3. Data
3.4 Risk Neutral Moments of VIX Options
5-minute return in one day, and the h-day realized volatility is the root mean square of daily realized volatility. In formulas:
2
The dynamic of S&P 500 realized volatility is similar to VIX, but the daily S&P 500 realized volatility is more volatile by comparing with the previous period. Since S&P 500 realized volatility reflects the intraday volatility, while VIX reflects the 30 days expectation, S&P 500 realized volatility has weaker autoregression property than VIX.1
3.3 VIX Options Data
In the data period, we have total 82,738 call and 44,188 put options of VIX come from every different date, expiration date, and exercise prices. Figure 2 shows the moneyness and daily frequency of VIX options in the sample period. The distribution of the VIX options moneyness is quietly different from the S&P 500 options - VIX options have wider range of moneyness.
3.4 Risk Neutral Moments of VIX Options
The near-month options data is chosen between 5- and 37-day time-to- maturity options and remove the options which have arbitrage opportunity. We have the 20,914 call and 13,611 put options of VIX to calculate the risk neutral moments. In the
1 See Long memory and nonlinearities in realized volatility: A Markov switching approach, Raggia and Bordignon (2012)
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frequency. The call options have longer tail on the right hand side (moneyness>2), and have heavy body in the 1.5 to 2 times the spot price. The positive skewness means that there are more investors trade in out-of-the- money VIX call options.calculation of the risk neutral momentum of VIX Options, we use the methodology from Bakshi, Kapadia and Madan (2003), and then get the daily risk neutral volatility, skewness, and kurtosis of VIX options. The formula of risk neutral moments is in Appendix II.
The risk neutral volatility of VIX is a kind of implied volatility of VIX, and the correlation between risk neutral volatility and VVIX is 0.84. The risk neutral skewness of VIX is usually positive. This fact reflects the property that the volatility smile of VIX options is a curve from lower left to the upper right side as the moneyness get higher.
If we compare the risk neutral skewness of VIX options to the risk neutral skewness of S&P 500 options, we can find that S&P 500 options have negative skewness. The reason is that people expect S&P 500 index is easy to fall down and hard to have a huge rise, while VIX usually increase shapely and return to normal slower.
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When VIX get higher and higher, the risk neutral skewness of VIX will close to zero and the risk neutral kurtosis of VIX will be low (See Figure 3). Hence, the VIX options volatility smile curve will tend to be a U-shape when VIX is high. Since the range of exercise price goes wider and the volatility of VIX increase, the implied volatility of the deep out-of-the-money and deep in-the-money VIX options will be higher than the normal situation.2
Figure 3 The scatter plots of risk neutral a) volatility, b) skewness, and c) kurtosis of VIX options
to VIX. High risk neutral volatility occurs when VIX is higher than 30. As VIX increase, the risk neutral skewness tends to be zero and the risk neutral kurtosis also tends to be lower.
2 See Consistent Modeling of SPX and VIX options, Gatheral (2008) 0
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VIX VVIX
We take the VVIX data from February 1, 2007 to 16 December, 2011 in the VIX forecasting model. Figure 4 presents the historical VIX and VVIX data in 2007-2011.
From Figure 4, we can see that VVIX is more sensitive to the sharp decreasing of S&P 500 index. We use VVIX as a variable in the regime switching model to predict the VIX change rate.
Table 1 shows the descriptive statistics of our data. We choose the 1-day, 5-day and 21-day future S&P 500 realized volatility as the dependent variables in the model to predict the future S&P 500 realized volatility. The data period is from 15 May, 2006 to 16 December, 2011. For the VIX forecasting model, we choose the 5-day and 21-day VIX change rate as the dependent variables. Since the time range of the VVIX data started from 2007, we reduce the data period in the VIX forecasting model. The data period for VIX forecasting is from5 May, 2006 to 16 December, 2011.
