- Email: [email protected]
Directional predictability of implied volatility: From crude oil to developed and emerging stock markets
Accepted Manuscript
Directional predictability of implied volatility: from crude oil to developed and emerging stock markets Elie Bouri , Donald Lien , David Roubaud , Syed Jawad Hussain Shahzad PII: DOI: Reference:
S1544-6123(17)30521-4 10.1016/j.frl.2018.02.022 FRL 875
To appear in:
Finance Research Letters
Received date: Revised date: Accepted date:
30 August 2017 12 December 2017 22 February 2018
Please cite this article as: Elie Bouri , Donald Lien , David Roubaud , Syed Jawad Hussain Shahzad , Directional predictability of implied volatility: from crude oil to developed and emerging stock markets, Finance Research Letters (2018), doi: 10.1016/j.frl.2018.02.022
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Highlights
Examine the directional predictability of implied volatility from crude oil to some major developed and emerging stock markets. Apply the cross-quantilograms via Han et al. (2016).
Identify predictability around the extreme end of oil volatility.
Rolling window analysis validates results.
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Directional predictability of implied volatility: from crude oil to developed and emerging stock markets
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Elie Bouri USEK Business School, Holy Spirit University of Kaslik Jounieh, Lebanon. Email: [email protected] Donald Lien College of Business, University of Texas at San Antonio, One University Circle, San Antonio, TX 78249, USA. Email: [email protected]
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David Roubaud Center for Energy and Sustainable Development, Montpellier Business School Montpellier, France. Email: [email protected]
Syed Jawad Hussain Shahzad Montpellier Business School, Montpellier France. E-mail: [email protected]
Abstract
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This paper investigates whether the implied volatility of crude oil improves the directional predictability of the implied volatility index for some major developed and emerging stock
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markets. Using cross-quantilograms via Han et al (2016), we find strong and persistent quantile predictability when the crude oil implied volatility is low. The effect remains significant but a bit weaker when the oil implied volatility is high. There is no improvement in directional
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predictability when the implied volatility of oil is at the medium level. The rolling window
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analysis indicates the above results are robust when the global financial crisis period is excluded. Keywords: Directional predictability; oil implied volatility; stock implied volatility; VIX;
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quantile dependence; developed and emerging stock markets JEL classification: C10; G10; G15
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1. Introduction
The predictability in financial markets is central to the international finance and the efficient market theory, especially the predictability of volatility is recognized to be more important than
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that of return. However, market participants are more concerned with future volatility than current volatility. Implied volatility is such a statistic derived from the options market. It reflects investors’ expectations of future market conditions and is informationally superior to historical volatility (Becker et al., 2007).
In the puzzling oil-stock nexus, the predictive power of oil market for stock indices continues to
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generate considerable debate (see Degiannakis et al. (2017) for a detailed literature review), especially with the use of implied volatility data. Liu et al. (2013) uncover the short- and long-
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term volatility spillovers. Kang et al. (2015) use a structural VAR and consider the impact of oil shocks on the covariance of US stock market volatility. Maghyereh et al. (2016) apply Granger
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causality and connectedness measures of Diebold and Yilmaz (2012). Raza et al. (2016) uncover the short- and long-run asymmetric relationships. Campos et al. (2017) include traditional macro-
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finance variable within Heterogeneous Autoregressive models. Bouri et al. (2017) focus on cointegration relationships and nonlinear causality. Based on the above, we notice that the predictability of oil volatility for stock market volatility in the different quantiles remains
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unexplored although the reaction of stock indices to crude oil is found to be heterogeneous across quantiles (Sim and Zhou, 2015; Reboredo and Ugolini, 2016). To address this gap, we examine the directional predictability of the volatility of stock markets based on oil implied volatility in low, median, and high quantiles via the cross-quantilogram of Han et al. (2016). In addition to its capability to detect the extreme quantiles dependence between two stationary series beyond the center and to address the stylized facts of data such as non-normality and long memory (Lee et al., 2017), the approach of Han et al. (2016) offers several advantages. First, it
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captures the magnitude, duration, and direction of the relationship between two stationary series. Second, it allows for an arbitrary selection of quantiles. Finally, the bootstrap element of Han et al. (2016) approach permits for the use of large lags in testing the directional predictability. Our main analysis shows that when the oil is at low risk, there is a greater probability that the stock index is of low risk as well. If the oil is at high risk, there is a lower probability for the
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stock index to be at high risk. When the oil is at average risk level, it provides no directional predictability for the implied volatility of stock index. Finally, our sample period starts with the global financial crisis (GFC) period (owing to the data availability) and therefore invites the concerns for structural breaks. We conduct the rolling-window analysis and report that the results
2. The bivariate cross-quantilogram We define two stationary time series as {
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validate the above conclusions.
}, i = 1, 2 where
and
represent the
implied volatility of country stock market (VIX) and oil implied volatility (OVX), respectively. The density functions and distribution of series -quantile of
is
( )
*
+ for
( )
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The
( ) and
are denoted by
dimensional series of quantiles are represented by ( (
)
(
(
( ), respectively.
). The expression of two-
)) , for α ≡(
) where the
lags is (
√ [
(
(
))
(
))] √ [
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[
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superscript T denotes the vector transpose operator. The cross-qunatilogram for α quantile with k
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( )
for k = 0, ±1, ±2, …, where
(
))]
(
( ) ≡ 1,
(
-
(1)
))]
α , 1( ) denotes the indicator function, and
( )] is called a quantile-hit. For the purpose of our study, we measure the directional
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1[
predictability through OVX by t.
(
( )
( ) below or above a quantile
(
(
) or
indicates that the oil VIX being below or above the quantile
(
) at time ) at time t
does not provide useful information for predicting whether the country VIX will be below or above the quantile
(
( )
) on the next trading day (t + 1). Conversely,
one-day directional predictability from OVX to country VIX at The sample analog of the cross-quantilogram is
=
(
) or
indicates a (
).
