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Predicting volatility of the Shanghai silver futures market: What is the role of the U.S. options market?
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Finance Research Letters 000 (2015) 1–10
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Predicting volatility of the Shanghai silver futures market: What is the role of the U.S. options market? Xingguo Luo a,∗, Zinan Ye b a b
College of Economics and Academy of Financial Research, Zhejiang University, Hangzhou 310027, China College of Economics, Zhejiang University, Hangzhou 310027, China
a r t i c l e
i n f o
Article history: Received 11 June 2015 Accepted 12 August 2015 Available online xxx JEL classification: G14 G13 Keywords: Silver futures Volatility forecasting Volatility index
a b s t r a c t This paper investigates whether the CBOE Silver ETF Volatility Index (VXSLV), which is the implied volatility calculated from the U.S. options market, contains information for predicting the volatility of the Shanghai silver futures market. In particular, we compare its performance with variables observed in the Chinese market. We find both in-sample and out-of-sample evidence that the VXSLV significantly improves daily and weekly volatility forecasts based on realized volatilities. Moreover, when market variables including trading volume, open interest and momentum are included, the VXSLV remains significant and enhances the forecasting performance of the market variables. © 2015 Elsevier Inc. All rights reserved.
1. Introduction Measuring and forecasting volatility has long been the core content of financial research (see, e.g., Andersen and Bollerslev, 1998; Bollerslev, 1986; Engle, 1982). There is an extensive literature on realized and option-implied volatility in equity markets. However, few studies focus on these topics in commodity markets. Martens and Zein (2004) compare the forecasting performance of realized volatility based on intraday returns and implied volatility for the WTI futures market. In a recent contribution, Haugom et al. (2014) document that the CBOE Crude Oil Volatility Index (OVX) has predictive power for realized
∗
Corresponding author. Tel.: +86 571 87953210; fax: +86 571 87953937. E-mail address: [email protected] (X. Luo).
http://dx.doi.org/10.1016/j.frl.2015.08.005 1544-6123/© 2015 Elsevier Inc. All rights reserved.
Please cite this article as: X. Luo, Z. Ye, Predicting volatility of the Shanghai silver futures market: What is the role of the U.S. options market? Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.08.005
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X. Luo, Z. Ye / Finance Research Letters 000 (2015) 1–10 Table 1 Summary statistics of RV calculated for 25-min intervals. DF is the Dickey Fuller test and PP is the Phillips–Perron test for the unit root. Values below –3.430 indicate rejection at the 1% significance level for both tests. AC(1) and AC(10) denote the autocorrelation for one and 10 lags, respectively.
RV
Mean
Max
Min
DF
PP
AC(1)
AC(10)
0.571
2.375
0.103
−16.161
−20.536
0.404
0.254
volatility in the U.S. oil market. Furthermore, they find a V-shaped leverage effect and call for further research on the effect for other commodities. In this paper, we investigate whether the CBOE Silver ETF Volatility Index (VXSLV), which is the implied volatility calculated from the U.S. options market, contains information for predicting the volatility of the Shanghai silver futures market1 . We find both in-sample and out-of-sample evidence that the VXSLV significantly improves daily and weekly volatility forecasts based on realized volatilities. Furthermore, the VXSLV remains significant when market variables including trading volume, open interest and momentum are included. In fact, the VXSLV enhances the forecasting performance of the market variables. Overall, our findings indicate that the U.S. options market plays an important role in predicting the volatility of the Shanghai silver futures market and have particular implications for international investors interested in this market. 2. The data We obtain the data from the Monopoly Financial Data Center, which has information on highfrequency trading in silver futures contracts conducted through the SHFE trading platform from May 10, 2012 to December 31, 2014, yielding a total of 620 trading days and 139,500 observations. 2.1. Realized volatility We follow Andersen and Bollerslev (1998) and define realized volatility (RV) as follows:
RVtd
=
M
2 , t = 1, . . . , T , ri,t
(1)
i=1
ri,t = 100 × ( ln Pi,t − ln Pi−1,t ),
(2)
where RVtd is the daily RV, M denotes the sampling frequency and Pi, t stands for the closing price for period i on day t. The choice of sampling frequency affects the accuracy of estimation. A higher sampling frequency is desirable because it provides more useful information, but the results of such an approach contain more microstructure noise. In this study, we apply the method proposed by Zhang et al. (2005) and choose a sampling frequency of 25 min.2 Table 1 provides the descriptive statistics for the RV. Both the Dickey Fuller test and the Phillips–Perron test show that the RV time series over the sample period is stationary. However, the autocorrelation over 10 lags indicates that the RV does not exhibit long memory. 2.2. Implied volatility Option-implied volatility is proven to contain information on future volatility (e.g., Blair et al., 2001; Busch et al., 2011; Benavides and Capistrán, 2012). On March 16, 2011, the CBOE began calculating the 1 Although silver futures were first listed on the Shanghai Futures Exchange (SHFE) in May 2012, this exchange had the largest trading volume in 2013 and 2014 as reported by the Futures Industry Association (FIA). 2 Based on the “two-scales estimator” approach proposed by Zhang et al. (2005), we choose the sampling frequency with lowest sum of variance and bias, from among the RV calculated for 1-, 5-, 15-, 25- and 45-min intervals.
Please cite this article as: X. Luo, Z. Ye, Predicting volatility of the Shanghai silver futures market: What is the role of the U.S. options market? Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.08.005
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Fig. 1. This figure plots time series of RV for Shanghai silver futures and IV (VXSLV), which is the CBOE Silver ETF Volatility Index calculated from the U.S. options market. RV is calculated for 25-min intervals. The sample period is from May 10, 2012 to December 31, 2014.
