ 2 ) − log(1 − √ )] ) 2 n m n σ [(µ − 12 σ 2 ) n1 − log(1 − Mn − cn > =P ( sn sn ≈P (ξ > [(µ − 21 σ 2 ) n1 − log(1 − √ n k√ n )] σ m n √ k√ n )] σn m n sn − cn − cn ) ) for large n. 1 For kn = 2mσ(2 log n) 2 , the above probability goes to 0 as n → ∞. A parallel argument gives the same conclusion for P (max1≤i≤n (exp{ √σn Zi + (µ − 12 σ 2 ) n1 } − 1) > k√ n ) m n 1 when kn = 2mσ(2 log n) 2 . In summary, P ( max 1≤i≤n √ 1 n|Si/n − S(i−1)/n | > kn , St {t∈[0,1]} ∈ [1/m, m]) → 0 as n → ∞ for kn = 2mσ(2 log n) 2 , i.e. P (Am Lemma 2. Given \ c Bm,n ) → 0 as n → ∞. √ 1 nαn → β ∈ [0, ∞), on Am , there exists N, cm ∈ (0, m ], such that for all (α ) n ≥ N, i = 0, 1, 2, · · · , n, Si/nn ≥ cm . Proof: (α ) ∀i = 0, 1, 2, · · · , n, Si/nn ≥ Si/n − αn ; and Si/n ≥ 1 on Am , and αn → 0 as n → ∞, m Determining the Volatility of a Price Process in The Presence of Rounding Errors 11 hence the conclusion. √ Lemma 3. Given βn = nαn → β ∈ [0, ∞), ∀m > 0, √ ³ Yi,n (αn ) nS(i−1)/n =O 2 log n n ´1/2 on Am T Bm,n . Proof: On Bm,n , Yi,n = √ √ (α ) (αn ) n|Si/nn − S(i−1)/n | ≤ n(|Si/n − S(i−1)/n | + 2αn ) ≤ 2mσ(2 log n)1/2 + 2βn . 1 By lemma 2, one can find a cm ∈ (0, m ] such that for large n, on Am T Bm,n , Yi,n 2mσ(2 log n)1/2 + 2βn √ ≤ . √ (αn ) ncm nS(i−1)/n Since βn → β < ∞, the above inequality implies that √ ³ Yi,n (α ) n nS(i−1)/n is O 2 log n n ´1/2 or smaller. Lemma 4. Z 0 1Z ∞ 2 2 1 β2 β2 X 1 βbu + yσx/βc 2 2 2 2σ x ) dydu = σ + 2 ( − 2 h(y)( exp{−2π k }). x x 6 π k2 β2 k=1 Determining the Volatility of a Price Process in The Presence of Rounding Errors Proof: Z 1Z βbu + yσx/βc 2 ) dydu x 0 µ ¶ βbU + Y σx/βc 2 =E , U ∼ unif [0, 1], Y ∼ N (0, 1) x β2 = 2 E(bU + Y σx/βc)2 x β2 σ 2 x2 = 2 E(bU + Zc)2 , Z ∼ N (0, 2 ) x β Z l+1 2 β X = 2 h σ2 x2 [l2 (l + 1 − z) + (l + 1)2 (z − l)]dz x β2 l l Z β 2 X l+1 = 2 h σ2 x2 (z 2 + {z} − {z}2 )dz x β2 l h(y)( l β2 σ 2 x2 = 2 [EZ 2 + E({Z}(1 − {Z}))], Z ∼ N (0, 2 ) x β ∞ 2 2 2 2 X β 1 1 β σ x =σ 2 + 2 ( − 2 exp{−2π 2 k 2 2 }), 2 x 6 π k β k=1 where {z} = z − bzc, is the fractional part of z. The last equality above is proved by using the Fourier expansion: Let f (z) = {z} − {z}2 for z ∈ R. Let L = 1/2, f (z) has Fourier coefficients bk (f ) = 0; 1 a0 (f ) = L Z L 1 f (z)dz = ; 3 −L 12 Determining the Volatility of a Price Process in The Presence of Rounding Errors Z 1 L kπz f (z) cos( )dz ak (f ) = L −L L Z 1 2 =2 z(1 − z) cos(2kπz)dz − 12 Z 1 =2 z(1 − z) cos(2kπz)dz 0 =− Therefore, 1 k2 π2 . ∞ 1 1 X 1 f (z) = − 2 cos2kπz 6 π k2 k=1 and E(f (Z)) = ∞ 1 X 1 1 − 2 RφZ (2kπ), 6 π k2 k=1 where RφZ (·) represents the real part of the characteristic function of Z. 2 2 For Z ∼ N (0, σβx2 ), one has, E(f (Z)) = E({Z}(1 − {Z})) = ∞ 2 2 1 X 1 1 2 2σ x − 2 exp{−2π k }, 6 π k2 β2 k=1 which finishes the proof of lemma 4. 