STOCKS IN THE SHORT RUN

Transcription

1 STOCKS IN THE SHORT RUN Bryan Ellickson, Benjamin Hood, Tin Shing Liu, Duke Whang and Peilan Zhou Department of Economics, UCLA November 11, 2011 Abstract This paper examines stock-price volatility in the very short run. Using transactions data from 2001 through 2009, we use a Mykland-Zhang (2009) block estimator to estimate the path of integrated volatility in 5-minute increments day by day for the 30 stocks that comprise the DJIA and for an ETF (the SPDR) that tracks the S&P 500. Using a Heston (1993) model, we estimate that 80% of the gap between the level of the volatility process and its asymptotic mean is eliminated within 5-minutes. Roughly two-thirds of daily realized volatility can be explained by a deterministic version of the Heston model that begins the trading day far above its equilibrium and converges to a constant. The remaining third reflects stochastic shocks to volatility arriving after trade begins. The asymptotic mean of the SPDR behaves much like the closing value of the VIX, a volatility index based on the S&P 500 stock index. Keywords: intraday volatility, Heston, quadratic variation, realized volatility, TAQ data, DJIA, SPDR, VIX. JEL Classification Numbers: G12 (Financial economics Asset pricing), C58 (Mathematical and quantitative methods Financial econometrics) Corresponding author: Bryan Ellickson. Department of Economics, UCLA, 405 Hilgard, Los Angeles, CA , ellickson@econ.ucla.edu. We thank Michael Brennan, Paul Ellickson, Jinyong Hahn and Francis Longstaff for useful comments. Support from the UCLA Academic Senate is gratefully acknowledged.

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3 1 Introduction This paper investigates stock market volatility. What distinguishes our approach from the literature is our focus on the path behavior of volatility within the course of a trading day. Stock prices can be very volatile. This suggests to many observers that stock exchanges perform badly, falling victim to investor irrationality or manipulation by high-frequency traders. Although our results will not put such concerns entirely to rest, we show that volatility processes within the trading day behave much more sensibly than one might expect. We find that volatility processes, although often out of equilibrium, can be approximated very well by a stationary Markov process that reverts with astonishing speed to its asymptotic mean. The volatility process is often out of equilibrium because it starts the trading day that way and because, during the day, information arrives that pushes volatility far away from the mean. However, we estimate that 80% of the gap between volatility and its mean is eliminated within five minutes. Higher volatility at the market open makes good economic sense. When trade begins, the exchange has been closed for many hours, perhaps for an entire weekend. If exchanges promote the revelation of private information, it stands to reason that it matters when they close. If so, once the markets open, we should expect to see a rapid decline in volatility until the cost of acquiring more information exceeds its benefit. The same should happen during the market day if news of substantial import crosses traders screens: volatility should surge and then, as the information is assimilated, revert toward its natural level. Our findings are consistent with this interpretation. Our study uses transactions data for the 30 individual stocks included in the Dow Jones Industrial Average (DJIA) and for an exchange-traded fund (ETF) that tracks the S&P 500 stock index. Beginning when the market opens and ending at the market close, we sample stock prices as often as we can, once every second. The data span nearly nine years from the start of decimal pricing early in 2001 through the end of We assume that the asset-price process X is an Itô process, a continuous semimartingale. 2 Because X is a semimartingale, its quadratic variation process ŒX; X 1 The trades part of the NYSE TAQ (Trades and Quotes) dataset includes transactions on all U.S. exchanges (not just the trading floor of the NYSE) as well as trades off the exchanges such as those in dark pools. Most trade on these exchanges occurs between the open and close of trade on the NYSE. 2 Protter (2004) provides a general treatment of the theory of semimartingales. Mykland and Zhang (2010) is a very accessible introduction to semimartingale methodology within the context 1

4 is well defined. The increment ŒX; X t ŒX:X s to quadratic variation over any time interval Œs; t provides a natural measure of the volatility of the asset-price process X over that interval, and there is a consistent estimator of this increment, realized volatility. If X t D log S t, the logarithm of the stock price S t at time t, and prices are sampled once a second, then the realized volatility over the interval Œs; t is simply the sum of squared one-second log returns over the interval. 3 The existing literature on volatility estimation using high-frequency data focuses on the increment ŒX; X 1 ŒX; X 0 to quadratic variation over the entire trading day, which we represent by the interval Œ0; 1. We estimate increments to quadratic variation over much shorter intervals. In Section 2 we partition the trading day into 78 five-minute intervals and use the increment to realized volatility to estimate the increment ŒX; X i ŒX; X i 1 to quadratic variation over each interval Œ i 1 ; i, a procedure applied separately to each of the 69,220 trading days in our data set. When plotted with 5-minute volatility estimates on the vertical axis and time on the horizontal axis, a pattern emerges that is quite uniform across stocks. Almost every day, volatility starts high and then descends, eventually settling down to a level that is sustained (with some interruptions) for the rest of the day. To convert this qualitative characterization into something more quantitative, we define a settled region that extends from the beginning of the 15th block (10:40 AM, 70 minutes into the trading day) to the end of the 69th block (3:15 PM, 45 minutes before the market close). For each trading day, we compute the median of the 5-minute volatility estimates for the 55 blocks in the settled region, using the median rather than the mean to diminish the influence of outliers (large positive shocks to volatility, which are quite common). Three features emerge that hold across stocks and time: Volatility in the first 5-minute block is much higher than the settled value: four times higher for the median day in 2001, rising to a peak 13.6 times higher by 2007, and diminishing somewhat thereafter. Once the settled region is reached, volatility remains fairly close to the settled value within that region. Within the settled region, the fluctuation from block to block is too large to be consistent with geometric Brownian motion volatility is stochastic. of financial markets. 3 Anderson and Benzoni (2009) review the literature on realized volatility. 2

