Modeling Volatility of S&P 500 Index Daily Returns:

Transcription

1 Modeling Volatility of S&P 500 Index Daily Returns: A comparison between model based forecasts and implied volatility Huang Kun Department of Finance and Statistics Hanken School of Economics Vasa 2011

2 HANKEN SCHOOL OF ECONOMICS Department of: Finance and Statistics Type of work: Thesis Author: Huang Kun Date: April, 2011 Title of thesis: Modeling Volatility of S&P 500 Index Daily Returns: A comparison between model based forecasts and implied volatility Abstract: The objective of this study is to investigate the predictability of model based forecasts and the VIX index on forecasting future volatility of S&P 500 index daily returns. The study period is from January 1990 to December 2010, including 5291 observations. A variety of time series models were estimated, including random walk model, GARCH (1,1), GJR(1,1) and EGARCH (1,1) models. The study results indicate that GJR (1,1) outperforms other time series models for out-of-sample forecasting. The forecast performance of VIX, GJR(1,1) and RiskMetrics were compared using various approaches. The empirical evidence does not support the view that implied volatility subsumes all information content, and the study results provide strong evidence indicating that GJR (1,1) outperforms VIX and RiskMetrics for modeling future volatility of S&P 500 index daily returns. Additionally, the results of the encompassing regression for future realized volatility at 5-, 10-, 15-, 30- and 60-day horizons, and the results of the encompassing regression for squared return shocks suggest that the joint use of GJR (1,1) and RiskMetrics can produce the best forecasts. By and large, our finding indicates that implied volatility is inferior for future volatility forecasting, and the model based forecasts have more explanatory power for future volatility. Keywords: volatility, S&P 500, GARCH, GJR, RiskMetrics, implied volatility

3 CONTENTS 1 Introduction 2 2 Literature Review.6 3 The CBOE Volatility Index VIX Implied Volatility The VIX Index 17 4 Time Series Models for Volatility Forecasting Random Walk Model The ARCH(q) Model The GARCH (p,q) Model The Stylized Facts of Volatility The GJR (p,q) Model The EGARCH (p,q) Model RiskMetrics Approach 25 5 Practical Issues for Model-building Test ARCH Effect Information Criterion Evaluating the Volatility Forecasts Out-of-sample Forecast Traditional Evaluation Statistics Data S&P 500 Index Daily Returns Autocorrelation of S&P 500 Index Daily Returns Testing ARCH Effect of S&P 500 Index Daily Returns Properties of the VIX Index Study on S&P 500 Index and the VIX Index Cross-correlation between S&P 500 Index and the VIX Index.34

4 6.3.2 S&P 500 Index Daily Returns and the VIX Index Estimation and Discussion Model Selection Test Numerical Accuracy of GARCH Estimates Estimates of Models BDS Test Graphical Diagnostic.51 8 Forecast Performance of Model Based Forecasts and VIX Out-of-sample Forecast Performance of GARCH Models In-sample Forecast Performance of VIX Comparing Predictability of Time Series Models and VIX Correlation between Realized Volatility and Volatility Forecasts Regression for In-sample Realized Volatility Residual Tests for Regression of In-sample Realized Volatility Regression for Out-of-sample Realized Volatility Residual Tests for Regression of Out-of-sample Realized Volatility Encompassing Regression for Realized Volatility Average Squared Deviation Regression for Squared Return Shocks Encompassing Regression for Squared Daily Return Shocks Conclusion.80 References.81 Appendix A. VIX and Future Realized Volatility.86 Appendix B. Out-of-sample Forecast Performance on Realized Volatility..89 Appendix C. Residuals from Regression for Out-of-sample Realized Volatility 91