13 Determining the Volatility of a Price Process in The Presence of Rounding Errors 3.3 14 Proof of Theorem 2 Recall that V n is defined in (2). For large n, V n IAm ∩Bm,n n X (α ) (αn ) = (log Si/nn − log S(i−1)/n )2 IAm ∩Bm,n i=1 (αn ) n (αn ) Si/n − S(i−1)/n 1X√ = [ n log( + 1)]2 IAm ∩Bm,n (α ) n S n i=1 n X (4) (i−1)/n √ Yi,n [ n log( √ (α ) + 1)]2 IAm ∩Bm,n n nS(i−1)/n i=1 = 1 n = Yi,n 1X√ [ n( √ (α ) n nS n n i=1 (i−1)/n Yi,n Yi,n 1 1 − ( √ (α ) )2 + θ3 )]2 IAm ∩Bm,n , θ ∈ (0, √ (α ) ). n n 2 nS 3 nS(i−1)/n (i−1)/n By lemma 2, one can find cm ∈ (0, Define 1 (α ) ] such that for large n, Si/nn ≥ cm for all i = 0, 1, 2, · · · , n. m y ( )2 , when x ≥ cm ; x φcm (x, y) = 3 8 6 ( 4 x2 − 3 x + 2 )y 2 , cm cm cm (6) when x < cm . For n large enough, by lemma 2 and lemma 3, (4) can be further written as n V n IAm ∩Bm,n = 1X (2 log n)3/2 (αn ) φcm (S(i−1)/n , Yi,n )IAm ∩Bm,n + O( )IAm ∩Bm,n n n1/2 i=1 =U (n, φcm )IAm ∩Bm,n + O( where U (·, ·) is defined in (3). (5) (2 log n)3/2 )IAm ∩Bm,n , n1/2 Determining the Volatility of a Price Process in The Presence of Rounding Errors Furthermore, c V n IAm = V n IAm ∩Bm,n + V n IAm ∩Bm,n (2 log n)3/2 c )IAm ∩Bm,n + V n IAm ∩Bm,n 1/2 n (2 log n)3/2 c + (V n − U (n, φcm ))IAm ∩Bm,n + O( )IAm ∩Bm,n n1/2 = U (n, φcm )IAm ∩Bm,n + O( = U (n, φcm )IAm = U (n, φcm )IAm + op (1) (by Lemma 1). By Delattre and Jacod (1997), Z U (n, φcm ) →P 1Z 1Z 0 Z h(y)φcm (St , βbu + yσSt /βc)dydudt, if β > 0; 0 1Z 0 h(y)φcm (St , yσSt )dydt, if β = 0. Note that cm ≤ 1/m, µ φcm (S(i−1)/n , Y ) = Y S(i−1)/n ¶2 IAm + φcm (S(i−1)/n , Y )IAcm . Lemma 4 gives, when β > 0, Z U (n, φcm )IAm →P 0 1 ∞ 1 2 2 β2 β2 X 1 σ2S 2 (σ S + − exp{−2π 2 k 2 2 t })dtIAm t 2 2 2 6 π k β St k=1 It is easy to check that the above convergence is also true when β = 0. 15 Determining the Volatility of a Price Process in The Presence of Rounding Errors Therefore, for β ∈ [0, ∞), V n IAm = U (n, φcm )IAm + op (1) Z 1 ∞ 2 2 1 2 2 β2 β2 X 1 2 2 σ St (σ S + →P − exp{−2π k })dtIAm . t 2 6 π2 k2 β2 0 St k=1 That is to say, for any δ > 0, ² > 0, there exists N, such that for all n > N, Z n P (|V IAm − 1 0 ∞ 2 2 1 2 2 β2 β2 X 1 2 2 σ St (σ S + − exp{−2π k })dtIAm | > δ) < ². t 6 π2 k2 β2 St2 k=1 On the other hand, since Am % Ω, there exists M large, such that P (AcM ) < ². Therefore, for n > N, Z ∞ 2 2 1 2 2 β2 β2 X 1 2 2 σ St P (|V − (σ St + − 2 exp{−2π k })dt| > δ) 2 6 π k2 β2 0 St k=1 Z 1 ∞ 2 2 1 2 2 β2 β2 X 1 c n 2 2 σ St ≤P (AM ) + P (|V IAM − (σ S + − exp{−2π k })dtIAM | > δ) t 2 6 π2 k2 β2 0 St n 1 k=1 <2². This proves Theorem 2. 