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∑
̂ ( )
̂ (
(
√∑
̂ (
(
))
̂ (
(
))√∑
)) ̂ (
(
(2)
))
for k = 0, ±1, ±2, …. In Eq. (2), ̂ ( ) indicates the unconditional sample quantile of proposed by Han et al. (2016). Furthermore, for
, Han et a. (2016) suggest a quantile ( ) = 0 for all k,
( )
against the
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version of the Ljung-Box-Pierce statistic to test alternative hypothesis
, as
0 for at least one k,
, under the below portmanteau
( ) test statistic, ̂ , for directional predictability from one time series to another for up to p lags
over the quantile pair )
(
)∑
).
̂ ( )
(3)
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̂(
(
Han et al. (2016) point the need to approximate the null distribution and conduct inferences. Accordingly, they use the stationary bootstrap of Politis and Romano (1994), which allows for {(
handling inherent serial dependence in data. Let
)}
the i-th block
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with length Li starting from ki.; Li indicates an independent and identically distributed variable with Prob (Li = s) = (
)
uniform distribution *
+ In case the upper limit (
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, s = 1,2,… for
t > T, we replace the pair (
) by (
(0,1); ki is an IID sequence drawn from a ) exceeds the sample size (T), when ) with j = k + (t mod(T-k)). The
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bootstrapped confidence intervals are constructed through pseudo re-sampling based on the sequence of blocks and the associated portmanteau test statistic. The approach of Han et al.
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(2016) was recently applied by Jiang et al. (2016) and Baumöhl and Lyócsa (2017).
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3. Empirical results 3.1 The data set
We use daily closing levels of the implied volatility index of crude oil (OVX) and the implied volatility indices of developed and emerging stock markets (France, Germany, India, Japan, Mexico, Netherland, Russia, South Africa, Sweden, Switzerland, UK, and USA), extracted from DataStream. The volatility series are plotted in Appendix Figure 1A. Owing to the data availability, our sample period is from June 3, 2008 – December 30, 2016. Importantly, it covers
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the GFC period as in Han et al. (2016)1. As shown in Appendix Table 1A, both the mean and standard deviation of the Russian VIX are higher than that of VIXs for the other countries and of the OVX. The Jarque-Bera test statistics confirm the non-normality of the series. All series are not stationary at levels, but are stationary at the first differences. We therefore conduct our analyses with the first difference of indices.
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Before applying the bivariate cross-quantilogram, we conduct the linear VAR Granger causality analysis (see Appendix Table 2A). There is a significant linear causality from OVX to each of the country VIXs at the 5% level.
We also examine the existence of non-linearity by applying the Brock–Dechert–Scheinkman (BDS) test of Brock et al. (1996) on the residuals of the OVX and country VIX equations in the
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VAR model. The p-values of the BDS test reject the null of no serial dependence in the residuals, providing evidence of non-linearity in the data (See Appendix Table 3A). This finding suggests the non-appropriateness of using a linear causality framework to examine predictability. We consider the presence of structural breaks by applying the tests of multiple structural breaks (Bai and Perron, 2003) to VAR models. Unreported results suggest the presence of at least one
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structural break around the GFC in all series. We will further examine this issue with the rollingwindow method.
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3.2 Return predictability from OVX to country VIXs
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Figure 1 presents the results of the daily price changes predictability for the 12 country VIXs for lag k =1, 2,….., 60 from the price changes of the OVX. We report the results of the cross-
Insert Figure 1 here
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quantilogram when both the OVX and country VIX are in three different quantiles.
For the case of France, there is generally a long-lived directional positive predictability in the low quantile (0.10) indicating that when the changes in the OVX are negative and very large in magnitude, it is very likely that the French VIX will also experience very large negative changes in the following 10 days. In contrast, in the high quantile (0.90), the cross-quantilogram is not significant at the next day, but significantly negative from day two to day five, suggesting that 1
Results for the sample period exclusive of the GFC are available from the authors.
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when the changes in the OVX are very low it is very likely that the French VIX will experience a very large positive changes in the following five days, except for the first day. In the median quantiles, the cross-quantilogram does not exhibit consistency in either the sign or the pattern. In fact, it oscillates very close to zero between positive and negative signs, except for day 15, and 40 for which there is a positive predictability. Similar results are reported for Japan, Mexico,
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Netherland, Russia, South Africa, Sweden, Switzerland, UK, and USA. In all these countries, the directional predictability of VIX is significant when OVX is at the low or high quantile. If the OVX is at the median quantile, no predictability is offered by the oil market. Germany and India present the other case. The next day effect is stronger in Germany than in France in the low quantile. For the high quantile (0.90), all the values of the cross-quantilogram are insignificant,
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except for lag three. This evidence of negative predictability implies that it is more likely for the German VIX to have a large positive change at day three when the changes in the OVX are very low. There is no predictability at the median quantile. Similar results are found for India. In other words, for these two countries, oil improves directional predictability only when it is at low risk. These results add to that of Bouri (2017) and Maghyereh et al. (2016) by showing that the
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dominance of oil volatility over the volatility of stock indices is only present in the tails and not
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around the mean, confirming the appropriateness of using a quantile-based approach.
Insert Figure 2 here
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We further examine the directional causality from all quantiles of OVX to all quantiles of VIXs. These quantiles are displayed on x- and y- axis, respectively, in Figure 2. The magnitude
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(positive/negative significant spillovers) is shown through colour combination from blue (high negative) to red (high positive), which is also shown through the color bar provided at the end of
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figure.