Figure 4 Volatility index (VIX) of S&P 500 index and volatility of volatility index (VVIX) form
2007 to 2011. The tags are the local maximums of VVIX.‧ 國
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Table 1 The Descriptive Statistics of Data
The Descriptive Statistics of 1-day, 5-day, 21-day S&P 500 realized volatilities (RVs), risk neutral skewness of S&P 500 options (SK), VIX, 1-day, 5-day, 21-day VIX change rates (CRVs) , risk neutral volatility, skewness, and kurtosis of VIX options (VOLV, SKV, KURV), and VVIX.
The data period is from 15 May, 2006 to 16 December, 2011, except the data of VVIX, which is from February 1, 2007 to 16 December, 2011.
RVt−1,t RVt−5,t RVt−21,t SK VIX CRVt−1,t CRVt−5,t CRVt−21,t VOLV SKV KURV VVIX
Mean 1.1096 1.1386 1.1599 -1.9213 24.3123 0.0471 0.2649 1.3232 0.9233 1.1452 7.5535 86.1197 Std. Error 0.0216 0.0204 0.0193 0.0134 0.3033 0.1981 0.3661 0.6360 0.0055 0.0187 0.1253 0.3824 Std. Dev. 0.8110 0.7649 0.7258 0.5025 11.3900 7.4476 13.757 23.9006 0.2068 0.7031 4.7080 13.4044 Skewness 2.6851 7.5927 5.6174 -0.2033 1.7241 4.1438 2.0295 2.1598 1.8867 0.6181 2.7799 0.9895
Kurtosis 11.3956 2.3696 2.1404 -0.1086 3.7736 0.7284 0.5813 1.1234 6.5465 1.9327 10.4052 1.2769 Min 0.2283 0.2626 0.4202 -3.6930 9.8900 -35.0588 -43.3592 -61.0001 0.5618 -1.3093 2.7487 59.7400 Max 8.6813 5.7764 4.6438 -0.6430 80.8600 49.6007 70.7415 110.1742 2.3016 4.8172 39.8771 145.1200
Numbers 1411 1411 1411 1411 1411 1411 1411 1411 1411 1411 1411 1229
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4.1 Single Variable Regression Testing
Before the introduce of the S&P 500 realized volatility forecasting models, we use single-variable linear regression to analysis the simple relationship between each variable and the 1-, 5-, 21-day S&P 500 realized volatility. From Table 2, the high adjusted R-square of RV terms show the autocorrelation property of the S&P 500 realized volatility. Also, the VIX change rate terms can provide some information to the future realized volatility.
Table 2 Single Variable Regression for S&P 500 Realized Volatility Forecasting Model
Single Variable Regression of the future 1-, 5-, 21-day S&P 500 realized volatility and each independent variable. All of the coefficients are 1% significant (except the α term of the VIX variable in 21-day forecasting). The regression function isRVt t h,
xt
t t h, .𝑅𝑉𝑡,𝑡+ℎ 𝑅𝑉𝑡−1,𝑡 𝑅𝑉𝑡−5,𝑡 𝑅𝑉𝑡−21,𝑡 VIX 𝐶𝑅𝑉𝑡−1,𝑡 𝐶𝑅𝑉𝑡−5,𝑡 𝐶𝑅𝑉𝑡−21,𝑡 𝑡 𝑉𝑡
h = 1
α 0.1966 0.1025 0.0983 - 0.3325 1.1092 1.1057 1.0904 1.9791 1.7616 0.8230 0.8831 0.8705 0.0593 0.1572 0.1584 0.0144 0.4525 - 0.5692 Adj.R2 0.6770 0.6950 0.6070 0.6940 0.0201 0.0715 0.18 0.078 0.243
h = 5
α 0.2715 0.1661 0.1470 - 0.2428 1.1416 1.1384 1.1227 2.0206 1.7681 0.7850 0.8560 0.8573 0.0570 0.0114 0.1270 0.0137 0.4573 - 0.5473 Adj.R2 0.6920 0.7330 0.6610 0.7190 0.0116 0.0513 0.1810 0.0895 0.2520
h=21
α 0.4123 0.3079 0.2814 - 0.0255 1.1673 1.1644 1.1520 1.9902 1.7049 0.6841 0.7578 0.7678 0.0493 0.0080 0.0103 0.0110 0.4280 - 0.4689 Adj.R2 0.5820 0.6390 0.5890 0.5980 0.0059 0.0369 0.1310 0.0869 0.2060
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The risk neutral skewness of VIX options has higher adjusted R-square value to the S&P 500 realized volatility than the risk neutral skewness of S&P 500 options:
The SK variable has adjusted R-square of approximately 8%, while SKV variable has 20% to 25%. The relationship between risk neutral skewness of VIX options and the future S&P 500 realized volatility is negative, and the S&P 500 realized volatility has strong positive correlation with VIX.