5 TABLES Table 1.Summary statistics for S&P 500 index daily returns 31 Table 2 Test for ARCH effect in S&P 500 daily index returns 33 Table 3. Summary statistics of the VIX index 34 Table 4. Cross-correlation between S&P 500 index daily returns and implied volatility index 35 Table 5. Regression results for VIX changes and S&P 500 index daily returns 38 Table 6. Information criteria for estimated GARCH (p,q) models 44 Table 7. The summary statistics of estimated volatility models 47 Table 8. BDS test for serial independence in residuals 50 Table 9. Forecast Performance of GARCH models 53 Table 10. In-sample forecast performance of VIX and GARCH specifications 55 Table 11 Correlation between Realized Volatility and Alternative Forecasters 59 Table 12. Performance of regression for in-sample realized volatility 61 Table 13. Forecast performance on out-of-sample realized volatility 63 Table 14. Residual tests for regression for in-sample realized volatility 66 Table 15. Performance of regression for out-of-sample realized volatility 68 Table 16. Residual tests for regression for out-of-sample realized volatility 71 Table 17. Encompassing regression for realized volatility 74 Table 18. The average squared deviation from alternative approaches 76 Table 19. Regression results for squared return shocks 77 Table 20. Encompassing regression results for squared return shocks 78 FIGURES Figure 1.Daily returns, squared daily returns and absolute daily returns for the S&P 500 index 32 Figure 2. Autocorrelation of, and for S&P 500 index 33 Figure 3. S&P 500 Index (logarithm) and the VIX Index 36 Figure 4. S&P 500 index daily returns and the VIX index 41 Figure 5 S&P 500 index absolute daily returns and the VIX index 42 Figure 6. Estimates from various GARCH (p,q) models 45 Figure 7. Graphical residual diagnostics from GARCH (1,1) to S&P 500 returns 52

6 2 1 Introduction Volatility is computed as the standard deviation of equity returns. Modeling volatility in financial market is important because volatility is often perceived as a significant element for the evaluation of assets, the measurement of risk, the investment decision making, the valuation of security and the monetary policy making. The stock market volatility is virtually time-varying. The empirical evidence dates back to the well-known pioneering studies of Mandelbrot (1963) and Fama (1965) demonstrated that large price (small price) changes tend to be followed by large price (small price) changes, implying that there are some periods which display pronounced volatility clustering. It is widely accepted that volatility changes in financial market are predictable. The various models have been applied by extensive empirical studies for future volatility forecasting and measuring the predictability of volatility forecasts. However, there is little consensus in terms of which model or family of models is the best for describing assets returns. To date the two most popular approaches for future volatility forecasting are considered to be the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model and the RistMetrics approach introduced by Robert Engle (1982) and J. P. Morgan (1992), respectively. The forecasts of these two approaches are derived on the basis of historical data. Additionally, the volatility implied from the actual observed option price is thought to be an efficient volatility forecasts and becoming more and more popular for volatility forecasting, particularly in the U.S market. A large number of empirical evidence documented that, under the efficient option market, implied volatility subsume forward-looking information contained in all other variables in the market s information set that help measure volatility of option s lifetime. By and large, the conventional approaches for volatility forecasting are classified into two categories, and they are time series models based on historical data and volatility implied from observed option price. The GARCH model is the natural extension of autoregressive conditional heteroscedasticity (ARCH) model which was thought to be the good description of