5 The pattern in our graphs resembles to some extent a u-shape that has been identified in the literature, a pattern sufficiently well established for Taylor (2005) to characterize as stylized fact. 4 In this literature, volatility is estimated for each 5- minute interval using the sample variance of 5-minute log returns over a period of several months. The variation in volatility over the course of the day is treated as a diurnal effect, analogous to a seasonal effect when dealing with daily or monthly data. In contrast, we compute our estimates day by day, using the 300 one-second log returns within a 5-minute interval to estimate the volatility in that interval on that particular day. We attribute higher volatility at the beginning of the day to disequilibrium in the volatility process and low volatility in the middle to a process near or at equilibrium. There is little or no evidence for a rise in volatility at the end of the day. However, the non-parametric approach of Section 2 cannot distinguish disequilibrium from a deterministic diurnal effect. That requires a structural model. Section 3 estimates a Heston (1993) model of the volatility process that implies the change in volatility from one five-minute block to the next is proportional to the gap between volatility and its asymptotic mean. The diurnal effect can be nested within an extended version of the Heston model. The Heston model has three parameters: the asymptotic mean N of the volatility process, the speed of mean reversion and the volatility of the volatility. If we turn off mean reversion ( D 0) and stochastic volatility ( D 0) and replace the asymptotic mean by a deterministic time varying process. N t / t2œ0;1 with a u-shape, then we obtain the alternative model. However, when we estimate the Heston model, with N assumed fixed, we find strong evidence across stocks and across time of very rapid mean reversion ( > 0) and substantial stochastic volatility ( > 0), in sharp contrast to the alternative model. The speed of mean reversion caught us by surprise. Our daily graphs of 5- minute realized volatility suggest a slower convergence to equilibrium at the beginning of each day, on the order of a hour or so. We attribute the difference to jumps in volatility arriving after the market opens, excluded in our specification of the Heston model. We identify blocks that appear to contain volatility jumps, upward movements in volatility that seem too large to be consistent with the Brownian shocks of the Heston model. On a typical day 9 or 10 blocks contain a volatility jump, a number quite uniform across stocks and time. Volatility jumps occur at much higher rates in the region of price discovery (blocks 1 14) 4 See Wood, McInish and Ord (1985), Harris (1986), Kawaller, Koch and Koch (1987), Andersen and Bollerslev (1997), and Areal and Taylor (2002). 3

6 and the end of day (blocks 70 78) than in the settled region (blocks 15 69). When the observations associated with those jumps are excluded, the regression used to estimate the Heston model behaves much more uniformly across stocks and time, consistent with our belief that failing to allow for these shocks results in specification error. The settled value N of volatility is roughly 60% of daily realized volatility in our sample. If we interpret the value of N on a particular trading day as a measure of equilibrium volatility, it is natural to ask whether its level is at all reasonable. In Section 4 we compare our estimate of the settled value of volatility for SPDR (an exchange-traded fund that tracks the S&P 500) with the daily closing value of the VIX (the CBOE s volatility index, based on the prices of option contracts on the S&P 500) over the period The two measures are closely related, providing further evidence for the validity of our analysis. Section 5 discusses some sampling issues related to our study. Section 6 offers some conclusions. 2 The quadratic variation process We begin by introducing a few concepts needed for our analysis, which applies the block estimator of Mykland and Zhang (2009) over a trading day. 5 Let X D.X t / t2œ0;1 denote a log stock price process over a single trading day, where X t D log S t and S t is the stock price at time t. Time t D 0 corresponds to 9:30 AM and t D 1 to 4:00 PM Eastern time, the opening and close of trading on the NYSE. We assume that the log price process X is an Itô process with time set Œ0; 1 adapted to the stochastic basis. ; F ; F; P/. 6 5 More precisely, we sample prices once a second and use 5-minute blocks, without taking out the mean. Mykland and Zhang also consider another estimator that modifies realized volatility by taking out the mean, which in our context amounts to computing the variance of 1-second log returns for each block and multiplying this variance by the length i of the 5-minute block to obtain an estimate of the increment to quadratic variation (scaled to the trading day). As they observe, it is easy to see that the sum of 300 one-second log returns over a block is the same as the 5-minute return over the entire block. If the 5-minute return over a block is very far from zero, taking out the mean might have a large effect. Because volatility is very high at the beginning of the trading day, 5-minute returns are often far from zero. For that reason, we initially used this alternative estimator. We subsequently re-estimated all of our results using realized volatility, expecting to find much higher estimates of the increment to realized volatility, especially early in the trading day. Instead we found the estimators yield virtually the same results. For a detailed comparison of the two estimators, see Hood (2011). 6 The triple. ; F ; P/ is a probability space. ; F / with probability measure P. The filtration F D.F t / t2œ0;1 is a collection of sub-sigma-algebras F t F satisfying the property F s F t if 4