VXSLV by applying the “VIX methodology” to options on the iShares silver trust fund.3 The VXSLV measures the market expectation of volatility over the next 30 days. Because China has not introduced silver options, we attempt to use the VXSLV as a proxy for implied volatility (IV) and investigate its information content with respect to the forecasting volatility of the Chinese silver futures market. Fig. 1 displays the daily RV based on 25-min intervals and the VXSLV closing price from May 12, 2012 to December 31, 2014. The correlation between the RV and the VXSLV is 0.49, and the Pearson test shows that it is significant at the 5% level. However, the RV seems to be noisier than the VXSLV. This should be expected because the RV is the instantaneous daily volatility while the VXSLV reflects the 30-day expected volatility. 2.3. Volume, open interest and momentum Previous studies find that trading volume, open interest and daily returns are also important determinants of volatility (Andersen, 1996; Doran and Ronn, 2005; Gallant et al. 1992; Girma and Mougoue, 2002). We include trading volume and open interest in the following empirical analysis. Particularly, we also include a momentum variable, rather than the positive and negative returns that are considered in Haugom et al. (2014).4 3
The historical data for the VXSLV can be obtained from the CBOE website: http://www.cboe.com/micro/vixetf/vxslv/. Originally, we include positive and negative returns and find that the two returns are insignificant in in-sample fitting. Moreover, the momentum variable performs better than the two returns in out-of-sample forecasting. 4
Please cite this article as: X. Luo, Z. Ye, Predicting volatility of the Shanghai silver futures market: What is the role of the U.S. options market? Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.08.005
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3. Empirical results In this section, we first conduct in-sample analysis by comparing the predictive power of the RV, the IV and other market variables at the daily and weekly horizons. Then, we evaluate the out-of-sample performance of the variables at the two horizons.5 3.1. Model framework We employ predictive regressions to investigate the information content of various variables. The RV0 model, which uses the lagged RV, only includes the following.6
RVt+1 = β0 + β1 RVt + εt+1 ,
(3)
the RV can be daily RV(RVtd ) or weekly RV(RVtw ) defined as a simple average of five trading days: d d d d d 1 5 × (RVt + RVt−1 + RVt−2 + RVt−3 + RVt−4 ). By including the IV and other exogenous variables
where RVtw = (EX), we make three extensions: RV–IV:
RVt+1 = β0 + β1 RVt + β2 IVt + εt+1
(4)
RV–EX:
RVt+1 = β0 + β1 RVt + β2 SIZEt + β3 OIt + β4 MOMt + εt+1
(5)
RV–IV–EX:
RVt+1 = β0 + β1 RVt + β2 IVt + β3 SIZEt +
β4 OIt + β5 MOMt + εt+1
(6)
3.2. In-sample fitting The models in Eqs. (3)–(6) are estimated for daily and weekly data. Tables 2 and 3 present the regression results at the daily and weekly horizons, respectively. Table 2 presents the results for the day-ahead fitting. It shows that the coefficient of RVtd in the RV0 model is highly significant. Furthermore, when the IV is added into the RV0 model (RV–IV), it is significant at the 1% level and dominates the lagged RV. Moreover, the R-squared value is improved significantly, from 0.167 to 0.226. The RV–EX model shows that trading volume and open interest are significant at the 1% level while the momentum variable is insignificant. Additionally, the EX variables improve the R-squared value from 0.167 to 0.227. In addition, we find that open interest decreases RV, which differs from the observations in Haugom et al. (2014). Bessembinder and Seguin (1993) and Chan et al. (2004) document similar relationships. Bessembinder and Seguin (1993) propose that open interest could measure the real demand for hedging. Hedgers may hold their positions longer than speculators. Thus, the level of hedging activities, reflected in the open interest of futures contracts, is likely to mitigate volatility. More important, when all of the variables are considered in the RV–IV–EX model, the IV remains significant at the 1% level while OI and SIZE become insignificant. Moreover, the R-squared value in the RV–IV–EX model can be further improved to be 0.293 compared with that in the RV–EX model. Table 3 shows the results for next 5-day average levels of RV. Again, we find that the IV remains significant at the 1% level when it is combined with the lagged RV and the EX variables in the RV–IV–EX model. Moreover, the IV improves the R-squared values. For example, the IV increases the R-squared values from 0.896 to 0.9 and from 0.898 to 0.901 for the RV0 and the RV–EX models, respectively. It is interesting to note that the IV improves volatility forecasts more at the daily horizon than at the weekly horizon, which is consistent with the observations in Haugom et al. (2014) for oil futures. For example, compared with 5 As a robustness check, we alternatively employ the RV calculated for 1-, 5-, 15- and 45-min intervals in both in-sample fitting and out-of-sample forecasting. The results show that our conclusion remains the same as the sampling frequency changes. 6 We also employ the Heterogeneous Auto Regressive-Realized Volatility (HAR-RV) model of Corsi (2009) and its extensions to capture the asymmetric behavior of volatility for traders with different time-horizons. We obtain quantitatively similar results.
Please cite this article as: X. Luo, Z. Ye, Predicting volatility of the Shanghai silver futures market: What is the role of the U.S. options market? Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.08.005
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Table 2 Daily horizon: in-sample fitting of different models for levels of RV from t to t + 1, from May 10, 2012 to December 31, 2014. Model (1) is the RV0 model:RVt+1 = β0 + β1 RVt + εt+1 . Models (2)– (4) denote its extensions. The RV–IV model is written as RVt+1 = β0 + β1 RVt + β2 IVt + εt+1 ; the RV–EX model is written as RVt+1 = β0 + β1 RVt + β2 SIZEt + β3 OIt + β4 MOMt + εt+1 ; and the RV–IV– EX model is written as RVt+1 = β0 + β1 RVt + β2 IVt + β3 SIZEt + β4 OIt + β5 MOMt + εt+1 . IV represents the VXSLV, which is the CBOE Silver ETF Volatility Index. EX represents exogenous variables including SIZE as the log values of trading volume, OI as the log values of open interest and MOM as the momentum variable.∗ ,∗∗ and ∗∗∗ indicate significance at the 10%, 5% and 1% levels, respectively. d RVt+1
RVtd
RV0 (1)
RV-IV (2)
RV-EX (3)
∗∗∗
∗∗∗
∗∗∗
0.411
IVt SIZEt OIt MOMt
β0 R2
0.334∗∗∗ 0.167
0.226 2.302∗∗∗
−0.257∗∗∗ 0.226
0.168
0.205∗∗∗ −0.155∗∗∗ 0.012 −0.191 0.227
RV-IV-EX (4) 0.143∗∗∗ 2.555∗∗∗ 0.066 −0.041∗∗∗ 0.189∗∗ −1.511∗∗∗ 0.293
Table 3 Weekly horizon: in-sample fitting of different models from the last five days to the next five days, from May 10, 2012 to December 31, 2014. Model (1) is the RV0 model: RVt+1 = β0 + β1 RVt + εt+1 . Models (2)–(4) denote its extensions. The RV–IV model is written as RVt+1 = β0 + β1 RVt + β2 IVt + εt+1 ; the RVEX model is written as RVt+1 = β0 + β1 RVt + β2 SIZEt + β3 OIt + β4 MOMt + εt+1 ; and the RV–IV–EX model is written as RVt+1 = β0 + β1 RVt + β2 IVt + β3 SIZEt + β4 OIt + β5 MOMt + εt+1 . IV represents the VXSLV, which is the CBOE Silver ETF Volatility Index. EX represents exogenous variables including SIZE as the log values of trading volume, OI as the log values of open interest, and MOM as the momentum variable. ∗ ,∗∗ and ∗∗∗ indicate significance at the 10%, 5% and 1% levels, respectively. w RVt+5
RVtw
RV0 (1)
RV-IV (2)
RV-EX (3)
∗∗∗
∗∗∗
∗∗∗
0.947
IVt SIZEt OIt MOMt
β0 R2
0.030∗∗ 0.896
0.892 0.371∗∗∗
−0.051∗∗ 0.900
0.886
0.034∗∗∗ −0.024∗∗ 0.001 −0.065 0.898
RV-IV-EX (4) 0.857∗∗∗ 0.378∗∗∗ 0.019∗ 0.002 0.007∗ −0.294∗∗∗ 0.901
the RV0 model, the R-squared value in the RV–IV model increases by 0.059 and 0.004 at the daily and weekly horizons, respectively. Overall, these in-sample fitting results show strong evidence that the IV contains additional information beyond what is embedded in historical volatilities and other market variables for predicting RV at the two horizons.7
7 We also evaluate these models at the monthly horizon. It turns out that the IV remains significant when it is combined with market variables in in-sample fitting. Furthermore, the IV helps to improve out-of-sample forecasts.
