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2. Literature
In this Section, we introduce the studies of VIX, VIX futures and risk-neutral skewness about future volatility forecasting. Furthermore, owing to the nonlinear feature of volatility, some researches that adopt Markov regime-switching model to capture this characteristic is also included in this Section.
2.1 VIX for Future Volatility Forecasting
The CBOE Volatility Index (VIX), introduced by the Chicago Board Options Exchange (CBOE) in 1993, is designed to measure the expected volatility of the S&P 500 index over the next 30 calendar days. When introduced in 1993, the VIX was originally based on implied volatilities of eight S&P 100 at-the-money put and call options. Since 2003, the VIX has been calculated, based on a model-free formula, by a wide range of S&P 500 out-of-the-money call and put option prices. This change is to reflect a more accurate view of market volatility. Some earlier studies find that implied volatility has predictive power for future volatility. For example, Latane and Rendleman (1976) demonstrate that the weighted average of Black and Scholes (B-S) call option implied volatilities is typically a better predictor of future volatility than volatility based on the historical return data. Fleming (1998) indicates that VIX has dominated predictive power for future volatility compared to historical volatility. Poon and Granger (2003) reviewed studies related to volatility forecasting and conclude that VIX is the best predictor for future volatility, although it may be a bias one. Moreover, Jiang and Tian (2005) suggest that their model-free volatility, which does not depend on option pricing model, subsumes all the information contained in B-S implied volatility and historical volatility and is a more efficient forecast for future volatility.
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2.2 VIX Futures for Future Volatility Forecasting
VIX futures were listed by the CBOE in March 26, 2004. They are exchange-traded futures contracts on volatility, and may be used to trade and hedge volatility. Since the VIX is untradable, a number of different studies involving forecast accuracy have been applied to VIX futures markets. Konstantinidi and Skiadopoulus (2011) demonstrate that VIX futures are predictable by their historical patterns, however the coefficients are too small to attain abnormal trading profits. Chung et al. (2011) investigate the informational role of S&P 500 index option and VIX option on the prediction of return, volatility and density in the S&P 500 index. They find that the information content implied from these two option market is not identical and all the predictions significantly improved by the information recover from VIX option. They apply the put-call parity to recover the information from VIX option, named implied VIX, which is similar to VIX futures. Shu and Zhang (2012) apply traditional linear Engle-Granger cointegration test and find that VIX futures prices have predictive ability on the underlying VIX. Furthermore, Frijns et al. (2013) document that VIX futures dominance VIX when the index returns is negative and the value of VIX is high. This finding suggests that on those days investors use VIX futures to hedge their positions rather than trading in the S&P 500 options.
Regarding to tests of Expectation Hypothesis, Nossman and Wilhelmsson (2009) test the expectation hypothesis, whether the VIX futures price is an unbiased estimator of the changes in the VIX index. They find that if the futures price is not adjusted by a risk premium, the expectation hypothesis is rejected. They report that risk premium adjusted futures prices predict the direction of one-day ahead VIX index correctly in 73 percent of the times.
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2.3 Risk-Neutral Skewness for Future Volatility Forecasting
The curve of S&P 500 implied volatility, also known as the smile or “skew”, has been one of the most studied features of S&P 500 option prices. Therefore, the Chicago Board Exchange (CBOE) introduced the CBOE Skew Index (SKEW) to measure the slope of the implied volatility curve that increases as the curve tends to steepen. This indicator can also measure the tail risk of the S&P 500 returns. Bakshi et al. (2003) show that the risk-neutral skewness can be expressed on the basis of option prices. Dennis and Mayhew (2002) describe a negative relation between the risk-neutral skewness and systematic risk, beta. They suggest that market risk is important in pricing individual stock options and indicate that market risk is reflected in the risk-neutral skewness extracted from the option prices. Similarly, Doran et al. (2007) report the evidence that risk-neutral skewness has strong predictive power in short-term crash/spike of the stock market. They find that large jump premium in the short term is the best explanation of significant negative skew for short maturity options. More recently, Byun and Kim (2013) investigate that risk-neutral skewness has incremental information content for future volatility in the S&P 500 index. Particularly, they concentrate on Heterogeneous Autoregressive model of Realized Volatility and Implied Volatility (HAR-RV-IV). They find that risk-neutral skewness significantly improve the accuracy of volatility forecasting at only daily and weekly horizons.
2.4 Markov Regime-Switching Model
Hamilton (1989) pioneered in the use of a Markov process to model the regime-changes, also termed “Markov switching model”. Since the introduction of Markov switching models to econometrics, there are considerable studies on using GARCH type with regime-switching models to capture the volatility dynamics of financial time series, in part because they give rise to a believable interpretation of nonlinearities associated
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with time-varying. Gray (1996) develops a generalized regime-switching (GRS) model using the conditional expectation of the past variance and the model can be regarded as the first MRS-GARCH. Marcucci (2005) compares different standard GARCH models and Markov regime-switching GARCH (MRS-GARCH) in terms of their ability to forecast the US stock market volatility at horizons that range from one day to one month.
The empirical analysis demonstrates that MRS-GARCH models outperform all standard GARCH models in forecasting volatility at horizons shorter than one week. However, at forecast horizons longer than one week, the asymmetric GARCH models are superior.
Relative to nonlinearities of realized volatility (RV), many previous researches evidence that RV exhibits high persistence or long-memory. Heterogeneous Autoregressive (HAR) model and Autoregressive Fractionally Integrated Moving Average (ARFIMA) models are generally used to capture this feature. Corsi (2009) utilizes past daily, weekly and monthly realized volatility to capture the long-memory property of realized volatility. Raggi and Bordignon (2012) adopt MRS-ARFIMA to capture long-memory and nonlinearities characteristic simultaneously. The out-sample results of volatility forecasting at several forecast horizons reveal that introducing these nonlinearities produces superior forecasts.
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intraday returns and the method to extract the implied volatility, risk-neutral skewness from VIX futures prices and S&P 500 option prices. In Section 3.3 and 3.4, we will specify the volatility forecasting model with Markov regime-switching and the methods to compare out-of-sample forecasting performance, respectively.3.1 Measuring Realized Volatility
compounded intraday return over the trading day t is denoted by, , , 1
t i t i t t
r ≡ p −p − (2)
where i=1, 2, ⋯, N
The daily realized volatility for the time t is defined by the sum of the square root of the daily realized variance.