We construct our measure of daily realized volatility for the S&P 500 index using high- frequency futures data. S&P 500 index futures trade on the Chicago Mercantile Exchange (CME) on the trading floor from 8:30AM to 3:15PM (Eastern Standard Time minus 1 hour, EST-1). Since January 3, 1994, these contracts also trade overnight on GLOBEX, the electronic trading system of the CME, from 3:30PM to 8:00AM (8:15AM from February 26, 1996, onwards). As a result, S&P 500 futures trade almost round the clock, providing a similar opportunity to construct realized volatilities as for the 24-hour foreign exchange market. Martens (2002) tested various measures of S&P 500 realized volatility, finding that the sum of squared 30-minute intranight and 5-minute intraday returns is a more accurate measure of volatility than using only the intraday returns, or the sum of squared intraday returns and the squared close-to-open return, showing that it is useful to incorpo- rate overnight trading prices. Hence, we will use the following measure of daily “realized
volatility”, s2t = nN X j=1 rNt,j2+ nD X j=1 rDt,j2, (3.1) where rN
t,j is the intranight (30-minute) return on day t in intranight period j (j =
1, . . . , nN = 33), and rDt,j is the intraday (five-minute) return on day t for intraday pe-
riod j (j = 1, . . . , nD = 91). Both rNt,j and rt,jD are continuously compounded returns.
Figure 3.1 shows a time series plot for the daily S&P 500 log realized volatility for the sample period from January 3, 1994, until December 29, 2003 (2521 daily observations). Table 3.1 contains descriptive statistics of the log realized volatility measure, as well as for daily returns rt=Pnj=1N rt,jN +
PnD
j=1rt,jD, for squared daily returns, and for daily returns
standardized with the realized standard deviation, rt/st. A number of interesting features
emerge from this table, which closely correspond with the distributional characteristics for
Figure 3.1: Realized S&P 500 volatility
1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 -3 -2 -1 0 1 2 3 4
(a) Log realized variance
Notes: Daily log realized volatility for S&P 500 returns for the period from January 3, 1994, until
Table 3.1: Descriptive statistics for daily S&P 500 return and realized volatility
Mean Med Min Max Std.dev. Skew Kurt
Returns 0.022 0.058 −7.811 5.737 1.193 −0.164 6.435
Standardized returns 0.086 0.060 −2.725 3.105 0.999 0.073 2.730
Squared returns 1.423 0.400 0.000 61.005 3.314 7.788 100.921
Realized variance 1.421 0.927 0.066 33.220 1.907 6.006 62.907
Log realized variance −0.107 −0.076 −2.721 3.503 0.934 0.133 3.026
Notes: The table contains summary statistics for daily S&P 500 return and realized volatility measures. The sample period covers January 3, 1994 until December 31, 2003 (2521 observations). Standardized returns are obtained by dividing the raw returns by the realized standard deviation.
realized exchange rate volatility documented in Andersen, Bollerslev, Diebold, and Labys (2001b). First, comparing the daily squared returns with the realized variance shows that these have an all but identical mean (1.423% and 1.421%, respectively). We would expect this to be the case, as both are unbiased measures of the true volatility. However, the stan- dard deviation of the realized variance is at 1.907 much smaller than the standard deviation of the squared returns, which equals 3.314. It is precisely this characteristic that shows that realized variance is a less noisy estimate of true volatility than the daily squared return. Second, the realized variance is heavily skewed and exhibits excess kurtosis. By contrast, the logarithm of realized volatility, log(s2
t), is much more symmetrically distributed and
has much lower kurtosis. It is for this reason that we will consider time series models for the log realized volatility. Third, the daily S&P 500 returns are skewed and leptokurtic. Standardized returns rt/st, however, exhibit much less skewness and excess kurtosis and
are in fact very close to being normally distributed.
Fourth, as documented in other studies (and therefore not shown here explicitly), the sample autocorrelation functions of the realized volatility measures exhibit a slow hyper- bolic decay, indicative for the presence of long memory. A further point is that the persis- tence in the autocorrelation functions is much stronger for the realized volatility measures than for the daily squared returns.
Fifth, returning to Figure 3.1, realized volatility appears to be higher on average in the second half of the sample period than during the first few years. It is difficult to pin
down when exactly this level shift occurred, and it appears that it is most adequately characterized as a gradual increase of volatility during 1996-1997. After this transition period volatility seems to remain high until the last two years of the sample during which it declines sharply. An alternative possibility is that multiple structural breaks have occurred, as suggested by Andreou and Ghysels (2002a).
