4. S&P 500 Realized Volatility Forecasting
4.2 Methodology
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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
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a) 1-day RV forecasting model
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c) 21-day RV forecasting model
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20060512 20070301 20071213 20080930 20090717 20100504 20110216 20111201
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 adjusted R-square, it does not have so many informations about the future dynamic of VIX if we only use the past VIX change rates as our variables. However, in model A2.3 we predict the 5-day VIX change rate by using the risk neutral moments of VIX options. The result shows that SKV and KURV terms are significant in both regimes.
Also, in Model A2.4, VVIX can provide more information than VIX change rate.
Hence the risk neutral moments of VIX options and VVIX can improve the predict power of the VIX change rate forecasting under the two-state regime switching model.
After we join the risk neutral moments of VIX options and VVIX, the variance of the regime 2 become smaller. So the definite separation of two regimes via adding effective variables can make higher accuracy in the forecasting of VIX change rate.
When we combine those variables above into one model, the explanatory power is increase. Model A2.5 ha adjusted R-square of 0.4651, which is about 10% higher than model A2.3 and A2.4, but some of the variables do not present well. The VOLV and VVIX variables are useful in the regime 1, while they are insignificant in the regime 2. KURV term is insignificant in the both regime. We stay the VVIX term in order to capture the volatility of VIX and VIX change rate.
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skewness of VIX options (SKV), and VVIX in the regime switching model to forecast the 5-day forward VIX change rate. All variables in the model are significant in two regimes, and we can see the obviously differences of the variables to the 5-day future VIX change rate.
Comparing with model A2.5, model A2.6 has more significant variables, and more simple. The coefficients of the same terms in the regime 1 and regime 2 imply the different impact of the variables to the 5-day future VIX change rate. The 21-day past VIX change rate term is a special case in model 2.6: In the regime 1, it has negative sign, which means the mean-reverting property of VIX. While in the regime 2, it become significantly positive. Just like the volatility clustering property of VIX, the higher VIX change rate will make more higher VIX in the future in the extreme situation.
For the 21-day future VIX change rate model, the differences between the two regimes are more obvious. Model B2.1 has 0.5237 adjusted R-square. So we can easily separate the two regimes by just give them the conditional mean and variance in each regime. Hence, most of the explanatory power of the 21-day VIX change rate forecasting model is contributed by the regime switching model.
In model B2.2, the 5-day and 21-day CRV terms are significant, but the addition of CRV decrease the distance between the means two regimes. So the effect of regime separation becomes weaker, and we have lower explanatory power. In model B2.3 and B2.4, the VOLV and VVIX variables have strong significance in the two regimes.
However, most of the variables are only significant in regime 1, the normal situation.
This situation presents an important fact that VVIX and the risk neutral volatility of VIX options play critical rules in the 21-day VIX forecasting.
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立 政 治 大 學
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Model B2.5 and B2.6 show the same result that VOLV and VVIX terms are the deterministic variables in regime 2. If we look back to the 21-day VIX change rate, risk neutral volatility of VIX options and VVIX, we can find that they have similar pattern. Since VOLV and VVIX are more sensitive to the change of VIX, these variables will sharply increase because a shock occurs. Therefore, the 21-day VIX change rate will be affected in the same time. After the shock, VOLV and VVIX become the important indicators of the recovering rate. The negative coefficients of VOLV and VVIX in the regime 2 in model B2.5 and B2.6 implies the strong mean reverting property of 21-day VIX change rate under the extreme regime.
5.5 Model Analysis
Figure 8 shows the results of the predicted VIX change rate, real VIX change rate, and the regimes of the 5-day and 21-day VIX change rate forecasting. Since the predicted value is constrained by the two regimes, the fitting of extreme changes of VIX are not so perfect. We cannot know how much percentage VIX will increase in the 5 or 21 future day. Nevertheless, we capture the shock when the regime goes from regime 1 to regime 2, and the change rate turn back to normal as the regime goes back to regime 1.
From the scatter graph of 5-day VIX change rate in Figure 9, the two regimes separate the whole data into the low change rate regime (regime1) and the high change rate regime (regime2). The regime separation of the 21-day model becomes much wider than the 5-day model. Regime switching model does not fit well if we look to the situation of high VIX change rate. Even if the prediction is not very precise, we still predict the overall direction of the VIX change rate in the longer term.