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Historic risk and implied volatility ⁎

Mehmet F. Dicle , John Levendis Loyola University New Orleans, College of Business, 6363 St. Charles Avenue, Box 15, New Orleans, LA 70118, United States of America

A R T IC LE I N F O

ABS TRA CT

JEL classiﬁcation: G12 G14 G17

This study extends the volatility prediction literature with (1) new intraday realized volatility measures and (2) various implied volatility indexes for commodities, currencies, and equities. Predicting volatility is important for academics, investors, and regulators. Applications range from forecasting stock and option returns to constructing early warning systems. Using twentythree Chicago Board Options Exchange VIX indexes, as opposed to the common S&P 100 and S&P 500 equity indexes, we ﬁnd a bidirectional lead-lag relationship between implied volatility and realized volatility. The lead-lag relationships are more robust and stronger using suggested intraday volatility measures than using the interday volatility measures that are common in the literature.

Keywords: Volatility Implied volatility S&P 500 VIX CBOE

1. Introduction While market eﬃciency theory (e.g., Fama, 1965a, 1965b, 1998; Malkiel & Fama, 1970) suggests that future returns are not predictable, it may be possible to predict future volatility (e.g., Becker, Clements, Doolan, & Hurn, 2015; Blair, Poon, & Taylor, 2010; Bollerslev, Patton, & Quaedvlieg, 2016; Celik & Ergin, 2014; Charoenwong, Jenwittayaroje, & Low, 2009; Chkili, Hammoudeh, & Nguyen, 2014; Christensen & Prabhala, 1998; Forsberg & Ghysels, 2007; Prokopczuk & Simen, 2014; Prokopczuk, Symeonidis, & Wese Simen, 2016; Sridharan, 2015). The lead-lag relationship between realized volatility and implied volatility has many implications for academics, for professionals, and for regulators. Given the risk-return relation, forecasting volatility would help to predict returns. As implied volatilities are calculated using market prices for options, forecasting implied volatility would help to predict option returns. Forecasting future volatility, whether implied or realized, would help to construct early warning systems. Existing early warning systems with long and short horizons have exhibited mixed results (Berg, Borensztein, & Pattillo, 2005). While most of the early warning systems seem to be associated with economic crises, Bates (1991, 2000) shows that option prices may help predict ﬁnancial market crashes. It is important to note that option prices depend on the volatility expectations of the market participants (see, e.g., Black & Scholes, 1973; Cox, Ross, & Rubinstein, 1979). Thus, the implied volatility for the entire market1 that is calculated using traded option prices can be used to predict ﬁnancial market crashes. Speciﬁcally, implied volatility forecasts future volatility, for the equity indexes as well as currencies, bonds, interest rates, and commodities (see, e.g., Busch, Christensen, & Nielsen, 2011; Charoenwong et al., 2009; Christensen & Prabhala, 1998; Fernandes, Medeiros, & Scharth, 2014; Hattori, 2017; Haugom, Langeland, Molnár, & Westgaard, 2014; Kourtis, Markellos, & Symeonidis, 2016; Pilbeam & Langeland, 2015; Prokopczuk & Simen, 2014; Seo & Kim, 2015; Shaikh & Padhi, 2014; Szakmary, Ors, Kim, & Davidson, 2003; Wong & Heaney, 2017). A natural extension would use the implied volatilities of individual securities for the same purpose. However, implied volatility indexes—such as those

⁎

Corresponding author. E-mail addresses: [email protected] (M.F. Dicle), [email protected] (J. Levendis). 1 One might use the CBOE VIX, as it is an implied volatility index based on options written on S&P 500 stocks. https://doi.org/10.1016/j.gfj.2019.100475 Received 17 June 2018; Received in revised form 5 May 2019; Accepted 8 May 2019 1044-0283/ © 2019 Elsevier Inc. All rights reserved.

Please cite this article as: Mehmet F. Dicle and John Levendis, Global Finance Journal, https://doi.org/10.1016/j.gfj.2019.100475

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that are available for the S&P 500 and other major indexes (and a few stocks and a few commodities)—are generally not available for individual securities. The notion of market eﬃciency posits that current prices reﬂect all available information. This publicly available information includes, naturally, implied volatility, since it is calculated using publicly available option prices. Lundblad (2007) shows that, if a long enough time period is used with the proper method, the relationship between risk and return is positive, whereas previous studies found mostly statistically insigniﬁcant or weak results (e.g., Baillie & DeGennaro, 1990). If future volatility can be predicted using current implied volatility, it would mean either that current option prices (on which implied volatilities are calculated) reﬂect future volatility or that it may be possible to forecast future returns, given the positive relationship between risk and return established by Lundblad (2007). This would prove quite useful for traders. Such predictability of returns would be a clear violation of market eﬃciency and the random walk theory. While predicting future volatility is important, it is also important to be able to predict implied volatility using past volatility measures. Predicted implied volatility (derived using the option pricing model of Black & Scholes, 1973) can be used to predict option prices (again using Black and Scholes's model). Also, if past volatility can predict implied volatility, and implied volatility can predict future volatility, then by transitivity past volatility would be expected to predict future volatility. As we report in the next section, the evidence for the lead-lag relationship between implied volatility and realized volatility is mixed, even though it is common to limit studies' focus to major equity indexes such as the S&P 100 and S&P 500. We extend the literature by evaluating not only the major equity indexes but also indexes for commodities, international equities, and currencies. We posit that the mixed results in the literature are due mainly to model misspeciﬁcations and diﬀerences in the way volatility is measured. As a remedy, we introduce two simple intraday realized volatility measures (intraday standard deviation and intraday high-low range) in addition to the interday realized volatility measure (rolling standard deviation) for their lead/lag relationships with implied volatility. Since high-frequency intraday data (by transaction or by minute) are not commonly available to all market participants, our simple intraday volatility measures are relevant to most traders and researchers. In order to estimate the lead-lag relationships between implied and actual volatility, we perform Granger noncausality analysis using VIX data for twenty-three securities with sample periods starting as early as year 2000. Our analysis is based on publicly available VIX indexes, their corresponding securities, and simple volatility measures. A description of our sample, its source, and the list of notations are provided in the data section. The Granger causality models for each of the volatility measures, and their results, are discussed in the section on models and empirical results. Concluding remarks summarize the study and present results, implications, and contributions to the related literature. Tables of estimation results are provided at the end of the study. 2. Literature review Before we proceed, it will be useful to deﬁne “actual volatility,” “expected volatility,” and “implied volatility.” By “actual volatility” we mean volatility that has already occurred (historical volatility). By “expected” we mean anticipated or forecasted volatility. By “implied volatility” we mean variability as implied by Black and Scholes's (1973) option pricing model. Today's option price depends upon today's “implied” volatility, which is an expectation of future actual volatility in the underlying asset. That is, we use historical data to predict expected volatility, whether it is implied volatility or actual volatility. Among all volatility measures implied volatility is unique in that it is future looking. Since its calculation is based on the current prices of options, implied volatility is a direct reﬂection of investors' expectations of future volatility. In other words, implied volatility is the expected future volatility implied by the currently traded option prices (Hull & White, 1987; Merton, 1973). Since it is a current measure of expected volatility, implied volatility has enjoyed signiﬁcant attention in the literature for its power to predict future volatility. In one of the earliest studies in this line of literature, Latane and Rendleman (1976) calculate the implied standard deviations for twenty-four companies that traded on the Chicago Board Options Exchange during 1973 and 1974. Their pioneering method for “weighted average implied standard deviation” is quite similar to how VIX indexes are calculated today. They ﬁnd signiﬁcant correlation between implied volatility and realized volatility. Furthermore, they report that implied volatility predicts future volatility better than realized volatility does. Pilbeam and Langeland (2015) provide supporting evidence that implied volatility predicts foreign exchange rate volatility. Szakmary et al. (2003) analyze equities, interest rates, currencies, and commodities in 35 markets. They report that implied volatility yields more accurate better forecasts than the other traditional models. For commodities, Haugom et al. (2014) and Prokopczuk and Simen (2014) show that implied volatility improves the predictive power of forecasting models, especially for short-term predictions. While it is common to evaluate implied volatility for its predictive power, Seo and Kim (2015) report that forecasting with implied volatility is aﬀected by both time and investor sentiment. From an international equities perspective, Shaikh and Padhi (2014) ﬁnd that implied volatility has predictive power for equities traded in India; Frijns, Tallau, and Tourani-Rad (2010) for equities in Australia; and Cheng and Fung (2012) for equities in Hong Kong. Analyzing equities from ten countries, Kourtis et al. (2016) report similar results in favor of implied volatility. Charoenwong et al. (2009) provide evidence to the same eﬀect for currencies. Several studies contradict these ﬁndings and report little or no predictive power for implied volatility. Rather than relying on the cross-sectional regressions popular in previous work, Day and Lewis (1992) use a GARCH model (Bollerslev, 1986; Engle, 1982) on the S&P 100 index and ﬁnd no predictive power of implied volatility. Lamoureux and Lastrapes (1993), analyzing ten individual stocks, conﬁrm these ﬁndings: “Implied variance tends to underpredict realized variance” (p. 321). Canina and Figlewski (1993), analyzing S&P 100 index options, “strongly refute” (pp. 676) the view that “an option's implied volatility is a good estimate of the market's expectation of the asset's future volatility” (p. 676). On the other hand, Christensen and Prabhala (1998) reevaluate Canina 2