Please cite this article as: X. Luo, Z. Ye, Predicting volatility of the Shanghai silver futures market: What is the role of the U.S. options market? Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.08.005
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X. Luo, Z. Ye / Finance Research Letters 000 (2015) 1–10 Table 4 The values of the loss function from various models, which are shown in Eqs. (3)–(6). MSE represents mean squared error and MAE represents mean absolute error. MAE and MSE are both calculated using daily and weekly predictions. d RVt+1
RV0 RV–IV RV–EX RV–IV–EX
w RVt+5
MAE
MSE
MAE
MSE
0.2561 0.2174 0.2338 0.2149
0.1251 0.1008 0.1140 0.1044
0.0554 0.0550 0.0543 0.0555
0.0066 0.0062 0.0066 0.0064
3.3. Out-of-sample forecasting In this section, we examine the out-of-sample forecasting performance of various variables. Particularly, we evaluate the relative performance using both a loss function and the Mincer and Zarnowitz (1969) regression based on a 450-day rolling window. Furthermore, we compare historical average forecasts against the competing predictive regression factors by following Campbell and Thompson (2008) and Goyal and Welch (2008). First, the loss function used here includes mean squared error (MSE) and mean absolute error (MAE), which are calculated as follows:
MSE =
n 1 (Observedt − Predictedt )2 n
(7)
n 1 |Observedt − Predictedt | n
(8)
t=1
MAE =
t=1
where n denotes the number of days contained in the forecasting period. Observedt represents the observed value of volatility and Predictedt represents the predictions. A lower value of MAE or MSE indicates higher predictive accuracy. Table 4 presents the values of the loss function from various models at the daily and weekly horizons. It shows that the IV is able to generate lower MSEs and MAEs in the RV–IV model at the two horizons. Furthermore, we employ the Mincer–Zarnowitz regression to compare the relative performance of the volatility models:
RVt+1 = β0 + β1 RV Model 1,t + β2 RVModel 2,t + εt+1
(9)
where RVt+1 is the observed RV at t+1, RV Model 1,t represents the predicted value from one of the models, w is the forecast from another model. By exchanging RVt+1 with RVt+5 , the method can be and RV Model 2,t applied to evaluation at the weekly horizon. The results of the Mincer–Zarnowitz regressions are given in Tables 5 and 6. When the Mincer–Zarnowitz regressions are applied to these models separately, the R-squared values are improved in daily and weekly forecasts in the RV–IV model. Particularly, Table 5 shows that the Rsquared value increases from 0.019 in the RV0 model to 0.149 in the RV–IV model. Furthermore, the RV–IV model is significant at the 1% level, while the RV–EX model is insignificant at the two horizons when the two models are combined in the regressions. Furthermore, the RV–IV–EX model outperforms the RV–EX model at the two horizons, which indicates that the IV improves forecasts even after including other EX variables. To further investigate the out-of-sample predictive performance of the various variables, we follow Campbell and Thompson (2008) and calculate out-of-sample R-squared statistics:
q R2OS
2
(rm+k − rˆm+k ) , r − r¯m+k )2 k=1 ( m+k
= 1 − k=1 q
(10)
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Table 5 The comparison of 1-day ahead predictions of volatility from RV model and its extensions using the Mincer-Zarnowitz regression (1969): RVt+1 = β0 + β1 RV Model 1,t + β2 RV Model 2,t + εt+1 . The 1st–4th columns evaluate the out-of-sample forecasting performance for each model separately. The 5th column compares the relative performance of the RV–EX model and the RV–IV–EX model. The last column compares the performance of the RV–IV model and the RV–EX model. The four models are shown in Eqs. (3)–(6) and estimated using a 450-day rolling window. MZD1 RV0 RV–IV RV–EX RV–IV–EX
β0 R2
0.330
MZD2
MZD3
MZD4
MZD5
0.605∗∗∗ 0.209∗∗∗ 0.149
−0.737∗∗ 1.237∗∗∗ 0.277∗∗∗ 0.175
MZD6
∗∗∗
0.622∗∗∗ 0.475∗∗∗ 0.280∗∗∗ 0.019
0.167∗∗∗ 0.149
0.245∗∗∗ 0.077
0.874∗∗∗ −0.304 0.184∗∗∗ 0.152
Table 6 The comparison of the 1-week ahead prediction of volatility from the RV model and its extensions using the Mincer–Zarnowitz regression (1969):RVt+1 = β0 + β1 RV Model 1,t + β2 RV Model 2,t + εt+1 . The 1st–4th columns evaluate the out-of-sample forecasting performance for each model separately. The 5th column compares the relative performance of the RV–EX model and the RV–IV–EX model. The last column compares the performance of the RV–IV model and the RV–EX model. The four models are shown in Eqs. (3)–(6) and estimated using a 450-day rolling window. MZW1 RV0 RV–IV RV–EX RV–IV–EX
β0
R2
MZW2
MZW3
MZW4
MZW5
0.914∗∗∗ 0.050∗∗∗ 0.846
-0.740 1.645∗∗∗ 0.060∗∗∗ 0.847
MZW6
∗∗∗
0.956
0.924∗∗∗ 0.919∗∗∗ 0.0141 0.835
0.035∗ 0.847
0.040∗∗ 0.839
0.930∗∗∗ -0.006 0.035∗ 0.846
Table 7 This table presents the out-of-sample forecasting results, the R2OS statistics, defined by R2OS = 1 −
q
k=1
q
k=1
(rm+k −ˆrm+k )2 , where rˆm+k (rm+k −r¯m+k )2
are forecasts obtained by the VXSLV (IV), the log values of open interest (OI), the log values of trading volume (SIZE) and the momentum factor (MOM) and r¯m+k are forecasts based on the historical average. The results for the daily and weekly data are reported in Panels A and B, respectively. Note that a positive R2OS indicates that the predictive regression forecast is better than the historical average. The out-of-sample period is May 16, 2014 to December 31, 2014.
