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Empirical bias in intraday volatility measures
Finance Research Letters 9 (2012) 231–237
Contents lists available at SciVerse ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
Empirical bias in intraday volatility measures q Yan Fang a, Florian Ielpo b,⇑, Benoît Sévi c a
MUREX, 8 rue Bellini, 75782 Paris cedex 16, France Banque Cantonale Vaudois, Place Saint François 14, P.O. Box 300, 1001 Lausanne, Switzerland c Aix-Marseille University (Aix-Marseille School of Economics), CNRS & EHESS, Château La Farge, Route des Milles, 13290 Les Milles, Aix-en-Provence, France b
a r t i c l e
i n f o
Article history: Received 5 May 2012 Accepted 21 August 2012 Available online 2 September 2012 JEL classification: C32 G1
a b s t r a c t Intraday volatility measures have recently become the norm in risk measurement and forecasting. This article empirically investigates the unbiasedness of three of these measures over four different datasets. We find that the three measures are significantly biased and that the bias can have either sign. Ó 2012 Elsevier Inc. All rights reserved.
Keywords: Volatility models Jumps Realized volatility Bipower variation
1. Introduction Over the past 10 years, the development of intraday datasets has greatly influenced the way both academics and practitioners estimate volatility. GARCH-related models (e.g., Bollerslev and Engle, 1986) are increasingly being replaced with models based on intraday realized estimators, following the seminal paper by Barndorff-Nielsen and Shephard (2002). One of the key questions regarding these intraday volatility measures is their unbiasedness, which has been challenged both theoretically and empirically. Various sources of bias have been identified in the literature, including microstructure noise, as discussed in Hansen and Lunde (2006). The presence of jumps, i.e. both a continuous and a discontinuous component in the financial returns, introduces an additional source of bias into
q Disclaimer: The views and opinions expressed in this article are those of the authors and do not necessarily reflect the views and opinions of BCV-AM. ⇑ Corresponding author. Address: The Centre d’Économie de la Sorbonne, 106 Bd de l’Hôpital, 75013 Paris, France. E-mail addresses: [email protected] (Y. Fang), fl[email protected] (F. Ielpo), [email protected] (B. Sévi).
1544-6123/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.frl.2012.08.001
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Y. Fang et al. / Finance Research Letters 9 (2012) 231–237
such volatility measures. The development of alternative sampling methodologies together with jump robust volatility measures has helped to mitigate the impact of these issues. As a result, these nonparametric measures based on intraday data have become a highly competitive solution to an important problem in financial markets: the fact that the volatility is not directly observable. However, outstanding issue regarding these intraday measures remains. When volatilities are constructed from an intraday dataset, using various cleaning schemes to improve the information content of the dataset, they still aim to measure the volatility of daily returns. Given the modifications applied to the intraday dataset, such as the trimming of jumps or the elimination of microstructure noise, combining the intraday and daily returns datasets is a non-trivial task. In this brief article, we propose a simple method to empirically test for bias in three intraday volatility measures using four different datasets. We find that the measures are significantly biased and that the sign of the bias depends on the measure and the series under investigation. Jin and Maheu (2010) proposed a similar investigation of the bias of the realized covariance. 2. Volatility and jump estimation using intraday data We consider three different volatility measures that are widely used in empirical applications: the realized volatility (RV) introduced in Barndorff-Nielsen and Shephard (2002), the bipower variation (BPV) presented in Barndorff-Nielsen and Shephard (2004) and the median realized volatility (MedRV) measure introduced in Andersen et al. (2012). Let r d;ti be the intraday return over the period ti for day d. The RV for day d is given by
RV d;M ¼
M X
r2d;ti ;
ð1Þ
i¼1
where M is the number of time intervals in one day.1 A potential problem with this estimate is that jumps in returns are treated as variance components, thereby overestimating the actual volatility of the returns. The following two measures were designed to address this problem. The BPV is defined as:
BPV d;M ¼ n1
M 1 X
jr d;tiþ1 jjrd;ti j;
ð2Þ
i¼1
where np 2p=2 C
1=2ðpþ1Þ Cð1=2Þ 2
¼ EðjZjp Þ denotes the mean absolute value of the standard normally distrib-
uted random variable, Z. The BPV is a consistent estimator of the integrated volatility and enables a decomposition of the realized volatility into its diffusive and non-diffusive parts. As the sampling frequency increases, the presence of jumps should have no impact because the return representing the jump is multiplied by a non-jump return that tends to zero asymptotically. This mechanism works in the case of rare jumps or at least when the probability of two consecutive jumps is negligible. In our empirical work, we use a staggered version of the BPV estimator:
BPV d;M;stag ¼ n1
M2 X
jr d;tiþ2 jjrd;ti j:
ð3Þ
i¼1
The staggered version is more robust to microstructure noise, as the serial correlation owing to the noise should be alleviated when considering products of returns that are not consecutive. Nevertheless, the BPV may be upward or downward biased in empirical applications as the sampling frequency is not high enough to eliminate the influence of jumps, and the sample may include a number of zero returns. This phenomenon has motivated the development of alternative estimators 1 A common extension of the RV measure, designed to reduce the impact of microstructure noise, is developed in Zhang et al. (2005). We also experimented with this measure and obtained results very similar to those obtained with the RV. For brevity, we do not report these results here but they are available from the authors upon request. 2 This notation is used throughout the paper.