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& Figlewski's, 1993 study and “ﬁnd that implied volatility does predict future realized volatility” of S&P 100 index options (p. 148). The main diﬀerences between these two studies are that Christensen and Prabhala use a monthly sampling frequency, a longer sample period, and nonoverlapping data. Their results are in line with the ﬁndings of Harvey and Whaley (1992) and Fleming (1998) for S&P 100 index options and the ﬁndings of Jorion (1995) for options on foreign currency futures. Still, in an extensive analysis of S&P 100 volatility, Koopman, Jungbacker, and Hol (2005) report that “models based on realised volatility outperform models with implied volatility when forecasting the volatility in the S&P 100 index series” (p. 23). For commodities Agnolucci (2009) and for foreign currencies Pong, Shackleton, Taylor, and Xu (2004) ﬁnd implied volatilities to be unimpressive compared to historical forecasts. Jorion (1995) attributes the weak predictive power of implied volatility for future volatility to various model speciﬁcations or inherent biases within the information eﬃciency of the option markets as well as within the option pricing models used. More recently, Jiang and Tian (2005) develop a model-free implied volatility measure and show it to be more eﬃcient in predicting future volatility. Their study addresses various model speciﬁcation biases pointed out by the earlier literature. Our study uses the a traditional implied volatility measure and provides evidence for its strong predictive power using intraday standard deviation measures. In other words, we argue that lack of evidence for such power stems not from the ineﬃciency and bias that are claimed to be inherent in implied volatility but from the ineﬃciency of the realized volatility measures commonly used. To our knowledge, we are the ﬁrst investigators to test that predictive power using the intraday standard deviation and daily high-low range. We believe that using these two simple realized volatility measures within a simple econometric model makes our study less vulnerable to biases such as ﬁshing or overestimation. Ultimately, we provide evidence that implied volatility can forecast future actual volatility, and that actual volatility can forecast future implied volatility. Contrasting the various realized volatility measures proves to be an important contribution to the volatility forecasting literature. Among the realized volatility measures we use, the high-low range measure best predicts implied volatility. 3. The data Empirical analysis of the lead-lag relationship between securities' actual and implied volatilities requires daily prices (high, low, opening, and closing values) of securities as well as implied volatilities. Since historic prices for options are not easily available to the general public (investors and most academics), we focus on publicly available data. While many studies in the literature use hand collected data from the ﬁnancial media (e.g., Aggarwal, Inclan, & Leal, 1999; Christensen & Prabhala, 1998), ﬁnancial data in electronic form have become much more available to the public within the past two decades. The Chicago Board Options Exchange (CBOE), for example, provides many VIX indexes for the major market indexes (as well as ﬁve individual stocks), currency indexes, and commodity indexes.2 Implied volatility indexes provided by CBOE are based on either index options or options on ETFs and stocks. For instance, VIX is based on the S&P 500 Index (SPXSM), and VXN is based on the NASDAQ 100 Index (NDX). Similarly, VXEFA is based on an ETF iShares MSCI EAFE Index Fund (EFA), and VXAZN is based on AMZN stock. We extend the method employed for the general VIX index to other implied volatility indexes, which are intended to predict an expected volatility for the subsequent 30 days. We use the mid-prices of bid and ask prices for both call and put options in index calculations. We obtain daily index values from Nasdaq (http://nasdaq.com) and daily implied volatility index values from the Chicago Board Options Exchange (CBOE, https://www.cboe.com). Daily currency values are from the Federal Reserve Bank of St. Louis (https:// fred.stlouisfed.org). While daily securities values and implied volatility index values have intraday high, low, opening, and closing values, currencies have only daily closing values. Table 1 summarizes the notations used within the empirical analysis. Descriptive statistics for the VIX indexes and for the securities are provided in Tables A.1 and A.2, respectively. Augmented Dickey-Fuller tests indicate that all returns and volatility measures are stationary. It is interesting to note that four of the securities included in our sample have positive mean daily change in VIX, namely, RVX, OVX, VXEWZ, and VXAPL. With respect to RVX and OVX, these unusual daily averages seem to be due to increased VIX levels before the year 2009. When we calculate the means for 2000–2009 and 2010–2018, we ﬁnd that the mean for 2000–2009 is positive and the mean for 2010–2018 is negative. With respect to VXEWZ and VXAPL, the culprit for the positive mean daily change in VIX seems to be 2018. For years before 2018, means are negative. However, for the year 2018, both means turn positive. Each VIX index in Table A.1 corresponds to a security in Table A.2. For example, the CBOE volatility index (VIX) in Table A.1 corresponds to the S&P 500 (GSPC) in Table A.2. The empirical analysis of lead-lag relationships evaluates these pairs, since implied volatilities are based on the securities on which the options are written. The availability of CBOE VIX indexes determined the scope of our analysis. We investigate the universe of VIX indexes provided by the CBOE: ﬁve major equity indexes, ﬁve major commodity indexes, three diﬀerent currencies, four foreign equity indexes, and ﬁve individual U.S. stocks.3 Most major equity market implied volatility indexes extend back to year 2000. The remaining VIX data sets have varying starting dates ranging between years 2004 and 2011. The sample for our study ends with October 2018. Our analyses evaluate three diﬀerent volatility measures: (1) the twenty-day rolling standard deviation of daily returns based on daily closing values (σit20); (2) the daily standard deviation of intraday high, low, opening, and closing values (σitI); and (3) the log diﬀerence of intraday high and intraday low (σitHL). To conserve space, we do not report the descriptive statistics on these measures 2 3

More information about VIX indexes and their calculations is available at http://www.cboe.com/products/vix-index-volatility. We used all available CBOE VIX indexes except those involving interest rates, as these are outside the scope of our study. 3