4.2 Methodology
Our model of S&P 500 realized volatility follows the HAR-RV-IV-SK model proposed by Byun and Kim (2013). The HAR-RV-IV-SK model is as Model 1.1.
For the next model, we add the 1-, 5- and 21-day change rate of VIX (CRV) into the HAR-RV-IV-SK model. We hope the new CRV variables can increase the explanation power of the future realized volatility of S&P 500 index. Then we have the second S&P 500 realized volatility forecasting model in model 1.2. The change rate of VIX comes from:
, ln / - ,
t h t t t h
CRV VIX VIX h
In model 1.3, we add the risk neutral skewness of VIX options (SKV) into the regression. We want to know if the additions of risk neutral skewness of VIX options (SKV) variables can provide more information on the original HAR-RV-IV-SK model.The calculation of SKV is the same formula of risk neutral skewness of S&P 500 options (see Appendix II).
In the final model for S&P 500 realized volatility forecasting, model 1.4 discusses the synergy of CRV and SKV variables in the HAR-RV-IV-SK model. The results are in Table 3.
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Model 1.2 HAR-RV-IV-CRV-SK Model
1,
1, 2Model 1.3 HAR-RV-IV-SK-SKV Model
1, 2Model 1.4 HAR-RV-IV-CRV-SK-SKV Model
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Table 3 Results of S&P 500 Realized Volatility Forecasting Model A) 1-day Future Realized Volatility of S&P 500 Index ( 𝑹𝑽
𝒕,𝒕+𝟏 )The model for 1-day S&P 500 realized volatility forecasting. Model (1.1) is the HAR-RV-IV-SK model, and (1.2) is HAR-RV-IV-CRV-SK model, which contains the change rate of VIX into the HAR-RV-IV-SK model. (1.3) adds risk neutral skewness of VIX options, i.e. the HAR-RV-IV-SK-SKV model. (1.4) is the HAR-RV-IV-CRV-SK-SKV model. The number in the parentheses is the t-statistics, and the stars behind the coefficients are the significant levels of 1%, 5% and 10%.