Y. Fang et al. / Finance Research Letters 9 (2012) 231–237
233
that do not suffer from these weaknesses. Andersen et al. (2012) recently proposed the median realized volatility (MedRV) estimator:
MedRV d;M ¼
M1 X p M pffiffiffi medðjr d;ti1 j; jDr d;ti j; jDr d;tiþ1 jÞ2 : 6 4 3 þ p M 2 i¼2
ð4Þ
The MedRV estimator has two main advantages: first, the impact of jumps completely vanishes except in the case of two consecutive jumps (which is rare at the sampling frequencies used in our empirical applications); and, second, the estimator is more robust to the occurrence of zero returns. We extract jumps from the intraday returns series using the methodology developed in Andersen et al. (2010). The authors demonstrate how the repeated application of a jump detection test, e.g., one of those examined in Huang and Tauchen (2005), enables the extraction of the intraday returns that constitute statistically significant jumps. The idea is first to run the jump detection test on all of the intraday returns from the day of interest, as usual. If the null hypothesis of no jump is rejected at a standard significance level, then the econometrician may consider the highest (absolute) intraday return as a jump, remove it from the original set of intraday returns and run the jump detection test again until it accepts the null hypothesis of no jump. In this way, it is possible to extract the number of jumps, Nd, and their sizes, xi,d, for each day. We use this approach with the ratio test of Huang and Tauchen (2005), which has been shown to offer promising results using extensive Monte Carlo tests (see Theodosiou and Z˘ikes˘, 2009; Dumitru and Urga, 2012). Using the BPV as an estimator of the integrated variance, the ratio test statistic is as follows:
ZJðM; dÞ ¼
pffiffiffiffiffi ðRV d;M BPV d;M;stag ÞRV 1 d;M M 1=2 : 4 2 n1 þ 2n1 5 maxf1; TQ d;M;stag BPV 2 d;M;stag g
ð5Þ
Here TQd,M,stag is the realized tripower quarticity which is computed in staggered form as for the BPV.3 This procedure may be generalized to the MedRV case by simply replacing the BPV with the MedRV in the previous equation and rescaling the test statistic. The rescaling factor is deduced from the limit theory of the MedRV estimator (see Andersen et al. (2012) and Theodosiou and Z˘ikes˘ (2009) for applications). The ratio-statistic in Eq. (5) has reasonable power against several empirically realistic calibrated stochastic volatility jump diffusion models (Andersen et al., 2007).4 3. Empirical methodology This section describes how we test for bias in each of the volatility measures. We use a discrete time framework, as our empirical investigation will make use of daily data once the intraday dataset has been converted into a daily dataset of estimated volatilities and returns. The following formula should provide a sufficiently flexible data generating process:
rd ¼ l þ rd;V d þ
Nd X xd;i ;
ð6Þ
j¼1
where rd is the return on day d, l is a drift, rd,V is the volatility of the return on day d as estimated by P the volatility measure V. d is a Gaussian perturbation and Nj¼1 xd;i is the jump component. Nd is the number of jumps on day d and xd,i is the size of the ith jump on date d. This process may be viewed as a discrete time version of the widely used continuous-time procedure in which a diffusive component is mixed with a jump component, allowing the volatility to vary with time.5 To test for bias in the various volatility measures, we suggest the following testable form of the above equation: P 1 3 The realized tripower quarticity is computed as: TQ d;M;stag ¼ M n4=3 M4 i¼1 jr d;tiþ4 jjr d;t iþ2 jjr d;ti j. See also Andersen et al. (2007) for a formal definition of ‘‘significant’’ jumps. 5 We focus on the return equation, which is central to our present analysis. More complex procedures considering stochastic volatility of a diffusive or jump form can be found in Chernov et al. (2003). However, these procedures are of little interest for our analysis. 3 4
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Y. Fang et al. / Finance Research Letters 9 (2012) 231–237
Table 1 Estimation results for the four datasets and the three volatility measures. The null hypothesis of the t-stats reported here is b = 1. BPV
MedRV
RV
b
St. Dev.
t-Stat
b
St. Dev.
t-Stat
b
St. Dev.
S&P500 Jumps observed All data Subdata1 Subdata2
1.18 1.18 1.176
0.015 0.021 0.021
12.05 8.533 8.38
1.199 1.197 1.197
0.015 0.021 0.021
13.12 9.225 9.211
1.144 1.15 1.134
0.014 0.021 0.02
Jumps unobserved All data Subdata1 Subdata2
1.201 1.197 1.202
0.0025 0.006 0.003
81.28 30.68 66.22
1.223 1.201 1.225
0.003 0.033 0.004
71.4 6.042 63.18
1.159 1.162 1.152
0.002 0.008 0.002
79.98 19.86 62.51
WTI Jumps observed All data Subdata1 Subdata2
1.281 1.206 1.351
0.019 0.025 0.028
14.76 8.13 12.38
1.305 1.224 1.382
0.019 0.026 0.029
15.75 8.715 13.16
1.233 1.165 1.297
0.018 0.024 0.027
12.71 6.738 10.91
Jumps unobserved All data Subdata1 Subdata2
1.21 1.11 1.322
0.036 0.057 0.059
5.832 1.928 5.457
1.224 1.08 1.359
0.042 0.062 0.053
5.285 1.295 6.774
1.192 1.135 1.278
0.033 0.049 0.056
5.832 2.725 4.958
MCD Jumps observed All data Subdata1 Subdata2
1.157 1.118 1.196
0.024 0.033 0.036
6.428 3.52 5.468
1.191 1.155 1.226
0.025 0.035 0.037
7.578 4.482 6.15
1.067 1.141 1.187
0.023 0.034 0.036
2.956 4.133 5.266
Jumps unobserved All data Subdata1 Subdata2
1.099 1.074 1.123
0.006 0.006 0.004
1.129 1.109 1.149
0.003 0.004 0.003
42.48 25.18 48.38
1.025 1.016 1.04
0.027 0.003 0.1
0.921 5.188 0.402
USDJPY Jumps observed All data Subdata1 Subdata2
0.955 0.954 0.956
0.013 0.018 0.018
3.431 2.527 2.406
0.894 0.899 0.887
0.002 0.003 0.003
51.74 35.43 39.65
0.881 0.878 0.882
0.012 0.017 0.017
9.944 7.226 6.931
Jumps unobserved All data Subdata1 Subdata2
0.979 0.981 0.976
0.013 0.019 0.019
1.58 1.014 1.304
0.913 0.921 0.903
0.002 0.003 0.003
44.93 28.18 36.28
0.836 0.838 0.832
0.002 0.003 0.003
r d ¼ l þ brd;V d þ
Nd X xd;i ;
17.93 12.48 33.82
t-Stat
9.944 7.32 6.607
87.27 56.02 66.9
ð7Þ
j¼1
where b is expected to be equal to one in the case of no bias and to be greater (less) than one when the volatility measures incorporate a downward (upward) bias.6 Our empirical strategy closely follows Jin and Maheu (2010), who investigate the same issue for realized covariances. We make two different assumptions regarding the jump component. First, we assume that jumps are unobserved and must therefore adopt assumptions regarding the underlying process determining Nd and xi,d. The number of jumps for a given day d is assumed to 6 Before turning to the estimation and tests presented in the next section, we began by running Monte Carlo tests assessing the potential bias in the estimation of b when the true data generation process is that of Eq. (4). The results demonstrate that we can consistently estimate the value of b. We do not report these results here, but they are available upon request.
