Realized Measures. Eduardo Rossi University of Pavia. November Rossi Realized Measures University of Pavia / 64

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1 Realized Measures Eduardo Rossi University of Pavia November 2012 Rossi Realized Measures University of Pavia / 64

2 Outline 1 Introduction 2 RV Asymptotics of RV Jumps and Bipower Variation 3 Realized range RRG with discretely sampled data RRG with microstructure noise Rossi Realized Measures University of Pavia / 64

3 Introduction Continuous-time SV models Martingale Representation Theorem For any univariate, square-integrable, continuous sample path logarithmic price process, which is not locally riskless, there exists a representation such that 0 t T r(t, h) = µ(t, h) + M(t, h) = h 0 µ(t h + u)du + h 0 σ(t h + u)dw (u) where µ(s) is an integrable, predictable and finite variation stochastic process, σ(s) is a strictly positive cadlag stochastic process satisfying [ h Pr 0 ] σ(t h + u)du < = 1 Rossi Realized Measures University of Pavia / 64

4 Introduction Continuous-time SV models The integral representation is equivalent to the standard (short-hand) SDE specification of the logarithmic price process, dp(t) = µ(t)dt + σ(t)dw (t) Within the class of continuous sample path semi-martingale (diffusion) models, there are no consequential restrictions involved in stating the model directly through a SDE. The volatility coefficient process in this formulation is termed the Instantaneous Volatility Process. {σ 2 (t)} t [0,T ] σ 2 (t) = lim σ 2 1 (t h) = lim h 0 h 0 h h 0 σ 2 (t h + u)du The notional volatility (or the increment to the QV process) equals the so-called Integrated Volatility, υ 2 (t) = [M, M] t [M, M] t h = h 0 σ 2 (t h + u)du Rossi Realized Measures University of Pavia / 64

5 Introduction Black and Scholes model The simplest possible case is provided by the time-invariant diffusion (continuous-time random walk), dp(t) = µdt + σdw (t) which underlies the Black-Scholes option pricing formula. This process has a deterministic mean return ( µ ) so the expected return volatility trivially equals the expected notional volatility E[ϑ 2 t F t h ] = E[υ 2 t F t h ] Because the volatility is also constant, the expected notional volatility is identical to the notional volatility. The notional volatility is equal to υ 2 t = h 0 σ 2 (t h + s)ds = h σ 2 Rossi Realized Measures University of Pavia / 64

6 Black and Scholes model Introduction This model is also straightforward to estimate from discretely sampled data by, e.g., maximum likelihood, as the returns are i.i.d. and normally distributed. Suppose that observations are only available at n + 1 equally spaced points over the [t h, t] time interval, where 0 h < t T ; i.e., t h, t h + (h/n),..., t h + (n 1)(h/n), t. By the definition of the process, the corresponding sequence of i = 1, 2,..., n discrete (h/n)-period returns, ( r t h + i h n, h ) ( = p t h + i h ) ( p t h + (i 1) h ) n n n ( r t h + i h n, h ) ( i.i.d.n µ h n n, h ) σ2. n Rossi Realized Measures University of Pavia / 64

7 Introduction Black and Scholes model The MLE of the drift is simply given by the sample mean of the (scaled) returns, ˆµ n = 1 n n ( h ) 1r ( t h + i h n n, h ) = n i=1 It follows immediately that E[ˆµ n ] = µ r(t, h) h = p(t) p(t h) h This fixed-interval, or fill-in asymptotic, estimator for the drift only depends on h (the length of the interval) and not n. For a fixed interval the in-fill asymptotics, obtained by increasing the number of intraday observations, are irrelevant for estimating the expected return. The estimator of the drift is independent of the sampling frequency, given by n, and depends only on the span of the data, h. Rossi Realized Measures University of Pavia / 64

8 Black and Scholes model Introduction The variance of the estimator [( ) 2 ] Var[ˆµ n ] = E ˆµ n µ [( p(t) p(t h) ) 2 ] = E µ h [( µ h + σ(w (t) W (t h)) = E h = σ2 h ) 2 ] µ Thus, although ˆµ n is an unbiased estimator for µ, it is not consistent as n. Rossi Realized Measures University of Pavia / 64