The scatter plot of log(s2
t) against rt−1 in Figure 3.2 reveals a rather pronounced re-
lationship between current volatility and lagged returns beyond that already captured in contemporaneous realized volatility. To examine the possible presence of a leverage effect, we estimate the “news impact curve” (Engle and Ng, 1993)
log(s2t) = β0+ β1|rt−1| + β2I[rt−1 < 0] + β3|rt−1|I[rt−1< 0], (3.2)
where I[A] is an indicator function for the event A, being equal to 1 if A occurs, and 0 otherwise. The fit from this regression is included in Figure 3.2 as well, along with the fit
Figure 3.2: Leverage effects in realized volatility
8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 -3 -2 -1 0 1 2 3 4 5 Lagged Return
Log Realized Variance
Notes: Scatter plot of daily log realized variance and lagged returns, based on observations for the period from January 3, 1994, until December 29, 2003 (2521 observations). The solid line is the fit of the news impact curve (3.2), where log realized volatility is regressed on a constant, the lagged absolute return, a dummy for negative returns and an interaction term of this dummy with the lagged absolute return. The
dashed line is the fit of a symmetric news impact curve, i.e. (3.2) with β2= β3= 0. The dot-dashed line
from a symmetric version of this news impact curve, obtained by setting β2 = β3 = 0 in
(3.2). It is clearly seen by the solid line in the graph that the impact of negative lagged returns is larger than the effect of positive returns of equal magnitude. Also, the parametric form in (3.2) appears to be quite reasonable, as can be seen by comparing the fit from this regression with a nonparametric regression of log realized volatility on the lagged return, also shown by the dash-dotted line in Figure 3.2.
Sixth, Table 3.2 shows the overall mean for all return and volatility measures on differ- ent types of days by distinguishing between regular days, holidays and days with macro- economic news announcements. It is clear that returns and volatility are both higher after holidays and that volatility during the Christmas period is only roughly half of its level during regular days. Volatility is in particular substantially higher on announcement days, especially on days when decisions regarding the federal funds rate are released when real- ized variance is on average 2.379%, substantially above its non-announcement day mean of 1.352%.
Finally, Table 3.3 shows the overall mean and the mean on different days-of-the-week. The top panel in the table confirms the common finding based on daily returns that Mon- days and Fridays exhibit higher volatility than other days by the S&P data. Interestingly,
Table 3.2: Descriptive statistics for daily S&P 500 return and realized volatility
ALL NONE HOL XMS ANN EMP PPI CPI FF
Returns 0.022 −0.005 0.051 0.054 0.130 0.204 0.129 0.036 0.265
Standardized returns 0.086 0.061 0.157 0.113 0.187 0.250 0.092 0.235 0.243
Squared returns 1.423 1.325 2.694 0.761 1.817 2.042 1.710 1.650 2.166
Realized variance 1.421 1.352 1.721 0.602 1.816 1.927 1.800 1.523 2.379
Log realized variance −0.107 −0.140 −0.058 −0.755 0.106 0.271 0.037 −0.031 0.156
Number of obs. 2521 1977 76 41 430 117 118 120 85
Notes: The table contains sample averages of daily S&P 500 returns and realized volatility measures. The sample period covers January 3, 1994 until December 31, 2003 (2521 observations). ALL indicates all days in the sample period; NONE indicates days without announcements, not following a holiday, and not in the Christmas period. HOL indicates days following a holiday. XMS denotes days during the Christmas period; ANN indicates all days with one or more macroeconomic announcements; EMP, PPI, CPI and FF indicate days with an announcement of employment, PPI, CPI, and the Federal Funds target rate, respectively. Standardized returns are obtained by dividing the raw returns by the realized standard deviation.
this pattern is quite different for the realized variance. Thursdays and Fridays exhibit the highest volatility but Mondays no longer have an above average volatility. In fact, for realized variance the mean is lowest on Mondays.
The observed patterns can to a large extent be explained by making the distinction between days with and without macro releases, similar as in Andersen and Bollerslev (1998b). The middle and bottom panels in Table 3.3 allow for a direct comparison. Squared returns and realized variance are both clearly higher on announcement days. From the
Table 3.3: Day-of-the-week effects in S&P 500 return and realized volatility
Overall MON TUE WED THU FRI
All days Returns 0.022 0.034 0.049 -0.008 0.000 0.038 Standardized returns 0.086 0.128 0.094 0.038 0.074 0.101 Squared returns 1.423 1.581 1.492 1.229 1.359 1.464 Realized variance 1.421 1.295 1.347 1.422 1.506 1.530 Non-announcement days Returns 0.000 0.052 0.001 -0.024 -0.022 -0.013 Standardized returns 0.065 0.137 0.039 0.031 0.060 0.049 Squared returns 1.342 1.514 1.597 1.183 1.281 1.038 Realized variance 1.339 1.247 1.361 1.354 1.436 1.292 Announcement days Returns 0.130 -2.967 0.261 0.093 0.172 0.113 Standardized returns 0.187 -1.323 0.332 0.087 0.182 0.176 Squared returns 1.817 12.276 1.035 1.534 1.946 2.089 Realized variance 1.816 8.975 1.284 1.867 2.036 1.880 Number of obs. 430 3 96 68 59 204
Notes: The table contains daily means for S&P 500 returns and realized vari-
ance. The sample period covers January 3, 1994 until December 31, 2003 (2521 observations). Standardized returns are obtained by dividing the raw returns by the realized standard deviation. The three panels distinguish be- tween statistics computed using all days (top panel), days without any macro news announcements (middle panel) and days on which at least one macro figure is released (bottom panel). The final row in the table shows the total number of announcement days and how these are dispersed across the days of the week.
bottom panel it becomes clear that most macro releases occur on Fridays (204 out of the total of 430 announcements). Squared returns and realized variance are both higher on Friday than on other announcement days. This explains the on average higher values for realized variance on Fridays in the top panel.
After correcting for announcement effects, squared returns are still higher at the begin- ning of the week with averages of 1.514% and 1.597% for Mondays and Tuesdays respec- tively, something which is not the case for realized variance.