7 The log price process X is a semimartingale, which allows its quadratic variation to be defined. The quadratic variation of a semimartingale X is itself a stochastic process, denoted ŒX; X D.ŒX; X t / t2œ0;1 (1) It is adapted to the same stochastic basis as the semimartingale X. Realized volatility converges uniformly in probability to quadratic variation on compact intervals. Let G N D ft 0 ; t 1 ; : : : ; t N g denote a grid that partitions the trading day Œ0; 1 with 0 D t 0 < t 1 < t 2 < t N D 1. Define the mesh mesh.g N / D supft j W j D 1; : : : ; Ng j where t j D t j t j 1 is the gap between adjacent points t j 1 and t j of the grid G N. Throughout this paper we set N D 23400, the number of one-second intervals in the trading day, and t j D 1=23400 for all j. Because our focus is on estimating volatility within the trading day, we define realized volatility for arbitrary closed intervals Œs; t Œ0; 1, including the entire trading day Œ0; 1 as a special case. Consider an interval Œs; t Œ0; 1, s < t, with s; t 2 G N. The realized volatility (RV) over the interval Œs; t with grid G N is defined by ŒX; X GN Œs;t WD X.X tj / 2 D X Œlog.S tj =S tj 1 / 2 (2) t j 2.s;t t j 2.s;t where X tj D X tj X tj 1 is the increment to the log price and log.s tj =S tj 1 / is the log return over the interval Œt j 1 ; t j. A standard result of semimartingale theory establishes that realized volatility is a consistent estimator of the increment to quadratic variation over Œs; t : i.e., if N! 1 and mesh.g N /! 0, then in probability. 7 In particular, ŒX; X GN Œs;t! ŒX; X t ŒX; X s ŒX; X GN Œ0;t D X t j t.x tj / 2 (3) s < t. 7 See, for example, Theorem 2.22 in Protter (2004). 5

8 gives the realized volatility over the interval Œ0; t. Table 1 lists the ticker symbols for each of the stocks in our sample, the components of the DJIA as of September SPDR, the exchange-traded fund that tracks the S&P 500 stock index, has ticker symbol SPY. In our tables we refer to the Dow Jones stocks and the ETF by their ticker symbols. AIG, Citibank and General Motors, casualties of the financial crisis and Great Recession of , are no longer included in the DJIA. Pfizer and Verizon were added to the DJIA in April 2004; Bank of America and Chevron joined in February Figure 1 plots the realized volatility process ŒX; X GN Œ0;t / t2œ0;1 for Alcoa for each of the 5 trading days in the last full week of trading in The pattern we observe for these 5 trading days is typical: most days the slope of the realized volatility process starts high but tapers off quickly, eventually approaching a slope that remains nearly constant for the rest of the trading day. This suggests an equilibrating process where volatility (the slope of the quadratic variation process) starts high but converges to a constant. Figure 1 suggests that volatility varies in a systematic way over the course of the trading day. To examine this structure more precisely we estimate the increments to quadratic variation over intervals Œs; t shorter than the entire trading day. We divide the trading day into 5-minute intervals (blocks), using the sub-grid H D f 0 ; 1 ; : : : ; 78 g G N to establish the boundaries of the blocks, where 0 D 0, 78 D 1 and i i 1 D 1=78 is the length of a block. We estimate the increment to quadratic variation over each 5-minute block. For block Œ i 1 ; i Œ0; 1, the increment to quadratic variation is ŒX; X Œi 1 ; i WD ŒX; X i ŒX; X i 1 If we divide this increment to quadratic variation by the length i WD i i 1 of the 5-minute interval, it becomes a rate of increase of quadratic variation per unit time (where a unit of time is a 6.5 hour market day): i 1 WD ŒX; X i ŒX; X i 1 i (4) 6

9 RV AA, Time Figure 1: The RV process ŒX; X GN (10 4 ), 12/14/ /18/2009 We also refer to i 1 as the slope of the quadratic variation process over the interval Œ i 1 ; i. Because i D 1=78, the slope (4) is a scalar multiple of the increment to quadratic variation, scaled to a rate per trading day. The realized volatility over the block Œ i 1 ; i with grid G N is ŒX; X GN Œ i 1 ; i D X Œlog.S tj =S tj 1 / 2 t j 2. i Dividing by the length of the block, we obtain an estimator of the slope of the volatility process, which we call the slope of the RV process over the interval Œ i 1 ; i : ŒX; X GN b Œ i 1 D i 1 ; i (5) i Because i D 1=78, this slope is a scalar multiple of realized volatility over the interval, scaled to a rate per trading day. 8 1 ; i 8 Equation (5) corresponds to equation (61) of Mykland and Zhang (2009) because the normalizing constant c 300;2 D 1. 7

10 ζ i 1 (x 10 4 ) AA, Time ζ 0 = 50.26, ζ = 12.69, GBM SD = 1.02 Figure 2: The slope process O (10 4 ) on 12/16/2009 Figure 2 plots the slope of realized volatility for Alcoa for each of the 78 fiveminute blocks on Wednesday 12/16/2009. The horizontal axis measures time and the vertical axis the estimates of the slopes (multiplied by 10 4 ), with each circle positioned at the midpoint of the corresponding interval. 9 To streamline notation, we use i 1 rather b i 1 to label the vertical axis. Figure 2 exhibits a pattern that turns out to be typical of our data: volatility starts high at the beginning of the day ( b 0 D 50: ), declines rapidly over the first hour or so of trade and then levels off. Apart from a few sharp upward movements, volatility fluctuates around this level for the rest of the trading day, with a tendency to move upward slightly toward the end. Our data yield 69,220 graphs like Figure 2, one graph for every trading day for each of the 30 Dow Jones stocks and for the ETF. The most striking features are (1) the high value of volatility at the beginning of the day, 9 For example, the horizontal coordinate for the first circle is located at t D 1=156, the midpoint of the interval Œ0; 1=78. 8