3.3 Rolling VIX predictability Given our sample period covers the GFC that potentially lead to structural changes in the VIX series, we conduct a rolling-window analysis to assess the robustness of our main results for the entire sample period (Figure 3). The recursive forward window size is maintained at 500 days and the step size is 22 days, approximately a month. Cross-quantilogram of daily spillovers from
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the OVX to country VIX when both markets take same quantiles are calculated with one day lag (i.e., p = 1).
Insert Figure 3 here
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For Germany and India, Today’s VOX helps predict tomorrow’s VIX if is of the low quantile, which is consistent with the complete sample result. Moreover, the high quintile VOX also improves the predictability of VIX if the GFC observations are removed. To be precise, consider the window that ends after May 1, 2012. For these windows, the starting point is after May 1, 2010 passing the GFC period. Figure 3 displays the significant predictability of VOX for the
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post-GFC period but not during the GFC period. Similar observation regarding the high quantile VOX appears in Netherlands, Russia, Sweden, Switzerland, and the UK. Thus, the GFC period is indeed different from the post period. Nevertheless, the results obtained from the complete sample remain valid for these countries for the post-GFC period.
Although the high quantile VOX helps predict VIX in the post GFC period, the low quantile
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VOX loses its predictive power in Sweden. The situation is much worse in the USA where, VOX has no predictability at all quantiles. These results are not consistent with the complete sample
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results. It is mostly due to the difference in the sample size. Also, the complete sample results are based upon the predictability with several possible lagged responses whereas the rolling-window method is restricted to the one-day-ahead predictability.
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Taken together, if the crude oil price is stable in the future, the market would expect the current economy to move forward smoothly, and hence the future stock index level to be stable as well.
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Accordingly, the higher probability of low oil implied volatility in the future leads to higher probability of low stock index implied volatility. A positive lead-lag relationship prevails when
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the oil volatility lies in the low quantile. Conversely, when the future oil price is very unstable, there is likely turmoil in the economy and subsequently the government would intervene. Any effective government policies help reduce the volatility of the stock market. Therefore, when the probability of large oil volatility is high today, the chance of large stock index volatility will be lower tomorrow in anticipation of the government interference. Herein a negative lead-lag relationship prevails when the oil volatility lies in the high quantile. On the other hand, during the great financial crisis period, the market does not have confidence in the effectiveness of the
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government policies. As a consequence, the above negative lead-lag relationship does not apply. Finally, when the oil volatility is at the median level, it has no predictable power for the implied volatility in the stock index, suggesting that the risk in the stock market would be determined by other factors.
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4. Conclusion We contributed to the literature on the ability of the crude oil to predict stock market conditions using implied volatility data and the cross-quantilograms approach. While the linear Granger causality test in the mean documents a lead-lag relationship in implied volatilities from oil to stock indices, the cross-quantilograms via Han et al (2016) shows a different and more nuanced
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picture. Specifically, when the risk of the oil is at the average level, there is no directional predictability at all. In contrast, the lead-lag relationship mainly holds at the tails of the oil implied volatility, in particular the left tails. The result is further validated through a rollingwindow analysis. In spite of possible structural change in the data structure such as time-varying covariance, all the samples constructed by the rolling windows display the same lead-lag
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relationships.
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References
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Campos, I., Cortazar, G. and Reyes, T., 2017. Modeling and Predicting Oil VIX: Internet Search Volume versus Traditional Macro-finance Variables. Energy Economics. https://doi.org/10.1016/j.eneco.2017.06.009
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Maghyereh, A.I., Awartani, B., Bouri E., 2016. The directional volatility connectedness between crude oil and equity markets: new evidence from implied volatility indexes. Energy Economics, 57, 78-93.
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Politis, D. N., Romano, J. P. 1994. The Stationary Bootstrap. Journal of the American Statistical Association, 89(428), 1303-1313. Raza, N., Shahzad, S.J.H., Tiwari, A.K. and Shahbaz, M., 2016. Asymmetric impact of gold, oil prices and their volatilities on stock prices of emerging markets. Resources Policy, 49, 290-301. Reboredo, J.C. and Ugolini, A., 2016. Quantile dependence of oil price movements and stock returns. Energy economics, 54, 33-49.
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Figure 1: Daily volatility spillovers from OVX to VIX. Panel A:
=0.10
Panel B:
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b). Germany
c). India
Panel C:
=0.90
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a). France
=0.50
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d). Japan
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e). Mexico
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f). Netherland
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g). Russia
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h). South Africa
i). Sweden
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j). Switzerland
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k). UK
l). USA
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Note: This Figure reports the sample cross-quantilogram of daily spillovers from the OVX to stock market volatilities Results for the low quantiles (αovx = αvix = 0.1) in Panel A, the median quantiles (αovx = αvix = 0.5) in Panel B, and the high quantiles (αovx = αvix = 0.9) in Panel C are given in the upper panels, from left to right respectively. Lag p is 1. Lower panels report the Ljung-Box type test statistics. Red-dashed lines represent the 95% bootstrapped confidence intervals for no directional predictability with 1,000 bootstrapped replicates. For each country, the upper chart indicates the cross-quantilogram results, where the lag k is on the horizontal axis; the lower chart presents the portmanteau test statistics.
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Figure 2: Directional causality in quantiles from OVX to country VIX Panel A: Lags=1
Panel A: Lags=5
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a). France
Panel A: Lags=22
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b). Germany
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controlc). India
d). Japan
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e). Mexico
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f). Netherland
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h). South Africa
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i). Sweden
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k). UK
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l). USA
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Note: x-axis corresponds to a quantile of the OVX and y-axis to a quantile of the country VIX. Color-bar at the end shows the intensity of the causal-flows.