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Table 1 Notations. Δ t i oit hit lit cit rit ΔVIXit σit20 σitI σitHL

Daily log diﬀerence Trading day Each index or security Daily opening price for security i at time t Daily highest price for security i at time t Daily lowest price for security i at time t Daily closing price for security i at time t Daily return for security i (ri, t = Δci, t = log (ci, t) − log (ci, t−1)) Daily percent change in CBOE volatility index i at time t Running standard deviation of daily returns over a twenty-day window for security i at time t Daily intraday standard deviation as proxied by the standard deviation of high, low, opening, and closing values of security i at time t Daily intraday standard deviation as proxied by the log diﬀerence of intraday high and intraday low for security i at time t

other than to say that they are all stationary across all securities. Instead, we report the descriptive statistics on the daily returns of the securities that these volatility measures are based on. For the daily returns for the securities, sample dates start and end with the corresponding VIX sample dates. While most daily mean returns are positive, there are a few that are negative: crude oil, Brazil equities, silver, and foreign currencies. While gold, as a commodity, has a positive daily mean return, Gold Miners ETF has a negative daily mean return. 4. Models and empirical results In order to estimate the lead and lag relationship between implied volatility and actual volatility, we estimate a set of Granger (1969) noncausality models. Each model provides us with the lead and lag relationship between a speciﬁc VIX index and the actual volatility in its corresponding security. Empirical results are presented separately for each of the three volatility measures. As is the common practice, we report only the results of the Wald tests for causality rather than include all of the estimated coeﬃcients for each lag.4 4.1. Twenty-day rolling standard deviation and VIX Several measures of return volatility are used in the literature. A rolling standard deviation of past returns is a common measure of past volatility (e.g., Fama, 1976; Mele, 2007; Merton, 1980; Oﬃcer, 1973; Schwert, 1989). Our analysis evaluates a rolling standard deviation with a window of twenty days (σit20), which approximates the number of trading days in a month and is a window length commonly used in the technical analysis literature (e.g., Chang & Osler, 1999; Leigh, Purvis, & Ragusa, 2002; Levich & Thomas, 1993; Sullivan, Timmermann, & White, 1999). Many popular technical analysis tools also use twenty days as the short term while 200 days is the long term. While we present the results for a twenty-day rolling standard deviation, we estimated several other window lengths and found quite similar results.5 The twenty-day rolling standard deviation is calculated as

⎛1 σit20 = ⎜ 20 ⎝

1/2

19

∑ (ri − ri,t )2⎞⎟

where ri =

⎠

t=0

1 20

19

∑ ri,t .

(1)

t=0

The Granger noncausality models have the following form:

ΔVIXit = α1i +

Δσit20 = α2i +

2

∑j=1 2

∑j=1

β1ij ΔVIXi, t − j +

2

∑k =1

γ1ik Δσi20 , t − k + ϵit ;

(2)

2

β2ij Δσi20 ,t−j +

∑ γ2ik ΔVIXi,t−k + eit .

(3)

k=1

The Wald test for noncausality tests the following restrictions:

Δσit20 → ΔVIXit : γ1i1 = γ1i2 = 0;

(4)

ΔVIXit → Δσit20: γ2i1 = γ2i2 = 0.

(5)

Table A.6 provides the Wald test results for Eqs. (4) and (5). The VIXs of all major equity indexes lead the actual volatilities of their underlying assets. These results support and extend those reported by Christensen and Prabhala (1998). Implied volatility leads the large domestic equity indexes, but this relationship is weak or nonexistent for individual equities or international equity markets. Implied volatilities for commodities generally do not lead 4 5

Estimation results for all of the parameters and models are available upon request. Available upon request. 4

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realized volatility. For currencies, statistically signiﬁcant (5% or better) VIX leads exist for the USD/GBP and USD/EUR foreign exchange rates. The weak predictability of currency volatility is in line with the ﬁndings by Pong et al. (2004). Does the rolling twenty-day standard deviation have a lead over implied volatilities (Δσit20 → ΔVIXit)? The results are mixed. A statistically signiﬁcant (at the 5% level or better) lead over implied volatility exists for nine of the securities, including the S&P 500 and S&P 100. As a robustness check for our results, we estimate Eqs. (2) and (3) with S&P 500 daily returns as an exogenous variable. Table B.9 shows the Wald test results, which provide even stronger evidence in favor of the suggested lead-lag relationships. There is no compelling evidence of bidirectional causality between actual and implied volatility. We hypothesize that this is due to the speciﬁc volatility measure being used. The rolling standard deviation includes information for the past nineteen trading days along with each trading day. In other words, the impact of each passing day on the twenty-day standard deviation is small (one trading day out of a total of twenty). To test this hypothesis, we turn to two diﬀerent intraday measures of volatility. 4.2. Intraday standard deviation and VIX Our intraday standard deviation measure contains information regarding only the most recent volatility: that found within the speciﬁc trading day. For this reason, we expect it to be more eﬃcient than the rolling standard deviation. Accordingly, we expect to detect a lead-lag relationship with the VIX that is more evident not only for the major equity indexes but also for other securities. An investor (or trader) can use the current day's high, low, opening, and closing prices to calculate the current volatility. One of the earliest researchers to suggest the use of daily high and low values instead of closing prices only was Parkinson (1980). Garman and Klass (1980) ﬁnd that volatility estimators that use daily low, high, opening, and closing prices are more eﬃcient. Similarly, Rogers and Satchell (1991) and Rogers, Satchell, and Yoon (1994) suggest using daily low, high, opening, and closing prices for estimating volatility. We provide a measure of volatility using intraday high, low, opening, and closing prices for each trading day (σitI). Accordingly, the intraday standard deviation is calculated as follows: 1/2 1 σitI = ⎛ [(ri − oit )2 + (ri − hit )2 + (ri − lit )2 + (ri − cit )2] ⎞ , ⎝4 ⎠

(6)

ri = (oit + hit + lit + cit )/4.

(7)

where

The Granger noncausality model is estimated as follows:

ΔVIXit = α1i +

2

∑j=1

β1ij ΔVIXi, t − j +

2

ΔσitI = α2i +

2

∑k =1

γ1ik ΔσiI, t − k + ϵit ;

(8)

2

∑ β2ij ΔσiI,t−j + ∑ γ2ik ΔVIXi,t−k + eit . j=1

(9)

k=1

The Wald test for noncausality tests the following restrictions:

ΔσitI → ΔVIXit : γ1i1 = γ1i2 = 0;

(10)

ΔσitI : γ2i1

(11)

ΔVIXit →

= γ2i2 = 0.