11 (2) the rapid decline over the first hour or so of trade, and (3) the tendency to settle down to some sort of equilibrium for most of the rest of the day. For the remainder of this section we focus on the first and third features, deferring investigation of the rapidity of decline to Section 3. Based on the examination of thousands of graphs like Figure 3, we define three regions of the trading day: A region of price discovery: Œ0; 14=78 Œ0; 0:179, the first 14 blocks (70 minutes) of the trading day. The settled region: Œ14=78; 69=78 Œ0:179; 0:885, the 4-hour 35-minute interval spanned by blocks 15 through 69. The end of the day: Œ69=78; 1 Œ0:885; 1, the 45-minute interval spanned by blocks 70 through 78. Our hypothesis is that, once price discovery is complete, the slope process settles down to its equilibrium distribution. However, as illustrated in Figure 2, this equilibrium is sometimes perturbed by large positive shocks to volatility followed by what appears to be quick reversion to the settled value. To reduce the effect of these outliers, we estimate the central tendency N of the slope process in the settled region using the median of the slope estimates for the 55 blocks that constitute the settled region. Table 2 reports the median ratiob 0 =N and Table 3 the ratio of the interquartile range of these ratios to their median for each of our stocks for each year. The final row of each table gives the column medians. For example, in Table 2 the column median is 10.5 in 2009, corresponding to the median ratio reported for Walmart. This means that, for Walmart, the slope of the volatility process over the first 5-minutes was at least 10.5 times the settled value for half of the trading days in For Alcoa on 12/16/2009, the ratio b 0 N D 50:26 12:69 D 3:96 is somewhat below the median 5.1 for that year. The column medians in Table 2 rise to 7.7 by 2006, nearly double to 13.6 in 2007 and decline to 10.5 by Only five entries are below 2. A more detailed 9

12 examination of the data, presented in Hood (2011) and Liu (2011), reinforces the conclusion that for every stock for almost every trading day the volatility process begins the trading day far above its settled value. Because volatility is constant within each 5-minute block, asymptotic standard errors and confidence intervals of the slope estimates in Figure 2 are easily computed: 10 for block Œ i 1 ; i with slope estimate b i 1 and prices sampled once a second, the asymptotic 95% confidence intervals are given by r b 2 i 1 1: b i 1 b i 1 0:16 b i 1 or, equivalently, Œ0:84 b i 1 ; 1:16 b i 1. These confidence bounds are quite tight. It is easy to see, for example, that the difference between the initial volatility increment b 0 and the settled value is highly significant. Figure 2 suggests that, by the time the slope process reaches the settled region, the process remains near the settled value. N It is reasonable to wonder whether, after the shock at the market open wears off, the volatility process settles down to a geometric Brownian motion (GBM), a process with constant volatility equal to the settled value N with the fluctuations around the settled value due to sampling error. Tables 4 and Table 5 address this question. Assuming that the true volatility t D N throughout the settled region, we compute the 95% confidence interval around N (using the median rather than the mean to reduce the effect of outliers): i.e., an interval of the form Œ0:84; N 1:16, N which yields Œ11:67; 13:71 for Alcoa on 12/16/2009. We also compute 8 confidence bounds around, N plotted as the horizontal dashed lines in Figure 2 along with the settled value. N Almost all the slopes in the settled region are inside the 8 bounds, but the 95% confidence bounds are much too narrow to contain 95% of the slope estimates. Table 4 reports for each year the median percentage of slopes in the settled region that are within 95% confidence bounds around, N and Table 5 does the same using 8 confidence bounds. The bottom row of each table reports the column medians. The column medians increase steadily from year to year in both tables, increasing from 15% to 35% in Table 4 and from 56% to 89% in Table 5 over the nine years covered by our sample. The results are surprisingly uniform across stocks within a given year. In Table 4 the percentage is far short of the 95% expected if the process had settled down to a geometric Brownian motion. However, as Figure 2 illustrates, the 8 confidence bounds are quite narrow but 10 See Mykland and Zhang (2009) and Mykland and Zhang (2010), pp

13 still manage to capture a large fraction of slopes within the settled region. This is true for all our stocks, especially from 2005 onward. We conclude that volatility is in fact stochastic, not just at the market open but throughout the trading day. 3 Mean-reverting volatility The volatility estimates in Section 2 are non-parametric, establishing several features of intraday volatility assuming only that the log price process and its volatility process are Itô processes. Volatility starts high at the beginning of the trading day and (after an hour or so) settles down to a much lower level. Although the settled process exhibits no trend away from the settled value N (except perhaps for a slight rise at the end of the day), the fluctuations around N are too large to be compatible with geometric Brownian motion. This strongly suggests that volatility is stochastic. We interpret high volatility at the beginning of the day as disequilibrium in the volatility process. Distinguishing disequilibrium from a diurnal effect requires a structural model. Our structural model also allows us to show that gaps between the level of the volatility process and its asymptotic mean tend to be eliminated quickly. In this section, we assume the volatility process follows a mean-reverting Heston (1993) model. Because we assume X is an Itô process, the slope of the quadratic variation process is well-defined infinitesimally as well as over finite intervals. Suppose X is an Itô process with stochastic differential dx t D t dt C p t dw t.t 2 Œ0; 1 / (6) where t is the drift and t the volatility at time t. 11 Then ŒX; X i ŒX; X i 1 D Z i i 1 t dt If volatility t were constant over the interval Œ i quadratic variation would be 1 ; i /, then the increment to Z i i 1 t dt D i 1 i 11 In the literature t WD p t is commonly referred to as the volatility of the Itô process. 11