Panel A:
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Figure 3: Rolling daily spillovers from OVX to country VIX =0.10
Panel B:
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a). France
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c). India
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Panel C:
=0.90
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d). Japan
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g). Russia
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h). South Africa
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i). Sweden
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k). UK
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l). USA
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Notes: This Figure reports the rolling-window (500 days recursive forward window with a step size of 22 days, approximately a month) cross-quantilogram of daily spillovers from the OVX to country VIX when both markets take same quantiles. Results for low quantile (OVX =0.1 to˛VIX =0.1), median quantile (OVX =0.5 to˛VIX =0.5), and high quantile (OVX =0.9 to˛VIX =0.9) are given in the left, middle, and right panels, respectively. Lag p is 1. Starting year of the rolling window is marked on the horizontal axis. Blue lines are the rolling cross-quantilogram for 1 day spillovers, dashed red lines are 95% bootstrapped confidence intervals for no predictability based on 1,000 bootstrapped replicates.
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Appendix
b). Germany
c). India
d). Japan
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a). France
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Figure 1A: Time series plots of the OVX (red) and country stock market volatility (blue)
f). Netherland
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e). Mexico
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g). Russia
h). South Africa
j). Switzerland
k). UK
l). USA
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Table 1A: Statistical properties of the sample data ADF -2.777* -4.702*** -3.589*** -3.424** -4.783*** -2.311 -4.048*** -3.384** -2.487 -2.957** -3.388** -4.299*** -2.819*
ADF(D) -31.79*** -39.02*** -25.14*** -62.72*** -30.83*** -39.44*** -30.54*** -38.76*** -36.30*** -26.71*** -25.12*** -22.87*** -22.70***
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J-B 811.97*** 4516.92*** 6041.86*** 2817.86*** 11552.84*** 5377.84*** 4060.77*** 25992.18*** 3258.73*** 3745.30*** 11872.85*** 6519.55*** 5773.66***
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Mean Std. Dev. Skewness Kurtosis OVX 38.23 14.93 1.15 4.85 France 24.06 8.70 1.97 8.73 Germany 24.26 9.28 2.20 9.74 India 23.04 9.89 1.85 7.06 Japan 26.93 10.43 2.74 12.68 Mexico 20.99 9.89 2.31 9.02 Netherland 23.10 10.38 2.03 8.21 Russia 39.23 22.87 3.41 18.24 South Africa 22.77 7.06 1.86 7.59 Sweden 21.45 9.63 1.95 8.00 Switzerland 19.47 8.50 2.71 12.89 UK 20.27 9.04 2.27 10.02 USA 20.93 10.17 2.29 9.40
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Note: This table reports the descriptive statistics and unit root properties of the data. Std. Dev. stand for standard deviations. J-B stand for Jarque-Bera test with the null hypothesis of normality. ADF and ADF(D) are the test statistics for augmented Dickey Fuller unit root test for the level and first differenced series, respectively. ***, ** and * stand for significance at 1%, 5% and 10% levels, respectively.
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Table 2A: Linear Granger causality tests Order of the VAR(p) 6 3 5 3 6 7 2 3 6 6 5 3
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France Germany India Japan Mexico Netherland Russia South Africa Sweden Switzerland UK USA
H0: OVX does not Granger cause stock market volatility 7.93973*** 49.4326*** 58.0192*** 6.91128*** 4.88568*** 9.88880*** 18.7009*** 20.5281*** 21.2855*** 10.1683*** 6.59082*** 3.25757**
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Note: This table reports the F-statistics for the no Granger causality restrictions imposed on a linear vector autoregressive (VAR) model under the null hypotheses H0. The order (p) of the VAR is selected by the Bayesian Information Criterion (BIC). *** and ** indicate rejection of the null hypothesis of no Granger causality at 1% and 5% levels of significance, respectively.
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Table 3A: Results from the Brock et al. (1996) – BDS - test m=5
m=6
18.375*** 21.627*** 23.187*** 15.848*** 20.872*** 25.138*** 15.832*** 20.524*** 19.236*** 18.804*** 23.687*** 20.261***
20.137*** 24.068*** 25.272*** 17.095*** 22.783*** 27.413*** 16.759*** 22.016*** 20.944*** 20.726*** 26.707*** 22.569***
21.936*** 26.412*** 27.341*** 18.226*** 24.786*** 29.791*** 17.782*** 24.108*** 22.685*** 22.416*** 29.277*** 25.169***
19.775*** 22.682*** 22.432*** 20.625*** 20.252*** 22.423*** 22.858*** 21.781*** 22.645*** 20.849*** 19.890*** 22.420***
21.504*** 24.692*** 24.488*** 22.630*** 22.180*** 24.404*** 24.841*** 23.723*** 24.615*** 22.892*** 21.802*** 24.359***
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m=4
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France Germany India Japan Mexico Netherland Russia South Africa Sweden Switzerland UK USA
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France Germany India Japan Mexico Netherland Russia South Africa Sweden Switzerland UK USA
m=2 m=3 OVX equation 13.285*** 15.893*** *** 14.624 18.734*** 16.276*** 20.453*** 11.226*** 14.403*** 13.752*** 18.244*** *** 19.154 22.608*** 11.033*** 14.358*** 15.702*** 18.747*** *** 14.509 17.101*** 13.200*** 16.464*** 14.461*** 19.825*** 13.110*** 17.375*** Stock volatility equation 14.237*** 16.260*** 15.150*** 18.274*** *** 15.502 18.469*** 13.999*** 16.600*** 12.697*** 16.239*** 15.934*** 18.564*** *** 15.660 18.737*** 15.230*** 17.801*** 15.764*** 18.752*** 14.233*** 17.038*** *** 13.908 16.269*** 15.903*** 18.659***
18.184*** 20.716*** 20.595*** 18.780*** 18.376*** 20.589*** 21.047*** 20.086*** 20.854*** 19.028*** 18.204*** 20.704***
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Note: The entries indicate the BDS test based on the residuals of OVX series and stock volatility series in a VAR for various countries. m denotes the embedded dimension of the BDS test. *** indicates the rejection of the null of residuals being iid at1 % level of significance.
