5. VIX Forecasting
5.5 Model Analysis
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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.
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changing of regime. The data is from Feb. 2, 2007 to Dec. 8, 2011.1
A) 5-day VIX change rate (model A2.6)
B) 21-day VIX change rate (model B2.6)
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rate and B) 21-day VIX change rate in model 2.6. The data near by the original point of each graph is in regime 1, and the data with high VIX change rate (real and predicted) is in regime 2R² = 0.5937
Real 21-day VIX change rate
Predicted 21-day VIX change rate
R² = 0.4702
Real 5-day VIX change rate
Predicted 5-day VIX change rate
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In the first part of this paper, we add the VIX change rate and risk neutral skewness of VIX options into the Heterogeneous Autoregressive model of Realized Volatility, Implied Volatility and Skewness (HAR-RV-IV-SK) model to capture the future dynamic of S&P 500 index realized volatility.
The result shows that the past 1-day, 5-day, and 21-day VIX change rate have information content to the future realized volatility of S&P index. The VIX change rate variables also increase the explanatory power of the model. In the same model, the risk neutral skewness of VIX options is significant when join with the VIX change rate, but provides less information about the future volatility of S&P 500 index.
From the residual analysis of VIX and 5-day VIX change rate, we find that the model has better predict power when VIX is low and the past VIX change rates are away from zero. The risk neutral skewness of S&P 500 options and VIX options are more useful when SK is strong negative and SKV is strong positive.
In the second model for VIX forecasting, we use the regime switching model to the capture dynamic of future 5-day and 21-day VIX. The regime switching model makes the forecasting model become more useful and flexible. Furthermore, we join the past VIX change rate, risk neutral volatility, skewness, kurtosis of VIX options, and VVIX variables in the VIX forecasting. The result indicates that risk neutral skewness of VIX options and VVIX is useful to forecast the 5-day future VIX change rate in the both regimes.
While in the 21-day VIX forecasting, the regime switching model can be more powerful, because the differences between the two regimes become wider. VVIX still contain strong information about the future 21-day VIX, while risk neutral skewness
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of VIX options contain less information relative to VVIX in the 21-day VIX forecasting model. In extreme situation, it is difficult to know how much VIX will increase. Nevertheless, our model provides a direct tool to forecast the future change rate of VIX.
This paper finds the properties of the VIX change rate, risk neutral moments of VIX options and VVIX variables. Also, we point out the information contents of these variables on the dynamics of S&P 500 realized volatility and VIX change rate. We can apply the S&P 500 realized volatility and VIX forecasting model in the detection of the abnormal change of S&P 500 index, VIX, or financial crisis.
Still, there are some details we need to think more in this paper. The first consideration is about the calculation of the risk neutral moments of VIX options. We take the 5-day to 21-day time-to-maturity VIX options to evaluate the risk neutral moments of VIX options variables in this paper. But the results may vary from the different expiration date of VIX options contracts.
Second, in the VIX forecasting model, the structure of regime switching can provide a preliminary observation to the dynamic of VIX. Nevertheless, the regime switching model is not so robust and reasonable enough. We need time to verify its accuracy and sustainability.
For the further research, we can apply the model into the different index, like the Dow Jones Industrial Average index (DJIA) and its volatility index (VXD). Also, we can use different kinds of model and take more useful variables to capture the dynamics of volatility index.
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F: Forward index level desired from index option prices : First strike below the forward index level F
: Strike price of the ith out-of-the-money option; a call if ; and a put if ; both put and call if =
: Interval between strike prices - half the difference between the strike on either side of
(Note: for the lowest strike is simply the difference between the lowest strike and the next higher strike. Likewise, for the highest strike is the difference between the highest strike and the next lower strike.)
R: Risk-free interest rate to expiration
( ): The midpoint of the bid-ask spread for each option with strike
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Appendix II – The formula of risk neutral moments of VIX options
𝑘𝑒𝑤(𝑡, 𝜏) = 𝑒𝑟𝑡𝑊(𝑡, 𝜏) − 𝜇(𝑡, 𝜏)𝑒𝑟𝑡𝑉(𝑡, 𝜏) + 𝜇(𝑡, 𝜏)3
And we set the risk neutral volatility as:
𝑉𝑜𝑙(𝑡, 𝜏) = 1 [𝑡𝑉(𝑡, 𝜏)]1/2
𝐶(𝑡, 𝜏; ) is the market prices of call and 𝑃(𝑡, 𝜏; ) is the market prices of put options with strike price K, maturity 𝜏 from time 𝑡.
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