Model Intercept 𝑅𝑉𝑡−1,𝑡 𝑅𝑉𝑡−5,𝑡 𝑅𝑉𝑡−21,𝑡 𝑉 𝑡 𝐶𝑅𝑉𝑡−1,𝑡 𝐶𝑅𝑉𝑡−5,𝑡 𝐶𝑅𝑉𝑡−21,𝑡 𝑡 𝑉𝑡 Adj.𝑅2
A1.1 -0.0502 0.2693*** 0.3205*** -0.1914*** 0.0342*** 0.0588*** 0.750
(-0.8855) (8.3764) (7.1068) (-4.0270) (11.1077) (2.6170)
A1.2 0.0353 0.1955*** 0.2515*** 0.0093 0.0281*** 0.0051*** 0.0032*** 0.0030*** 0.0670*** 0.763 (0.6288) (5.9936) (5.1122) (0.1760) (8.5399) (3.1258) (3.0595) (4.9185) (3.0153)
A1.3 -0.0608 0.2693*** 0.3209*** -0.1915*** 0.0344*** 0.0611*** 0.0079 0.749
(-0.9741) (8.3748) (7.1118) (-4.0275) (10.9905) (2.6359) (0.4066)
A1.4 -0.0148 0.1929*** 0.2454*** 0.0178 0.0293*** 0.0050*** 0.0034*** 0.0032*** 0.0803*** 0.0409** 0.764 (-0.2435) (5.9161) (4.9841) (0.3342) (8.7834) (3.0295) (3.1897) (5.2179) (3.4805) (2.1114)
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B) 5-day Future Realized Volatility of S&P 500 Index ( 𝑹𝑽
𝒕,𝒕+𝟓 )The model for 5-day S&P 500 realized volatility forecasting. Model (1.1) is the HAR-RV-IV-SK model, and (1.2) is HAR-RV-IV-CRV-SK model, which contains the change rate of VIX into the HAR-RV-IV-SK model. (1.3) adds risk neutral skewness of VIX options, i.e. the HAR-RV-IV-SK-SKV model. (1.4) is the HAR-RV-IV-CRV-SK-SKV model. The number in the parentheses is the t-statistics, and the stars behind the coefficients are the significant levels of 1%, 5% and 10%.
Model Intercept 𝑅𝑉𝑡−1,𝑡 𝑅𝑉𝑡−5,𝑡 𝑅𝑉𝑡−21,𝑡 𝑉 𝑡 𝐶𝑅𝑉𝑡−1,𝑡 𝐶𝑅𝑉𝑡−5,𝑡 𝐶𝑅𝑉𝑡−21,𝑡 𝑡 𝑉𝑡 Adj.𝑅2
B1.1 0.1175** 0.2253*** 0.3216*** -0.0219 0.0241*** 0.0791*** 0.776
(2.3215) (7.8389) (7.9770) (-0.5157) (8.7537) (3.9398)
B1.2 0.2000*** 0.1679*** 0.2238*** 0.1381*** 0.0212*** 0.0029** 0.0017* 0.0036*** 0.0938*** 0.789 (3.9882) (5.7560) (5.0899) (2.9132) (7.2001) (1.9723) (1.8018) (6.6281) (4.7241)
B1.3 0.0983* 0.2254*** 0.3223*** -0.0220 0.0245*** 0.0833*** 0.0143 0.776
(1.7612) (7.8402) (7.9916) (-0.5171) (8.7557) (4.0198) (0.8187)
B1.4 0.1426*** 0.1650*** 0.2167*** 0.1478*** 0.0225*** 0.0027* 0.0019** 0.0038*** 0.1090*** 0.0468*** 0.790 (2.6242) (5.6638) (4.9328) (3.1168) (7.5660) (1.8518) (1.9708) (7.0110) (5.2924) (2.7052)
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Table 3 Results of S&P 500 Realized Volatility Forecasting Model (Continued) C) 21-day Future Realized Volatility of S&P 500 Index ( 𝑹𝑽
𝒕,𝒕+𝟐𝟏 )The model for 21-day S&P 500 realized volatility forecasting. Model (1.1) is the HAR-RV-IV-SK model, and (1.2) is HAR-RV-IV-CRV-SK model, which contains the change rate of VIX into the HAR-RV-IV-SK model. (1.3) adds risk neutral skewness of VIX options, i.e. the HAR-RV-IV-SK-SKV model. (1.4) is the HAR-RV-IV-CRV-SK-SKV model. The number in the parentheses is the t-statistics, and the stars behind the coefficients are the significant levels of 1%, 5% and 10%.