Panel A: daily Panel B: weekly
IV
OI
SIZE
MOM
0.278 0.660
−0.075 −0.167
0.081 0.315
−0.046 −0.154
where rˆm+k are forecasts obtained by the various variables and r¯m+k are forecasts based on the historical average. The results for daily and weekly data are reported in Panels A and B, respectively, in Table 7. The out-of-sample R2OS statistics of the VXSLV are 0.278 and 0.66 for daily and weekly data, respectively. Note that a positive R2OS indicates that the predictive regression forecast is better than the historical average. Therefore, we provide further out-of-sample evidence on the predictive ability of the VXSLV. Please cite this article as: X. Luo, Z. Ye, Predicting volatility of the Shanghai silver futures market: What is the role of the U.S. options market? Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.08.005
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Fig. 2. Out-of-sample performance of daily predictive regressions. In particular, these are the cumulative squared prediction errors of the historical average minus the cumulative squared prediction error of the alternative. The alternative is a model that relies on predictive variables: the VXSLV (IV), the log values of open interest (OI), the log values of trading volume (SIZE) and the momentum factor (MOM). An increase in a line indicates that the model in question has better performance; a decrease in a line indicates that the historical average has better performance. The out-of-sample period is May 16, 2014 to December 31, 2014.
Fig. 3. Out-of-sample performance of weekly predictive regressions. In particular, these are the cumulative squared prediction errors of the historical average minus the cumulative squared prediction error of the alternative. The alternative is a model that relies on predictive variables: the VXSLV (IV), the log values of open interest (OI), the log values of trading volume (SIZE) and the momentum factor (MOM). An increase in a line indicates that the model in question has better performance; a decrease in a line indicates that the historical average has better performance. The out-of-sample period is May 16, 2014 to December 31, 2014.
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Moreover, we are able to determine the out-of-sample performance of the predictive regressions by plotting the cumulative differences in squared forecast errors (Goyal and Welch, 2008). The cumulative differences in squared forecast errors are calculated as the cumulative squared prediction errors of the historical average minus the cumulative squared prediction error of the alternative. The alternative is a model that relies on predictive variables including the VXSLV, open interest, trading volume and the momentum factor. Figs. 2 and 3 show the results. Note that an increase in a line indicates that model in question has better performance and a decrease in a line indicates that the historical average has better performance. The VXSLV outperforms the historical average in both daily and weekly forecasts. 4. Conclusions This paper studies the forecasting of volatility in the Shanghai silver futures market. In particular, we investigate the role of the U.S. options market, represented by the VXSLV. We compare the predictive power of the VXSLV with that of exogenous variables including trading volume, open interest, and momentum. Both the in-sample and out-of-sample results show that the VXSLV can significantly improve predictions at daily and weekly horizons. Furthermore, the VXSLV remains significant when the exogenous variables are included. More important, the VXSLV helps to improve the forecasts obtained from the exogenous variables at daily and weekly horizons. Our study is the first to highlight the important role of the U.S. options market in predicting the volatility of the Shanghai silver futures market. A similar effect could be observed in other markets, e.g., gold futures market, by using the CBOE Gold ETF Volatility index (GVZ). Additionally, it would be interesting to examine asymmetric causality by separately considering positive and negative macroeconomic shocks (Onan et al., 2014; Nguyen et al., 2015). Nevertheless, we leave these topics for future research. Acknowledgments We would like to thank the Editor, Brian Lucey and two anonymous referees for their very useful comments and suggestions. The research reported in this paper was supported by the National Natural Science Foundation of China (project no. 71301143), the Natural Science Foundation of Zhejiang Province (project no. LQ13G030001), and the Academy of Financial Research, Zhejiang University and Zhejiang Province (project no. XK14001). References Andersen, T.G., 1996. Return volatility and trading volume: an information flow interpretation of stochastic volatility. J. Finance 51 (1), 169–204. Andersen, T.G., Bollerslev, T., 1998. Deutsche mark-dollar volatility: intraday activity patterns, macroeconomic announcements, and longer run dependencies. J. Finance 53 (1), 219–265. Benavides, G., Capistrán, C., 2012. Forecasting exchange rate volatility: the superior performance of conditional combinations of time series and option implied forecasts. J. Empirical Finance 19 (5), 627–639. Bessembinder, H., Seguin, P.J., 1993. Price volatility, trading volume, and market depth: evidence from futures markets. J. Financial Quant. Anal. 28 (1), 21–39. Blair, B.J., Poon, S.H., Taylor, S.J., 2001. Forecasting S&P 100 volatility: the incremental information content of implied volatilities and high-frequency index returns. J. Econ. 105 (1), 5–26. Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. J. Econ. 31 (3), 307–327. Busch, T., Christensen, B.J., Nielsen, M., 2011. The role of implied volatility in forecasting future realized volatility and jumps in foreign exchange, stock, and bond markets. J. Econ. 160 (1), 48–57. Campbell, J.Y., Thompson, S.B., 2008. Predicting excess stock returns out of sample: can anything beat the historical average? Rev. Financial Stud. 21 (4), 1509–1531. Chan, K.C., Fung, H.G., Leung, W.K., 2004. Daily volatility behavior in Chinese futures markets. J. Int. Financial Markets, Institutions Money 14 (5), 491–505. Corsi, F., 2009. A simple approximate long-memory model of realized volatility. J. Financial Econ. 7 (2), 174–196. Doran, J.S., Ronn, E.I., 2005. The bias in Black-Scholes/Black implied volatility: an analysis of equity and energy markets. Rev. Derivatives Res. 8 (3), 177–198. Engle, R.F., 1982. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 (4), 987–1007. Gallant, A.R., Rossi, P.E., Tauchen, G., 1992. Stock prices and volume. Rev. Financial Stud. 5 (2), 199–242.