Directional predictability of implied volatility: from crude oil to developed and emerging stock markets Elie Bouri , Donald Lien , David Roubaud , Syed Jawad Hussain Shahzad PII: DOI: Reference:
S1544-6123(17)30521-4 10.1016/j.frl.2018.02.022 FRL 875
To appear in:
Finance Research Letters
Received date: Revised date: Accepted date:
30 August 2017 12 December 2017 22 February 2018
Please cite this article as: Elie Bouri , Donald Lien , David Roubaud , Syed Jawad Hussain Shahzad , Directional predictability of implied volatility: from crude oil to developed and emerging stock markets, Finance Research Letters (2018), doi: 10.1016/j.frl.2018.02.022
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Highlights
Examine the directional predictability of implied volatility from crude oil to some major developed and emerging stock markets. Apply the cross-quantilograms via Han et al. (2016).
Identify predictability around the extreme end of oil volatility.
Rolling window analysis validates results.
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Directional predictability of implied volatility: from crude oil to developed and emerging stock markets
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Elie Bouri USEK Business School, Holy Spirit University of Kaslik Jounieh, Lebanon. Email: [email protected] Donald Lien College of Business, University of Texas at San Antonio, One University Circle, San Antonio, TX 78249, USA. Email: [email protected]
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David Roubaud Center for Energy and Sustainable Development, Montpellier Business School Montpellier, France. Email: [email protected]
Syed Jawad Hussain Shahzad Montpellier Business School, Montpellier France. E-mail: [email protected]
Abstract
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This paper investigates whether the implied volatility of crude oil improves the directional predictability of the implied volatility index for some major developed and emerging stock
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markets. Using cross-quantilograms via Han et al (2016), we find strong and persistent quantile predictability when the crude oil implied volatility is low. The effect remains significant but a bit weaker when the oil implied volatility is high. There is no improvement in directional
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predictability when the implied volatility of oil is at the medium level. The rolling window
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analysis indicates the above results are robust when the global financial crisis period is excluded. Keywords: Directional predictability; oil implied volatility; stock implied volatility; VIX;
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quantile dependence; developed and emerging stock markets JEL classification: C10; G10; G15
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1. Introduction
The predictability in financial markets is central to the international finance and the efficient market theory, especially the predictability of volatility is recognized to be more important than
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that of return. However, market participants are more concerned with future volatility than current volatility. Implied volatility is such a statistic derived from the options market. It reflects investors’ expectations of future market conditions and is informationally superior to historical volatility (Becker et al., 2007).
In the puzzling oil-stock nexus, the predictive power of oil market for stock indices continues to
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generate considerable debate (see Degiannakis et al. (2017) for a detailed literature review), especially with the use of implied volatility data. Liu et al. (2013) uncover the short- and long-
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term volatility spillovers. Kang et al. (2015) use a structural VAR and consider the impact of oil shocks on the covariance of US stock market volatility. Maghyereh et al. (2016) apply Granger
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causality and connectedness measures of Diebold and Yilmaz (2012). Raza et al. (2016) uncover the short- and long-run asymmetric relationships. Campos et al. (2017) include traditional macro-
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finance variable within Heterogeneous Autoregressive models. Bouri et al. (2017) focus on cointegration relationships and nonlinear causality. Based on the above, we notice that the predictability of oil volatility for stock market volatility in the different quantiles remains
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unexplored although the reaction of stock indices to crude oil is found to be heterogeneous across quantiles (Sim and Zhou, 2015; Reboredo and Ugolini, 2016). To address this gap, we examine the directional predictability of the volatility of stock markets based on oil implied volatility in low, median, and high quantiles via the cross-quantilogram of Han et al. (2016). In addition to its capability to detect the extreme quantiles dependence between two stationary series beyond the center and to address the stylized facts of data such as non-normality and long memory (Lee et al., 2017), the approach of Han et al. (2016) offers several advantages. First, it
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captures the magnitude, duration, and direction of the relationship between two stationary series. Second, it allows for an arbitrary selection of quantiles. Finally, the bootstrap element of Han et al. (2016) approach permits for the use of large lags in testing the directional predictability. Our main analysis shows that when the oil is at low risk, there is a greater probability that the stock index is of low risk as well. If the oil is at high risk, there is a lower probability for the
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stock index to be at high risk. When the oil is at average risk level, it provides no directional predictability for the implied volatility of stock index. Finally, our sample period starts with the global financial crisis (GFC) period (owing to the data availability) and therefore invites the concerns for structural breaks. We conduct the rolling-window analysis and report that the results
2. The bivariate cross-quantilogram We define two stationary time series as {
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validate the above conclusions.
}, i = 1, 2 where
and
represent the
implied volatility of country stock market (VIX) and oil implied volatility (OVX), respectively. The density functions and distribution of series -quantile of
is
( )
*
+ for
( )
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The
( ) and
are denoted by
dimensional series of quantiles are represented by ( (
)
(
(
( ), respectively.
). The expression of two-
)) , for α ≡(
) where the
lags is (
√ [
(
(
))
(
))] √ [
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[
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superscript T denotes the vector transpose operator. The cross-qunatilogram for α quantile with k
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( )
for k = 0, ±1, ±2, …, where
(
))]
(
( ) ≡ 1,
(
-
(1)
))]
α , 1( ) denotes the indicator function, and
( )] is called a quantile-hit. For the purpose of our study, we measure the directional
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1[
predictability through OVX by t.
(
( )
( ) below or above a quantile
(
(
) or
indicates that the oil VIX being below or above the quantile
(
) at time ) at time t
does not provide useful information for predicting whether the country VIX will be below or above the quantile
(
( )
) on the next trading day (t + 1). Conversely,
one-day directional predictability from OVX to country VIX at The sample analog of the cross-quantilogram is
=
(
) or
indicates a (
).