9 Black and Scholes model Introduction Consider now the (unadjusted) estimator for σ 2 defined by the sum of the (scaled) squared returns, ˆσ n 2 1 n ( h ) 1r ( t h + i h n n n, h ) 2 n Because it follows that i=1 [ ( E r t h + i h n, h ) 2 ] n = µ 2( h n E[ˆσ n] 2 = σ 2 + µ 2( h ) n ) ( 2 + σ 2 h ) n Hence, the drift induces only a second order bias, or O(n 1 ) term, in the estimation of σ 2 as n. Rossi Realized Measures University of Pavia / 64

10 Black and Scholes model Introduction This estimator for the diffusion coefficient is consistent as n [ ( E r t h + i h n, h ) 3 ] = 3µσ 2( h ) ( 2 + µ 3 h ) 3 n n n [ ( E r t h + i h n, h ) 4 ] = 3σ 4( h ) 2 + 6µ 2 σ 2( h ) ( 3 + µ 4 h ) 4 n n n n the terms involving the drift coefficient are an order of magnitude smaller, for n large, than those that pertain only to the diffusion coefficient. This feature allows us to estimate the return variation with a high degree of precision even without specifying the underlying mean drift. Rossi Realized Measures University of Pavia / 64

11 Black and Scholes model Introduction This (along with the second moment given above) and the fact that the returns are i.i.d., implies that Var[ˆσ n] 2 = E[(ˆσ n) 2 2 ] E[ˆσ n] 2 2 = 2σ4 n + 4µ2 σ 2 n 2 By a standard Law of Large Numbers, p lim ˆσ n 2 = σ 2. n The realized variation measure is a biased but consistent estimator of the underlying (squared) volatility coefficient. Increasing the number of (scaled) squared return observations over the interval then produces an increasing number of unbiased and uncorrelated measures of σ 2, and simply averaging these yields a consistent estimator. Rossi Realized Measures University of Pavia / 64

12 RV The data-driven, or nonparametric volatility measurements afford direct ex-post empirical appraisals of the notional volatility, υ 2 (t, h), without any specific functional form assumptions. The nonparametric measurements more generally achieve consistency by measuring the volatility as (weighted) sample averages of increasingly finer sampled squared (or absolute) returns over (and possible outside) the [t h, t] interval. The realized volatility (RV ) measures build on the idea of an increasing number of observations over fixed length time intervals. Rossi Realized Measures University of Pavia / 64

13 RV Set up No price jumps and frictionless market. The asset s logarithmic price process p(t) must be a semi-martingale to rule out arbitrage opportunities: dp(t) = µ(t)dt + σ(t)dw (t) 0 t T µ(t) and σ(t) are predictable processes, µ(t) is of finite variation, σ(t) is strictly positive and square integrable, i.e., E [ ] t 0 σ2 (s)ds < the processes µ(t) and σ(t) signify the instantaneous conditional mean and volatility of the return. Rossi Realized Measures University of Pavia / 64

14 RV Quadratic variation The continuously compounded return over the time interval from t h to t, t t r(t, h) = p(t) p(t h) = µ(s)ds + σ(s)dw (s) t h t h and its quadratic variation QV (t, h) is QV (t, h) = t t h σ 2 (s)ds innovations to the mean component µ(t) do not affect the sample path variation of the return. Intuitively, this is because the mean term, µ(t)dt, is of lower order in terms of second order properties than the diffusive innovations, σ(t)dw (t). When cumulated across many high-frequency returns over a short time interval of length h they can effectively be neglected. Rossi Realized Measures University of Pavia / 64

15 RV Realized Volatility The diffusive sample path variation over [t h, t] is also known as the integrated variance IV (t, h), In this setting, the quadratic and integrated variation coincide. This is however no longer true for more general return process like, e.g., the stochastic volatility jump-diffusion model. Absent microstructure noise and measurement error, the return quadratic variation can be approximated arbitrarily well by the corresponding cumulative squared return process. Rossi Realized Measures University of Pavia / 64

16 Realized Volatility RV Consider a partition {t h + j, j = 1,..., n h} n The corresponding sequence of i = 1, 2,..., n discrete (h/n)-period returns, ( r t h + i h n ; h ) = p(t h + i h n n ) p(t h + (i 1)h n )). The RV is simply the second (uncentered) sample moment of the return process over a fixed interval of length h, scaled by the number of observations n (corresponding to the sampling frequency 1/n), so that it provides a volatility measure calibrated to the h-period measurement interval. RV (t, h; n) = n r i=1 ( t h + h i n ; h ) 2 n Rossi Realized Measures University of Pavia / 64