Model Intercept 𝑅𝑉𝑡−1,𝑡 𝑅𝑉𝑡−5,𝑡 𝑅𝑉𝑡−21,𝑡 𝑉 𝑡 𝐶𝑅𝑉𝑡−1,𝑡 𝐶𝑅𝑉𝑡−5,𝑡 𝐶𝑅𝑉𝑡−21,𝑡 𝑡 𝑉𝑡 Adj.𝑅2
C1.1 0.4109*** 0.1759*** 0.3266*** 0.1714*** 0.0073* 0.0954*** 0.664
(6.9197) (5.1404) (6.8560) (3.4457) (2.2679) (4.0464)
C1.2 0.4691*** 0.1139*** 0.2921*** 0.3359*** 0.0016 0.0031* 0.0040*** 0.0014** 0.0979*** 0.673 (7.8578) (3.2300) (5.5283) (5.9656) (0.4503) (1.7420) (3.5335) (2.2266) (4.1333)
C1.3 0.3754*** 0.1759*** 0.3281*** 0.1713*** 0.0081** 0.1032*** 0.0265 0.664
(5.7455) (5.1412) (6.8864) (3.4450) (2.4664) (4.2442) (1.3043)
C1.4 0.4083*** 0.1105*** 0.2847*** 0.3464*** 0.0030 0.0029 0.0042*** 0.0017*** 0.1142*** 0.0499** 0.674 (6.3171) (3.1353) (5.3886) (6.1449) (0.8501) (1.6338) (3.6835) (2.6184) (4.6482) (2.4306)
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In the S&P 500 1-day future realized volatility forecasting model (Table 3A), all the coefficients of model A1.1, HAR-RV-IV-SK model, are significant except the interception term. Comparing to the original HAR-RV-IV-SK model by Byun and Kim (2013), our research has higher significance. The differences may come from the different data period.
In model A1.2, we add the past 1-day, 5-day, 21-day VIX change rate terms in the model, and the model become a HAR-RV-IV-CRV-SK model. The adjusted R-square of model A1.2 is 0.763 that increases 1.3% from model A1.1. We attribute the increase of adjusted R-square to the information content of VIX change rate.
However, the coefficient of past 21-day S&P 500 realized volatility in A1.2 increases from -0.1914 to 0.0093 and then become both positive and insignificant.
The adding of VIX change rate variables reduces the coefficient values of 1-day, 5-day RV and VIX terms, but increase the coefficient values of 21-day and S&P 500 skewness term. VIX change rate also makes a structural change of the model. This situation can also be observed in the model A1.4 and the 5-day and 21-day S&P 500 realized volatility forecasting model.
Model A1.3 shows that the VIX skewness cannot promote the explanatory power, and the VIX skewness term is insignificant. However, in Model A1.4, the VIX change rate terms increase the power of the S&P 500 realized volatility forecasting model and make the coefficient of VIX skewness becomes significant but has little impact on the explanatory power of the model. From Model 1.3 to Model 1.4, the t-statistic of both S&P500 skewness and VIX skewness term increase that means the VIX change rate can increase the significance of both variables.
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For the 5-day S&P500 realized volatility forecasting, the 21-day RV term is negative and insignificant. While we add 1-day, 5-day, and 21-day VIX change rates as new variables the 21-day RV term is positive and significant. The 1-day and 5-day VIX change rate terms in Model B1.2 and B1.4 provide less information to the 5-day future S&P500 realized volatility. Also, the adding of VIX change rates increases the significance of S&P500 skewness and VIX skewness and raises the explanatory power of the model.
In the model of 21-day S&P500 realized volatility forecasting, the VIX and 1-day VIX terms become less informative in the models. Still, the adding of VIX change rate makes the forecasting model becomes more powerful and redistributes the weight of each variable in the model. The changing of the model structure increases the importance of the past 21-day realized volatility, VIX change rates, S&P500 skewness and VIX skewness terms.
Note that the 5-day S&P500 realized volatility forecasting model has the highest adjusted R-square of the models. In the data period, the HAR-RV-IV-CRV- SK-SKV model provides more accuracy to the 5-day S&P500 realized volatility, comparing to the 1-day and 21-day models. The reasons may be that 1-day S&P500 realized volatility has too much noise to make the prediction less accurate and the 21-day time horizon contains more uncertainty in the future. We conclude that the 5 trading days is a more appropriate time horizon to forecast the S&P500 realized volatility.