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Finance Research Letters 000 (2015) 1–10
Contents lists available at ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
Predicting volatility of the Shanghai silver futures market: What is the role of the U.S. options market? Xingguo Luo a,∗, Zinan Ye b a b
College of Economics and Academy of Financial Research, Zhejiang University, Hangzhou 310027, China College of Economics, Zhejiang University, Hangzhou 310027, China
a r t i c l e
i n f o
Article history: Received 11 June 2015 Accepted 12 August 2015 Available online xxx JEL classification: G14 G13 Keywords: Silver futures Volatility forecasting Volatility index
a b s t r a c t This paper investigates whether the CBOE Silver ETF Volatility Index (VXSLV), which is the implied volatility calculated from the U.S. options market, contains information for predicting the volatility of the Shanghai silver futures market. In particular, we compare its performance with variables observed in the Chinese market. We find both in-sample and out-of-sample evidence that the VXSLV significantly improves daily and weekly volatility forecasts based on realized volatilities. Moreover, when market variables including trading volume, open interest and momentum are included, the VXSLV remains significant and enhances the forecasting performance of the market variables. © 2015 Elsevier Inc. All rights reserved.
1. Introduction Measuring and forecasting volatility has long been the core content of financial research (see, e.g., Andersen and Bollerslev, 1998; Bollerslev, 1986; Engle, 1982). There is an extensive literature on realized and option-implied volatility in equity markets. However, few studies focus on these topics in commodity markets. Martens and Zein (2004) compare the forecasting performance of realized volatility based on intraday returns and implied volatility for the WTI futures market. In a recent contribution, Haugom et al. (2014) document that the CBOE Crude Oil Volatility Index (OVX) has predictive power for realized
∗
Corresponding author. Tel.: +86 571 87953210; fax: +86 571 87953937. E-mail address: [email protected] (X. Luo).
http://dx.doi.org/10.1016/j.frl.2015.08.005 1544-6123/© 2015 Elsevier Inc. All rights reserved.
Please cite this article as: X. Luo, Z. Ye, Predicting volatility of the Shanghai silver futures market: What is the role of the U.S. options market? Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.08.005
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X. Luo, Z. Ye / Finance Research Letters 000 (2015) 1–10 Table 1 Summary statistics of RV calculated for 25-min intervals. DF is the Dickey Fuller test and PP is the Phillips–Perron test for the unit root. Values below –3.430 indicate rejection at the 1% significance level for both tests. AC(1) and AC(10) denote the autocorrelation for one and 10 lags, respectively.
RV
Mean
Max
Min
DF
PP
AC(1)
AC(10)
0.571
2.375
0.103
−16.161
−20.536
0.404
0.254
volatility in the U.S. oil market. Furthermore, they find a V-shaped leverage effect and call for further research on the effect for other commodities. In this paper, we investigate whether the CBOE Silver ETF Volatility Index (VXSLV), which is the implied volatility calculated from the U.S. options market, contains information for predicting the volatility of the Shanghai silver futures market1 . We find both in-sample and out-of-sample evidence that the VXSLV significantly improves daily and weekly volatility forecasts based on realized volatilities. Furthermore, the VXSLV remains significant when market variables including trading volume, open interest and momentum are included. In fact, the VXSLV enhances the forecasting performance of the market variables. Overall, our findings indicate that the U.S. options market plays an important role in predicting the volatility of the Shanghai silver futures market and have particular implications for international investors interested in this market. 2. The data We obtain the data from the Monopoly Financial Data Center, which has information on highfrequency trading in silver futures contracts conducted through the SHFE trading platform from May 10, 2012 to December 31, 2014, yielding a total of 620 trading days and 139,500 observations. 2.1. Realized volatility We follow Andersen and Bollerslev (1998) and define realized volatility (RV) as follows:
RVtd
=
M
2 , t = 1, . . . , T , ri,t
(1)
i=1
ri,t = 100 × ( ln Pi,t − ln Pi−1,t ),
(2)
where RVtd is the daily RV, M denotes the sampling frequency and Pi, t stands for the closing price for period i on day t. The choice of sampling frequency affects the accuracy of estimation. A higher sampling frequency is desirable because it provides more useful information, but the results of such an approach contain more microstructure noise. In this study, we apply the method proposed by Zhang et al. (2005) and choose a sampling frequency of 25 min.2 Table 1 provides the descriptive statistics for the RV. Both the Dickey Fuller test and the Phillips–Perron test show that the RV time series over the sample period is stationary. However, the autocorrelation over 10 lags indicates that the RV does not exhibit long memory. 2.2. Implied volatility Option-implied volatility is proven to contain information on future volatility (e.g., Blair et al., 2001; Busch et al., 2011; Benavides and Capistrán, 2012). On March 16, 2011, the CBOE began calculating the 1 Although silver futures were first listed on the Shanghai Futures Exchange (SHFE) in May 2012, this exchange had the largest trading volume in 2013 and 2014 as reported by the Futures Industry Association (FIA). 2 Based on the “two-scales estimator” approach proposed by Zhang et al. (2005), we choose the sampling frequency with lowest sum of variance and bias, from among the RV calculated for 1-, 5-, 15-, 25- and 45-min intervals.
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Fig. 1. This figure plots time series of RV for Shanghai silver futures and IV (VXSLV), which is the CBOE Silver ETF Volatility Index calculated from the U.S. options market. RV is calculated for 25-min intervals. The sample period is from May 10, 2012 to December 31, 2014.