16 Determining the Volatility of a Price Process in The Presence of Rounding Errors 3.4 17 Proof of Theorem 3 and Theorem 4 By (4), for large n, √ n nV IAm ∩Bm,n n √ 1X √ Yi,n = n [ n( √ (α ) n nS n i=1 (i−1)/n Yi,n 1 − ( √ (α ) 2 nS n (i−1)/n Yi,n 1 )2 + θ3 )]2 IAm ∩Bm,n , θ ∈ (0, √ (α ) 3 nS n ). (7) (i−1)/n 1 Using the cm ∈ (0, m ] as in (5), we define y ( )3 , when x ≥ cm ; x ψcm (x, y) = 3x 4 ( 3 − 4 )y 3 , when x < cm . cm cm (7) can be further written as √ n √ (2 log n)2 nV IAm ∩Bm,n = nU (n, φcm )IAm ∩Bm,n − U (n, ψcm )IAm ∩Bm,n + O( )IAm ∩Bm,n ; n1/2 and √ n nV IAm √ √ c = nV n IAm ∩Bm,n + nV n IAm ∩Bm,n √ =( nU (n, φcm ) − U (n, ψcm ))IAm √ √ (2 log n)2 c + ( nV n − nU (n, φcm ) + U (n, ψcm ))IAm ∩Bm,n )IAm ∩Bm,n + O( n1/2 √ = nU (n, φcm )IAm − U (n, ψcm )IAm + op (1), where φcm is defined in (6), ψcm in (8) and U (·, ·) in (3). (8) Determining the Volatility of a Price Process in The Presence of Rounding Errors 18 Note that ψcm (St , σSt y) is an odd function of y, and β = 0; by Delattre and Jacod (1997), Z U (n, ψcm ) →P 0 1Z h(y)ψcm (St , σSt y)dydt = 0. Therefore, U (n, ψcm )IAm →P 0. As a consequence, √ n √ nV IAm = nU (n, φcm )IAm + op (1). (9) Also by Delattre and Jacod (1997), √ n[U (n, φcm ) − Z 0 Z 1 Γφcm (St , βn )dt] →stably in law 0 1 ∆(φcm , φcm )(St , 0)1/2 dBs , (10) where Γφcm (St , βn ) Z 1Z h(y)φcm (St , βn bu + yσSt /βn c)dydu = 0 µ ¶ Z 1Z βn bu + yσSt /βn c 2 = h(y) ( ) IAm + φcm (St , βn bu + yσSt /βn c)IAcm dydu St 0 ∞ β2 1 β2 1 X 1 σ2S 2 =(σ 2 + n 2 − n2 2 exp{−2π 2 k 2 2 t })IAm + 2 6 St π St k βn k=1 Z 1Z h(y)φcm (St , βn bu + yσSt /βn c)dyduIAcm (by Lemma 4) ; 0 (11) Determining the Volatility of a Price Process in The Presence of Rounding Errors 19 and ∆(φcm , φcm )(St , 0) Z Z = hσSt (y)φ2cm dy − ( hσSt (y)φcm dy)2 Z Z y y = hσSt (y)[( )4 IAm + φ2cm (St , y)IAcm ]dy − ( hσSt (y)[( )2 IAm + φcm (St , y)IAcm ]dy)2 St St Z Z y 4 y =[ hσSt (y)( ) dy − ( hσSt (y)( )2 dy)2 ]IAm + St St Z Z [ hσSt (y)φcm (St , y)2 dy − ( hσSt (y)φcm (St , y)dy)2 ]IAcm ; hence ∆(φcm , φcm )(St , 0)1/2 Z Z y 4 y =[ hσSt (y)( ) dy − ( hσSt (y)( )2 dy)2 ]1/2 IAm + St St Z Z [ hσSt (y)φcm (St , y)2 dy − ( hσSt (y)φcm (St , y)dy)2 ]1/2 IAcm Z Z =(2σ 4 )1/2 IAm + [ hσSt (y)φcm (St , y)2 dy − ( hσSt (y)φcm (St , y)dy)2 ]1/2 IAcm . Plugging (11) and (12) into (10), and note that √ βn2 R 1 n π2 0 1 St2 P∞ 1 k=1 k2 