14 and the slope of the quadratic variation process over the interval would be i 1. However, we assume that the volatility process D. t / t2œ0;1 is stochastic, an Itô process generated by the mean-reverting Heston (1993) model d t D. N t /dt C p t db t (7) where B is a Wiener process (possibly correlated with the Wiener process W that drives the price process) and the parameter, called the volatility of volatility, is strictly positive. We can make the correlation structure more explicit by adding a superscript to the process W appearing in the price equation, so that equation (6) becomes dx t D t dt C p t dw 1 t t 2 Œ0; 1 ; X 0 D x 0 (8) The Wiener process B is defined by B t D W 1 t C p 1 2 W 2 t t 2 Œ0; 1 (9) where W D.W 1 ; W 2 / is a 2-dimensional Wiener process with independent components and jj 1. We assume > 0, so will revert to N, its asymptotic mean. We also require 2 < 2 N, which guarantees that the process remains positive for all t. 12 Subject to these restriction, is a stationary ergodic Markov process. The first and second moments of t are easily computed. 13 The expected value of t is given by E t D N C e t. 0 N / (10) which implies that The variance of t is given by ( Var. t / D 2 0.e t which implies that lim E t D N t!1 e 2t / C N ) 2.1 2e t C e 2t / lim Var. t/ D N 2 t! The process characterized by equation (7) is also called a CIR process because of its use in the Cox, Ingersoll and Ross (1993) model of the short rate. The stochastic process was first introduced into the mathematics literature by Feller (1951). 13 See Shreve (2004), pp

15 Because the volatility process is not directly observable, these analytical results are not directly usable for empirical purposes: we are only able to estimate the slope of the quadratic variation process over intervals Œ i 1 ; i, not the infinitesimal slope. However, the continuous-time Heston model implies a linear relationship between the increment i D i i 1 (the change in volatility from one point of the grid H to the next) and the gap N i 1 (the distance between the asymptotic mean N and i 1 ). Consider a specific block Œ i 1 ; i. Equation (7) implies: 14 i D N C e h. i 1 N / C e h Z i i 1 e up u db u where h D 1=78 is the length of the 5-minute block Œ i 1 ; i. Subtracting i 1 from both sides and rearranging gives i i 1 D.1 e h /. N i 1 / C e h Z i i 1 e up u db u (11) Because the Wiener process B is a martingale, the expectation conditioned on F i 1 of the stochastic integral appearing on the right-hand side of this equation is zero, and so EŒ i i 1 j F i 1 D.1 e h /. N i 1 / (12) Thus, equation (11) defines a linear regression with slope and disturbance term i i 1 D ˇ. N i 1 / C " i 1 (13) ˇ D 1 e h 2.0; 1/ (14) " i 1 D e h Z i i 1 e up u db u Because is not observable, we cannot directly estimate the regression (13). Instead, in the spirit of Mykland-Zhang (2009), we assume the process can be approximated by a discrete-time process that is constant over the half-open 5- minute intervals Œ i 1 ; i /. Substituting b i 1 D ŒX; X GN Œ i 1 ; i and b i D i ŒX; X GN Œ i ; ic1 ic1 14 See Shreve (2004), p

16 for i 1 and i respectively in equation (13), we obtain the linear model b i b i 1 D ˇ. N b i 1 / C " i 1 (15) Although the volatility process for the Heston model is a stationary ergodic Markov process, we know that the volatility process starts almost every trading day far above its asymptotic mean. Because the trading day is relatively short, the effects of the initial condition will not have enough time to wear off before the market closes, so the sample mean of the 5-minute estimates will overstate the asymptotic mean. Furthermore, as Figure 2 suggests, the slope process for realized volatility often includes several outliers lying well above neighboring slope estimates. For all these reasons, we use as our estimate of the asymptotic mean N the median of the slope estimates for the 55 blocks that constitute the settled region from block 15 to 69, the same measure of central tendency used in Section 2. ζ i 1 (x 10 4 ) x x AA, x x x x x x x x x x x Time ζ 0 = 50.26, ζ = 12.69, Filter 1 = 22.69, # Vol.Jumps = 13 Figure 3: Volatility jumps for Alcoa on 12/16/2009 There is one more issue that must be addressed before we turn to the regression results: the possible presence of jumps in the volatility process. Figure 3 repeats 14