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Y. Fang et al. / Finance Research Letters 9 (2012) 231–237
Table 2 Robustness check: estimation results when no assumption is made for the distribution of jumps. The null hypothesis of the t-stats reported here is b = 1. BPV
MedRV
RV
b
St. Dev.
t-Stat
b
St. Dev.
t-Stat
b
St. Dev.
t-Stat
S&P500 All data Subdata1 Subdata2
1.196 1.198 1.195
0.015 0.021 0.021
12.996 9.240 9.122
1.216 1.216 1.217
0.015 0.022 0.022
14.086 9.941 9.969
1.157 1.166 1.148
0.015 0.021 0.021
10.758 7.960 7.226
WTI All data Subdata1 Subdata2
1.286 1.165 1.397
0.020 0.026 0.031
14.355 6.453 12.973
1.321 1.194 1.437
0.020 0.026 0.031
15.660 7.394 13.873
1.200 1.077 1.312
0.019 0.024 0.029
10.758 3.251 10.858
MCD All data Subdata1 Subdata2
1.159 1.118 1.199
0.025 0.033 0.036
6.484 3.524 5.543
1.192 1.155 1.228
0.025 0.035 0.037
7.624 4.486 6.216
1.068 1.045 1.091
0.023 0.031 0.033
3.025 1.451 2.786
USDJPY All data Subdata1 Subdata2
0.957 0.956 0.956
0.013 0.018 0.018
3.330 2.388 2.370
0.980 0.983 0.976
0.013 0.019 0.019
1.504 0.893 1.283
0.882 0.880 0.883
0.012 0.017 0.017
9.835 7.068 6.899
follow a Poisson distribution with intensity parameter k. The sizes of the jumps xi,d are assumed to follow a Gaussian distribution with expectation l and variance r. Within this simple setting, we are able to generate highly non Gaussian distributions for returns (see e.g. Merton, 1974; Bates, 1996) with both large kurtosis and non-zero skewness. The parsimony of the model, with the jump component containing only three parameters, should increase the estimation efficiency, which is a crucial feature for our approach. The parameters will be estimated using maximum likelihood. For the derivation of the likelihood function, see Bates and Craine (1999) and Jorion (1988). In the second case, we extract jumps using the sequential detection methodology described in Andersen et al. (2010) (see previous section). Jumps can then be considered as observed. The rest of the parameters will be estimated using maximum likelihood.7
4. Empirical results 4.1. Main results In our empirical analysis, we use tick-by-tick data from four different assets: S&P500 stock index futures, WTI futures, the MacDonald’s individual stock and the US Dollar vs. Japanese Yen exchange rate. All of these assets are very liquid and are therefore suitable for using realized estimators; days on which the trading activity was not sufficient to compute these estimators have been removed. The data were filtered with respect to three parameters: the length of the trading period in the day, the number of zero returns and the number of transactions. The S&P500 futures series begins on January 1st, 1996, and ends on July 31, 2008 (3,192 trading days, originally). The West Texas Intermediate (WTI) light sweet crude oil futures series starts on October 8, 2001, and ends on January 15, 2010. The USD/JPY exchange rate covers the period from 7 When both volatility and jumps are nonparametrically estimated using intraday volatility measures and the method in Andersen et al. (2010), the likelihood function can actually be split into two different parts: the pure diffusive part and the jump part that is assumed, as in the unobserved case, to be normally distributed with a Poisson arrival rate. These two parts can be optimized separately, as the parameters driving the jump part (resp. the diffusive part) do not enter into the expression of the loglikelihood of the diffusive part (resp. the jump part). As the derivation of these expressions is straightforward, we do not report them in this article.