7 3 stock returns and an efficient technique for estimating and analyzing time-varying volatility in stock returns. The seminal ARCH (q) model was pioneered by Engle (1982), representing a function of the squared returns of the past q periods and formulating the conditional variance of returns via maximum likelihood procedure rather than making use of the sample standard deviation. However, there are some limitations of ARCH (q) model. For example, how to decide the appropriate number of lags of the squared residual in the model; the large value of q may induce a nonparsimonious conditional variance model; non-negative constraints might be violated. Some problems of ARCH (q) model can be overcome by GARCH (p,q) model which incorporates the additional dependencies on p lags of the past volatility and the variance of residuals is modeled by an autoregressive moving average ARMA (p,q) process replacing the AR (q) process of ARCH (q) model. GARCH (p,q) model is widely used in practice. The extensive empirical evidence suggest that GARCH (p,q) model is a more parsimonious model than ARCH (q) model and provides a framework for deeper time-varying volatility estimation. One of outstanding features of the GARCH (p,q) model is that it can effectively remove the excess kurtosis in returns. Particularly, GARCH (1,1) model is widely recognized as the most popular framework for modeling volatilities of many financial time series. However, the standard symmetric GARCH (p,q) model also has some underlying limitations. For instance, the requirement that the conditional variance is positive may be violated for the estimated model. The only way to avoid this problem is to place the constraints for coefficients to force them to be positive. The second limitation is that it cannot explain the leverage effect, although it has good performance for explaining volatility clustering and leptokurtosis in a time series. Thirdly, the direct feedback between the conditional mean and conditional variance is not allowed by the standard GARCH (p,q) model. In order to overcome the limitations of the standard symmetric GARCH (p,q) model, a number of extensions have been introduced, such as the asymmetric GJR (p,q) and EGARCH (p,q) models which can better capture the dynamics of time series

8 4 and make the modeling more flexible. As another conventional approach for volatility forecasting, implied volatility is the volatility implied from observed option price and computed by option pricing formulas, such as the Black-Scholes formula which is widely used in practice. As we know, the required parameters for computing option price using Black-Scholes model are stock price, strike price, risk free interest rate, time to maturity, volatility as well as dividend. Being the unique unknown parameter, implied volatility is thought to be the representation of the future volatility by consensus because option is priced on the basis of future payoffs. Today, implied volatility indices have been constructed and published by stock exchange in many countries, and it is widely recognized that implied volatility index has superior predictability for future stock market volatility. A common question regarding to implied volatility is whether the option price subsumes all relevant information about future volatility. The large number of empirical evidence from previous studies (e.g., Fleming, Ostdiek and Whaley 1995, Christensen and Prabhala 1998, Giot 2005a, Giot 2005b, Corrado and Miller, JR. 2005, Giot and Laurent 2006, Frijns, Tallau and Tourani-Rad 2008, Becker, Clements and McClelland 2009, Becker, Clements and Coleman-Fenn 2009, Frijns, Tallau and Tourani-Rad 2010) demonstrate that implied volatility is a forward-looking measure of market volatility. However, the poor predictive power of implied volatility was also indicated by some studies, such as Day and Lewis (1992), Canina and Figlewski (1993), Becker, Clements and White (2006), Becker, Clements and White (2007) and Becker and Clements (2008). The objective of our study is to investigate whether the model based forecasts or the CBOE volatility index (the VIX index published by Chicago Board Options Exchange) is superior on forecasting future volatility of S&P 500 index daily returns The data used for our study ranges from January 1990 to December There are several reasons why we consider the use of the VIX index. First, it is on the basis of S&P 500 index which is considered to be the core index for the U.S equity market. Second, VIX is widely believed as the market s expectation of S&P 500 index. Third,

9 5 VIX has considerable data set of historical prices over 20-year. Finally, the information content and performance of VIX have been studied by a large number of empirical studies using various approaches, but the study results are conflict. Therefore, it is interesting to examine the performance of VIX by our own study. The time series model studied in this paper includes random walk model, ARCH (p) model, GARCH (p,q) model, GJR (p,q) model, EGARCH (p,q) model and RiskMetrics approach. We first estimated the parameters of respective time series model, and then examined their out-of-sample forecast performance. Our empirical evidence suggest that GJR (1,1) model performs best for modeling S&P 500 index future returns. Next, the predictive power between GJR (1,1), RiskMetrics approach and VIX were compared by different approaches. We performed the regression of future realized volatility at different forecasting horizons of both in-sample and out-of-sample periods, as well as the study of their forecasting performance on the average daily return shocks. To guard against spurious inferences, the diagnostic tests of residuals were conducted. Our study results are in line with Becker, Clements and White (2006), Becker, Clements and White (2007) and Becker and Clements (2008). The empirical evidence of our study does not support the view that implied volatility subsumes all information content, and the study results provide strong evidence indicating that GJR (1,1) is superior for modeling future volatility of S&P 500 index daily returns. Additionally, the results of encompassing regression for future realized volatility at 5-, 10-, 15-, 30- and 60-day horizons, and the results of the encompassing regression for squared return shocks suggest that the joint use of GJR(1,1) and RiskMetrics can produce the best forecasts. The rest of this paper is structured as follow. We reviewed literatures in section 2. In section 3, the implied volatility and the VIX index are introduced. The time series models and practical issues for modeling are detailed in section 4 and section 5, respectively. Section 6 outlines the data used for our study. The estimates of time series models are discussed in section 7. Section 8 presents the empirical results of comparison between VIX, RiskMetrics and GJR(1,1). Finally, section 9 concludes.

