Data

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立 政 治 大 學

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4. Empirical Analysis

We analyze the ability of the forecasting models with Markov switching approach to forecast future volatility of S&P 500 index based on five-minute intraday return. In Section 4.1, we describe the data of S&P 500 realized volatility, VIX, SKEW and Residual of VIX, and the filtering rule of VIX futures to ensure the confidence in empirical analysis. We also illustrate the economic implication of Residual of VIX in this Section. In Section 4.2 and Section 4.3, we evaluate the predictive power of the VIX futures and risk-neutral skewness from the empirical results of the in- and out-of-sample, respectively. Last, in Section 4.4, we will compare the differences of predictive accuracy between the MRS-HAR models and the corresponding HAR models.

4.1 Data

In this study, data can be simply divided into three parts. First, the index, including the S&P 500 intraday index and the implied volatility index measure (VIX), are provided by Chicago Mercantile Exchange (CME) and Chicago Board of Options Exchange (CBOE), respectively. For the S&P 500 intraday index, the period is from 08:30 a.m. to 03:00 p.m. To construct the realized volatility, we divide the intraday data of S&P 500 index at five-minute frequency into 78 intra-daily return groups.⁴ Second, SKEW, the risk-neutral skewness measure, is provided by Chicago Board of Options Exchange (CBOE). Last, VIX futures is obtained from Chicago Board of Options Exchange (CBOE). The data frequency is daily for VIX, SKEW and VIX futures, and we use the daily settlement prices for VIX futures. All of our data covers the period from 3 January 2006 to 31 October 2012, which consists of 1704 daily observations.

4 We use five-minute sampling frequency in order to keep the balance of accuracy and avoid the microstructure problems.

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立 政 治 大 學

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N a tio na

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We filter the VIX futures by the following rules to strengthen the reliability of the empirical results. First, trading volume less than five contracts are excluded. Second, we only consider the near-term contract and when time to maturity less than nine calendar days, move to the next-term contract.

Table 1 presents the summary statistics of the S&P 500 index daily realized volatility, VIX, SKEW and Residual of VIX. The average values of daily realized volatility and VIX are 1.03% and 1.20%, and the standard deviation are 0.76% and 0.57%, respectively.

The skewness and excess kurtosis of daily realized volatility are 2.90 and 13.29, respectively. However, the skewness and excess kurtosis of VIX are 1.90 and 4.66, respectively. The daily realized volatility is more positive skewness than VIX and both daily realized volatility and VIX are leptokurtic. The minimum and maximum of daily realized volatility are 0.23% and 8.68%, and the minimum and maximum of VIX are 0.52% and 4.23%, respectively. For SKEW and Residual of VIX, the average value are 119.59 and 0.00, and standard deviation are 5.06 and 1.88, respectively. The skewness of SKEW and Residual of VIX are 0.29 and -1.85, and the excess kurtosis of SKEW and Residual of VIX are 0.21 and 12.11, respectively. The minimum and maximum of SKEW are 106.43 and 142.02, and the minimum and maximum of Residual of VIX are -16.67 and 6.45, respectively. The Ljung-Box test statistic of daily realized volatility, VIX, SKEW and Residual of VIX are all high and the corresponding p-value is zero at 0.01 significance level, which means the time series of these four variables are high serial correlation. Lastly, Augmented Dickey-Fuller test statistic of these four variables are all significant which means they are stationary process.⁵

Figure 1 exhibits three time series plots during the whole sample period, from 3 January 2006 to 31 October 2012. First is the time series of daily realized volatility and

5 The ADF test statistic of weekly and monthly realized volatility are -3.6062 and -3.8857, respectively.

These variables are all significant at 5% which implies they are stationary process.