Table A.7 provides the Wald test results for Eqs. (10) and (11). Regarding ΔVIXit → ΔσitI, the results indicate that the intraday standard deviation is a more eﬃcient predictor of realized volatility than is the rolling standard deviation. Except for USD/EUR ETF, Brazil equity ETF, Gold Miners ETF, and Alphabet stock, all securities in our sample had statistically signiﬁcant (at 5% or better) implied volatility leads over realized volatility. These results conﬁrm that the most current trading day's volatility is much more important than past trading days' volatility, and in eﬀect endorse GARCH models, which weight the previous days' volatilities by their proximity to the current day. Regarding ΔσitI → ΔVIXit, the results are signiﬁcant only for China equity ETF, Gold Miners ETF, and Amazon and Apple stocks. Except for Gold Miners ETF, these are all bidirectional lead-lag relationships. The lack of statistically signiﬁcant leads by intraday volatility over VIX for most securities in our sample implies the importance of historic volatility versus the current day's volatility. As a robustness check for our results, we estimate Eqs. (8) and (9) with S&P 500 daily returns as an exogenous variable. Table B.10 provides the Wald test results, which are very similar to the results of the model without the exogenous variable. We expect, however, that the intraday standard deviation, while more eﬃcient than the rolling standard deviation, contains intraday noise. We posit that it is this noise that makes the intraday volatility lose its predictive power over implied volatility. 4.3. Intraday high-low range and VIX Alizadeh, Brandt, and Diebold (2002) suggest an alternative intraday volatility measure. Like a range, it is calculated as “the diﬀerence between the highest and lowest log security prices over a ﬁxed sampling interval” (p. 1048). In this respect their range measure is a slight deviation from the usual intraday high, low, opening, and closing measures in the literature. As they point out, the data required to calculate their range measure are publicly available and easy to compute. These data are It is also used frequently in 5

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ﬁnancial technical analysis. Our measure for daily range (σitHL) is the log diﬀerence of the daily high and daily low (high-low measure). Since daily high and daily low prices point to the extremes of price disagreement between traders, we expect that the range between these prices reﬂects current volatility more eﬃciently than σitI. That is, we believe that the daily opening and closing values are a source of noise in the intraday standard deviation as calculated above. The intraday high-low range measure is calculated as follows:

σitHL = log (hit ) − log (lit ).

(12)

The Granger noncausality model is estimated as follows:

ΔVIXit = α1i +

2

∑j=1

β1, j ΔVIXi, t − j +

2

σitHL = α2i +

2

∑k =1

γ1, k σiHL , t − k + ϵit ;

(13)

2

∑ β2,j σiHL , t − j + ∑ γ2, k ΔVIXi, t − k + eit . j=1

(14)

k=1

The Wald test for noncausality tests the following restrictions:

σitHL → ΔVIXit : γ1i1 = γ1i2 = 0;

(15)

ΔVIXit → σitHL: γ2i1 = γ2i2 = 0.

(16) 6

Table A.8 provides the Wald test results for Eqs. (15) and (16). The results for ΔVIXit → σitHL and for σitHL → ΔVIXit are both statistically signiﬁcant at the 1% level for almost all of the securities included in the study. Exceptions for ΔVIXit → σitHL include USD/EUR ETF, Brazil equity ETF, and Alphabet stock (and gold and silver ETFs at 5% signiﬁcance). Exceptions for σitHL → ΔVIXi, d include the Russell 2000, Brazil equity, and silver and gold ETFs. As a robustness check for our results, we estimate Eqs. (13) and (14) with S&P 500 daily returns as an exogenous variable. Table B.11 provides the Wald test results, which strengthen the evidence in favor of the suggested lead-lag relationships. Overall, these results point to a bidirectional lead-lag relationship between realized volatility and implied volatility. Also, these results emphasize the importance of using the most recent trading day's volatility (rather than a measure that averages over several days' volatility) to best predict future volatility. Any volatility measure that draws heavily from data that are more than one day old can be expected to be a less eﬃcient predictor of implied volatility. Similarly, implied volatility will predict the volatility of the nearest trading day. 5. Concluding remarks Predicting future volatility has received considerable attention in the literature, and implied volatility has been tested as the most likely candidate predictor. We suspect that the mixed results in the literature are due to the realized volatility measures used, model speciﬁcations, and possible biases within the information eﬃciency of the option markets and the option pricing models. Among the three measures of realized volatility we tested, the running standard deviation weights past volatility equally with the current day's volatility. The intraday standard deviation includes noise, as daily opening and closing prices do not reﬂect price volatility within a trading day. While previous studies have neglected the high-low range, we found this simple volatility measure to be the most eﬃcient predictor of implied volatility. With the high-low range volatility measure, we ﬁnd that implied volatility leads future volatility with only a few exceptions. This is important for predicting future volatility and constructing useful early warning systems. The same volatility measure also leads implied volatility. This is important for securities for which implied volatility is not available (i.e., individual stocks). It is also important for forecasting option prices as well as predicting future volatility. Appendix A. Tables and ﬁgures Table A.1 Descriptive statistics for VIX indexes. Title CBOE CBOE CBOE CBOE CBOE

Volatility Index NASDAQ Volatility Index DJIA Volatility Index S&P 100 Volatility Index Russell 2000 Volatility Index

Notation

First

Mean

Min.

Max.

Stdev.

DF-z

ΔVIXt ΔVXNt ΔVXDt ΔVXOt ΔRVXt

01/03/2000 10/12/2000 01/03/2000 01/03/2000 01/05/2004

−0.0001 −0.0002 −0.0000 −0.0000 0.0000

−0.3506 −0.3161 −0.4081 −0.3815 −0.3643

0.7682 0.4689 0.5281 0.7593 0.5404

0.0685 0.0583 0.0646 0.0769 0.0561

−74.4790 −69.7515 −76.9491 −78.2066 −65.1561

*** *** *** *** ***

(continued on next page)

6 As a robustness check, we reestimated the model using the standard deviation of the log of each intraday price (opening, high, low, and closing). The results are quite similar and are available upon request.

6

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Table A.1 (continued) Title

Notation

First

Mean

Min.

Max.

Stdev.

DF-z

CBOE Crude Oil ETF Volatility Index CBOE Gold ETF Volatility Index CBOE EuroCurrency ETF Volatility Index CBOE EFA ETF Volatility Index CBOE Emerging Markets ETF Volatility Index CBOE China ETF Volatility Index CBOE Brazil ETF Volatility Index CBOE Silver ETF Volatility Index CBOE Gold Miners ETF Volatility Index CBOE Energy Sector ETF Volatility Index CBOE Equity VIX on Amazon CBOE Equity VIX on Apple CBOE Equity VIX on Goldman Sachs CBOE Equity VIX on Google CBOE Equity VIX on IBM CBOE/CME FX Euro Volatility IndexSM CBOE/CME FX Yen Volatility IndexSM CBOE/CME FX British Pound Volatility IndexSM

ΔOVXt ΔGVZt ΔEVZt ΔVXEFAt ΔVXEEMt ΔVXFXIt ΔVXEWZt ΔVXSLVt ΔVXGDXt ΔVXXLEt ΔVXAZNt ΔVXAPLt ΔVXGSt ΔVXGOGt ΔVXIBMt ΔEUVIXt ΔJYVIXt ΔBPVIXt

05/14/2007 03/15/2010 03/15/2010 01/03/2008 03/17/2011 03/17/2011 03/17/2011 03/17/2011 03/17/2011 03/17/2011 06/02/2010 06/02/2010 06/02/2010 06/02/2010 06/02/2010 01/03/2007 01/03/2007 01/03/2007

0.0001 −0.0002 −0.0001 −0.0000 −0.0001 −0.0001 0.0001 −0.0004 −0.0001 −0.0001 −0.0001 0.0000 −0.0003 −0.0001 −0.0001 −0.0009 −0.0003 −0.0005

−0.4399 −0.3069 −0.3981 −0.6867 −0.2981 −0.2028 −0.6196 −0.2513 −0.6079 −0.3103 −1.1474 −0.4035 −0.2929 −0.6174 −0.5259 −0.7397 −0.2801 −0.4387

0.4250 0.4807 0.2891 0.5565 0.5049 0.3658 0.3240 0.5661 0.5707 0.2981 0.4217 0.4245 0.4438 0.4022 0.3476 0.4572 0.4123 0.3713

0.0474 0.0538 0.0456 0.0805 0.0624 0.0517 0.0505 0.0478 0.0565 0.0558 0.0757 0.0675 0.0627 0.0734 0.0643 0.0491 0.0493 0.0425

−59.5518 −50.8530 −48.9471 −60.0814 −46.0406 −46.1548 −45.0810 −46.3137 −59.6475 −44.7170 −49.7036 −46.0199 −51.1011 −50.4748 −47.9927 −56.8927 −53.7909 −49.2556

*** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** ***

Notes: Daily implied volatility index values are obtained from Chicago Board Options Exchange (CBOE) (https://www.cboe.com). The sample period ends with October 2018. *, **, and *** refer to statistical signiﬁcance at the 10%, 5%, and 1% levels respectively.