17 Figure 2, but with slope estimates marked with a cross inside the circle to indicate our candidates for blocks containing volatility jumps. 15 Thirteen volatility jumps are flagged, which is fairly typical. We identify volatility jumps in two ways. The first method (filter 1) sets a threshold level for volatility, equal to N plus 3 times the gap between the 75% percentile and the median of the empirical distribution of the 55 slope estimates in the settled region. 16 The lower dashed line in Figure 3 corresponds to N D 12:69, our estimate of the asymptotic mean for this trading day in The upper dashed line represents the threshold (22.69) for filter 1. Any slope estimate from block 15 onward is declared to contain a volatility jump if the estimate lies on or above the threshold. For i 2 f2; 3; : : : ; 14g an estimate b i 1 above the threshold is flagged if, in some block prior to block Πi 1 ; i, the slope process was below the threshold. The rationale for filter 1 is that, once the slope process reaches the vicinity of, N it tends to stay reasonably close to. N 17 However, on a typical day perhaps a dozen slopes will lie relatively far away from the rest, suggesting the arrival of a shock too great to be attributed to the Wiener process in equation (7). Ideally, of course, we would like to model these jumps explicitly, adding a jump component to the Heston model: d t D. N t /dt C p t db t C dj t where J is a pure jump process. This may well be worth doing, especially once time stamps are reported to the nearest millisecond or better and trading volumes increase to the point where slopes can be accurately estimated over intervals much shorter than 5 minutes. But for now, we settle for identifying candidates for jumps in this way. Filter 1 does not capture all of the volatility jumps before block 15 that we think it should. If the volatility process, typically beginning far above the threshold for filter 1, is still above the threshold and reverses its downward trend, then any upward movement from one block to the next also ought to qualify as a volatility jump. The rationale is that, because mean reversion is very strong, above the 15 In the online version of this paper, the volatility jumps flagged by filter 1 are colored purple and the jumps flagged by filter 2 are colored green. We have also plotted the expectation E t as a function of time as described by equation (10). It is the curved line that starts at 0 and converges to the asymptotic mean N. 16 We choose this gap rather than the interquartile range as a measure of the dispersion of the empirical distribution because the outliers are usually far above the median and rarely below. 17 Table 5 illustrates this for estimates in the settled region. Examining graphs like Figure 3 for many stocks on many different days provides convincing justification for this claim. The plots for all our stocks for every trading day are available on our website at 15

18 threshold the drift term of equation (7) dominates the diffusion term, so upward movements starting above the threshold must reflect a large increment to the diffusion term, too large to be generated by the Heston model without jumps. The first two volatility jumps in Figure 3 were identified using this second method, which we call filter 2. ζ i AA, x x x x x x x x x x x x ζ ζ i 1 β = 0.41, SE = 0.05, # Vol.Jumps = 13 Figure 4: Daily regression for Alcoa, 12/16/2009 We turn now to the regression results. Figure 4 shows the results of the regression for Alcoa on 12/16/2009. The change in slope b i D b i b i 1 is on the vertical axis and the gap N b i 1 is on the horizontal axis. These regressions can have as many as 77 observations, one for each pair of adjacent 5-minute blocks. For this trading day, only 64 observations are included in the regression. We exclude a pair of adjacent blocks from the regression if the second block in the pair contains a volatility jump (which the Heston model does not allow for). The excluded observations are marked with crosses in Figure 4. Because the regression implied by the Heston model has no constant term, the regression line passes through the point.0; 0/: if the gap N b i 1 D 0, then the expected increment b i is also 0. The slope of the regression, the coefficient of mean reversion, 16

19 is ˇ D 0:41, with a standard error of 0.05: i.e., 41% of the gap between N and b i 1 is eliminated within 5 minutes. These results are typical in the sense that the regression slopes are almost always above 0.40 and the standard errors for the coefficients are small relative to the estimates. However, the daily estimates of the mean-reversion coefficient vary a lot from day to day and from stock to stock. The variability in the slope estimates arises because these estimates are strongly influenced by a few highly leveraged values of the independent variable N b i 1, values that lie far from their sample mean. Most often the observation with the highest leverage comes from the interval Œ 0 ; 1, the first block, as in the case of Alcoa on 12/16/2009. From Figure 3 N b 0 D 12:69 50:26 38 and b 1 b 0 18 which corresponds to the pair. N b 0 ; b 1 b 0 / D. 38; 18/ that lies in the lower left-hand corner of Figure 4, slightly below the regression line. The slope of the regression line in Figure 4 is clearly influenced heavily by an outlier above the regression line with coordinates. N b i 1 ; b i b i 1 /. 27; 1/ Because N 13, this implies that b i 1 40, which corresponds to b 3 in Figure 3, a volatility jump. Because b 3 is a volatility jump, the pair. b 2 ; b 3 / is excluded from the regression. The pair. b 3 ; b 4 / is included in the regression, yielding the outlier. N b 3 ; b 4 b 3 /. 27; 1/ that lies above the regression line in Figure 4. Removing that outlier roughly doubles the slope of the regression. Fortunately, we can deal with this leverage problem by pooling our data. Table 6 reports the least-squares estimate of the mean-reversion parameter ˇ for each stock and the ETF, pooling all the observations for a year. Because there are approximately 250 trading days in a year, the number of observations in each of these regressions is around 16,000 rather than 64. Standard errors are tiny, so for appearance sake Table 7 presents t-statistics instead. The results are quite stunning. The column medians in Table 6 range from 0.77 to 0.82 over the years 17