VXSLV by applying the “VIX methodology” to options on the iShares silver trust fund.3 The VXSLV measures the market expectation of volatility over the next 30 days. Because China has not introduced silver options, we attempt to use the VXSLV as a proxy for implied volatility (IV) and investigate its information content with respect to the forecasting volatility of the Chinese silver futures market. Fig. 1 displays the daily RV based on 25-min intervals and the VXSLV closing price from May 12, 2012 to December 31, 2014. The correlation between the RV and the VXSLV is 0.49, and the Pearson test shows that it is significant at the 5% level. However, the RV seems to be noisier than the VXSLV. This should be expected because the RV is the instantaneous daily volatility while the VXSLV reflects the 30-day expected volatility. 2.3. Volume, open interest and momentum Previous studies find that trading volume, open interest and daily returns are also important determinants of volatility (Andersen, 1996; Doran and Ronn, 2005; Gallant et al. 1992; Girma and Mougoue, 2002). We include trading volume and open interest in the following empirical analysis. Particularly, we also include a momentum variable, rather than the positive and negative returns that are considered in Haugom et al. (2014).4 3
The historical data for the VXSLV can be obtained from the CBOE website: http://www.cboe.com/micro/vixetf/vxslv/. Originally, we include positive and negative returns and find that the two returns are insignificant in in-sample fitting. Moreover, the momentum variable performs better than the two returns in out-of-sample forecasting. 4
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3. Empirical results In this section, we first conduct in-sample analysis by comparing the predictive power of the RV, the IV and other market variables at the daily and weekly horizons. Then, we evaluate the out-of-sample performance of the variables at the two horizons.5 3.1. Model framework We employ predictive regressions to investigate the information content of various variables. The RV0 model, which uses the lagged RV, only includes the following.6
RVt+1 = β0 + β1 RVt + εt+1 ,
(3)
the RV can be daily RV(RVtd ) or weekly RV(RVtw ) defined as a simple average of five trading days: d d d d d 1 5 × (RVt + RVt−1 + RVt−2 + RVt−3 + RVt−4 ). By including the IV and other exogenous variables
where RVtw = (EX), we make three extensions: RV–IV:
RVt+1 = β0 + β1 RVt + β2 IVt + εt+1
(4)
RV–EX:
RVt+1 = β0 + β1 RVt + β2 SIZEt + β3 OIt + β4 MOMt + εt+1
(5)
RV–IV–EX:
RVt+1 = β0 + β1 RVt + β2 IVt + β3 SIZEt +
β4 OIt + β5 MOMt + εt+1
(6)
3.2. In-sample fitting The models in Eqs. (3)–(6) are estimated for daily and weekly data. Tables 2 and 3 present the regression results at the daily and weekly horizons, respectively. Table 2 presents the results for the day-ahead fitting. It shows that the coefficient of RVtd in the RV0 model is highly significant. Furthermore, when the IV is added into the RV0 model (RV–IV), it is significant at the 1% level and dominates the lagged RV. Moreover, the R-squared value is improved significantly, from 0.167 to 0.226. The RV–EX model shows that trading volume and open interest are significant at the 1% level while the momentum variable is insignificant. Additionally, the EX variables improve the R-squared value from 0.167 to 0.227. In addition, we find that open interest decreases RV, which differs from the observations in Haugom et al. (2014). Bessembinder and Seguin (1993) and Chan et al. (2004) document similar relationships. Bessembinder and Seguin (1993) propose that open interest could measure the real demand for hedging. Hedgers may hold their positions longer than speculators. Thus, the level of hedging activities, reflected in the open interest of futures contracts, is likely to mitigate volatility. More important, when all of the variables are considered in the RV–IV–EX model, the IV remains significant at the 1% level while OI and SIZE become insignificant. Moreover, the R-squared value in the RV–IV–EX model can be further improved to be 0.293 compared with that in the RV–EX model. Table 3 shows the results for next 5-day average levels of RV. Again, we find that the IV remains significant at the 1% level when it is combined with the lagged RV and the EX variables in the RV–IV–EX model. Moreover, the IV improves the R-squared values. For example, the IV increases the R-squared values from 0.896 to 0.9 and from 0.898 to 0.901 for the RV0 and the RV–EX models, respectively. It is interesting to note that the IV improves volatility forecasts more at the daily horizon than at the weekly horizon, which is consistent with the observations in Haugom et al. (2014) for oil futures. For example, compared with 5 As a robustness check, we alternatively employ the RV calculated for 1-, 5-, 15- and 45-min intervals in both in-sample fitting and out-of-sample forecasting. The results show that our conclusion remains the same as the sampling frequency changes. 6 We also employ the Heterogeneous Auto Regressive-Realized Volatility (HAR-RV) model of Corsi (2009) and its extensions to capture the asymmetric behavior of volatility for traders with different time-horizons. We obtain quantitatively similar results.
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Table 2 Daily horizon: in-sample fitting of different models for levels of RV from t to t + 1, from May 10, 2012 to December 31, 2014. Model (1) is the RV0 model:RVt+1 = β0 + β1 RVt + εt+1 . Models (2)– (4) denote its extensions. The RV–IV model is written as RVt+1 = β0 + β1 RVt + β2 IVt + εt+1 ; the RV–EX model is written as RVt+1 = β0 + β1 RVt + β2 SIZEt + β3 OIt + β4 MOMt + εt+1 ; and the RV–IV– EX model is written as RVt+1 = β0 + β1 RVt + β2 IVt + β3 SIZEt + β4 OIt + β5 MOMt + εt+1 . IV represents the VXSLV, which is the CBOE Silver ETF Volatility Index. EX represents exogenous variables including SIZE as the log values of trading volume, OI as the log values of open interest and MOM as the momentum variable.∗ ,∗∗ and ∗∗∗ indicate significance at the 10%, 5% and 1% levels, respectively. d RVt+1
RVtd
RV0 (1)
RV-IV (2)
RV-EX (3)
∗∗∗
∗∗∗
∗∗∗
0.411
IVt SIZEt OIt MOMt
β0 R2
0.334∗∗∗ 0.167
0.226 2.302∗∗∗
−0.257∗∗∗ 0.226
0.168
0.205∗∗∗ −0.155∗∗∗ 0.012 −0.191 0.227
RV-IV-EX (4) 0.143∗∗∗ 2.555∗∗∗ 0.066 −0.041∗∗∗ 0.189∗∗ −1.511∗∗∗ 0.293
Table 3 Weekly horizon: in-sample fitting of different models from the last five days to the next five days, from May 10, 2012 to December 31, 2014. Model (1) is the RV0 model: RVt+1 = β0 + β1 RVt + εt+1 . Models (2)–(4) denote its extensions. The RV–IV model is written as RVt+1 = β0 + β1 RVt + β2 IVt + εt+1 ; the RVEX model is written as RVt+1 = β0 + β1 RVt + β2 SIZEt + β3 OIt + β4 MOMt + εt+1 ; and the RV–IV–EX model is written as RVt+1 = β0 + β1 RVt + β2 IVt + β3 SIZEt + β4 OIt + β5 MOMt + εt+1 . IV represents the VXSLV, which is the CBOE Silver ETF Volatility Index. EX represents exogenous variables including SIZE as the log values of trading volume, OI as the log values of open interest, and MOM as the momentum variable. ∗ ,∗∗ and ∗∗∗ indicate significance at the 10%, 5% and 1% levels, respectively. w RVt+5
RVtw
RV0 (1)
RV-IV (2)
RV-EX (3)
∗∗∗
∗∗∗
∗∗∗
0.947
IVt SIZEt OIt MOMt
β0 R2
0.030∗∗ 0.896
0.892 0.371∗∗∗
−0.051∗∗ 0.900
0.886
0.034∗∗∗ −0.024∗∗ 0.001 −0.065 0.898
RV-IV-EX (4) 0.857∗∗∗ 0.378∗∗∗ 0.019∗ 0.002 0.007∗ −0.294∗∗∗ 0.901
the RV0 model, the R-squared value in the RV–IV model increases by 0.059 and 0.004 at the daily and weekly horizons, respectively. Overall, these in-sample fitting results show strong evidence that the IV contains additional information beyond what is embedded in historical volatilities and other market variables for predicting RV at the two horizons.7
7 We also evaluate these models at the monthly horizon. It turns out that the IV remains significant when it is combined with market variables in in-sample fitting. Furthermore, the IV helps to improve out-of-sample forecasts.