Table A.2 Descriptive statistics for securities. Title

Notation

First

Mean

Min.

Max.

Stdev.

DF-z

S&P 500 NASDAQ 100 Dow 30 S&P 100 Russell 2000 United States Oil ETF SPDR Gold ETF CurrencyShares Euro ETF iShares MSCI EAFE ETF iShares MSCI Emerging Markets ETF iShares China Large-Cap ETF iShares MSCI Brazil Capped ETF iShares Silver Trust VanEck Vectors Gold Miners ETF Energy Select Sector SPDR ETF Amazon.com, Inc. Apple Inc. The Goldman Sachs Group, Inc. Alphabet Inc. IBM U.S./Euro Foreign Exchange Rate Japan/U.S. Foreign Exchange Rate U.S./U.K. Foreign Exchange Rate

ΔGSPCt ΔNDXt ΔDJIt ΔOEXt ΔRUTt ΔUSOt ΔGLDt ΔFXEt ΔEFAt ΔEEMt ΔFXIt ΔEWZt ΔSLVt ΔGDXt ΔXLEt ΔAMZNt ΔAAPLt ΔGSt ΔGOOGt ΔIBMt ΔDEXUSEUt ΔDEXJPUSt ΔDEXUSUKt

01/03/2000 10/12/2000 01/03/2000 01/03/2000 01/05/2004 05/14/2007 03/15/2010 03/15/2010 01/03/2008 03/17/2011 03/17/2011 03/17/2011 03/17/2011 03/17/2011 03/17/2011 06/02/2010 06/02/2010 06/02/2010 06/02/2010 06/02/2010 01/03/2007 01/03/2007 01/03/2007

0.0001 0.0002 0.0002 0.0001 0.0003 −0.0004 0.0000 −0.0001 0.0000 0.0000 0.0001 −0.0002 −0.0005 −0.0005 0.0000 0.0012 0.0009 0.0003 0.0007 0.0001 −0.0000 −0.0000 −0.0001

−0.0947 −0.1111 −0.0820 −0.0919 −0.1261 −0.1130 −0.0919 −0.0273 −0.1184 −0.0871 −0.0744 −0.1782 −0.1525 −0.1131 −0.0889 −0.1353 −0.1319 −0.1064 −0.0875 −0.0864 −0.0300 −0.0522 −0.0817

0.1096 0.1185 0.1051 0.1066 0.0886 0.0917 0.0479 0.0312 0.1475 0.0605 0.0688 0.0848 0.0700 0.1065 0.0536 0.1462 0.0850 0.0905 0.1489 0.0849 0.0462 0.0334 0.0443

0.0120 0.0160 0.0113 0.0120 0.0149 0.0216 0.0100 0.0058 0.0151 0.0133 0.0153 0.0199 0.0177 0.0241 0.0136 0.0195 0.0156 0.0160 0.0151 0.0122 0.0062 0.0066 0.0062

−74.2399 −70.4322 −73.9234 −74.7394 −66.2949 −56.0663 −48.0669 −46.2824 −58.3170 −45.6320 −44.7668 −44.6633 −43.9313 −46.0267 −43.6003 −46.0136 −44.7623 −47.7434 −44.8670 −45.1362 −52.6688 −54.1503 −50.8140

*** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** ***

Notes: Daily security values are obtained from Nasdaq (http://nasdaq.com). The sample period ends with October 2018. *, **, and *** refer to statistical signiﬁcance at the 10%, 5%, and 1% levels respectively.

Table A.3 Descriptive statistics for securities' running standard deviation of daily returns over a twenty-day window (σit20). Title

Notation

First

Mean

Min.

Max.

Stdev.

DF-z

S&P 500 NASDAQ 100 Dow 30 S&P 100 Russell 2000 United States Oil ETF

ΔGSPCt ΔNDXt ΔDJIt ΔOEXt ΔRUTt ΔUSOt

02/01/2000 01/25/2001 02/01/2000 02/01/2000 02/03/2004 06/12/2007

−0.0000 −0.0001 −0.0000 0.0000 0.0002 0.0000

−0.5875 −0.5750 −0.6646 −0.6559 −0.4453 −0.4029

0.8237 0.6805 0.8096 0.8917 0.6952 0.4627

0.0709 0.0689 0.0705 0.0723 0.0627 0.0614

−66.7528 −64.6512 −65.4695 −66.3928 −59.9710 −52.6742

*** *** *** *** *** ***

(continued on next page) 7

Global Finance Journal xxx (xxxx) xxxx

M.F. Dicle and J. Levendis

Table A.3 (continued) Title

Notation

First

Mean

Min.

Max.

Stdev.

DF-z

SPDR Gold ETF CurrencyShares Euro ETF iShares MSCI EAFE ETF iShares MSCI Emerging Markets ETF iShares China Large-Cap ETF iShares MSCI Brazil Capped ETF iShares Silver Trust VanEck Vectors Gold Miners ETF Energy Select Sector SPDR ETF Amazon.com, Inc. Apple Inc. The Goldman Sachs Group, Inc. Alphabet Inc. IBM U.S./Euro Foreign Exchange Rate Japan/U.S. Foreign Exchange Rate U.S./U.K. Foreign Exchange Rate

ΔGLDt ΔFXEt ΔEFAt ΔEEMt ΔFXIt ΔEWZt ΔSLVt ΔGDXt ΔXLEt ΔAMZNt ΔAAPLt ΔGSt ΔGOOGt ΔIBMt ΔDEXUSEUt ΔDEXJPUSt ΔDEXUSUKt

04/13/2010 04/13/2010 02/01/2008 04/14/2011 04/14/2011 04/14/2011 04/14/2011 04/14/2011 04/14/2011 06/30/2010 06/30/2010 06/30/2010 06/30/2010 06/30/2010 02/02/2007 02/02/2007 02/02/2007

−0.0001 −0.0003 −0.0001 0.0004 0.0004 0.0005 −0.0001 0.0001 0.0001 0.0002 0.0000 0.0001 0.0002 0.0003 −0.0000 −0.0001 −0.0001

−0.7953 −0.6763 −0.6961 −0.4096 −0.4150 −0.8358 −0.7429 −0.4424 −0.6014 −1.1300 −0.7731 −0.4326 −1.0896 −0.9263 −0.4176 −0.9779 −0.5984

0.5828 0.5814 0.6204 0.4887 0.4101 0.9736 0.7253 0.5216 0.5856 1.2727 0.9692 0.4835 1.1381 1.3382 0.7089 0.6881 0.8102

0.0770 0.0639 0.0705 0.0627 0.0630 0.0681 0.0819 0.0619 0.0679 0.1030 0.0863 0.0702 0.0986 0.1037 0.0651 0.0756 0.0639

−46.7489 −47.1911 −49.1634 −40.1573 −42.7978 −40.6465 −41.4708 −44.6899 −42.0413 −43.4492 −43.6203 −44.7572 −44.0355 −44.0675 −54.0579 −51.4983 −52.2267

*** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** ***

Notes: Daily security values are obtained from Nasdaq (http://nasdaq.com). The sample period ends with October 2018. *, **, and *** refer to statistical signiﬁcance at the 10%, 5%, and 1% levels respectively.