17 Realized Volatility RV The RV provides a consistent nonparametric measure of the Notional Volatility. Although the definition is stated in terms of equally spaced observations, most results carry over to situations in which the RV is based on the sum of unevenly, but increasingly finely sampled squared returns. Rossi Realized Measures University of Pavia / 64

18 RV Realized Volatility The theoretical properties of RV have been discussed from different perspectives in a number of paper: Andersen, T.G., T. Bollerslev, F.X. Diebold and P. Labys (2001a), The Distribution of Realized Exchange Rate Volatility, Journal of the American Statistical Association, 96, Andersen, T.G., Bollerslev, T., Diebold, F.X. and Labys, P. (2003), Modeling and forecasting realized volatility, Econometrica 71, Andersen, T.G., T. Bollerslev, F.X. Diebold and P. Labys (2000b), Exchange Rate Returns Standardized by Realized Volatility are (Nearly) Gaussian, Multinational Financ Journal, 4, Barndorff-Nielsen, O.E. and N. Shephard (2001a), Non-Gaussian Ornstein-Uhlenbeck-Based Models and Some of Their Uses in Financial Economics, Journal of the Royal Statistical Society, Series B, 63, Barndorff-Nielsen, O.E. and N. Shephard (2002a), Econometric Analysis of Realised Volatility and its Use in Estimating Stochastic Volatility Models, Journal of the Royal Statistical Society, Series B, 64, Rossi Realized Measures University of Pavia / 64

19 RV Realized Volatility Realized Volatility as an Unbiased Volatility Estimator If the return process is square-integrable and µ(t) = 0 then for any value of n 1 and h > 0, ϑ 2 (t, h) = E[υ 2 (t, h) F t h ] = E[RV (t, h; n) F t h ] the ex-post RV is an unbiased estimator of ex-ante expected volatility. The result remains approximately true for a stochastically evolving mean return process over relevant horizons under weak auxiliary conditions, as long as the underlying returns are sampled at sufficiently high frequencies. The RV approach interprets RV (t, h; n) as a measure of the overall volatility for the [t h, t] time interval. Since RV is approximately unbiased for the corresponding unobserved QV, the RV measure is the natural benchmark against which to gauge the performance of volatility forecasts. The quadratic variation is directly related to the actual return variance and to the expected return variance. Rossi Realized Measures University of Pavia / 64

20 RV Realized Volatility The ex-ante expected notional volatility (E[υ 2 (t, h) F t h ]) is also the critical determinant of expected volatility, expressed as ϑ 2 (t, h) = E[{r(t, h) E(µ(t, h) F t h )} 2 F t h ] = E[υ 2 (t, h) F t h ] + Var[µ(t, h) F t h ] + 2 Cov[M(t, h), µ(t, h) F t h ] Any empirical measures of (ex-ante expected) notional volatility will necessarily depend on the assumed parametric model structure. Rossi Realized Measures University of Pavia / 64

21 RV Realized Volatility The RV concept is associated with the return variation measured over a discrete time interval rather than with the so-called spot or instantaneous volatility. This distinction separates the RV approach from a voluminous literature in statistics seeking to estimate spot volatility from discrete observations, predominantly in a setting with a constant diffusion coefficient. Rossi Realized Measures University of Pavia / 64

22 RV Asymptotics of RV Asymptotics of RV Consistency of RV The RV provides a consistent estimator of the Notional Volatility, p lim RV (t, h; n) = υ 2 (t, h) n 0 < h t T where the convergence is uniform in probability. in the limit for increasingly finely sampled returns, or n, RV is a consistent (nonparametric) estimator of the (realized) notional volatility over any fixed length time interval, h > 0. Rossi Realized Measures University of Pavia / 64

23 RV Asymptotics of RV The ex-post realized notional volatility provides an asymptotically (for n ) unbiased estimator of the ex-ante expected notional volatility, as long as the return is bounded over the relevant horizon. For example, one may imagine a bound on the return that prevents a small investment in the asset from ever producing a return that exceeds a (large) multiple of the expected value of all resources available in the world-wide economy. It is only a sufficient condition, and the proposition will apply in most cases, even when the boundedness condition is violated. Rossi Realized Measures University of Pavia / 64 Asymptotics of RV Consistency of Ex-Ante Expected Realized Volatility The Expected RV provides a consistent estimator of the Expected Notional Volatility, p lim E[RV (t, h; n) F t h ] = E[υ 2 (t, h) F t h ] n 0 < h t T provided the return process is bounded and the expected return, µ(s), is of integrable variation for s [t h, t].