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Y. Fang et al. / Finance Research Letters 9 (2012) 231–237
December 30, 1996, to June 1, 2007 and the MacDonald’s (MCD) stock price series goes from July 1, 2001, to December 30, 2005. We do not present here the computation of the volatility measures and of the jumps in detail; these are available in a companion article (Ielpo and Sévi, 2011).8 All volatility measures were computed using 5-min trading intervals, as this sampling frequency provides a good balance between noise and the number of observations used. Our estimation results are reported in Table 1. The table gives the estimated b, its standard deviation and the associated t-statistic. The theory behind these model-free measures of volatility predicts that b in Eq. (7) should be equal to 1. Our results demonstrate that there is an empirical bias in all of these measures for each of the datasets investigated. For example, in the S&P500 case with the MedRV measure and unobserved jumps, the volatility is underestimated by 22.3% on average. The results are fairly comparable regardless of or not the jumps are observed, lending some additional robustness to the reported results. In the S&P500, MCD and WTI cases, all volatility measures underestimate the actual data volatility, as b is estimated to be higher than 1. In the Japanese Yen case, the measured volatility overestimates the actual volatility of the daily returns. We also split each sample into two subperiods, in order to check the stability of our results across samples; the results are generally consistent for the different subsamples. Interestingly, we find that the Realized Volatility displays the lowest bias of all of the measures. In addition, in the MCD case when jumps are unobserved, we accept the null hypothesis that b = 1 in the RV case. 4.2. Alternative method The previous estimates are obtained using assumptions regarding the nature of the jumps.9 The jumps that we obtained using the nonparametric methodology in Andersen et al. (2010) can also be used to run a robustness check on our previous results given the potential implications of these hypotheses. In this section, we relax10 the assumption of Poisson distributed jumps and estimate the model parameters given in Eq. (8). Given that in the ‘‘observed jump’’ case we use model-free estimates for the jumps on each day, we simply subtract the daily jump part of the returns from the returns themselves before re-running the estimation of b.11 Let ~r d the jump-adjusted returns for day d:
~r d ¼ rd
b Nd X ^xd;i ¼ l þ brd;V d :
ð8Þ
j¼1
b d and ^ Using the notation defined previously. The values of N xd;i are estimated using the methodology presented in Section 2. The conditional distribution of the ~rd is Gaussian, making the estimation of b even easier than in previous cases. The resulting estimates are reported in Table 2, along with the t-stats. The results are qualitatively similar to the previous ones. The results are qualitatively similar, especially for the case of observed jumps; for most volatility measures, we find that the risk is underestimated. Only in the currency case is the risk overestimated, as before. Interestingly, we find that the risk is correctly estimated in the first subsample in the MacDonald’s shares case. More importantly, this robustness check shows that the previous results have a limited dependency on the specification of the jump distribution. We therefore obtain the same conclusions either using information regarding jumps from the intraday datasets (as in the present case or when estimating the parameters of the jump part from intraday data) or not (as in the estimation of Eq. (7) when 8 In particular, descriptive statistics about jumps provide evidence of the usefulness of sequential jump detection as a number of days with more than one jump are identified for each asset series. 9 We assumed that the number of jumps in a day was Poisson distributed and that the size of the jumps was normally distributed. 10 We thank the referee for suggesting this approach, which makes it possible to estimate the parameters of a simple diffusion model as in Andersen et al. (2003). 11 Andersen et al. (2010) call these returns ‘‘jump-adjusted returns’’. We compute them using Eq. (24) in their paper, which allows for several jumps in the same day. We do not differentiate between days with a single jump and days with more than one jump as the method in Eq. (24) is also relevant in the first case. We checked this assumption empirically.
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jumps are unobserved): the intraday volatility measures considered in this paper are biased estimates of the volatility of the daily returns for the considered samples. 5. Conclusion In this brief article, we test for bias in intraday volatility measures applied to four different datasets. Using the test proposed in this paper, the three measures considered are empirically found to be biased, despite their underlying theoretical properties. The magnitude and sign of the bias can vary, but for three of the series investigated in this paper, we find that the realized volatility, bipower variation and median realized volatility all underestimate the actual volatility of the returns. However, the underestimation does not affect the ability of these measures to provide an accurate description of the volatility dynamics, as described, e.g., in Andersen et al. (2007) and Maheu and McCurdy (2011). References Andersen, T.G., Bollerslev, T., Diebold, F.X., Labys, P., 2003. Modeling and forecasting realized volatility. Econometrica 71, 579– 625. Andersen, T.G., Bollerslev, T., Diebold, F.X., 2007. Roughing it up: including jump components in the measurement, modeling and forecasting of return volatility. Review of Economics and Statistics 89, 701–720. Andersen, T.G., Bollerslev, T., Frederiksen, P., Nielsen, M., 2010. Continuous-time models, realized volatilities, and testable distributional implications for daily stock returns. Journal of Applied Econometrics 25, 233–261. Andersen, T.G., Dobrev, D., Schaumburg, E., 2012. Jump-robust volatility estimation using nearest neighbor truncation. Journal of Econometrics 169, 75–93. Barndorff-Nielsen, O., Shephard, N., 2002. Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society, Series B 64, 253–280. Barndorff-Nielsen, O., Shephard, N., 2004. Power and bipower variation with stochastic volatility and jumps. Journal of Financial Econometrics 2, 1–37. Bates, D., 1996. Jumps and stochastic volatility: exchange rate processes implicit in Deutsche Mark options. Review of Financial Studies 9, 69–107. Bates, D., Craine, R., 1999. Valuing the futures market clearinghouse’s default exposure during the 1987 crash. Journal of Money, Credit, and Banking 31, 248–272. Bollerslev, T., Engle, R., 1986. Modelling the persistence of conditional variances. Econometric Reviews 5, 1–50. Chernov, M., Gallant, A.R., Ghysels, E., Tauchen, G., 2003. Alternative models for stock price dynamics. Journal of Econometrics 116, 225–257. Dumitru, A.-M., Urga, G., 2012. Identifying jumps in financial assets: a comparison between nonparametric jump tests. Journal of Business and Economic Statistics 30, 242–255. Hansen, P.R., Lunde, A., 2006. Realized variance and market microstructure noise. Journal of Business and Economic Statistics 24, 127–218. Huang, X., Tauchen, G., 2005. The relative price contribution of jumps to total price variance. Journal of Financial Econometrics 3, 456–499. Ielpo, F., Sévi, B., 2011. Do Jumps Help in Forecasting the Density of Returns? SSRN Working Papers. Jin, X., Maheu, J., 2010. Modelling Realized Covariances and Returns. Working Papers Tecipa-408. University of Toronto, Department of Economics. Jorion, P., 1988. On jump processes in the foreign exchange and stock markets. Review of Financial Studies 1, 427–445. Maheu, J.M., McCurdy, T.H., 2011. Do high-frequency measures of volatility improve forecasts of return distributions? Journal of Econometrics 160, 69–76. Merton, R.C., 1974. On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance 29, 449–470. Theodosiou, M., Z˘ikes˘, P., 2009. A Comprehensive Comparison of Alternative Tests for Jumps in Asset Prices. Unpublished manuscript. Graduate School of Business, Imperial College London. Zhang, L., Mykland, P.A., Aït-Sahalia, Y., 2005. A tale of two time scales: determining integrated volatility with noisy high frequency data. Journal of the American Statistical Association 100, 1394–1411.