20 The median estimates of ˇ for individual stocks are remarkably close to 0.8 in most cases, and the corresponding t-statistics are extremely high throughout Table 7, usually well above 200. The estimates of ˇ rise in the turbulent years 2008 and 2009, yielding a column median of 0.85 in 2008 and 0.88 in 2009, once again with extremely large t statistics. The main issue that undoubtedly troubles the reader is the exclusion of observations with volatility jumps in the second element of adjacent pairs of blocks. Table 8 gives the OLS estimates of the mean-reversion parameter ˇ for the pooled regression with volatility jumps included, and Table 9 reports the corresponding t-statistics. The estimates of ˇ are lower as are the t-statistics, but still quite high. Except for 2007, the column medians range from 0.62 to 0.75, which implies that around 2/3 of the gap between the slope and its asymptotic mean vanishes within 5 minutes. The corresponding column medians for the t-statistics range from 93.3 to The year 2007 is an exception, with a column median 0.45 for the ˇ estimate and 76.6 for the t-statistic. We much prefer to exclude the volatility jumps, a preference reinforced when we see what happens when we modify one of the procedures in Section 2. Recall that in Table 5 we computed the median percentage of slopes within the settled region that lie within 8 standard deviations of, N where the standard deviation is the one that would be obtained if sampling from a geometric Brownian motion with volatility. N In that computation, volatility jumps were not excluded. (The only time we exclude volatility jumps anywhere in this study is for the regressions we are now discussing.) The column medians of Table 5 show that over half of the slopes lie within these bounds in 2001, a percentage that steadily increases to 89% by If we repeat this exercise with volatility jumps excluded, the percentage included within the bounds increases substantially, reaching nearly 100% by Our plot of the slope process for Alcoa provides a nice illustration: eliminating volatility jumps excludes 5 of the volatility jumps outside the confidence bounds on 12/16/2009. This supports our view that the slopes labeled as volatility jumps are incompatible with the Heston model without jumps and that the slopes we call volatility jumps are in fact jumps. The structural model estimated in this section can be used to give a quantitative estimate of the relative importance of the high initial slope and other sources of stochastic volatility in accounting for daily realized volatility, the increment to quadratic variation that is the primary focus in the literature. Figure 5 illustrates the basic idea for Alcoa on 12/16/2009. The top curve is identical to one of the curves plotting the RV process ŒX; X GN in Figure 1. The straight line plots the 18

21 QV AA, Time Figure 5: Accounting for daily RV quadratic variation process ŒX; X t D N t.t 2 Œ0; 1 / that would obtain if the log stock-price process X were a geometric Brownian motion with constant volatility t D N. The curved line plots the function Z t 0 E s ds D N t C 1 1 e t 0 N.t 2 Œ0; 1 / where E s is the expectation at time s of the volatility process for the Heston model given by equation (10). Inverting equation (14), we can express the meanreversion parameter in terms of our preferred mean-reversion parameter ˇ: D 1 log.1 ˇ/ D 78 log.1 ˇ/ h The curve displayed in Figure 5 is plotted using the estimates for 0 and N for Alcoa on 12/16/2009 and the estimate of ˇ for the daily regression with volatility jumps excluded. 19

22 To construct Tables 10 and 11 we repeat these procedures for every trading day, but we use the pooled (annual) regression rather than the daily regression for the estimate of ˇ. Each trading day produces two ratios: R t N E 0 s ds ŒX; X GN 1 and ŒX; X GN 1 The first ratio represents the fraction of daily RV that could be explained by the settled process, if the settled process lasted the whole day and exhibited no stochastic volatility. The second ratio represents the fraction of daily RV that could be explained by the Heston model with no stochastic volatility (i.e., the volatility of volatility parameter D 0, which turns equation (7) into an ordinary rather than stochastic differential equation). The volatility process is deterministic (conditional on the initial condition 0, which we assume is measurable with respect to the -algebra F 0 ), declining toward N at an exponential rate from the initial value Tables 10 and 11 report the median value of these ratios for each stock for each year, multiplied by 100 to express the ratio as a percentage. The column medians in Table 10 rise from 51.9% to 66.7% from 2001 through 2006, decline to around 60% in 2007 and 2008 and then rise to 63.2% in Roughly speaking, for a typical stock around 60% of the daily RV could be accounted for by a geometric Brownian motion with volatility N. The column medians in Table 11 are higher than corresponding column medians in Table 10, rising from 53.5% in 2001 to a peak of 70.2% in 2006, then declining slowly to 67.3% by The modesty of the increase from Table 10 to Table 11 caught us by surprise. Using the column medians, in 2001 the nonstochastic Heston model explains only 53:5% 51:9% 100% 51:9% 3:3% of the gap between the daily volatility generated by a GBM with volatility N and daily realized volatility. The percent explained rises to 67:3% 63:2% 100% 63:2% 11:1% 18 If D 0, then t is given by equation (10) with E t replaced by t. Consequently, the deterministic process follows the curved line in Figure 3 that starts at 0 and approaches N asymptotically. 20

23 in 2009, still lower than we anticipated. Plots of the slope process like Figure 2 led us to expect that the effects of the initial high slope would be quite long lasting, taking an hour or more to settle down and contributing perhaps half of daily RV. However, as we have discovered in this section, mean reversion is very quick, eliminating around 80% of the shock within five minutes! As noted earlier, Figure 5 was constructed using the estimate of ˇ produced by the daily regression on 12/16/2009: ˇ D 0:41. The annual coefficient of mean reversion for Alcoa in 2009 was ˇ D 0:88 (see Table 6). When this value is used, the middle curve in Figure 5 is almost indistinguishable from the line below. We conclude that the primary explanation for the gap between N and daily realized volatility is stochastic volatility within the trading day rather than the shock that arrives at the market open. This stochastic volatility is a mixture of Brownian shocks (captured by the Heston model without jumps) and volatility jumps. We have no good way to assess the importance of volatility jumps relative to the diffusion component, but our sense is that volatility jumps are quite important. The median number of volatility jumps that arrive during a trading day was 10 in three of the years and 9 in the rest, and this rate is quite uniform across stocks. These jumps are concentrated in the price discovery region (blocks 1 14) and the end of day (blocks 71 79) rather than the settled region. Table 12 reports the column median of the fraction of volatility jumps in each of these regions for each year, where the column medians are constructed as in our other tables. Because there is so little variation across stocks, we have chosen to report only the column medians. 19 In the discovery region the median fraction is 4/14, or 28.6%, for every year in our sample except In the settled region the median fraction is much lower, 4/55 (7.3%) for six of the years and 3/55 (5.5%) in the other three years. The rate rises at the end of day, 1/9 (11.1%) for , 2/9 (22.2%) for and 3/9 (33.3%) in The SPDR and the VIX Up to this point we have paid little attention to the settled level of volatility N except for noting that it is considerably smaller than daily realized volatility, roughly 60% of its value. In this section we compare the settled value for SPDR with another measure of daily volatility of the S&P 500, the VIX. The VIX is the CBOE s index of market volatility, introduced by Whaley (1993). Originally based on 19 See Liu (2011) for the detailed tables reporting the fraction for each stock. 21