24 Asymptotics of RV RV Asymptotics of RV Decomposition of the RV measure into the separate terms associated with the potential sources of error and bias ( r t h + i n ; 1 ) = µ(i) + M(i) n In the frictionless arbitrage-free setting, the return on a risky asset over time intervals of length h/n has a martingale innovation of order (h/n) 1/2 while the corresponding mean component is at most of order h/n RV (t, h; n) = n [µ(i) 2 + 2µ(i)M(i) + M(i) 2 ] i=1 = O p (h 2 /n) + O p (h 3/2 /n 1/2 ) + n M(i) 2 [ n ] = υ 2 (t, h) + O p (n 1 ) + O p (n 1/2 ) + M(i) 2 υ 2 (t, h) i=1 i=1 Rossi Realized Measures University of Pavia / 64

25 Asymptotics of RV RV Asymptotics of RV The RV may differ from the notional volatility for two distinct reasons. The terms O p (n 1 ) + O p (n 1/2 ) reflect the mean returns which only truly vanishes in the limit for n. The expected return over short intervals (large n) are necessarily small, so the contribution from these terms will be empirically negligible. [ n ] The last term i=1 M(i)2 υ 2 (t, h) has the interpretation of a measurement error term, the cumulative squared martingale innovations provide unbiased estimators for the corresponding notional volatility (quadratic variation). This term has a zero expected value. For any given value of n it induces a measurement error that is unrelated to the mean return. This component is the source of empirically relevant deviations between realized volatility and (realized) notional volatility. The actual size and exact distribution of the errors obviously depend on the particular return process and must be analyzed on a case-by-case basis. Rossi Realized Measures University of Pavia / 64

26 RV Asymptotics of RV Asymptotics of RV RV is approximately (apart from minor biases induced by the mean component) unbiased for the corresponding notional volatility. It also follows from the local martingale property: If M is a locally square integrable martingale, then the associated (M 2 [M, M]) process is a local martingale, E[M(t, h) 2 ([M, M] t [M, M] t h ) F t h ] = 0 0 < h t T. and the decomposition of RV that the associated measurement errors are approximately uncorrelated. RV s constitute natural and convenient inputs into modeling and inference procedures concerning the expected notional volatility and, by extension, the expected return volatility. Rossi Realized Measures University of Pavia / 64

27 Asymptotics of RV RV Asymptotics of RV The fixed h, large n asymptotics, or realized volatility asymptotics is pivotal in practice. In spite of the theoretical desirability of letting interval size, h, shrink indefinitely as an increasing number of high-frequency return observations is used within each (vanishing) interval, this idea is difficult (impossible) to mimic in practice. The number of data points, n, that adhere (approximately) to the underlying no-arbitrage semi-martingale property over short time intervals is severely limited by various market microstructure frictions. This invariably puts an effective (asset and/or market specific) lower bound on the highest sampling frequency that is applicable in empirical work, say 1/n > 1/N. The RV approach exploiting intraday return observations allow for directly observable return volatility measures that are consistent, approximately unbiased and have uncorrelated measurement errors. It is natural to exploit these properties by building a time series model directly for the observed realized volatility measures through standard ARMA style modeling. Rossi Realized Measures University of Pavia / 64

28 Asymptotics of RV RV Asymptotics of RV The imposition of additional restrictions on the return process allows for important additional insight into the size and asymptotic distribution of the realized volatility errors. In particular, consider the class of continuous sample path diffusions, characterized by the SDE, dp(t) = µ(p(t), σ(t ))dt + σ(t )dw (t) 0 t T where µ(p(t), σ(t )) is predictable and of finite variation and the σ(t ) process is independent of the Brownian motion, W (t). Barndorff-Nielsen and Shephard (2002) showed that for n : where QV (t, h) υ 2 (t, h) and ) (RV (t, h; n) QV (t, h) n 2IQ(t, h) IQ(t, h) = t t h σ 4 (s)ds d N(0, 1) is the integrated quarticity, with IQ(t, h) independent from the limiting Gaussian distribution. Rossi Realized Measures University of Pavia / 64

29 RV Asymptotics of RV Asymptotics of RV To conduct ex-post inference regarding the actual realized return variation over a given period a consistent estimator of IQ(t, h) is required. IQ(t, h), is unobserved and is likely to display large period-to-period variation. Rossi Realized Measures University of Pavia / 64