Contents lists available at SciVerse ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
Empirical bias in intraday volatility measures q Yan Fang a, Florian Ielpo b,⇑, Benoît Sévi c a
MUREX, 8 rue Bellini, 75782 Paris cedex 16, France Banque Cantonale Vaudois, Place Saint François 14, P.O. Box 300, 1001 Lausanne, Switzerland c Aix-Marseille University (Aix-Marseille School of Economics), CNRS & EHESS, Château La Farge, Route des Milles, 13290 Les Milles, Aix-en-Provence, France b
a r t i c l e
i n f o
Article history: Received 5 May 2012 Accepted 21 August 2012 Available online 2 September 2012 JEL classification: C32 G1
a b s t r a c t Intraday volatility measures have recently become the norm in risk measurement and forecasting. This article empirically investigates the unbiasedness of three of these measures over four different datasets. We find that the three measures are significantly biased and that the bias can have either sign. Ó 2012 Elsevier Inc. All rights reserved.
Keywords: Volatility models Jumps Realized volatility Bipower variation
1. Introduction Over the past 10 years, the development of intraday datasets has greatly influenced the way both academics and practitioners estimate volatility. GARCH-related models (e.g., Bollerslev and Engle, 1986) are increasingly being replaced with models based on intraday realized estimators, following the seminal paper by Barndorff-Nielsen and Shephard (2002). One of the key questions regarding these intraday volatility measures is their unbiasedness, which has been challenged both theoretically and empirically. Various sources of bias have been identified in the literature, including microstructure noise, as discussed in Hansen and Lunde (2006). The presence of jumps, i.e. both a continuous and a discontinuous component in the financial returns, introduces an additional source of bias into
q Disclaimer: The views and opinions expressed in this article are those of the authors and do not necessarily reflect the views and opinions of BCV-AM. ⇑ Corresponding author. Address: The Centre d’Économie de la Sorbonne, 106 Bd de l’Hôpital, 75013 Paris, France. E-mail addresses: [email protected] (Y. Fang), fl[email protected] (F. Ielpo), [email protected] (B. Sévi).
1544-6123/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.frl.2012.08.001
232
Y. Fang et al. / Finance Research Letters 9 (2012) 231–237
such volatility measures. The development of alternative sampling methodologies together with jump robust volatility measures has helped to mitigate the impact of these issues. As a result, these nonparametric measures based on intraday data have become a highly competitive solution to an important problem in financial markets: the fact that the volatility is not directly observable. However, outstanding issue regarding these intraday measures remains. When volatilities are constructed from an intraday dataset, using various cleaning schemes to improve the information content of the dataset, they still aim to measure the volatility of daily returns. Given the modifications applied to the intraday dataset, such as the trimming of jumps or the elimination of microstructure noise, combining the intraday and daily returns datasets is a non-trivial task. In this brief article, we propose a simple method to empirically test for bias in three intraday volatility measures using four different datasets. We find that the measures are significantly biased and that the sign of the bias depends on the measure and the series under investigation. Jin and Maheu (2010) proposed a similar investigation of the bias of the realized covariance. 2. Volatility and jump estimation using intraday data We consider three different volatility measures that are widely used in empirical applications: the realized volatility (RV) introduced in Barndorff-Nielsen and Shephard (2002), the bipower variation (BPV) presented in Barndorff-Nielsen and Shephard (2004) and the median realized volatility (MedRV) measure introduced in Andersen et al. (2012). Let r d;ti be the intraday return over the period ti for day d. The RV for day d is given by
RV d;M ¼
M X
r2d;ti ;
ð1Þ
i¼1
where M is the number of time intervals in one day.1 A potential problem with this estimate is that jumps in returns are treated as variance components, thereby overestimating the actual volatility of the returns. The following two measures were designed to address this problem. The BPV is defined as:
BPV d;M ¼ n1
M 1 X
jr d;tiþ1 jjrd;ti j;
ð2Þ
i¼1
where np 2p=2 C
1=2ðpþ1Þ Cð1=2Þ 2
¼ EðjZjp Þ denotes the mean absolute value of the standard normally distrib-
uted random variable, Z. The BPV is a consistent estimator of the integrated volatility and enables a decomposition of the realized volatility into its diffusive and non-diffusive parts. As the sampling frequency increases, the presence of jumps should have no impact because the return representing the jump is multiplied by a non-jump return that tends to zero asymptotically. This mechanism works in the case of rare jumps or at least when the probability of two consecutive jumps is negligible. In our empirical work, we use a staggered version of the BPV estimator:
BPV d;M;stag ¼ n1
M2 X
jr d;tiþ2 jjrd;ti j:
ð3Þ
i¼1
The staggered version is more robust to microstructure noise, as the serial correlation owing to the noise should be alleviated when considering products of returns that are not consecutive. Nevertheless, the BPV may be upward or downward biased in empirical applications as the sampling frequency is not high enough to eliminate the influence of jumps, and the sample may include a number of zero returns. This phenomenon has motivated the development of alternative estimators 1 A common extension of the RV measure, designed to reduce the impact of microstructure noise, is developed in Zhang et al. (2005). We also experimented with this measure and obtained results very similar to those obtained with the RV. For brevity, we do not report these results here but they are available from the authors upon request. 2 This notation is used throughout the paper.