10 6 2 Literature Review The predictability of ARCH (q) model on volatility of equity returns has been studied by extensive literature. However, the empirical evidence indicating the good forcast performance of ARCH (q) model are sporadic. The previous studies by Franses and Van Dijk (1996), Braisford and Faff (1996) and Figlewski (1997) examined the out-of-sample forecast performance of ARCH (q) models, and their study results are conflict. However, the common ground of their studies is that the regression of realized volatility produce a quite low statistic of R 2. Since the average R 2 is smaller than 0.1, they suggested that ARCH (q) model has weak predictive power on future volatility. There is a variety of restrictions influencing the forecasting performance of ARCH models. The frequency of data is one of restrictions, and it is an issue widely discussed in preceding papers. Nelson (1992) studied ARCH model and documented that the ARCH model using high frequency data performs well for volatility forecasting, even when the model is severely misspecified. However, the out-of-sample forecasting ability of medium- and long-term volatility is poor. The existing literature regarding to the study on GARCH type models can be classified into two categories, and they are the investigation on the basic symmetric GARCH models and the GARCH models with various volatility specifications. Wilhelmsson (2006) investigated the forecast performance of the basic GARCH (1,1) model by estimating S&P 500 index future returns with nine different error distributions, and found that allowing for a leptokurtic error distribution leads to significant improvements in variance forecasts compared to using the normal distribution. Additionally, the study also found that allowing for skewness and time variation in the higher moments of the distribution does not further improve forecasts. Chuang, Lu and Lee (2007) studied the volatility forecasting performance of the standard GARCH models based on a group of distributional assumptions in the context of stock market indices and exchange rate returns. They found that the

11 7 GARCH model combined with the logistic distribution, the scaled student s t distribution and the Riskmetrics model are preferable both stock markets and foreign exchange markets. However, the complex distribution does not always outperform a simpler one. Franses and van Dijk (1996) examined the predictability of the standard symmetric GARCH model as well as the asymmetric Quadratic GARCH and GJR models on weekly stock market volatility forecasting, and the study results indicated that the QGARCH model has the best forecasting ability on stock returns within the sample period. Brailsford and Faff (1996) investigated the predictive power of various models on volatility of the Australia stock market. They tested the random walk model, the historical mean model, the moving average model, the exponential smoothing model, the exponential weighted moving average model, the simple regression model, the symmetric GARCH models and two asymmetric GJR models. The empirical evidence suggested that GJR model is the best for forecasting the volatility of Australia stock market returns. Chong, Ahmad and Abdullah (1999) compared the stationary GARCH, unconstrained GARCH, non-negative GARCH, GARCH-M, exponential GARCH and integrated GARCH models, and they found that exponential GARCH (EGARCH) performs best in describing the often-observed skewness in stock market indices and in out-of-sample (one-step-ahead) forecasting. Awartani and Corradi (2005) studied the predictability of different GARCH models, particularly focused on the predictive content of the asymmetric component. The study results show that GARCH models allowing for asymmetries in volatility produce more accurate volatility predictions. Evans and McMillan (2007) studied the forecasting performance of nine competing models for daily volatility for stock market returns of 33 economies. The empirical results show that GARCH models allowing for asymmetries and long-memory dynamics provide the best forecast performance.