30 Asymptotics of RV RV Asymptotics of RV Theorem (Barndorff-Nielsen and Shephard (2007)) Suppose that the log-price p(t) is a Brownian semimartingale and that for all t < t 0 µ2 (u; σ(u ))du <, then as n so 1 ) (RV (t, h) QV (t, h) 2 δ t 0 σ 2 (u )dw (u) where δ h/n and the convergence is in law stable as a process. The convergence in law stable means that if X n X in law stable this implies that for any random variable Z, the pair (Z, X n ) converges in law to (Z, X ). Rossi Realized Measures University of Pavia / 64

31 Asymptotics of RV RV Asymptotics of RV Now, let X n = ( ) n RV (t, h) QV (t, h) and t Z = σ 4 (u )du the convergence in law stably implies that ) (RV (t, h) QV (t, h) L n N(0, 2) t 0 σ4 (u )du without the convergence in law stably this result cannot be deduced. 0 Rossi Realized Measures University of Pavia / 64

32 Asymptotics of RV RV Asymptotics of RV Barndorff-Nielsen and Shephard (2002) showed that the IQ can be consistently estimated using RQ(t, h) = 1 n ( r t h + h i 3 n ; h ) 4 n it follows that i=1 ) δ (RV 1/2 (t, h) QV (t, h) 2 3 RQ(t, h) d N(0, 1) Rossi Realized Measures University of Pavia / 64

33 Asymptotics of RV RV Asymptotics of RV Barndorff-Nielsen and Shephard (2002c) find that an improved finite-sample (finite n) approximation may be obtained by the log-linearization. The improvement is related to the logarithm delivering a variance-stabilizing transformation: ) δ (log 1/2 RV (t, h) log QV (t, h) 2 RQ(t,h) 3 RV (t,h) 2 d N(0, 1) Rossi Realized Measures University of Pavia / 64

34 RV Asymptotics of RV If the instantaneous return is a continuous-time process and the return, mean, and volatility processes are uncorrelated (i.e., dw (t) and innovations to µ(t) and σ(t) are mutually independent) The discrete-time return r(t, h) is normally distributed conditional on the cumulative drift µ(t, h) = t µ(s)ds and the QV (t, h) (which in this t h setting equals the integrated variance υ 2 (t, h)): Normal Mixture Distribution The discrete-time returns r(t, h) over [t h, t], 0 < h t T, for the continuous sample path diffusion is distributed as a normal mixture, r(t, h) µ(t, h), υ 2 (t, h) N ( µ(t, h), υ 2 (t, h) ) The (ex-ante) mean return and the notional volatility is not directly observable. The return distribution is mixed Gaussian with the mixture governed by the realizations of the IV (and integrated mean) process. Rossi Realized Measures University of Pavia / 64

35 RV Asymptotics of RV Extreme realizations (draws) from the integrated variance process render return outliers likely while persistence in the integrated variance process induces volatility clustering. Rossi Realized Measures University of Pavia / 64

36 RV Asymptotics of RV The RV concept is associated with the return variation measured over a discrete time interval rather than with the so-called spot or instantaneous volatility. In principle, the RV measurement can be adapted to spot volatility estimation: as h 0, QV (t, h) σ 2 (t). RV converges to instantaneous volatility when both h and h/n shrink. For this to happen, however, h/n must converge at a rate higher than h, so as the interval shrinks we must sample returns at an ever increasing frequency. In practice, this is infeasible, because intensive sampling over tiny intervals magnifies the effects of microstructure noise. Rossi Realized Measures University of Pavia / 64

37 RV Jumps and Bipower Variation Jumps A broad class of SV models that allow for the presence of jumps in returns is defined by dp(t) = µ(t)dt + σ(t)dw (t) + κ(t)dq(t) 0 t T where q is a Poisson process uncorrelated with W and governed by the jump intensity λ(t): Pr{dq t = 1} = λ t dt with λ t positive and finite. This assumption implies that there can only be a finite number of jumps in the price path per time period. Common restriction in the finance literature, though it rules out infinite activity LÈvy processes. The scaling factor κ(t) denotes the magnitude of the jump in the return process if a jump occurs at time t. Rossi Realized Measures University of Pavia / 64