Table A.4 Descriptive statistics for securities' daily intraday standard deviation as proxied by the standard deviation of high, low, opening, and closing values (σitI). Title

Notation

First

Mean

Min.

Max.

Stdev.

DF-z

S&P 500 NASDAQ 100 Dow 30 S&P 100 Russell 2000 United States Oil ETF SPDR Gold ETF CurrencyShares Euro ETF iShares MSCI EAFE ETF iShares MSCI Emerging Markets ETF iShares China Large-Cap ETF iShares MSCI Brazil Capped ETF iShares Silver Trust VanEck Vectors Gold Miners ETF Energy Select Sector SPDR ETF Amazon.com, Inc. Apple Inc. The Goldman Sachs Group, Inc. Alphabet Inc. IBM

ΔGSPCt ΔNDXt ΔDJIt ΔOEXt ΔRUTt ΔUSOt ΔGLDt ΔFXEt ΔEFAt ΔEEMt ΔFXIt ΔEWZt ΔSLVt ΔGDXt ΔXLEt ΔAMZNt ΔAAPLt ΔGSt ΔGOOGt ΔIBMt

01/04/2000 10/13/2000 01/04/2000 01/04/2000 01/06/2004 05/15/2007 03/16/2010 03/16/2010 01/04/2008 03/18/2011 03/18/2011 03/18/2011 03/18/2011 03/18/2011 03/18/2011 06/03/2010 06/03/2010 06/03/2010 06/03/2010 06/03/2010

−0.0000 −0.0001 0.0001 −0.0000 0.0004 −0.0000 0.0001 −0.0003 0.0001 0.0005 0.0005 −0.0001 −0.0004 −0.0003 −0.0001 0.0013 0.0010 −0.0001 0.0005 −0.0003

−1.9090 −1.7516 −1.9894 −1.8794 −1.9091 −1.8469 −2.0250 −1.9333 −2.3219 −1.8747 −1.7790 −1.6632 −2.0733 −1.6200 −1.5462 −1.5966 −1.9377 −1.4552 −1.5190 −1.8606

2.0283 2.3515 2.0958 2.0601 2.1029 2.1165 2.3279 2.4048 2.3962 2.3411 2.0246 1.9954 2.7046 1.9340 1.7694 2.5274 2.2497 1.9636 2.0321 1.8550

0.6131 0.5686 0.6087 0.6030 0.6151 0.5613 0.6571 0.6280 0.5660 0.5555 0.5277 0.5260 0.6401 0.5618 0.4851 0.5283 0.5494 0.5195 0.5282 0.5111

−123.7677 −114.2311 −122.9195 −122.4791 −107.3403 −90.3977 −79.9548 −79.7470 −86.6733 −70.7257 −70.6575 −72.7798 −73.4957 −75.9289 −69.3688 −73.6106 −69.0719 −78.1154 −72.2884 −72.9069

*** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** ***

Notes: Daily security values are obtained from Nasdaq (http://nasdaq.com). The sample period ends with October 2018. *, **, and *** refer to statistical signiﬁcance at the 10%, 5%, and 1% levels respectively.

Table A.5 Descriptive statistics for securities' daily intraday standard deviation as proxied by the log diﬀerence of intraday high and intraday low values (σitHL). Title

Notation

First

Mean

Min.

Max.

Stdev.

DF-z

S&P 500 NASDAQ 100 Dow 30 S&P 100 Russell 2000 United States Oil ETF SPDR Gold ETF CurrencyShares Euro ETF iShares MSCI EAFE ETF

ΔGSPCt ΔNDXt ΔDJIt ΔOEXt ΔRUTt ΔUSOt ΔGLDt ΔFXEt ΔEFAt

01/03/2000 10/12/2000 01/03/2000 01/03/2000 01/05/2004 05/14/2007 03/15/2010 03/15/2010 01/03/2008

0.0132 0.0170 0.0130 0.0133 0.0156 0.0233 0.0092 0.0049 0.0119

0.0015 0.0021 −0.0034 0.0018 0.0022 0.0025 0.0017 0.0006 0.0017

0.1090 0.1141 0.1215 0.1152 0.1263 0.1792 0.0568 0.0243 0.1161

0.0101 0.0127 0.0098 0.0102 0.0110 0.0146 0.0058 0.0028 0.0105

−31.0189 −28.1564 −31.9229 −30.5714 −28.8022 −28.0496 −33.2285 −33.6469 −20.9647

*** *** *** *** *** *** *** *** ***

(continued on next page) 8

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M.F. Dicle and J. Levendis

Table A.5 (continued) Title

Notation

First

Mean

Min.

Max.

Stdev.

DF-z

iShares MSCI Emerging Markets ETF iShares China Large-Cap ETF iShares MSCI Brazil Capped ETF iShares Silver Trust VanEck Vectors Gold Miners ETF Energy Select Sector SPDR ETF Amazon.com, Inc. Apple Inc. The Goldman Sachs Group, Inc. Alphabet Inc. IBM

ΔEEMt ΔFXIt ΔEWZt ΔSLVt ΔGDXt ΔXLEt ΔAMZNt ΔAAPLt ΔGSt ΔGOOGt ΔIBMt

03/17/2011 03/17/2011 03/17/2011 03/17/2011 03/17/2011 03/17/2011 06/02/2010 06/02/2010 06/02/2010 06/02/2010 06/02/2010

0.0114 0.0122 0.0208 0.0162 0.0269 0.0161 0.0224 0.0182 0.0196 0.0170 0.0137

0.0027 0.0026 0.0053 0.0033 0.0058 0.0034 0.0037 0.0042 0.0050 0.0038 0.0036

0.0714 0.0844 0.0847 0.1135 0.1195 0.0846 0.1199 0.1677 0.1222 0.1164 0.0551

0.0069 0.0072 0.0098 0.0113 0.0144 0.0089 0.0121 0.0101 0.0106 0.0091 0.0066

−24.0956 −22.7372 −28.1528 −26.7333 −26.8190 −21.0326 −27.3136 −27.7347 −25.8583 −26.7679 −28.8834

*** *** *** *** *** *** *** *** *** *** ***

Notes: Daily security values are obtained from Nasdaq (http://nasdaq.com). The sample period ends with October 2018. *, **, and *** refer to statistical signiﬁcance at the 10%, 5%, and 1% levels respectively.