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∑
̂ ( )
̂ (
(
√∑
̂ (
(
))
̂ (
(
))√∑
)) ̂ (
(
(2)
))
for k = 0, ±1, ±2, …. In Eq. (2), ̂ ( ) indicates the unconditional sample quantile of proposed by Han et al. (2016). Furthermore, for
, Han et a. (2016) suggest a quantile ( ) = 0 for all k,
( )
against the
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version of the Ljung-Box-Pierce statistic to test alternative hypothesis
, as
0 for at least one k,
, under the below portmanteau
( ) test statistic, ̂ , for directional predictability from one time series to another for up to p lags
over the quantile pair )
(
)∑
).
̂ ( )
(3)
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̂(
(
Han et al. (2016) point the need to approximate the null distribution and conduct inferences. Accordingly, they use the stationary bootstrap of Politis and Romano (1994), which allows for {(
handling inherent serial dependence in data. Let
)}
the i-th block
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with length Li starting from ki.; Li indicates an independent and identically distributed variable with Prob (Li = s) = (
)
uniform distribution *
+ In case the upper limit (
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, s = 1,2,… for
t > T, we replace the pair (
) by (
(0,1); ki is an IID sequence drawn from a ) exceeds the sample size (T), when ) with j = k + (t mod(T-k)). The
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bootstrapped confidence intervals are constructed through pseudo re-sampling based on the sequence of blocks and the associated portmanteau test statistic. The approach of Han et al.
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(2016) was recently applied by Jiang et al. (2016) and Baumöhl and Lyócsa (2017).
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3. Empirical results 3.1 The data set
We use daily closing levels of the implied volatility index of crude oil (OVX) and the implied volatility indices of developed and emerging stock markets (France, Germany, India, Japan, Mexico, Netherland, Russia, South Africa, Sweden, Switzerland, UK, and USA), extracted from DataStream. The volatility series are plotted in Appendix Figure 1A. Owing to the data availability, our sample period is from June 3, 2008 – December 30, 2016. Importantly, it covers
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the GFC period as in Han et al. (2016)1. As shown in Appendix Table 1A, both the mean and standard deviation of the Russian VIX are higher than that of VIXs for the other countries and of the OVX. The Jarque-Bera test statistics confirm the non-normality of the series. All series are not stationary at levels, but are stationary at the first differences. We therefore conduct our analyses with the first difference of indices.
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Before applying the bivariate cross-quantilogram, we conduct the linear VAR Granger causality analysis (see Appendix Table 2A). There is a significant linear causality from OVX to each of the country VIXs at the 5% level.
We also examine the existence of non-linearity by applying the Brock–Dechert–Scheinkman (BDS) test of Brock et al. (1996) on the residuals of the OVX and country VIX equations in the
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VAR model. The p-values of the BDS test reject the null of no serial dependence in the residuals, providing evidence of non-linearity in the data (See Appendix Table 3A). This finding suggests the non-appropriateness of using a linear causality framework to examine predictability. We consider the presence of structural breaks by applying the tests of multiple structural breaks (Bai and Perron, 2003) to VAR models. Unreported results suggest the presence of at least one
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structural break around the GFC in all series. We will further examine this issue with the rollingwindow method.
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3.2 Return predictability from OVX to country VIXs
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Figure 1 presents the results of the daily price changes predictability for the 12 country VIXs for lag k =1, 2,….., 60 from the price changes of the OVX. We report the results of the cross-
Insert Figure 1 here
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quantilogram when both the OVX and country VIX are in three different quantiles.
For the case of France, there is generally a long-lived directional positive predictability in the low quantile (0.10) indicating that when the changes in the OVX are negative and very large in magnitude, it is very likely that the French VIX will also experience very large negative changes in the following 10 days. In contrast, in the high quantile (0.90), the cross-quantilogram is not significant at the next day, but significantly negative from day two to day five, suggesting that 1
Results for the sample period exclusive of the GFC are available from the authors.
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when the changes in the OVX are very low it is very likely that the French VIX will experience a very large positive changes in the following five days, except for the first day. In the median quantiles, the cross-quantilogram does not exhibit consistency in either the sign or the pattern. In fact, it oscillates very close to zero between positive and negative signs, except for day 15, and 40 for which there is a positive predictability. Similar results are reported for Japan, Mexico,
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Netherland, Russia, South Africa, Sweden, Switzerland, UK, and USA. In all these countries, the directional predictability of VIX is significant when OVX is at the low or high quantile. If the OVX is at the median quantile, no predictability is offered by the oil market. Germany and India present the other case. The next day effect is stronger in Germany than in France in the low quantile. For the high quantile (0.90), all the values of the cross-quantilogram are insignificant,
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except for lag three. This evidence of negative predictability implies that it is more likely for the German VIX to have a large positive change at day three when the changes in the OVX are very low. There is no predictability at the median quantile. Similar results are found for India. In other words, for these two countries, oil improves directional predictability only when it is at low risk. These results add to that of Bouri (2017) and Maghyereh et al. (2016) by showing that the
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dominance of oil volatility over the volatility of stock indices is only present in the tails and not
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around the mean, confirming the appropriateness of using a quantile-based approach.
Insert Figure 2 here
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We further examine the directional causality from all quantiles of OVX to all quantiles of VIXs. These quantiles are displayed on x- and y- axis, respectively, in Figure 2. The magnitude
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(positive/negative significant spillovers) is shown through colour combination from blue (high negative) to red (high positive), which is also shown through the color bar provided at the end of
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figure.