38 Jumps RV Jumps and Bipower Variation p(t) = = t 0 t 0 t µ(u)du + µ(u)du + 0 t 0 t σ(u)dw (u) + κ(u)dq(u) 0 σ(u)dw (u) + J(t) 0 t T It is useful to decompose, J(t): J(t) = J A (t) + J M (t) where, assuming J(t) has an absolutely continuous intensity J A (t) = t 0 c(u)du c(t) = E[dJ(t) F t ] then J M (t) is the compensated jump process so J M (t) is a martingale while J A (t) is a purely continuous local finite variation process. Rossi Realized Measures University of Pavia / 64

39 RV Jumps and Bipower Variation Jumps The log-price has the decomposition p(t) = A(t) + M(t) = t 0 t t (µ(u) + c(u))du + σ(u)dw (u) + J(t) c(u)du 0 0 In this case, the quadratic return variation process over the interval from t h to t, 0 h t T, is the sum of the diffusive integrated variance and the cumulative squared jumps: QV (t, h) = t t h σ 2 (s)ds + t s=t h J 2 (s) IV (t, h) + t s=t h where J(t) = κ(t)dq(t) is non-zero only if there is a jump at time t. J 2 (s) Rossi Realized Measures University of Pavia / 64

40 RV Jumps and Bipower Variation Jumps The RV estimator remains a consistent measure of the total QV in the presence of jumps. The diffusive and jump volatility components appear to have distinctly different persistence properties. It is useful both for analytic and predictive purposes to obtain separate estimates of these two factors in the decomposition of the QV implied by equation. Rossi Realized Measures University of Pavia / 64

41 Bipower Variation RV Jumps and Bipower Variation To this end, the m-skip bipower variation, BV, introduced by Barndorff-Nielsen OE, Shephard N (2006) Econometrics of testing for jumps in financial economics using bipower variation, Journal of Financial Econometrics 4:1-30. provides a consistent estimate of the IV component, BV (t, h; m, n) = π 2 n i=m+1 ( r t h + i h n ; 1 n ) r ( t h + (i m) h ; 1 ) n n Setting m = 1 yields the the Realized Bipower Variation RBV (t, h) BV (t, h; 1, n). The RBV is a consistent estimator of the Bipower Variation, defined as: π BV (t) = p lim n 2 n ( r t h + i h n ; 1 n i=2 ) r ( t h + (i 1) h ; 1 ) n n Rossi Realized Measures University of Pavia / 64

42 RV Jumps and Bipower Variation Bipower Variation The RBV is robust to the presence of jumps and therefore, in combination with RV, it yields a consistent estimate of the cumulative squared jump component. As n : RV (t, h) RBV (t, h) QV (t, h) IV (t, h) = t s=t h J 2 (s) RBV works because only a finite number of terms in the sum are affected by jumps, while each return which does not have a jump goes to zero in probability. since the probability of jumps in contiguous time intervals goes to zero as h/n 0 those terms which do include jumps do not impact the probability limit. Rossi Realized Measures University of Pavia / 64

43 Identification of Jumps RV Jumps and Bipower Variation One potential use of RBV is to test for the hypothesis that a set of data is consistent with a null hypothesis of continuous sample paths. We can do this by asking if RV (t, h) RBV (t, h) is statistically significantly bigger than zero. This demands a distribution theory for RBV, calculated under the null that p(t) is a Brownian semimartingale, and σ > 0. Barndorff-Nielsen, Graversen, Jacod, Podolskij, and Shephard (2006) have established a CLT which covers this situation ( ) (( ( ) ) δ 1/2 RBV (t, h) BV (t, h) d 0 (2 + θ) 2 t N, RV (t, h) IV (t, h) 0) σ4 (u)du where θ = (π 2 /4) + π Rossi Realized Measures University of Pavia / 64

44 RV Jumps and Bipower Variation Identification of Jumps Under the null of no jumps, the fact that asymptotically A Var(RV (t, h) IV (t, h)) = A Cov(RV (t, h) IV (t, h), BV (t, h) IV (t, h)) is no coincidence and reflects a situation exactly analogous to that of the Hausman (1978) test. Asymptotically, the situation is one with Gaussian errors, and RV (t, n) is the most efficient estimate of the integrated variance The RBV (t, n) is a less efficient estimator under the maintained assumption of no jumps, though it is also more robust. The following statistics can be used as the basis of a test of the null of no jumps (Barndorff-Nielsen and Shephard (2006) test) δ 1/2 ((2/π)RBV (t, h) BV (t, h)) θ t 0 σ4 (u)du d N(0, 1) Rossi Realized Measures University of Pavia / 64