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that do not suffer from these weaknesses. Andersen et al. (2012) recently proposed the median realized volatility (MedRV) estimator:
MedRV d;M ¼
M1 X p M pffiffiffi medðjr d;ti1 j; jDr d;ti j; jDr d;tiþ1 jÞ2 : 6 4 3 þ p M 2 i¼2
ð4Þ
The MedRV estimator has two main advantages: first, the impact of jumps completely vanishes except in the case of two consecutive jumps (which is rare at the sampling frequencies used in our empirical applications); and, second, the estimator is more robust to the occurrence of zero returns. We extract jumps from the intraday returns series using the methodology developed in Andersen et al. (2010). The authors demonstrate how the repeated application of a jump detection test, e.g., one of those examined in Huang and Tauchen (2005), enables the extraction of the intraday returns that constitute statistically significant jumps. The idea is first to run the jump detection test on all of the intraday returns from the day of interest, as usual. If the null hypothesis of no jump is rejected at a standard significance level, then the econometrician may consider the highest (absolute) intraday return as a jump, remove it from the original set of intraday returns and run the jump detection test again until it accepts the null hypothesis of no jump. In this way, it is possible to extract the number of jumps, Nd, and their sizes, xi,d, for each day. We use this approach with the ratio test of Huang and Tauchen (2005), which has been shown to offer promising results using extensive Monte Carlo tests (see Theodosiou and Z˘ikes˘, 2009; Dumitru and Urga, 2012). Using the BPV as an estimator of the integrated variance, the ratio test statistic is as follows:
ZJðM; dÞ ¼
pffiffiffiffiffi ðRV d;M BPV d;M;stag ÞRV 1 d;M M 1=2 : 4 2 n1 þ 2n1 5 maxf1; TQ d;M;stag BPV 2 d;M;stag g
ð5Þ
Here TQd,M,stag is the realized tripower quarticity which is computed in staggered form as for the BPV.3 This procedure may be generalized to the MedRV case by simply replacing the BPV with the MedRV in the previous equation and rescaling the test statistic. The rescaling factor is deduced from the limit theory of the MedRV estimator (see Andersen et al. (2012) and Theodosiou and Z˘ikes˘ (2009) for applications). The ratio-statistic in Eq. (5) has reasonable power against several empirically realistic calibrated stochastic volatility jump diffusion models (Andersen et al., 2007).4 3. Empirical methodology This section describes how we test for bias in each of the volatility measures. We use a discrete time framework, as our empirical investigation will make use of daily data once the intraday dataset has been converted into a daily dataset of estimated volatilities and returns. The following formula should provide a sufficiently flexible data generating process:
rd ¼ l þ rd;V d þ
Nd X xd;i ;
ð6Þ
j¼1
where rd is the return on day d, l is a drift, rd,V is the volatility of the return on day d as estimated by P the volatility measure V. d is a Gaussian perturbation and Nj¼1 xd;i is the jump component. Nd is the number of jumps on day d and xd,i is the size of the ith jump on date d. This process may be viewed as a discrete time version of the widely used continuous-time procedure in which a diffusive component is mixed with a jump component, allowing the volatility to vary with time.5 To test for bias in the various volatility measures, we suggest the following testable form of the above equation: P 1 3 The realized tripower quarticity is computed as: TQ d;M;stag ¼ M n4=3 M4 i¼1 jr d;tiþ4 jjr d;t iþ2 jjr d;ti j. See also Andersen et al. (2007) for a formal definition of ‘‘significant’’ jumps. 5 We focus on the return equation, which is central to our present analysis. More complex procedures considering stochastic volatility of a diffusive or jump form can be found in Chernov et al. (2003). However, these procedures are of little interest for our analysis. 3 4
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Table 1 Estimation results for the four datasets and the three volatility measures. The null hypothesis of the t-stats reported here is b = 1. BPV
MedRV
RV
b
St. Dev.
t-Stat
b
St. Dev.
t-Stat
b
St. Dev.
S&P500 Jumps observed All data Subdata1 Subdata2
1.18 1.18 1.176
0.015 0.021 0.021
12.05 8.533 8.38
1.199 1.197 1.197
0.015 0.021 0.021
13.12 9.225 9.211
1.144 1.15 1.134
0.014 0.021 0.02
Jumps unobserved All data Subdata1 Subdata2
1.201 1.197 1.202
0.0025 0.006 0.003
81.28 30.68 66.22
1.223 1.201 1.225
0.003 0.033 0.004
71.4 6.042 63.18
1.159 1.162 1.152
0.002 0.008 0.002
79.98 19.86 62.51
WTI Jumps observed All data Subdata1 Subdata2
1.281 1.206 1.351
0.019 0.025 0.028
14.76 8.13 12.38
1.305 1.224 1.382
0.019 0.026 0.029
15.75 8.715 13.16
1.233 1.165 1.297
0.018 0.024 0.027
12.71 6.738 10.91
Jumps unobserved All data Subdata1 Subdata2
1.21 1.11 1.322
0.036 0.057 0.059
5.832 1.928 5.457
1.224 1.08 1.359
0.042 0.062 0.053
5.285 1.295 6.774
1.192 1.135 1.278
0.033 0.049 0.056
5.832 2.725 4.958
MCD Jumps observed All data Subdata1 Subdata2
1.157 1.118 1.196
0.024 0.033 0.036
6.428 3.52 5.468
1.191 1.155 1.226
0.025 0.035 0.037
7.578 4.482 6.15
1.067 1.141 1.187
0.023 0.034 0.036
2.956 4.133 5.266
Jumps unobserved All data Subdata1 Subdata2
1.099 1.074 1.123
0.006 0.006 0.004
1.129 1.109 1.149
0.003 0.004 0.003
42.48 25.18 48.38
1.025 1.016 1.04
0.027 0.003 0.1
0.921 5.188 0.402
USDJPY Jumps observed All data Subdata1 Subdata2
0.955 0.954 0.956
0.013 0.018 0.018
3.431 2.527 2.406
0.894 0.899 0.887
0.002 0.003 0.003
51.74 35.43 39.65
0.881 0.878 0.882
0.012 0.017 0.017
9.944 7.226 6.931
Jumps unobserved All data Subdata1 Subdata2
0.979 0.981 0.976
0.013 0.019 0.019
1.58 1.014 1.304
0.913 0.921 0.903
0.002 0.003 0.003
44.93 28.18 36.28
0.836 0.838 0.832
0.002 0.003 0.003
r d ¼ l þ brd;V d þ
Nd X xd;i ;
17.93 12.48 33.82
t-Stat
9.944 7.32 6.607
87.27 56.02 66.9
ð7Þ
j¼1
where b is expected to be equal to one in the case of no bias and to be greater (less) than one when the volatility measures incorporate a downward (upward) bias.6 Our empirical strategy closely follows Jin and Maheu (2010), who investigate the same issue for realized covariances. We make two different assumptions regarding the jump component. First, we assume that jumps are unobserved and must therefore adopt assumptions regarding the underlying process determining Nd and xi,d. The number of jumps for a given day d is assumed to 6 Before turning to the estimation and tests presented in the next section, we began by running Monte Carlo tests assessing the potential bias in the estimation of b when the true data generation process is that of Eq. (4). The results demonstrate that we can consistently estimate the value of b. We do not report these results here, but they are available upon request.