12 8 By and large, the extensive empirical studies and evidence demonstrated that GARCH models allowing for asymmetries perform very well for modeling future volatility. EWMA model is also a widely used technique for modeling and forecasting volatility of equity returns in financial markets, and the well-known RiskMetrics approach is virtually the variation of EWMA. A great deal of existing studies using EWMA model on various markets demonstrated that EWMA model has different performance. Akgiray (1989) first examined the forecast performance of EWMA technique on volatility forecasting for stocks on the NYSE. The study also examined predictability of ARCH and GARCH models. The finding indicated that EWMA model is useful for forecasting time series, however, the GARCH model performs best for forecasting volatility. Tse (1991) studied volatility of stock returns of Japanese market during the period of 1986 to 1989 using ARCH, GARCH and EWMA models. The study results revealed that the EWMA model outperforms ARCH and GARCH models for volatility forecasting of stock returns in Tokyo Stock Exchange during the sample period. Tse and Tung (1992) investigated monthly volatility movements in Singapore stock market using three different volatility forecasting models which are the naive method based on historical sample variance, EWMA and GARCH models. The study results suggested that EWMA model is the best for predicting volatility of monthly returns for Singapore market. Wash and Tsou (1998) investigated the volatility of Australian index from January 1, 1993 to December 31, 1995 using a variety of forecasting techniques, and they are historical volatility, an improved extreme-value method, the ARCH/GARCH class of models, and EWMA model. The hourly data, daily data and weekly data were used, respectively. The finding indicated that the EWMA model outperforms other volatility forecasting techniques within the sample period. Galdi and Pereira (2007) examined and compared efficiency of EWMA model, GARCH model and stochastic volatility (SV) for Value at Risk (VaR). The empirical

13 9 results domonstrated that VaR calculated by EWMA model was less violated than by GARCH models and SV for a sample with 1500 observations. Patev, Kanaryan and Lyroudi (2009) studied volatility forecasting on the thin emerging stock markets, and their study primarily focused on Bulgaria stock market. Three different models which are RiskMetrics, EWMA with t-distribution and EWMA with GED distribution were employed for investigation. The study results suggested that both EWMA with t-distribution and EWMA with GED distribution have good performance for modeling and forecasting volatility of stock returns of Bulgaria market. They also concluded that EWMA model can be effectively used for volatility forecasting on emerging markets. Implied volatility is another popular issue which has attracted a great deal of attention by empirical research. Particularly, the information content of implied volatility is the subject of many studies and it has been well documented that implied volatility is an efficient volatility forecast and it subsumes all information contained in other variables. The predictability of model based forecasts and implied volatility have been compared by a number of studies, and the objective is to find out the answer for whether implied volatility or model based forecasts is superior for future volatility forecasting. The implied volatility from index option has been widely studied but the study results are conflict. The studies by Day and Lewis (1992), Canina and Figlewski (1993), Becker et al. (2006), Becker et al. (2007) and Becker and Clements (2008) demonstrated that historical data subsumes important information that is not incorporated into option prices, suggesting that implied volatility has poor performance on volatility forecasting. However, the empirical evidence from the studies by Poterba and Summers (1986), Sheikh (1989), Harvey and Whaley (1992), Fleming, Ostdiek and Whaley (1995), Christensen and Prabhala (1998), Blair, Poon and Taylor (2001), Poon and Granger (2001), Mayhew and Stivers (2003), Giot (2005 a), Giot (2005 b), Corrado and Miller, JR. (2005), Giot and Laurent (2006), Frijns et al. (2008), Becker, Clements and McClelland (2009), Becker, Clements and Coleman-Fenn (2009) and Frijns et al. (2010) documented that the implied