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VIX; Second is the time series of the difference between daily realized volatility and VIX (RV-VIX) and SKEW; Last is the time series of RV-VIX and Residual of VIX. From the first plot, we can observe that both volatility track each other considerable closely. From the second and third plot, we could observe that SKEW and Residual of VIX move to the opposite direction against RV-VIX simultaneously when RV-VIX has large movement.⁶

Figure 2 displays the economic implication of Residual of VIX during the whole sample period, from 3 January 2006 to 31 October 2012. First is the time series of daily VIX and VIX futures; Second is the time series of RV-VIX and Residual of VIX; Last is the time series of RV-VIX and Spread.⁷ From the first plot, we could observe that the values of VIX futures generally below the values of VIX, especially during the Financial Crisis.⁸ This may imply that the investors actively take positions in the VIX futures to hedge volatility rather than trading in the S&P 500 index options when the market undergoes severe volatile. Interestingly, we can observe that the path of Residual of VIX and Spread are analogous from the second and third plots.⁹ This may suggest that the economic implication of Residual of VIX is similar to Spread. However, the Residual of VIX is uncorrelated to VIX while Spread is highly correlated to VIX. Therefore, the Residual of VIX, relative to VIX, contains incremental information for future volatility forecasting.

6 Here, we roughly regard the Financial Crisis as large movement or the periods of high volatility state.

During the whole sample period, the correlation between the difference of daily realized volatility and VIX and SKEW (corr(RV-VIX, SKEW)) is -0.1454. The correlation between the difference of daily realized volatility and VIX and Residual of VIX (corr(RV-VIX, reVIX)) is -0.4514. During the Financial Crisis, corr(RV-VIX, SKEW) is -0.2394 and corr(RV-VIX, reVIX) is -0.4789.

7 Spread is defined as the differences between VIX futures and VIX.

8 The proportion of VIX futures below the VIX is 0.2411 during the whole sample period, and the proportion is 0.3780 during the Financial Crisis.

9 During the whole sample period, corr(RV-VIX, Spread) is -0.4847 and corr(reVIX, Spread) is 0.8076.

During the Financial Crisis, corr(RV-VIX, Spread) is -0.5080 and corr(reVIX, Spread) is 0.8859.

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Table 1: Summary Statistics of Daily Realized Volatility, VIX, SKEW and Residual of VIX

The table reports the summary statistics of the S&P 500 index daily realized volatility, VIX, SKEW and Residual of VIX from 3 January 2006 to 31 October 2012. Note: (1) The VIX here is daily VIX which is calculated by VIX divided by square root of 365. (2) LB test for lag=10 represents Ljung-Box test statistic for ten lags serial correlation. (3) ADF test represents Augmented Dickey-Fuller test statistic for testing the stationary process. (4) ^** and ^*** denote Significant at 5%

and 1%, respectively.

Figure 1: Time series of daily realized volatility, VIX, SKEW and Residual of VIX.

Figure 1 exhibits three time series plots from 3 January 2006 to 31 October 2012. First is the time series of daily realized volatility (left scale) and VIX (right scale); Second is the time series of the difference between daily realized volatility and VIX (RV-VIX) (left scale) and SKEW (right scale); Third is the time series of RV-VIX (left scale) and Residual of VIX (right scale).

Gray area denotes the Financial Crisis from November 2007 to June 2009. (Here, we roughly regard the Financial Crisis as the periods of high volatility state.)

2006 2007 2008 2009 2010 2011 2012

0.01

2006 2007 2008 2009 2010 2011 2012

60

2006 2007 2008 2009 2010 2011 2012

-20

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Figure 2: Time series of daily VIX, VIX futures, Residual of VIX and Spread.

Figure 2 exhibits three time series plots from 3 January 2006 to 31 October 2012. First is the time series of daily VIX and VIX futures; Second is the time series of the difference between daily realized volatility and VIX (RV-VIX) (left scale) and Residual of VIX (right scale); Third is the time series of RV-VIX (left scale) and Spread (right scale). Gray area denotes the Financial Crisis from November 2007 to June 2009 and Spread denotes the difference of VIX futures and VIX. (Here, we roughly regard the Financial Crisis as the periods of high volatility state.)