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Table 2 Robustness check: estimation results when no assumption is made for the distribution of jumps. The null hypothesis of the t-stats reported here is b = 1. BPV
MedRV
RV
b
St. Dev.
t-Stat
b
St. Dev.
t-Stat
b
St. Dev.
t-Stat
S&P500 All data Subdata1 Subdata2
1.196 1.198 1.195
0.015 0.021 0.021
12.996 9.240 9.122
1.216 1.216 1.217
0.015 0.022 0.022
14.086 9.941 9.969
1.157 1.166 1.148
0.015 0.021 0.021
10.758 7.960 7.226
WTI All data Subdata1 Subdata2
1.286 1.165 1.397
0.020 0.026 0.031
14.355 6.453 12.973
1.321 1.194 1.437
0.020 0.026 0.031
15.660 7.394 13.873
1.200 1.077 1.312
0.019 0.024 0.029
10.758 3.251 10.858
MCD All data Subdata1 Subdata2
1.159 1.118 1.199
0.025 0.033 0.036
6.484 3.524 5.543
1.192 1.155 1.228
0.025 0.035 0.037
7.624 4.486 6.216
1.068 1.045 1.091
0.023 0.031 0.033
3.025 1.451 2.786
USDJPY All data Subdata1 Subdata2
0.957 0.956 0.956
0.013 0.018 0.018
3.330 2.388 2.370
0.980 0.983 0.976
0.013 0.019 0.019
1.504 0.893 1.283
0.882 0.880 0.883
0.012 0.017 0.017
9.835 7.068 6.899
follow a Poisson distribution with intensity parameter k. The sizes of the jumps xi,d are assumed to follow a Gaussian distribution with expectation l and variance r. Within this simple setting, we are able to generate highly non Gaussian distributions for returns (see e.g. Merton, 1974; Bates, 1996) with both large kurtosis and non-zero skewness. The parsimony of the model, with the jump component containing only three parameters, should increase the estimation efficiency, which is a crucial feature for our approach. The parameters will be estimated using maximum likelihood. For the derivation of the likelihood function, see Bates and Craine (1999) and Jorion (1988). In the second case, we extract jumps using the sequential detection methodology described in Andersen et al. (2010) (see previous section). Jumps can then be considered as observed. The rest of the parameters will be estimated using maximum likelihood.7
4. Empirical results 4.1. Main results In our empirical analysis, we use tick-by-tick data from four different assets: S&P500 stock index futures, WTI futures, the MacDonald’s individual stock and the US Dollar vs. Japanese Yen exchange rate. All of these assets are very liquid and are therefore suitable for using realized estimators; days on which the trading activity was not sufficient to compute these estimators have been removed. The data were filtered with respect to three parameters: the length of the trading period in the day, the number of zero returns and the number of transactions. The S&P500 futures series begins on January 1st, 1996, and ends on July 31, 2008 (3,192 trading days, originally). The West Texas Intermediate (WTI) light sweet crude oil futures series starts on October 8, 2001, and ends on January 15, 2010. The USD/JPY exchange rate covers the period from 7 When both volatility and jumps are nonparametrically estimated using intraday volatility measures and the method in Andersen et al. (2010), the likelihood function can actually be split into two different parts: the pure diffusive part and the jump part that is assumed, as in the unobserved case, to be normally distributed with a Poisson arrival rate. These two parts can be optimized separately, as the parameters driving the jump part (resp. the diffusive part) do not enter into the expression of the loglikelihood of the diffusive part (resp. the jump part). As the derivation of these expressions is straightforward, we do not report them in this article.