14 10 volatilities from index options can capture most of the relevant information in the historical data. The implied volatility index (VIX) from CBOE is a widely used index option for empirical research on implied volatility in practice. The VIX index was the volatility implied from the option price of S&P 100 index, and the calculation method has been changed since Today, the VIX index is computed by the option price from S&P 500 index. Therefore, the literature regarding to the empirical studies on VIX can be classified into two categories: VIX based on S&P 100 index and VIX based on S&P 500 index. Most studies found that the volatility implied by S&P 100 index option prices to be a biased and inefficient forecast of future volatility and to contain little or no incremental information beyond that in past realized volatility. Day and Lewis (1992) examined the volatility implied from the call option prices of S&P 100 index of the period from 1985 to 1989 by the use of the cross-sectional regression. The information content of implied volatility was compared to the conditional volatility of GARCH and EGARCH models of both in-sample and out-of-sample periods. The information content of implied volatility of in-sample period was examined by the likelihood ratio of the nested conditional volatility GARCH and EGARCH models augmented with implied volatility as an exogenous variable. The out-of-sample forecast performance of implied volatility and GARCH and EGARCH models was studied by running the regression for the ex post volatility on implied volatility and the volatility forecasts from GARCH and EGARCH models. The study results show that implied volatility is biased and inefficient. The drawback of their study may be the use of overlapping samples to predict one-week ahead volatility of options which have the remaining life up to 36-day. Canina and Figlewski (1993) showed that implied volatility has no virtual correlation with future return volatility and does not incorporate information contained in recent observed volatility. According to the analysis by Canina and Figlewski (1993), one reason for producing their study results could be the use of S&P 100 index options (OEX) and the index option markets process volatility information

15 11 inefficiently. The second reason is that the Black-Scholes option pricing model may be not suitable for pricing index options since prohibitive transaction costs associated with hedging of options in the cash index market. However, the Black-Scholes model does not require continuous trading in cash markets. Christesen and Prabhala (1998) mentioned that Constantinides (1994) have argued that transaction costs have no first-order effect on option prices. Therefore, transaction costs cannot interpret the apparent failure of the Black-Scholes model for the OEX options market. It seems that the study results of Canina and Figlewski (1993) refute the basic principle of option pricing theory. (Christesen and Prabhala 1998) The study by Christensen and Prabhala (1998) was the development of the study by Canina and Figlewski (1993). They reinvestigated the relation between implied volatility and realized volatility of the OEX options market, and they found the different study results. Their finding indicates that implied volatility outperforms past volatility in forecasting future volatility and subsumes the information content of past volatility in some of their specifications. Christensen and Prabhala (1998) argued that the reason causing their study results to be different from Canina and Figlewski s (1993) is that they used a longer volatility series, and this increases statistical power and allows for evolution in the efficiency of the market for OEX index options since their introduction in Their sample data ranges from November 1983 to May 1995 which equals to 11.5 year. However, the data used by Canina and Figlewski (1993) was from March 15, 1983 to March 28, 1987, and this period preceded the October 1987 crash. Christensen and Prabhala (1998) documented that there was a regime shift around the crash period, and implied volatility is more biased before the crash. The second reason is that they used monthly data to sample the implied and realized volatility series, while the daily data was used by Canina and Figlewski (1993). The lower frequency of data enables them to construct volatility series with nonoverlapping data with exactly one implied and one realized volatility coving each time period, and their nonoverlapping sample yields more reliable regression estimates relative to less precise and potentially inconsistent estimates obtained from overlapping samples used in previous work.