45 RV Jumps and Bipower Variation Relative Jump Huang and Tauchen (2005) a measure of jumps, the Relative Jump: RJ(t, h) = RV (t, h) RBV (t, h) RV (t, h) (1) It is an indicator of the contribution (if any) of jumps to the total within-day variance of the process. 100 RJ is a direct measure of the percentage contribution of jumps, if any, to total price variance. Under the maintained assumptions of no jumps, then asymptotically RV (t, h) RBV (t, h) is independent of RV (t, h) conditional on the volatility path, and thus RJ(t, h) is asymptotically the ratio of two conditionally independent random variables. To determine the scale of RV (t, h) and RBV (t, h) in units of conditional standard deviation, one needs to estimate the IQ. Rossi Realized Measures University of Pavia / 64

46 RV The p-th Realized Power Variation Jumps and Bipower Variation The realized power variation of order p, V (p; t, h; n), is the (scaled) cumulative sum of the absolute p-th power of the high-frequency returns V (p; t, h; n) n p/2 1 µ 1 p where µ p = E z p with z N(0, 1). n (t r h + jh n ; 1 ) p n It converges, as n, to the corresponding power variation of order p, V (p; t, h): V (p; t, h; n) p t t h j=1 σ p (s)ds V (p; t, h). V (4; t, h; n) is a natural choice as a consistent estimator for the IQ(t, h). This conclusion is heavily dependent on the absence of jumps in the price process. Rossi Realized Measures University of Pavia / 64

47 Multipower Variation RV Jumps and Bipower Variation The standard estimator of IQ is not robust to jumps. One way of overcoming this problem is to use a multipower variation (MPV ) measure introduced by Barndorff-Nielsen and Shephard (2006). ( m ) 1 n MP(t, h, r; n) = δ (1 r+/2) µ ri i=1 j=m+1 { m 1 i=0 ( r t h + where r i > 0, r = (r 1, r 2,..., r m ) for all i. RBV (t, h) = MP(t, h, 1, 1; n), r + = m i=1 r i (j i)h ; 1 ) } r i n n Rossi Realized Measures University of Pavia / 64

48 RV Jumps and Bipower Variation Andersen, Bollerslev, and Diebold (2004) test Andersen, Bollerslev, and Diebold (2004) suggest using the jump-robust realized TP(t, h), the Realized Tri-Power Quarticity statistic, which is a special case of the multipower variations studied in Barndorff-Nielsen and Shephard (2004a) with TP(t, h) δ 1 µ 3 4/3 even in the presence of jumps. n r t 1,i 2 4/3 r t 1,i 1 4/3 r t 1,i 4/3 i=3 r t 1,j r TP(t, h) p ( t 1 + j n ; 1 ) n t t 1 σ 4 (s)ds Rossi Realized Measures University of Pavia / 64

49 RV Jumps and Bipower Variation Andersen, Bollerslev, and Diebold (2004) test Based on Barndorff-Nielsen and Shephard s (2006) theoretical results, Andersen, Bollerslev, and Diebold (2004) use the time series where RV (t, h) BV (t, h) z TP,t = (v bb v qq ) 1 n TP(t, h) v bb = (µ 1 1 1) + 2(µ 2 1 1) v qq = µ 4 µ 2 2 d For each t, z t N(0, 1) as n, on the assumption of no jumps. Thus the sequence {z t } T t=1 provides evidence on the daily occurrence of jumps in the price process. The extensive simulation-based evidence for specific parametric continuous time diffusions reported in Huang and Tauchen (2005) suggests that the z t statistic tends to over-reject the null hypothesis of no jumps for large critical values. Rossi Realized Measures University of Pavia / 64

50 RV Jumps and Bipower Variation Barndorff-Nielsen and Shephard (2006) test The BN-S test infers whether jumps occur during a time interval (usually a trading day) by comparing RV, as an estimator for the quadratic variance, with RBV : z TP,r,t = 1 RBV (t, h)/rv (t, h) 0.61δ max{1, TP(t, h)/rbv (t, h)2 } Rossi Realized Measures University of Pavia / 64

51 RV Jumps and Bipower Variation There is a scale normalizing constant n in front of the summation because each absolute return is of order t, so the product is of order ( t) 2, and the summation t. n is 1/ t, which cancels out the summation order, and the whole expression approaches a well defined limit. Rossi Realized Measures University of Pavia / 64