exp{−2π 2 k 2 σ 2 St2 2 }dt βn (12) point- wise goes to zero on the set Am as n → ∞, one has, Z 1 √ 1 Γφcm (St , βn )dt]IAcm 2 dt)]IAm + n[U (n, φcm ) − 0 St 0 Z 1Z Z 4 2 → stably in law N (0, 2σ )IAm + [ hσSt (y)φcm (St , y) dy − ( hσSt (y)φcm (St , y)dy)2 ]1/2 dBs IAcm . √ β2 n[U (n, φcm ) − (σ 2 + n 6 Z 1 0 For any continuous function g that vanishes outside a compact set, the above stable convergence implies √ β2 E[g( n[U (n, φcm ) − (σ 2 + n 6 Z 0 1 1 dt)]IAm )IAm ] → E[g(N (0, 2σ 4 )IAm )IAm ]. St2 (13) Determining the Volatility of a Price Process in The Presence of Rounding Errors 20 And by defining ηcm (·, ·) to be 1 ( )2 , when x ≥ cm ; x ηcm (x, y) = 3 8 6 ( 4 x2 − 3 x + 2 ), when x < cm , cm cm cm one has, V0n IAm = V n IAm − βn2 U (n, ηcm )IAm . 6 (14) Again, by Delattre and Jacod (1997), Z U (n, ηcm ) →P 1Z 0 h(y)ηcm (St , σSt y)dydt. As a consequence, n U (n, ηcm )IAm = 1X 1 I →P (αn ) 2 Am n (S ) i=1 i/n Z 0 1 1 dtIAm . St2 By the assumption that √ 2 nβn → γ < ∞, one has √ βn2 β2 n( U (n, ηcm ) − n 6 6 Z 1 0 1 dt)IAm →P 0. St2 By (9), (14) and (15), √ n √ β2 nV0 IAm = n(U (n, φcm ) − n 6 Z 0 1 1 dt)IAm + op (1). St2 (15) Determining the Volatility of a Price Process in The Presence of Rounding Errors 21 Also since that g is uniformly continuous, √ lim E[g( n(V0n − σ 2 )IAm )IAm ] n→∞ √ β2 = lim E[g( n[U (n, φcm ) − (σ 2 + n n→∞ 6 Z 0 1 1 dt)]IAm )IAm ] St2 =E[g(N (0, 2σ 4 )IAm )IAm ] (by (13)), which implies, for any ² > 0, there exists N , such that ∀n ≥ N , √ |E[g( n[V0n − σ 2 ]IAM )IAM ] − E[g(N (0, 2σ 4 )IAM )IAM ]| < ². Note also that g is bounded, suppose |g| ≤ Mg . Recall that P (AcM ) → 0 , one can choose M such that P (AcM ) < ²/Mg . So for n ≥ N, √ |E[g( n[V0n − σ 2 ])] − E[g(N (0, 2σ 4 ))]| √ ≤|E[g( n[V0n − σ 2 ]IAM )IAM ] − E[g(N (0, 2σ 4 )IAM )IAM ]|+ √ |E[g( n[V0n − σ 2 ]IAcM )IAcM ]| + |E[g(N (0, 2σ 4 )IAcM )IAcM ]| √ ≤|E[g( n[V0n − σ 2 ]IAM )IAM ] − E[g(N (0, 2σ 4 )IAM )IAM ]| + 2M g ∗ P (AcM ) ≤3² Hence we’ve proved that for all continuous g that vanishes outside a compact set, √ lim E[g( n[V0n − σ 2 ])] = E[g(N (0, 2σ 4 ))], n→∞ Determining the Volatility of a Price Process in The Presence of Rounding Errors 22 i.e., √ n[V0n − σ 2 ] →L N (0, 2σ 4 ). This finishes the proof of Theorem 4. The proof of Theorem 3 is basically contained in the proof above. 