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December 30, 1996, to June 1, 2007 and the MacDonald’s (MCD) stock price series goes from July 1, 2001, to December 30, 2005. We do not present here the computation of the volatility measures and of the jumps in detail; these are available in a companion article (Ielpo and Sévi, 2011).8 All volatility measures were computed using 5-min trading intervals, as this sampling frequency provides a good balance between noise and the number of observations used. Our estimation results are reported in Table 1. The table gives the estimated b, its standard deviation and the associated t-statistic. The theory behind these model-free measures of volatility predicts that b in Eq. (7) should be equal to 1. Our results demonstrate that there is an empirical bias in all of these measures for each of the datasets investigated. For example, in the S&P500 case with the MedRV measure and unobserved jumps, the volatility is underestimated by 22.3% on average. The results are fairly comparable regardless of or not the jumps are observed, lending some additional robustness to the reported results. In the S&P500, MCD and WTI cases, all volatility measures underestimate the actual data volatility, as b is estimated to be higher than 1. In the Japanese Yen case, the measured volatility overestimates the actual volatility of the daily returns. We also split each sample into two subperiods, in order to check the stability of our results across samples; the results are generally consistent for the different subsamples. Interestingly, we find that the Realized Volatility displays the lowest bias of all of the measures. In addition, in the MCD case when jumps are unobserved, we accept the null hypothesis that b = 1 in the RV case. 4.2. Alternative method The previous estimates are obtained using assumptions regarding the nature of the jumps.9 The jumps that we obtained using the nonparametric methodology in Andersen et al. (2010) can also be used to run a robustness check on our previous results given the potential implications of these hypotheses. In this section, we relax10 the assumption of Poisson distributed jumps and estimate the model parameters given in Eq. (8). Given that in the ‘‘observed jump’’ case we use model-free estimates for the jumps on each day, we simply subtract the daily jump part of the returns from the returns themselves before re-running the estimation of b.11 Let ~r d the jump-adjusted returns for day d:
~r d ¼ rd
b Nd X ^xd;i ¼ l þ brd;V d :
ð8Þ
j¼1
b d and ^ Using the notation defined previously. The values of N xd;i are estimated using the methodology presented in Section 2. The conditional distribution of the ~rd is Gaussian, making the estimation of b even easier than in previous cases. The resulting estimates are reported in Table 2, along with the t-stats. The results are qualitatively similar to the previous ones. The results are qualitatively similar, especially for the case of observed jumps; for most volatility measures, we find that the risk is underestimated. Only in the currency case is the risk overestimated, as before. Interestingly, we find that the risk is correctly estimated in the first subsample in the MacDonald’s shares case. More importantly, this robustness check shows that the previous results have a limited dependency on the specification of the jump distribution. We therefore obtain the same conclusions either using information regarding jumps from the intraday datasets (as in the present case or when estimating the parameters of the jump part from intraday data) or not (as in the estimation of Eq. (7) when 8 In particular, descriptive statistics about jumps provide evidence of the usefulness of sequential jump detection as a number of days with more than one jump are identified for each asset series. 9 We assumed that the number of jumps in a day was Poisson distributed and that the size of the jumps was normally distributed. 10 We thank the referee for suggesting this approach, which makes it possible to estimate the parameters of a simple diffusion model as in Andersen et al. (2003). 11 Andersen et al. (2010) call these returns ‘‘jump-adjusted returns’’. We compute them using Eq. (24) in their paper, which allows for several jumps in the same day. We do not differentiate between days with a single jump and days with more than one jump as the method in Eq. (24) is also relevant in the first case. We checked this assumption empirically.
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jumps are unobserved): the intraday volatility measures considered in this paper are biased estimates of the volatility of the daily returns for the considered samples. 5. Conclusion In this brief article, we test for bias in intraday volatility measures applied to four different datasets. Using the test proposed in this paper, the three measures considered are empirically found to be biased, despite their underlying theoretical properties. The magnitude and sign of the bias can vary, but for three of the series investigated in this paper, we find that the realized volatility, bipower variation and median realized volatility all underestimate the actual volatility of the returns. However, the underestimation does not affect the ability of these measures to provide an accurate description of the volatility dynamics, as described, e.g., in Andersen et al. (2007) and Maheu and McCurdy (2011). References Andersen, T.G., Bollerslev, T., Diebold, F.X., Labys, P., 2003. Modeling and forecasting realized volatility. Econometrica 71, 579– 625. Andersen, T.G., Bollerslev, T., Diebold, F.X., 2007. Roughing it up: including jump components in the measurement, modeling and forecasting of return volatility. Review of Economics and Statistics 89, 701–720. Andersen, T.G., Bollerslev, T., Frederiksen, P., Nielsen, M., 2010. Continuous-time models, realized volatilities, and testable distributional implications for daily stock returns. Journal of Applied Econometrics 25, 233–261. Andersen, T.G., Dobrev, D., Schaumburg, E., 2012. Jump-robust volatility estimation using nearest neighbor truncation. Journal of Econometrics 169, 75–93. Barndorff-Nielsen, O., Shephard, N., 2002. Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society, Series B 64, 253–280. Barndorff-Nielsen, O., Shephard, N., 2004. Power and bipower variation with stochastic volatility and jumps. Journal of Financial Econometrics 2, 1–37. Bates, D., 1996. Jumps and stochastic volatility: exchange rate processes implicit in Deutsche Mark options. Review of Financial Studies 9, 69–107. Bates, D., Craine, R., 1999. Valuing the futures market clearinghouse’s default exposure during the 1987 crash. Journal of Money, Credit, and Banking 31, 248–272. Bollerslev, T., Engle, R., 1986. Modelling the persistence of conditional variances. Econometric Reviews 5, 1–50. Chernov, M., Gallant, A.R., Ghysels, E., Tauchen, G., 2003. Alternative models for stock price dynamics. Journal of Econometrics 116, 225–257. Dumitru, A.-M., Urga, G., 2012. Identifying jumps in financial assets: a comparison between nonparametric jump tests. Journal of Business and Economic Statistics 30, 242–255. Hansen, P.R., Lunde, A., 2006. Realized variance and market microstructure noise. Journal of Business and Economic Statistics 24, 127–218. Huang, X., Tauchen, G., 2005. The relative price contribution of jumps to total price variance. Journal of Financial Econometrics 3, 456–499. Ielpo, F., Sévi, B., 2011. Do Jumps Help in Forecasting the Density of Returns? SSRN Working Papers. Jin, X., Maheu, J., 2010. Modelling Realized Covariances and Returns. Working Papers Tecipa-408. University of Toronto, Department of Economics. Jorion, P., 1988. On jump processes in the foreign exchange and stock markets. Review of Financial Studies 1, 427–445. Maheu, J.M., McCurdy, T.H., 2011. Do high-frequency measures of volatility improve forecasts of return distributions? Journal of Econometrics 160, 69–76. Merton, R.C., 1974. On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance 29, 449–470. Theodosiou, M., Z˘ikes˘, P., 2009. A Comprehensive Comparison of Alternative Tests for Jumps in Asset Prices. Unpublished manuscript. Graduate School of Business, Imperial College London. Zhang, L., Mykland, P.A., Aït-Sahalia, Y., 2005. A tale of two time scales: determining integrated volatility with noisy high frequency data. Journal of the American Statistical Association 100, 1394–1411.