52 Realized range Realized range The Realized Range, RRG, is a rivaling approach to the estimation of IV, based on the aggregation of intra-daily ranges. Martens and Van Dijk (2007), and Christensen and Podolskij (2007) and Christensen, Podolskij and Vetter (2009). The RRG is appealing when MN prevents from the use of the whole record of high-frequency prices to compute RV since it exploits a larger amount of information and it is in principle able to attain a higher precision. Rossi Realized Measures University of Pavia / 64

53 Parkinson estimator Realized range The daily range estimator of Parkinson (1980) developed under the assumption that the log-price follows a scaled Brownian motion: the daily range is defined as s p = dp(t) = σdw (t) sup {p t p s }. 0 t,s 1 On the [0, 1] interval: E[s r p] = λ r σ r. Rossi Realized Measures University of Pavia / 64

54 Parkinson estimator Realized range Under the assumption of a fully observed continuous time log-price path, Parkinson s estimator of the daily integrated volatility: RG P = 1 λ 2 s 2 p = s2 p 4 log(2). where λ r = E[s r W ] = E[sup {W 1 W 0 } r ] Rossi Realized Measures University of Pavia / 64

55 Realized range Realized range Consider an equidistant partition 0 = t 0 < t 1 <... < t n = 1, where t i = i/n, and = 1/n. The intraday range at sampling times t i 1 and t i (i = 1, 2,..., n) is s pi, = sup t i 1 t,s t i {p t p s }. The RRG estimator for the interval [0, 1] is defined as: n RRG = 1 λ 2 i=1 s 2 p i, Christensen and Podolskij (2007) show that RRG p IV. Result obtained for very general continuous time processes, including models with leverage, long-memory, diurnal effects or jumps (in σ(t)). Rossi Realized Measures University of Pavia / 64

56 Realized range RRG Asymptotic distribution RRG converges in law to a mixed normal with σ governing the mixture, i.e.: n(rrg IV ) d MN(0, ΛIQ) with IQ = 1 0 σ(u)4 du. For RRG Λ is approximately 0.4 while for RV is 2. RRG uses all the data, whereas RV is based on high-frequency returns sampled at fixed points in time. It follows that n(rrg IV ) ΛRRQ d N(0, 1) where RRQ = n n λ 4 i=1 s4 p i, consistently estimates the IQ. Rossi Realized Measures University of Pavia / 64

57 Realized range RRG with discretely sampled data RRG with discretely sampled data When the inference is based on a finite sample, the intraday high-low statistic will be progressively more downward biased as n gets larger, since the number of prices in each decreases. The true range is not observed! The source of bias is λ 2, which is constructed on the presumption that p is fully observed. Christensen and Podolskij (2007) assume that mn + 1 equally spaced price observations are available. n intervals each with m returns. The log-price for each time in the interval (0, 1) is p i 1 n +, i = 1..., n and t mn t = 0,..., m. The observed range over the i-th interval is: { s pi,,m = max 0 s,t m p i 1 n + t mn p i 1 n + s mn }. Rossi Realized Measures University of Pavia / 64

58 Realized range RRG with discretely sampled data RRG with discretely sampled data and s W,m = { } max Wt/m W s/m 0 s,t m λ r,m = E[s r W,m]. λ r,m is the r-th moment of the range of a sbm over [0, 1] when only m increments of the underlying continuous time process are observed. Numerical simulation to compute λ r,m The RRG estimator based on discrete observations is RRG m = 1 λ 2,m n i=1 RRG m is a consistent estimator of IV as n. s 2 p i,,m. Rossi Realized Measures University of Pavia / 64

59 Realized range RRG with discretely sampled data RRG with discretely sampled data RRGm is a consistent estimator of IV as n. If we assume that the log-price follows the SV process and m c N : n(rrg m IV ) d N(0, 1) Λm RRQm with Λ m = λ4,m λ2 2,m, and λ 2 2,m RRQ m = n λ 4,m n i=1 s 4 p i,,m. Rossi Realized Measures University of Pavia / 64

60 Realized range RRG with microstructure noise RRG with microstructure noise Christensen, Podolskij and Vetter (2009): With an i.i.d. MN, the RRG estimator of IV is RRGm,BC = 1 n (s pi,,m 2 ω N ) 2 λ 2,m where λ r,m = E i=1 Var[ω 2 ] can be consistently estimated with [ ( ) r ] max W t W s. t:η t m =ω,s:η s m = ω m m ω 2 N = RV N 2N, N = nm is the total number of log-returns. Rossi Realized Measures University of Pavia / 64