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X. Luo, Z. Ye / Finance Research Letters 000 (2015) 1–10 Table 4 The values of the loss function from various models, which are shown in Eqs. (3)–(6). MSE represents mean squared error and MAE represents mean absolute error. MAE and MSE are both calculated using daily and weekly predictions. d RVt+1
RV0 RV–IV RV–EX RV–IV–EX
w RVt+5
MAE
MSE
MAE
MSE
0.2561 0.2174 0.2338 0.2149
0.1251 0.1008 0.1140 0.1044
0.0554 0.0550 0.0543 0.0555
0.0066 0.0062 0.0066 0.0064
3.3. Out-of-sample forecasting In this section, we examine the out-of-sample forecasting performance of various variables. Particularly, we evaluate the relative performance using both a loss function and the Mincer and Zarnowitz (1969) regression based on a 450-day rolling window. Furthermore, we compare historical average forecasts against the competing predictive regression factors by following Campbell and Thompson (2008) and Goyal and Welch (2008). First, the loss function used here includes mean squared error (MSE) and mean absolute error (MAE), which are calculated as follows:
MSE =
n 1 (Observedt − Predictedt )2 n
(7)
n 1 |Observedt − Predictedt | n
(8)
t=1
MAE =
t=1
where n denotes the number of days contained in the forecasting period. Observedt represents the observed value of volatility and Predictedt represents the predictions. A lower value of MAE or MSE indicates higher predictive accuracy. Table 4 presents the values of the loss function from various models at the daily and weekly horizons. It shows that the IV is able to generate lower MSEs and MAEs in the RV–IV model at the two horizons. Furthermore, we employ the Mincer–Zarnowitz regression to compare the relative performance of the volatility models:
RVt+1 = β0 + β1 RV Model 1,t + β2 RVModel 2,t + εt+1
(9)
where RVt+1 is the observed RV at t+1, RV Model 1,t represents the predicted value from one of the models, w is the forecast from another model. By exchanging RVt+1 with RVt+5 , the method can be and RV Model 2,t applied to evaluation at the weekly horizon. The results of the Mincer–Zarnowitz regressions are given in Tables 5 and 6. When the Mincer–Zarnowitz regressions are applied to these models separately, the R-squared values are improved in daily and weekly forecasts in the RV–IV model. Particularly, Table 5 shows that the Rsquared value increases from 0.019 in the RV0 model to 0.149 in the RV–IV model. Furthermore, the RV–IV model is significant at the 1% level, while the RV–EX model is insignificant at the two horizons when the two models are combined in the regressions. Furthermore, the RV–IV–EX model outperforms the RV–EX model at the two horizons, which indicates that the IV improves forecasts even after including other EX variables. To further investigate the out-of-sample predictive performance of the various variables, we follow Campbell and Thompson (2008) and calculate out-of-sample R-squared statistics:
q R2OS
2
(rm+k − rˆm+k ) , r − r¯m+k )2 k=1 ( m+k
= 1 − k=1 q
(10)
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Table 5 The comparison of 1-day ahead predictions of volatility from RV model and its extensions using the Mincer-Zarnowitz regression (1969): RVt+1 = β0 + β1 RV Model 1,t + β2 RV Model 2,t + εt+1 . The 1st–4th columns evaluate the out-of-sample forecasting performance for each model separately. The 5th column compares the relative performance of the RV–EX model and the RV–IV–EX model. The last column compares the performance of the RV–IV model and the RV–EX model. The four models are shown in Eqs. (3)–(6) and estimated using a 450-day rolling window. MZD1 RV0 RV–IV RV–EX RV–IV–EX
β0 R2
0.330
MZD2
MZD3
MZD4
MZD5
0.605∗∗∗ 0.209∗∗∗ 0.149
−0.737∗∗ 1.237∗∗∗ 0.277∗∗∗ 0.175
MZD6
∗∗∗
0.622∗∗∗ 0.475∗∗∗ 0.280∗∗∗ 0.019
0.167∗∗∗ 0.149
0.245∗∗∗ 0.077
0.874∗∗∗ −0.304 0.184∗∗∗ 0.152
Table 6 The comparison of the 1-week ahead prediction of volatility from the RV model and its extensions using the Mincer–Zarnowitz regression (1969):RVt+1 = β0 + β1 RV Model 1,t + β2 RV Model 2,t + εt+1 . The 1st–4th columns evaluate the out-of-sample forecasting performance for each model separately. The 5th column compares the relative performance of the RV–EX model and the RV–IV–EX model. The last column compares the performance of the RV–IV model and the RV–EX model. The four models are shown in Eqs. (3)–(6) and estimated using a 450-day rolling window. MZW1 RV0 RV–IV RV–EX RV–IV–EX
β0
R2
MZW2
MZW3
MZW4
MZW5
0.914∗∗∗ 0.050∗∗∗ 0.846
-0.740 1.645∗∗∗ 0.060∗∗∗ 0.847
MZW6
∗∗∗
0.956
0.924∗∗∗ 0.919∗∗∗ 0.0141 0.835
0.035∗ 0.847
0.040∗∗ 0.839
0.930∗∗∗ -0.006 0.035∗ 0.846
Table 7 This table presents the out-of-sample forecasting results, the R2OS statistics, defined by R2OS = 1 −
q
k=1
q
k=1
(rm+k −ˆrm+k )2 , where rˆm+k (rm+k −r¯m+k )2
are forecasts obtained by the VXSLV (IV), the log values of open interest (OI), the log values of trading volume (SIZE) and the momentum factor (MOM) and r¯m+k are forecasts based on the historical average. The results for the daily and weekly data are reported in Panels A and B, respectively. Note that a positive R2OS indicates that the predictive regression forecast is better than the historical average. The out-of-sample period is May 16, 2014 to December 31, 2014.