Table A.6 Granger causality estimation summary results for Eqs. (2) and (3). VIX

Security

γ1i1

γ1i2

χ2

γ2i1

γ2i2

χ2

Δσit20 → ΔVIXit VIX VXN VXD VXO RVX OVX GVZ EVZ VXEFA VXEEM VXFXI VXEWZ VXSLV VXGDX VXXLE VXAZN VXAPL VXGS VXGOG VXIBM EUVIX JYVIX BPVIX

GSPC NDX DJI OEX RUT USO GLD FXE EFA EEM FXI EWZ SLV GDX XLE AMZN AAPL GS GOOG IBM DEXUSEU DEXJPUS DEXUSUK

−0.0144 −0.0083 0.0092 −0.0118 −0.0039 0.0119 −0.0268 0.0077 0.047 0.0013 0.01 −0.0149 −0.0072 0.0755 0.0004 −0.0606 −0.014 −0.0085 −0.001 −0.0494 −0.002 −0.0108 0.0163

* **

*** ***

***

−0.0364 −0.0253 −0.0275 −0.0443 −0.024 0.0036 −0.0273 −0.0246 −0.0157 −0.031 −0.0474 0.0193 −0.0411 0.0091 −0.0252 −0.0189 −0.0598 0.004 −0.0271 −0.0353 0 −0.0125 0.0066

** ** ** ***

*

** ***

*** * **

7.6292 4.4016 4.5699 8.8284 2.6612 0.7209 6.0695 2.8363 4.9939 1.7731 6.2992 1.8553 8.7879 13.803 1.7295 15.8869 13.1034 0.2264 2.7617 19.946 0.019 1.7836 1.8653

**

**

** * ** ** *** *** ***

***

ΔVIXit → Δσit20 0.0422 0.0418 0.0458 0.0414 0.0531 0.0143 0.0213 0.0154 0.0291 0.0161 0.0144 0.0335 0.0305 0.0196 0.0287 0.0473 0.0439 0.0353 −0.0133 0.0587 0.038 0.0607 0.0152

*** ** *** *** ***

*

**

0.0818 0.0784 0.0842 0.067 0.051 0.0618 0.0659 0.0277 0.0383 0.0602 0.093 0.02 0.0282 0.0292 0.0659 0.0585 0.0391 0.0222 −0.0219 0.1192 0.0562 0.0313 0.0783

*** *** *** *** *** ** ** ** *** ***

** *

*** ** ***

33.8107 23.5359 32.3115 28.5279 14.3794 6.334 4.4975 1.0571 6.9876 7.0565 10.5819 1.5044 0.9468 1.3427 6.4373 5.7825 4.4204 2.5856 0.6933 12.7892 6.8611 5.3813 7.7305

*** *** *** *** *** **

** ** ***

** *

*** ** * **

Notes: VIX refers to individual implied volatility indexes. Securities are the paired securities with the corresponding VIX index. Δ refers to log diﬀerence change, t to each trading day, i to each index or security, and σit20 to the running standard deviation of daily returns with a twenty-day window for security i. *, **, and *** refer to statistical signiﬁcance at the 10%, 5%, and 1% levels respectively.

Table A.7 Granger causality estimation summary results for Eqs. (8) and (9). VIX

Security

γ1i1

γ1i2

χ2

γ2i1

γ2i2

χ2

ΔσitI → ΔVIXit VIX VXN VXD VXO RVX OVX GVZ EVZ

GSPC NDX DJI OEX RUT USO GLD FXE

0.0005 0.0024 0.0026 0.0027 0.0003 0.0012 −0.0021 0.0013

0.0013 0.0017 0.0018 0.002 0.0012 −0.0022 −0.0004 −0.0027

0.4492 1.8333 2.0734 1.5501 0.5307 3.5597 1.1215 5.1739

ΔVIXit → ΔσitI

*

1.2018 1.1451 1.0311 1.0734 1.214 0.6023 0.6208 0.571

*** *** *** *** *** *** *** **

0.6306 0.1871 0.6379 0.5135 0.275 0.3897 −0.1264 0.1053

*** *** *** * **

147.9574 87.7311 103.366 141.3361 67.1265 12.9606 7.9051 5.4582

*** *** *** *** *** *** ** *

(continued on next page) 9

Global Finance Journal xxx (xxxx) xxxx

M.F. Dicle and J. Levendis

Table A.7 (continued) VIX

Security

γ1i1

γ1i2

χ2

γ2i1

γ2i2

χ2

ΔσitI → ΔVIXit VXEFA VXEEM VXFXI VXEWZ VXSLV VXGDX VXXLE VXAZN VXAPL VXGS VXGOG VXIBM

EFA EEM FXI EWZ SLV GDX XLE AMZN AAPL GS GOOG IBM

0.0069 0.0054 0.0063 0.0047 −0.0005 0.0089 0.0073 −0.0108 0.0061 0.0071 −0.0029 −0.0033

** * ** * *** ** *** ** **

0.0064 0.0017 0.001 0.0044 0.0012 0.0066 0.0032 −0.0066 −0.0018 0.0031 0.0016 −0.0049

**

* *** *

6.2216 3.4798 6.8476 4.4893 0.707 12.9317 5.882 9.9875 6.7264 5.4713 1.5965 2.7852

ΔVIXit → ΔσitI ** **

*** * *** ** *

0.5862 0.6612 0.7482 0.1164 0.4804 0.2079 0.7495 0.4257 1.0198 0.4958 0.0552 0.4508

*** *** *** * *** *** *** *** ***

0.1704 0.0947 0.073 −0.2655 −0.6465 −0.2296 0.1655 0.3951 0.0072 −0.0407 0.1809 0.3449

**

***

**

25.5377 13.7281 13.1862 2.0919 9.4615 3.3873 18.7682 18.0949 41.0409 10.4003 1.8215 13.8668

*** *** *** *** *** *** *** *** ***

Notes: VIX refers to individual implied volatility indexes. Securities are the paired securities with the corresponding VIX index. Δ refers to log diﬀerence change, t to each trading day, i to each index or security, and σitI to the daily standard deviation of intraday high, low, opening, and closing values for security i. *, **, and *** refer to statistical signiﬁcance at the 10%, 5%, and 1% levels respectively.

Table A.8 Granger causality estimation summary results for Eqs. (13) and (14). VIX

Security

γ1i1

γ1i2

χ2

γ2i1

γ2i2

χ2

ΔσitHL → ΔVIXit VIX VXN VXD VXO RVX OVX GVZ EVZ VXEFA VXEEM VXFXI VXEWZ VXSLV VXGDX VXXLE VXAZN VXAPL VXGS VXGOG VXIBM

GSPC NDX DJI OEX RUT USO GLD FXE EFA EEM FXI EWZ SLV GDX XLE AMZN AAPL GS GOOG IBM

−0.1173 0.154 0.139 0.0748 −0.0731 0.0498 −0.6214 0.0177 0.5733 0.0458 0.2791 −0.0162 −0.0894 0.3394 0.1204 −0.7299 −0.2821 −0.0133 −0.5956 −0.5556

*** ***

*** *** * *** **

−0.3407 −0.3454 −0.421 −0.5562 −0.1711 −0.247 −0.613 −1.0017 −0.7783 −0.8568 −0.7764 −0.1761 −0.1684 −0.1954 −0.6209 −0.0734 −0.9804 −0.5721 −0.1641 −0.4961

** *** *** *** *** *** *** *** *** ***

** *** *** *** **

18.7917 14.3512 13.1857 21.7674 7.1898 13.5599 24.6169 7.977 13.5149 15.7514 15.7253 2.3728 5.0422 12.3807 14.4356 32.3704 61.2825 20.3909 15.8573 18.0434

*** *** *** *** ** *** *** ** *** *** *** * *** *** *** *** *** *** ***

ΔVIXit → ΔσitHL 0.0138 0.0143 0.0114 0.0123 0.0199 0.0142 0.0081 0.0027 0.0085 0.0109 0.0102 0.0139 0.0104 0.0128 0.0175 0.0108 0.022 0.0133 0.0016 0.0075

*** *** *** *** *** *** *** ** *** *** *** *** ** ** *** *** *** *** ***

0.0015 −0.0035 0.0038 0.0011 −0.0023 0.0128 −0.0013 0.0006 0 0.0018 0.0041 0.0013 −0.0109 0.0027 −0.0007 0.0048 0.0004 0.003 0.0017 0.0025

* **

***

**

83.4374 47.3882 54.9308 82.5162 73.7481 16.5838 13.5755 4.9582 26.408 26.7082 17.349 12.0738 9.8307 5.7923 38.4734 14.6705 56.6495 18.3506 0.9404 14.9716

*** *** *** *** *** *** *** * *** *** *** *** *** * *** *** *** *** ***

Notes: VIX refers to individual implied volatility indexes. Securities are the paired securities with the corresponding VIX index. Δ refers to log diﬀerence change, t to each trading day, i to each index or security, and σitHL to the log diﬀerence of intraday low and intraday high for security i. *, **, and *** refer to statistical signiﬁcance at the 10%, 5%, and 1% levels respectively.