24 prices for S&P 100 index options, since 9/22/2003 the VIX has been calculated using index options for the S&P 500. (See Whaley (2009).) Figure 6 compares the daily closing value of the VIX (the top panel) with our estimate of the daily settled value for SPDR (the bottom panel). The hash marks on the horizontal axis mark the beginning of each calendar year (so that, for example, 01 is shorthand for 1/1/2001). SPDR VIX /17 7/24 10/ Figure 6: The settled value of the SPDR and the closing value of the VIX Because the VIX is an annualized standard deviation while N is a daily variance, we re-scale our estimates appropriately. Assuming 250 trading days in a year, the bottom panel plots the time series for q q N D N We have also added matching vertical lines to the top and bottom panel to mark days when the settled value for the SPDR reaches a local maximum. Since N is quite volatile, the choice of where to place these lines is somewhat subjective. Table 13 compares the daily time series for the VIX and N. The resemblance is quite striking. Over the period , the correlation coefficient is The 22

25 mean and medians are quite close. The SPDR N has a larger range, with both a lower minimum and a higher maximum, and its standard deviation is higher. The local peaks for the settled value of the SPDR, marked by the vertical lines, match reasonably closely local peaks for the VIX. In Figure 6 we have labeled some of the local peaks with their date: 9/17 (short for 9/17/2001) marks the day markets reopened after the 9/11 disaster, the peak on 7/24/2002 occurred three days after Worldcom filed for bankruptcy and on 10/10/2008 the trading range of the Dow exceeded 1000 for the first time in its history. The larger range and standard deviation of the SPDR N relative to the VIX is what we should expect. As Whaley (2009) notes: In attempting to understand VIX, it is important to emphasize that it is forward-looking, measuring volatility that the investors expect to see. It is not backward-looking, measuring volatility that has been recently realized, as some commentators sometimes suggest. [Whaley (2009), p. 98.] In contrast, N is backward looking, but not very far back: it measures the settled value of volatility for a trading day once the trading day has ended, which we associate with the asymptotic mean of the volatility process. It is a local measure of volatility, not an indicator of volatility anticipated sometime in the future. This comparison of our estimates with the VIX provides convincing evidence that estimating volatility using all available prices makes sense, contrary to claims in the literature on market microstructure noise. 20 If those claims invalidate our estimates of the settled value of SDPR, there must be something wrong with the VIX as well. It is also clear that our measure has something different to offer than the VIX, a local measure of volatility closely tied to a structural model of the volatility process. Like the VIX, the progress of realized volatility throughout the trading day can be calculated in real time, with RV plots like those in Figure 1 updated trade by trade. Plots of the slope process, like Figure 2, could also be updated at 5-minute intervals, supplemented by forecasts using the Heston model and identification of volatility jumps that might impede progress toward an equilibrium. All of this is possible not just for the SPDR but for any asset with a market liquid enough to justify the methods we have used: certainly for those in the DJIA and we suspect for many more. 20 See Aît-Sahalia and Mykland (2009) for a review of this literature. 23

26 5 Observing prices Our implementation of the Mykland-Zhang (1999) block estimator samples stock prices once a second. However, a price process is not a physical process (such as air temperature) that can be observed as often as we like. As Engle (2000) remarks, [M]uch of the movement to higher frequency econometrics was a consequence of the availability of higher frequency measurements of the economy. It is natural to suppose that this will continue and that we will have ever increasing frequencies of observations. However, a moment s reflection will reveal that this is not the case. The limit in nearly all cases is achieved when all transactions are recorded. These transactions may occur in the supermarket, on the internet or in financial markets. It is difficult to think of economic variables that really are measurable at arbitrarily high frequencies. [Engle (2000), pp. 1 2.] Engle argues for modeling a high-frequency process as a marked-point process (MPP), a process that describes (1) the arrival time T k (the point) and (2) the characteristic X Tk (the mark) of each trade. If we take the log stock price as the mark, then.x Tk / k0 is the sequence of log stock prices we observe. We can interpret this sequence as a sample from a càdlàg 21 process X on Œ0; 1 that is constant between arrivals. This process X is a semimartingale with quadratic variation ŒX; X t D X Tk / T k t.x 2 (16) at time t where X Tk D X Tk X Tk 1, and this is also the realized volatility at time t. Thus, the realization of ŒX; X t can be computed exactly because we see all the trades, which is exactly Engle s point. Unfortunately this is impossible in practice because the TAQ data does not report transaction times exactly, but rounded to the nearest second. In our analysis, we order transactions by time stamp. If more than one transaction shares the same time stamp t j, we arbitrarily select one of the transactions to represent the last transaction in the interval.t j 1 ; t j. 22 This is what we mean when we say that prices are sampled once a second. 21 A stochastic process with paths that are continuous from the right with limits from the left. 22 More precisely, we sort the transactions by time stamp and, for each set of transactions with the same time stamp, select the last transaction. 24