1.4.1 Data description
We use the tick-by-tick quoted data on S&P 500 index (SPX) options from CBOE’s Market Data Report (MDR) tapes over the time period from January 1996 to December 2010. The underlying SPX prices are also provided in the tapes. We obtain daily data from OptionMetrics for equity options and S&P 500 index options. We use the Zero Curve file, which contains the current zero-coupon interest rate curve, and the Index Dividend file, which contains the current dividend yield, from OptionMetrics to calculate the implied volatility for each tick-by-tick data from CBOE’s MDR tapes. Daily and monthly stock return data are from CRSP while intraday transactions data are from TAQ data sets.
Financial statement data are from COMPUSTAT. Fama and French (1993) factors and their momentum UMD factor are obtained from the online data library of Ken French.9 VIX index is obtained from the website of CBOE.10 While we use the ‘new’ VIX index to calculate the market variance risk premium as proposed by Bollerslev, Tauchen, and Zhou (2009), we also use the ‘old’ VIX, which is based on the S&P 100 options and Black–
Scholes implied volatilities, as our volatility factor, following Ang, Hodrick, Xing, and
9 http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/
10 http://www.cboe.com/micro/vix/historical.aspx
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Zhang (2006). We use the index option prices from the Option Price file to replicate the market skewness factor and the market kurtosis factor of Chang, Christoffersen, and Jacobs (2013).
We follow the literature (see, for example, Jiang and Tian 2005; Chang, Christoffersen, and Jacobs, 2013; among others) to filter out index option prices that violate the arbitrage bounds.11 We also eliminate in-the-money options (e.g. put options with K/S>1.03 and call options with K/S<1.03) because prior study suggests that they are less liquid. We use the daily SPX low and high prices, downloaded from Yahoo Finance,12 to filter out the MDR data that are outside the [low, high] interval.
For the computation of the market volatility-of-volatility, we first partition the tick-by-tick S&P500 index options data into five-minute intervals. For each maturity within each interval, we linearly interpolate implied volatilities for a fine grid of one thousand moneyness levels (K/S) between 0.01% and 300%13 and use equations (1.26) and (1.27) to estimate the model-free implied variance. We then use linearly interpolate maturities to obtain the estimate at a fixed 30-day horizon. For each day, our measure for market volatility-of-volatility (VoV) is calculated by using the bipower variation formula of equation (1.32) with the 81 within-day five-minute annualized 30-day model-free implied variance estimates covering the normal CBOE trading hours from 8:30 a.m. to 3:15 p.m. Central Time.
The market variance risk premium ( , ), following Bollerslev, Tauchen, and Zhou (2009), is defined as the difference between the ex-ante implied variance ( , )
11 Moreover, we eliminate all observations for which the ask price is lower than the bid price, the bid price is equal to zero, or the average of the bid and ask price is less than 3/8.
12 http://finance.yahoo.com/q/hp?s=^GSPC+Historical+Prices
13 For moneyness levels below or above the available moneyness level in the market, we use the implied volatility of the lowest or highest available strike price.
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and the ex-post realized variance ( , ), i.e. , ≡ , , . We focus on a fixed maturity of 30 days. Market implied variance ( , ) is measured by the squared
‘new’ VIX index divided by 12. Summation of SPX five-minute squared logarithmic returns are used to calculate the market realized variance ( , ). With eighty five- minute intervals per trading day and the overnight return, we construct the daily market realized variance, using a rolling window of 22 trading days starting from the current day.
We construct the individual model-free implied variance ( , ) using equity options data from the Volatility Surface file that provides Black-Scholes implied volatilities for options with standard maturities and delta levels. The individual implied variance is estimated by applying the same methodology that we use for the index options on the equity options data with 30-day maturity.