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To know the behaviors of the new-adding variables, we do a simple residual analysis on the VIX, VIX change rate, the risk neutral skewness of S&P 500 Options and VIX options variables. Figure 5 is the scatter chart of VIX to the regression on when the VIX change rate residual in 1-day, 5-day, and 21-day in HAR-RV- IV-CRV-SK-SKV model. The residual become wider distributed as VIX increase.
Figure 6 is the scatter chart of the past 5-day VIX change rate to the error. Most of the high residuals occur in the situation of high VIX and close-to-zero VIX change rate. On the other hand, we can have a more precise prediction when the VIX change rates have previous patterns. The residuals distribution of the 1-day and 21-day historical VIX change rate are similar to the 5-day VIX change rate.
In Figure 7, we show the relationship of both the risk neutral skewness of S&P 500 options and risk neutral skewness of VIX options to the error term of the regression model. In the normal situation S&P 500 options have negative skewness, and VIX options have positive skewness.
While in the extreme events, the volatility of S&P 500 index become higher and VIX will increase. The degrees of volatility smile in the both S&P 500 options and VIX options tend to U-shapes. Therefore, the risk neutral skewness of VIX options will decrease toward zero and the risk neutral skewness of S&P 500 options increase toward zero when VIX is high. So the S&P500 realized volatility become unpredictable in the high volatile situation.
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21-Day Future RV 5-Day Future RV 1-Day future RV
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21-Day future RV 5-Day future RV 1-Day future RV
Figure 5 VIX verses the residual of model 1.4 for the 1-day, 5 -day, and 21-day S&P 500 future
realized volatility. The scale of VIX (x-axis) is the exponential type. The error is small if VIX is low, and become higher as VIX increase. The errors separate widely in the range of 40 to 80.Figure 6 5-day VIX change rate verses the error of model 1.4 for the 1-, 5 -, and 21-day S&P 500
future realized volatility. Most of the residuals lie on the range of -0.5 to 0.5. When 5-day CRV is closed to zero, the variables become less informative. It is more difficult to capture the future dynamic of S&P 500 realized volatility from the VIX change rate.‧
21-day S&P 500 realized volatility forecasting. S&P 500 options have positive skewness, and VIX options are usually positive. As the risk neutral skewness of both S&P 500 options and VIX options close to zero, the errors tent to be widely distributed.-2
b) 5-day RV forecasting model
-2
a) 1-day RV forecasting model
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c) 21-day RV forecasting model
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5. VIX Forecasting
In this section, we focus on the VIX forecasting by using regime switching model in the VIX change rate prediction. Figure 7 shows the historical VIX and the 1-day, 5-day, and 21-day VIX change rate (CRVs). From the 1-day VIX change rate, we observe the volatility clustering of VIX. However, we cannot affirm any information but its amplitude in the graph of 1-day VIX change. In the 5-day and 21-day change rate of VIX, we can clearly see the change direction and the accumulated amplitude. Hence, we use the 5-day and 21- day VIX change rate to be our dependent variables in the regression.
Figure 8 The relationship within VIX, 1-day, 5-day, and 21-day historical VIX change rate.
The formula of VIX change rate isCRVt h t , ln
VIXt /VIXt h-
. As the period of the change rate variable gets longer, the pattern of accumulative change of VIX become more apparent.VIX
1-Day CRV
5-Day CRV
21-Day CRV
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For the independent variables, we test the past 1-day, 5-day, 21-day VIX change rate (CRV), risk neutral momentum (including volatility, skewness, and kurtosis) of VIX options (VOLV, SKV, KURV), and VIX of VIX (VVIX) in the VIX forecasting model. Table 4 is the result of single-variable linear regression of the VIX change rate forecasting model.