4 CONCLUSIONS AND DISCUSSION In summary, we have proved the following convergence in probability: • If αn ≡ α ∈ (0, ∞): 1 1 √ V n →P σ n • If r ∞ 1 2 X log(kα) L1 (log(1 + ))2 . π k k=1 √ nαn → β ∈ [0, ∞): V n β2 →P σ + 6 Z 2 0 1 1 β2 dt − π2 St2 Z 1 0 ∞ 2 2 1 X 1 2 2 σ St exp{−2π k }dt. k2 β2 St2 k=1 And we have proved the central limit theorem that if • √ n[V n − σ 2 − 2 βn 6 R1 1 0 St2 dt] →L N (0, 2σ 4 ) and √ 2 nβn → γ < ∞, √ n[V n − α2n 6 Pn 1 i=1 (S (αn ) )2 i/n − σ 2 ] →L N (0, 2σ 4 ). We have used the later result to create a bias-correction that works for “small rounding”. Note that while we work with observations on a time interval [0, 1], results for the more general case of time interval [0, T ] is obtained by rescaling. The case of time varying volatility and/or unequal observation times can be studied using the methods of Jacod and Protter (1998) and Mykland and Zhang (2006). Determining the Volatility of a Price Process in The Presence of Rounding Errors 23 REFERENCES Aldous, D. (1989), Probability Approximations Via The Poisson Clumping Heuristic, New York: Springer. Delattre, S. and Jacod, J. (1997), “A Central Limit Theorem for Normalized Functions of the Increments of a Diffusion Process, in the Presence of Round-Off Errors,” Bernoulli, 3, 1–28. Ferguson, T. S. (1996), A Course in Large Sample Theory, London, U.K.: Chapman and Hall. Jacod, J. (1996), “La Variation Quadratique du Brownien en Présence d’Erreurs d’Arrondi,” Astérisque, 236, 155–161. Jacod, J. and Protter, P. (1998), “Asymptotic Error Distributions for the Euler Method for Stochastic Differential Equations,” Annals of Probability, 26, 267–307. Kolassa, J. and McCullagh, P. (1990), “Edgeworth series for lattice distributions,” Annals of Statistics, 18, 981–985. Li, Y. and Mykland, P. A. (2006), “Are Volatility Estimators Robust with Respect to Modeling Assumptions?” Tech. rep., The University of Chicago, Department of Statistics. Mykland, P. A. and Zhang, L. (2006), “ANOVA for Diffusions,” The Annals of Statistics, forthcoming, 34, # 4. Revuz, D. and Yor, M. (1999), Continuous Martingales and Brownian Motion, Berlin, Germany: Springer-Verlag, 3rd ed. Zeng, Y. (2003), “A partially-observed model fo0r micro-movement of asset process with Bayes esti- Determining the Volatility of a Price Process in The Presence of Rounding Errors 24 mation via filtering for a class of micro-movement models of asset price,” Mathematical Finance, 13, 411–444. Zhang, L., Mykland, P. A., and Aı̈t-Sahalia, Y. (2005a), “Edgeworth expansions for realized volatility and related estimators,” Tech. rep., University of Illinois at Chicago. — (2005b), “A Tale of Two Time Scales: Determining Integrated Volatility with Noisy HighFrequency Data,” Journal of the American Statistical Association, 472, 1394–1411.