3.3 Rolling VIX predictability Given our sample period covers the GFC that potentially lead to structural changes in the VIX series, we conduct a rolling-window analysis to assess the robustness of our main results for the entire sample period (Figure 3). The recursive forward window size is maintained at 500 days and the step size is 22 days, approximately a month. Cross-quantilogram of daily spillovers from
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the OVX to country VIX when both markets take same quantiles are calculated with one day lag (i.e., p = 1).
Insert Figure 3 here
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For Germany and India, Today’s VOX helps predict tomorrow’s VIX if is of the low quantile, which is consistent with the complete sample result. Moreover, the high quintile VOX also improves the predictability of VIX if the GFC observations are removed. To be precise, consider the window that ends after May 1, 2012. For these windows, the starting point is after May 1, 2010 passing the GFC period. Figure 3 displays the significant predictability of VOX for the
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post-GFC period but not during the GFC period. Similar observation regarding the high quantile VOX appears in Netherlands, Russia, Sweden, Switzerland, and the UK. Thus, the GFC period is indeed different from the post period. Nevertheless, the results obtained from the complete sample remain valid for these countries for the post-GFC period.
Although the high quantile VOX helps predict VIX in the post GFC period, the low quantile
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VOX loses its predictive power in Sweden. The situation is much worse in the USA where, VOX has no predictability at all quantiles. These results are not consistent with the complete sample
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results. It is mostly due to the difference in the sample size. Also, the complete sample results are based upon the predictability with several possible lagged responses whereas the rolling-window method is restricted to the one-day-ahead predictability.
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Taken together, if the crude oil price is stable in the future, the market would expect the current economy to move forward smoothly, and hence the future stock index level to be stable as well.
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Accordingly, the higher probability of low oil implied volatility in the future leads to higher probability of low stock index implied volatility. A positive lead-lag relationship prevails when
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the oil volatility lies in the low quantile. Conversely, when the future oil price is very unstable, there is likely turmoil in the economy and subsequently the government would intervene. Any effective government policies help reduce the volatility of the stock market. Therefore, when the probability of large oil volatility is high today, the chance of large stock index volatility will be lower tomorrow in anticipation of the government interference. Herein a negative lead-lag relationship prevails when the oil volatility lies in the high quantile. On the other hand, during the great financial crisis period, the market does not have confidence in the effectiveness of the
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government policies. As a consequence, the above negative lead-lag relationship does not apply. Finally, when the oil volatility is at the median level, it has no predictable power for the implied volatility in the stock index, suggesting that the risk in the stock market would be determined by other factors.
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4. Conclusion We contributed to the literature on the ability of the crude oil to predict stock market conditions using implied volatility data and the cross-quantilograms approach. While the linear Granger causality test in the mean documents a lead-lag relationship in implied volatilities from oil to stock indices, the cross-quantilograms via Han et al (2016) shows a different and more nuanced
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picture. Specifically, when the risk of the oil is at the average level, there is no directional predictability at all. In contrast, the lead-lag relationship mainly holds at the tails of the oil implied volatility, in particular the left tails. The result is further validated through a rollingwindow analysis. In spite of possible structural change in the data structure such as time-varying covariance, all the samples constructed by the rolling windows display the same lead-lag
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relationships.
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References
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Campos, I., Cortazar, G. and Reyes, T., 2017. Modeling and Predicting Oil VIX: Internet Search Volume versus Traditional Macro-finance Variables. Energy Economics. https://doi.org/10.1016/j.eneco.2017.06.009
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Maghyereh, A.I., Awartani, B., Bouri E., 2016. The directional volatility connectedness between crude oil and equity markets: new evidence from implied volatility indexes. Energy Economics, 57, 78-93.
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Politis, D. N., Romano, J. P. 1994. The Stationary Bootstrap. Journal of the American Statistical Association, 89(428), 1303-1313. Raza, N., Shahzad, S.J.H., Tiwari, A.K. and Shahbaz, M., 2016. Asymmetric impact of gold, oil prices and their volatilities on stock prices of emerging markets. Resources Policy, 49, 290-301. Reboredo, J.C. and Ugolini, A., 2016. Quantile dependence of oil price movements and stock returns. Energy economics, 54, 33-49.
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Figure 1: Daily volatility spillovers from OVX to VIX. Panel A:
=0.10
Panel B:
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b). Germany
c). India
Panel C:
=0.90
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a). France
=0.50
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d). Japan
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e). Mexico
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f). Netherland
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g). Russia
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h). South Africa
i). Sweden
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j). Switzerland
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k). UK
l). USA
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Note: This Figure reports the sample cross-quantilogram of daily spillovers from the OVX to stock market volatilities Results for the low quantiles (αovx = αvix = 0.1) in Panel A, the median quantiles (αovx = αvix = 0.5) in Panel B, and the high quantiles (αovx = αvix = 0.9) in Panel C are given in the upper panels, from left to right respectively. Lag p is 1. Lower panels report the Ljung-Box type test statistics. Red-dashed lines represent the 95% bootstrapped confidence intervals for no directional predictability with 1,000 bootstrapped replicates. For each country, the upper chart indicates the cross-quantilogram results, where the lag k is on the horizontal axis; the lower chart presents the portmanteau test statistics.
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Figure 2: Directional causality in quantiles from OVX to country VIX Panel A: Lags=1
Panel A: Lags=5
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a). France
Panel A: Lags=22
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d). Japan
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e). Mexico
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h). South Africa
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i). Sweden
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l). USA
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Note: x-axis corresponds to a quantile of the OVX and y-axis to a quantile of the country VIX. Color-bar at the end shows the intensity of the causal-flows.