In the single-variable regression, all the independent variables have low explanatory power (adjusted R2< 8%). However, if we use the two-state regime switching model to forecast the 5-day or 21-day futures VIX change rate, we can increase the power of these variables. We have different coefficients in the different states. This property will help us to capture the dynamic of VIX.
Table 4 Single Variable Regression for VIX Change Rate Forecasting Model
The single-variable regression of the future 5-day and 21-day VIX change rate with each independent variable. The regression function isCRVt t h,
xt
t t h, .𝐶𝑅𝑉𝑡,𝑡+ℎ 𝐶𝑅𝑉𝑡−1,𝑡 𝐶𝑅𝑉𝑡−5,𝑡 𝐶𝑅𝑉𝑡−21,𝑡 𝑉 𝑉𝑡 𝑉𝑡 𝑅𝑉𝑡 𝑉𝑉 𝑡
h = 5
α 0.3953 0.4646 0.4613 9.9189*** -3.4180*** -1.4849** 12.2485***
-0.2891*** -0.2404*** -0.0514 -10.2763*** 3.5099*** 0.2494*** -0.1380***
Adj.R2 0.0237 0.0569 0.00738 0.0226 0.0273 0.00583 0.0166
h = 21
α 1.7107 1.7541 1.8880*** 24.8897*** -9.3314*** -3.7705*** 37.1807***
-0.2984*** -0.1609*** -0.1131*** -25.0186*** 10.1982*** 0.7304*** -0.4136***
Adj.R2 0.00758 0.00758 0.0121 0.0442 0.0769 0.0178 0.0491
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The regime switching model is given by a structure of different regimes, we have:
, ~ (0, 2)
We use the following two steps to determine the log likelihood function:
First, consider the joint density of 𝑦𝑡and the unobserved 𝑡 variable, which is the product of the conditional and marginal densities:
1 1 1
( ,t t| t ) ( t | t, t ) ( t| t ) f y S F f y S F f S F where 𝐹𝑡−1 refers to information up to time 𝑡 − 1.
Second, to obtain the marginal density of 𝑦𝑡, integrate the 𝑡 variable out of the above joint density by summing over all possible values of 𝑡:
1 1
The log likelihood function is given by
1
3 The theorem comes from the book by Kim and Nelson, 1999. State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications. Page 59-95.
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In the 5-day and 21- day VIX forecasting model, we separate the time series into regime 1 and regime 2. By adopting the regime switching model, all of the variables in the regression will have different coefficient in each state. Note that distinguish of two regimes differs from the different input and output variables.
First, in model 2.1, we do a previous observation that if we can separate the time line into two regimes: the normal volatility regime (Regime 1) and the extreme volatility regime (Regime 2).Then we add the past 1-day, 5-day, and 21-day VIX change rates (CRVs) in model 2.2. The model becomes a Regime Switching Heterogeneous Autoregressive model of VIX Change Rate (RS-HAR-CRV model).
The informations of the VIX derivatives are important to VIX forecasting. Under the regime switching model, we use the risk neutral volatility, skewness, of VIX options (VOLV, SKV, KURV) variables in the model 2.3. Also, in model 2.4, we use the volatility of volatility index (VVIX) to predict the future VIX change rate.
Next we show the synergy of the all variables in model 2.5 by adding all the variables above into the regime switching model. Finally, we reduce the independent variables to VIX change rate (CRV), risk neutral skewness of VIX options (SKV) and VVIX in model 2.6.
The regression formulas of the models are as below. Each model has two groups of the beta coefficients and the different coefficient of the standard deviation in the residual term in the different regime. The results of 5-day VIX forecasting model is in Table 5 and the results of 21-day VIX forecasting model is in Table 6.
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Model 2.1 Simple Regime Switching Model (on VIX Change Rate)
2
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Table 5 Regime Switching Model for the 5-day VIX Change Rate Forecasting
The regime switching model for the 5-day VIX change rate forecasting. The number in the parentheses is the t-statistics, and the stars behind the coefficients are the significant levels of 1%, 5% and 10%. The transition probabilities are the probabilities that the regime change form i to j (i, j = 1, 2).