To compute the individual realized variance ( , ), we extract from TAQ database the intraday transaction and quote data within the normal trading hours from 9:30 a.m. to 4:00 p.m. Eastern Time. We first adopt the step-by-step cleaning procedures proposed by Bardorff-Nielsen, Hansen, Lunde, and Shephard (2009) to screen the TAQ high frequency data,14 and then we follow Sadka (2006) to remove quotes in which the quoted spread is more than 25% and remove trades in which the absolute value of one-tick return is more than 25%. The resulting 78 five-minute trades and quotes per trading day in a rolling window of 22 trading days are separately used to calculate the trade-based daily individual realized variance ( , ) and the quote-based daily individual realized variance ( , ). To avoid the effect from stale prices in trades or in quotes, we further require that the both the number of five-minute trades and that of quotes in the 22-day rolling window
14 We apply the rules of P1, P2, P3, Q1, Q2, T1, T2, and T3 as described in the section 3.1 of Bardorff-Nielsen, Hansen, Lunde, and Shephard (2009) to carry out the cleaning procedures.
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should be more than 78×11=858. To conserve space, we will focus on the trade-based realized variance, i.e. , , , while the results for the quote-based measure are available upon request.
We estimate the monthly expected individual variance risk premium ( , ) through a forecast model. We adopt a linear forecast model, following Drechsler and Yaron (2011) and Han and Zhou (2012), to estimate the expected realized variance ( , ) with the lagged realized variance and the lagged model-free implied variance measured at the end of the month.15 Thus, the expected individual variance risk premium is defined
as , , , .
To implement our empirical model, we construct innovations in market moments.
First, following Ang, Hodrick, Xing, and Zhang (2006), innovations in market volatility ( is measured by its first order difference, i.e. . Chang, Christoffersen, and Jacobs (2013) indicate that taking the first difference is appropriate for VIX, whereas an ARMA(1,1) model is need to remove the autocorrelation in the their skewness and kurtosis factors. Following their approach, the innovations in market volatility-of-volatility ( is computed as the ARMA(1,1) model residuals of the market volatility-of-volatility.
1.4.2 Descriptive statistics
The daily measure of VoV is plotted in Figure 1. 1. There are clear spikes on the graph—the Asian financial crisis in1997, the LTCM crisis in1998, September11, 2001,
15 Specifically, for stock i, we run the regression: , , , and defined the fitted
value as , , i.e. , ≡ , , , .
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the WorldCom and Enron bankruptcies in 2001 and 2002, subprime loan crisis in 2007, the recent financial crisis in 2008, and the flash crash in 2010.
Table 1.1 reports descriptive statistics for the daily factors used in this paper. In our sample, the mean of 30-day market variance risk premium (VRP) is 17.260 (in percentages squared), which is slightly smaller than 18.3 in Bollerslev, Tauchen, and Zhou’s (2010) sample. The mean of VoV is 0.054%, which is much smaller than its standard deviation, 0.563%. The mean of SKEW is -1.663 and the mean of KURT is 9.313.
Thus, the risk-neutral distribution of the market return is asymmetric and has fat tails.
Panel B reports the Spearman correlations between factors, including the excess market return (MKT), the Fama and French (1993) SMB and HML factors, the momentum factor (UMD), the changes in VIX (ΔVIX; Ang, Hodrick, Xing, and Zhang, 2006), innovations in VoV (ΔVoV), and Chang, Christoffersen, and Jacobs (2013) innovations in market skewness factor (ΔSKEW) and market kurtosis factor (ΔKURT). As expected, MKT is negatively correlated with both ΔVIX (-0.779) and ΔVoV (-0.044), supporting the
leverage effect predicted by our model. Moreover, VRP is positively correlated with ΔVoV (0.145), consistent with our theory that the variance risk premium and the market volatility-of-volatility are both driven by the economic volatility-of-volatility. ΔKURT and ΔSKEW are highly correlated with a correlation value of -0.863, which is comparable to -0.83 reported by Chang, Christoffersen, and Jacobs (2013). ΔVoV shows little correlation with ΔVIX (0.049), ΔSKEW (-0.017), and ΔKURT (-0.010), which suggests that ΔVoV should be an independent state variable that cannot be explained by these market moments studied in the literature.
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