Panel A: daily Panel B: weekly
IV
OI
SIZE
MOM
0.278 0.660
−0.075 −0.167
0.081 0.315
−0.046 −0.154
where rˆm+k are forecasts obtained by the various variables and r¯m+k are forecasts based on the historical average. The results for daily and weekly data are reported in Panels A and B, respectively, in Table 7. The out-of-sample R2OS statistics of the VXSLV are 0.278 and 0.66 for daily and weekly data, respectively. Note that a positive R2OS indicates that the predictive regression forecast is better than the historical average. Therefore, we provide further out-of-sample evidence on the predictive ability of the VXSLV. Please cite this article as: X. Luo, Z. Ye, Predicting volatility of the Shanghai silver futures market: What is the role of the U.S. options market? Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.08.005
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Fig. 2. Out-of-sample performance of daily predictive regressions. In particular, these are the cumulative squared prediction errors of the historical average minus the cumulative squared prediction error of the alternative. The alternative is a model that relies on predictive variables: the VXSLV (IV), the log values of open interest (OI), the log values of trading volume (SIZE) and the momentum factor (MOM). An increase in a line indicates that the model in question has better performance; a decrease in a line indicates that the historical average has better performance. The out-of-sample period is May 16, 2014 to December 31, 2014.
Fig. 3. Out-of-sample performance of weekly predictive regressions. In particular, these are the cumulative squared prediction errors of the historical average minus the cumulative squared prediction error of the alternative. The alternative is a model that relies on predictive variables: the VXSLV (IV), the log values of open interest (OI), the log values of trading volume (SIZE) and the momentum factor (MOM). An increase in a line indicates that the model in question has better performance; a decrease in a line indicates that the historical average has better performance. The out-of-sample period is May 16, 2014 to December 31, 2014.
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Moreover, we are able to determine the out-of-sample performance of the predictive regressions by plotting the cumulative differences in squared forecast errors (Goyal and Welch, 2008). The cumulative differences in squared forecast errors are calculated as the cumulative squared prediction errors of the historical average minus the cumulative squared prediction error of the alternative. The alternative is a model that relies on predictive variables including the VXSLV, open interest, trading volume and the momentum factor. Figs. 2 and 3 show the results. Note that an increase in a line indicates that model in question has better performance and a decrease in a line indicates that the historical average has better performance. The VXSLV outperforms the historical average in both daily and weekly forecasts. 4. Conclusions This paper studies the forecasting of volatility in the Shanghai silver futures market. In particular, we investigate the role of the U.S. options market, represented by the VXSLV. We compare the predictive power of the VXSLV with that of exogenous variables including trading volume, open interest, and momentum. Both the in-sample and out-of-sample results show that the VXSLV can significantly improve predictions at daily and weekly horizons. Furthermore, the VXSLV remains significant when the exogenous variables are included. More important, the VXSLV helps to improve the forecasts obtained from the exogenous variables at daily and weekly horizons. Our study is the first to highlight the important role of the U.S. options market in predicting the volatility of the Shanghai silver futures market. A similar effect could be observed in other markets, e.g., gold futures market, by using the CBOE Gold ETF Volatility index (GVZ). Additionally, it would be interesting to examine asymmetric causality by separately considering positive and negative macroeconomic shocks (Onan et al., 2014; Nguyen et al., 2015). Nevertheless, we leave these topics for future research. Acknowledgments We would like to thank the Editor, Brian Lucey and two anonymous referees for their very useful comments and suggestions. The research reported in this paper was supported by the National Natural Science Foundation of China (project no. 71301143), the Natural Science Foundation of Zhejiang Province (project no. LQ13G030001), and the Academy of Financial Research, Zhejiang University and Zhejiang Province (project no. XK14001). References Andersen, T.G., 1996. Return volatility and trading volume: an information flow interpretation of stochastic volatility. J. Finance 51 (1), 169–204. Andersen, T.G., Bollerslev, T., 1998. Deutsche mark-dollar volatility: intraday activity patterns, macroeconomic announcements, and longer run dependencies. J. Finance 53 (1), 219–265. Benavides, G., Capistrán, C., 2012. Forecasting exchange rate volatility: the superior performance of conditional combinations of time series and option implied forecasts. J. Empirical Finance 19 (5), 627–639. Bessembinder, H., Seguin, P.J., 1993. Price volatility, trading volume, and market depth: evidence from futures markets. J. Financial Quant. Anal. 28 (1), 21–39. Blair, B.J., Poon, S.H., Taylor, S.J., 2001. Forecasting S&P 100 volatility: the incremental information content of implied volatilities and high-frequency index returns. J. Econ. 105 (1), 5–26. Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. J. Econ. 31 (3), 307–327. Busch, T., Christensen, B.J., Nielsen, M., 2011. The role of implied volatility in forecasting future realized volatility and jumps in foreign exchange, stock, and bond markets. J. Econ. 160 (1), 48–57. Campbell, J.Y., Thompson, S.B., 2008. Predicting excess stock returns out of sample: can anything beat the historical average? Rev. Financial Stud. 21 (4), 1509–1531. Chan, K.C., Fung, H.G., Leung, W.K., 2004. Daily volatility behavior in Chinese futures markets. J. Int. Financial Markets, Institutions Money 14 (5), 491–505. Corsi, F., 2009. A simple approximate long-memory model of realized volatility. J. Financial Econ. 7 (2), 174–196. Doran, J.S., Ronn, E.I., 2005. The bias in Black-Scholes/Black implied volatility: an analysis of equity and energy markets. Rev. Derivatives Res. 8 (3), 177–198. Engle, R.F., 1982. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 (4), 987–1007. Gallant, A.R., Rossi, P.E., Tauchen, G., 1992. Stock prices and volume. Rev. Financial Stud. 5 (2), 199–242.
Please cite this article as: X. Luo, Z. Ye, Predicting volatility of the Shanghai silver futures market: What is the role of the U.S. options market? Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.08.005
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Please cite this article as: X. Luo, Z. Ye, Predicting volatility of the Shanghai silver futures market: What is the role of the U.S. options market? Finance Research Letters (2015), http://dx.doi.org/10.1016/j.frl.2015.08.005