Appendix B. Granger noncausality estimations with the S&P 500 as an exogenous variable Table B.9 Granger causality estimation summary results for Eqs. (2) and (3) with S&P 500 daily returns as an exogenous variable. VIX

VIX VXN VXD VXO RVX OVX

Security

GSPC NDX DJI OEX RUT USO

χ2

χ2

Δσit20 → ΔVIXit

ΔVIXit → Δσit20

15.1429 7.4408 9.1336 18.6703 4.3475 0.1105

*** ** ** ***

36.9428 25.5881 35.3302 32.1517 17.052 6.4331

*** *** *** *** *** **

(continued on next page) 10

Global Finance Journal xxx (xxxx) xxxx

M.F. Dicle and J. Levendis

Table B.9 (continued) VIX

GVZ EVZ VXEFA VXEEM VXFXI VXEWZ VXSLV VXGDX VXXLE VXAZN VXAPL VXGS VXGOG VXIBM EUVIX JYVIX BPVIX

Security

GLD FXE EFA EEM FXI EWZ SLV GDX XLE AMZN AAPL GS GOOG IBM DEXUSEU DEXJPUS DEXUSUK

χ2

χ2

Δσit20 → ΔVIXit

ΔVIXit → Δσit20

5.1165 2.1191 4.2457 2.3868 8.2951 2.6594 10.747 20.0627 2.0059 15.3268 14.0884 1.3968 1.0286 17.8163 0.019 1.7836 1.8653

*

** *** *** *** ***

***

4.7183 1.0483 8.0938 7.0657 10.9497 1.5183 1.0085 1.3262 5.839 5.8307 4.3839 2.9012 0.6979 12.8387 6.8611 5.3813 7.7305

* ** ** ***

* *

*** ** * **

Notes: VIX refers to individual implied volatility indexes. Securities are the paired securities with the corresponding VIX index. Δ refers to log diﬀerence change, t to each trading day, i to each index or security, and σit20 to the running standard deviation of daily returns with a twenty-day window for security i. *, **, and *** refer to statistical signiﬁcance at the 10%, 5%, and 1% levels respectively.

Table B.10 Granger causality estimation summary results for Eqs. (8) and (9) with S&P 500 daily returns as an exogenous variable. VIX

VIX VXN VXD VXO RVX OVX GVZ EVZ VXEFA VXEEM VXFXI VXEWZ VXSLV VXGDX VXXLE VXAZN VXAPL VXGS VXGOG VXIBM

Security

GSPC NDX DJI OEX RUT USO GLD FXE EFA EEM FXI EWZ SLV GDX XLE AMZN AAPL GS GOOG IBM

χ2

χ2

ΔσitI → ΔVIXit

ΔVIXit → ΔσitI

0.1166 3.6073 3.5205 2.2515 0.3437 4.3708 3.6282 5.8071 9.9096 4.1299 6.7383 6.7998 2.0781 9.9095 2.5538 10.5441 2.9996 3.4208 3.2014 5.723

158.228 95.1664 111.1952 154.0112 75.1367 13.3452 9.0908 5.4376 28.503 15.54 14.5446 2.5551 10.0907 4.5635 20.5973 18.6198 44.0706 11.9069 1.7318 14.6935

* *** ** ** *** ***

*

*** *** *** *** *** *** ** * *** *** *** *** *** *** *** *** ***

Notes: VIX refers to individual implied volatility indexes. Securities are the paired securities with the corresponding VIX index. Δ refers to log diﬀerence change, t to each trading day, i to each index or security, and σitI to the daily standard deviation of intraday high, low, opening, and closing values for security i. *, **, and *** refer to statistical signiﬁcance at the 10%, 5%, and 1% levels respectively.

Table B.11 Granger causality estimation summary results for Eqs. (13) and (14) with S&P 500 daily returns as an exogenous variable. VIX

VIX VXN

Security

GSPC NDX

χ2

χ2

ΔσitHL → ΔVIXit

ΔVIXit → ΔσitHL

27.2576 26.2371

*** ***

94.2238 52.5083

*** ***

(continued on next page) 11

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Table B.11 (continued) VIX

VXD VXO RVX OVX GVZ EVZ VXEFA VXEEM VXFXI VXEWZ VXSLV VXGDX VXXLE VXAZN VXAPL VXGS VXGOG VXIBM

Security

DJI OEX RUT USO GLD FXE EFA EEM FXI EWZ SLV GDX XLE AMZN AAPL GS GOOG IBM

χ2

χ2

ΔσitHL → ΔVIXit

ΔVIXit → ΔσitHL

21.3507 34.9834 10.7668 18.4558 27.3901 7.6505 14.4689 13.0576 10.8076 3.0851 6.3777 12.3095 10.238 33.1785 56.2041 8.1035 17.2224 14.7201

*** *** *** *** *** ** *** *** *** ** *** *** *** *** ** *** ***

64.1691 96.6948 85.5218 16.9809 15.0875 4.9136 30.0111 29.8492 19.5189 12.4691 10.5575 6.9308 43.9242 16.1531 61.9247 20.674 1.2287 16.7599

*** *** *** *** *** * *** *** *** *** *** ** *** *** *** *** ***

Notes: VIX refers to individual implied volatility indexes. Securities are the paired securities with the corresponding VIX index. Δ refers to log diﬀerence change, t to each trading day, i to each index or security, and σitHL to the log diﬀerence of intraday low and intraday high for security i. *, **, and *** refer to statistical signiﬁcance at the 10%, 5%, and 1% levels respectively.

Appendix C. Granger noncausality between GARCH volatilities and implied volatilities Table C.12 Granger causality estimation summary results for GARCH-based volatility measures and implied volatilities. VIX

VIX VXN VXD VXO RVX OVX GVZ EVZ VXEFA VXEEM VXFXI VXEWZ VXSLV VXGDX VXXLE VXAZN VXAPL VXGS VXGOG VXIBM EUVIX JYVIX BPVIX

Security

GSPC NDX DJI OEX RUT USO GLD FXE EFA EEM FXI EWZ SLV GDX XLE AMZN AAPL GS GOOG IBM DEXUSEU DEXJPUS DEXUSUK

χ2

χ2

hit → ΔVIXit

ΔVIXit → hit

4.17 4.4213 4.0853 5.1427 3.5939 4.66 14.9466 3.5105 2.397 12.2793 13.7232 15.8863 12.6549 10.032 6.8654 7.248 24.1338 9.6003 4.7745 16.6756 3.0188 7.7043 8.664

144.6606 63.7483 107.4833 111.4365 164.701 243.0305 415.5784 31.8015 37.3558 137.7391 241.8702 133.6061 489.836 139.341 142.7808 136.2302 21.9868 124.8601 99.2481 121.4021 103.8281 209.0134 25.1986

* * ***

*** *** *** *** *** ** ** *** *** * *** ** **

*** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** ***

Notes: VIX refers to individual implied volatility indexes. Securities are the paired securities with the corresponding VIX index. t refers to each trading day, i to each index or security, and hit to the estimated GARCH volatility model of daily returns for security i. *, **, and *** refer to statistical signiﬁcance at the 10%, 5%, and 1% levels respectively.

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