Transition Probabilities
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The regime switching model for the 21-day VIX change rate forecasting. The number in the parentheses is the t-statistics, and the stars behind the coefficients are the significant levels of 1%, 5% and 10%. The transition probabilities are the probabilities that the regime change form i to j (i, j = 1, 2).
Transition Probabilities
Model Intercept CRVt−1,t CRVt−5,t CRVt−21,t VOLVt SKVt KURVt VV 𝑡 Sigma To Regime 1 Regime 2 Adj.R2
B2.1 Regime 1 -8.2142*** 14.1073*** 0.99*** 0.01 0.5237
(-13.7799) (4.3609)
Regime 2 34.8365*** 22.9001*** 0.05*** 0.95
(17.7402) (3.3837)
B2.2 Regime 1 -9.8859*** -0.1112 -0.3172*** -0.2175*** 11.6646*** 0.98*** 0.02 0.4436
(-20.2621) (1.6210) (8.0508) (9.7098) (4.1347)
Regime 2 23.5203*** -0.4914** 0.8957*** -0.3233*** 27.5125*** 0.03*** 0.97
(13.1186) (2.6956) (5.6906) (5.0437) (3.0284)
B2.3 Regime 1 14.5879*** -30.6915*** 7.5713*** -0.4276*** 13.2329*** 0.98*** 0.02 0.5825
(4.3970) (-9.0758) (8.3000) (-3.4963) (3.9221)
Regime 2 72.0552*** -46.8926*** 2.7645 0.0632 21.2359*** 0.05*** 0.95
(8.7879) (-8.0913) (1.0446) (0.2576) (3.4901)
B2.4 Regime 1 28.4374*** -0.4700*** 12.2089*** 0.98*** 0.02 0.5682
(6.9334) (-9.1797) (3.9465)
Regime 2 104.0067*** -0.9147*** 21.0203*** 0.04*** 0.96
(14.1415) (-11.0338) (3.8271)
B2.5 Regime 1 -6.5782 -0.1940*** -0.0535 -0.1172*** 10.8242** 6.0724*** 0.2030* -0.2329*** 12.5265*** 0.98*** 0.02 0.5577 (-1.4416) (-3.2826) (1.2442) (-6.1684) (2.4963) (5.7531) (1.8174) (-3.1774) (5.2174)
Regime 2 -10.6672** -0.8905*** 0.0292 -0.1220 -54.7147*** 5.9532 0.2157 0.9991*** 24.3090*** 0.05*** 0.95 (-2.3895) (-9.9720) (0.3074) (-1.4170) (-3.9697) (1.5867) (0.4431) (5.0562) (3.6674)
B2.6 Regime 1 -2.9040 -0.0958 -0.3389*** -0.1317*** 5.0355*** -0.1423*** 11.9434*** 0.98*** 0.02 0.5930 (-0.7581) (-1.3666) (-8.1860) (-5.2262) (6.2701) (3.2195) (4.2267)
Regime 2 83.8596*** 0.0280 -0.0775 -0.0872 0.7063 -0.6361*** 22.2546*** 0.04*** 0.96 (6.6294) (0.1649) (-0.7092) (-1.1489) (0.3402) (4.5763) (6.3587)
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5.4 VIX Forecasting Results
In the 5-day VIX change rate forecasting, model A2.1 simply separate the data into regime 1 and regime 2. In the regime 1 of model A2.1, the 5-day VIX change rate follows a normal distribution with negative mean (μ1 = − ) and relatively lower variance (σ1 = ). This means that VIX tend to be stable and slowly decrease in the regime 1. In the regime 2, VIX increase (μ2 = ) and become more volatile (σ2 = 1 ), but the variance is too large to make sure the change direction of VIX.
Model A2.2 is the MS-HAR-CRV model. Although the CRV can increase the
Model A2.2 is the MS-HAR-CRV model. Although the CRV can increase the