Panel A:
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Figure 3: Rolling daily spillovers from OVX to country VIX =0.10
Panel B:
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a). France
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c). India
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Panel C:
=0.90
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d). Japan
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g). Russia
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h). South Africa
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i). Sweden
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k). UK
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l). USA
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Notes: This Figure reports the rolling-window (500 days recursive forward window with a step size of 22 days, approximately a month) cross-quantilogram of daily spillovers from the OVX to country VIX when both markets take same quantiles. Results for low quantile (OVX =0.1 to˛VIX =0.1), median quantile (OVX =0.5 to˛VIX =0.5), and high quantile (OVX =0.9 to˛VIX =0.9) are given in the left, middle, and right panels, respectively. Lag p is 1. Starting year of the rolling window is marked on the horizontal axis. Blue lines are the rolling cross-quantilogram for 1 day spillovers, dashed red lines are 95% bootstrapped confidence intervals for no predictability based on 1,000 bootstrapped replicates.
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Appendix
b). Germany
c). India
d). Japan
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a). France
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Figure 1A: Time series plots of the OVX (red) and country stock market volatility (blue)
f). Netherland
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e). Mexico
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h). South Africa
j). Switzerland
k). UK
l). USA
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Table 1A: Statistical properties of the sample data ADF -2.777* -4.702*** -3.589*** -3.424** -4.783*** -2.311 -4.048*** -3.384** -2.487 -2.957** -3.388** -4.299*** -2.819*
ADF(D) -31.79*** -39.02*** -25.14*** -62.72*** -30.83*** -39.44*** -30.54*** -38.76*** -36.30*** -26.71*** -25.12*** -22.87*** -22.70***
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J-B 811.97*** 4516.92*** 6041.86*** 2817.86*** 11552.84*** 5377.84*** 4060.77*** 25992.18*** 3258.73*** 3745.30*** 11872.85*** 6519.55*** 5773.66***
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Mean Std. Dev. Skewness Kurtosis OVX 38.23 14.93 1.15 4.85 France 24.06 8.70 1.97 8.73 Germany 24.26 9.28 2.20 9.74 India 23.04 9.89 1.85 7.06 Japan 26.93 10.43 2.74 12.68 Mexico 20.99 9.89 2.31 9.02 Netherland 23.10 10.38 2.03 8.21 Russia 39.23 22.87 3.41 18.24 South Africa 22.77 7.06 1.86 7.59 Sweden 21.45 9.63 1.95 8.00 Switzerland 19.47 8.50 2.71 12.89 UK 20.27 9.04 2.27 10.02 USA 20.93 10.17 2.29 9.40
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Note: This table reports the descriptive statistics and unit root properties of the data. Std. Dev. stand for standard deviations. J-B stand for Jarque-Bera test with the null hypothesis of normality. ADF and ADF(D) are the test statistics for augmented Dickey Fuller unit root test for the level and first differenced series, respectively. ***, ** and * stand for significance at 1%, 5% and 10% levels, respectively.
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Table 2A: Linear Granger causality tests Order of the VAR(p) 6 3 5 3 6 7 2 3 6 6 5 3
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France Germany India Japan Mexico Netherland Russia South Africa Sweden Switzerland UK USA
H0: OVX does not Granger cause stock market volatility 7.93973*** 49.4326*** 58.0192*** 6.91128*** 4.88568*** 9.88880*** 18.7009*** 20.5281*** 21.2855*** 10.1683*** 6.59082*** 3.25757**
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Note: This table reports the F-statistics for the no Granger causality restrictions imposed on a linear vector autoregressive (VAR) model under the null hypotheses H0. The order (p) of the VAR is selected by the Bayesian Information Criterion (BIC). *** and ** indicate rejection of the null hypothesis of no Granger causality at 1% and 5% levels of significance, respectively.
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Table 3A: Results from the Brock et al. (1996) – BDS - test m=5
m=6
18.375*** 21.627*** 23.187*** 15.848*** 20.872*** 25.138*** 15.832*** 20.524*** 19.236*** 18.804*** 23.687*** 20.261***
20.137*** 24.068*** 25.272*** 17.095*** 22.783*** 27.413*** 16.759*** 22.016*** 20.944*** 20.726*** 26.707*** 22.569***
21.936*** 26.412*** 27.341*** 18.226*** 24.786*** 29.791*** 17.782*** 24.108*** 22.685*** 22.416*** 29.277*** 25.169***
19.775*** 22.682*** 22.432*** 20.625*** 20.252*** 22.423*** 22.858*** 21.781*** 22.645*** 20.849*** 19.890*** 22.420***
21.504*** 24.692*** 24.488*** 22.630*** 22.180*** 24.404*** 24.841*** 23.723*** 24.615*** 22.892*** 21.802*** 24.359***
CR IP T
m=4
AN US
M
ED
France Germany India Japan Mexico Netherland Russia South Africa Sweden Switzerland UK USA
PT
France Germany India Japan Mexico Netherland Russia South Africa Sweden Switzerland UK USA
m=2 m=3 OVX equation 13.285*** 15.893*** *** 14.624 18.734*** 16.276*** 20.453*** 11.226*** 14.403*** 13.752*** 18.244*** *** 19.154 22.608*** 11.033*** 14.358*** 15.702*** 18.747*** *** 14.509 17.101*** 13.200*** 16.464*** 14.461*** 19.825*** 13.110*** 17.375*** Stock volatility equation 14.237*** 16.260*** 15.150*** 18.274*** *** 15.502 18.469*** 13.999*** 16.600*** 12.697*** 16.239*** 15.934*** 18.564*** *** 15.660 18.737*** 15.230*** 17.801*** 15.764*** 18.752*** 14.233*** 17.038*** *** 13.908 16.269*** 15.903*** 18.659***
18.184*** 20.716*** 20.595*** 18.780*** 18.376*** 20.589*** 21.047*** 20.086*** 20.854*** 19.028*** 18.204*** 20.704***
AC
CE
Note: The entries indicate the BDS test based on the residuals of OVX series and stock volatility series in a VAR for various countries. m denotes the embedded dimension of the BDS test. *** indicates the rejection of the null of residuals being iid at1 % level of significance.