16 12 Blair et.al (2001) compared ARCH models and VIX based on S&P 100 index using both daily index returns and intraday returns. The data ranges from November 1983 to May 1995, and it spans a time period of 139 months which is approximately 11.5 years. The study results indicate VIX performs very well on volatility forecasting and the volatility forecasts are unbiased. The technique for computing VIX was improved in Since the new computation is based on the option price of S&P 500 index rather than S&P 100 index, therefore, the evaluation of the performance of VIX on forecasting future volatility of S&P 500 index became the subject of most empirical research. However, the results of various studies are also conflict. Corrado and Miller, JR. (2005) studied implied volatility indices VIX, VXO as well as VXN which are based on S&P 500, S&P 100 and Nasdaq 100 indices, respectively. The study period spans 16 years from January 1988 to December They compared the results of OLS regression to the estimates derived from instrument variable regression, and the study results documented that implied volatility indices VIX, VXO and VXN dominate historical realized volatility. Particularly, VXN is nearly unbiased and it can produce more efficient forecasts than realized volatility. Giot and Laurent (2006) investigated information content of both VIX and VXO implied volatility indices. The data used for their study ranges from January 1990 to May The information content was evaluated by running an encompassing regression of the jump/continuous components of historical volatility, and implied volatility was augmented as an additional variable. The study results show that implied volatility subsumes most relevant volatility information. They also indicated that the addition of the jump/continuous components can hardly affect the explanatory power of the encompassing regression. Becker, Clements and McClelland (2009) examined information content of VIX by seeking the answers for two questions. First, whether the VIX index subsumes information regarding to how historical jump activity contributed to the price volatility; second, whether the VIX reflects any incremental information pertaining to

17 13 future jump activity relative to model based forecasts. The empirical results of their study provide the affirmative answers for these two questions. Becker, Clements and Coleman-Fenn (2009) compared model based forecasts and VIX. They argued that the unadjusted implied volatility is inferior. However, the transformed VIX augmented with the volatility risk-premium can have the same good performance as model based forecasts. The study results of Becker et al. (2006), Becker et al. (2007) and Becker and Clements (2008) refute the hypothesis of VIX being an efficient volatility forecast. The same data set was used for these three studies, ranging from January 1990 to October The study results indicate that there is significant and positive relationship between VIX and future volatility, but the VIX is an inefficient volatility forecast. There are several determinant variables for computing the implied volatility, such as the index level, risk free interest rate, dividends, contractual provisions of the option and the observed option price. The measurement errors of these variables may lead to the biased estimation of implied volatilities. Since the implied volatilities used by early studies contain relevant measurement errors whose magnitudes are unknown, therefore, this may be the primary reason leading to the conflicting study results of various studies. In addition, the biasness of implied volatility estimation can also be induced by some other factors. For example, the relatively infrequent trading of the stocks in the index; the use of closing prices which have different closing times of stock and options markets; the bid or ask price effects which may cause the first order autocorrelation of the implied volatility series to be negative. Comparing to index option, the study based on the individual stock options is sporadic. The studies by Latané and Rendleman (1976) was conducted with expectation of favoring implied volatility, however, the results are less overwhelming due to these studies predate the development of conditional heteroskedasticity models and applied naive models of historical volatility.

18 14 Lamoureux and Lastrapes (1993) examined implied volatility based on the option prices of 10 stocks of a 2-year short period from April 1982 to March They demonstrated that implied volatility is biased and inefficient, and the GARCH model performs better on modeling the conditional variance. Additionally, they also found that when implied volatility was included as a state variable in the GARCH conditional variance equation, historical return shocks still provided important additional information beyond that reflected in option prices. Their study results are difficult to interpret because they used overlapping samples to examine one day ahead forecasting ability of implied volatility computed from options that have a much longer remaining life which is up to 129 trading days. Based on the theory and methodology of the study by Lamoureux and Lastrapes (1993), Mayhew and Stivers (2003) examined 50 firms with the highest option volume traded on the CBOE between 1988 and 1995, and they used the daily time series of the volatility index (VIX) from CBOE. During this period, the VIX represented the implied volatility of an at-the-money option based on the S&P 100 Index with 22 trading days to expiration. Their study results show that the implied volatility outperforms GARCH specification. In addition, when implied volatility is added to the conditional variance equation, it captures most of all of the relevant information in past return shocks, at least for stocks with actively-traded options. Furthermore, they documented that return shocks from period 2 and older provide reliable incremental volatility information for only a few firms in the sample.finally, they also found that the implied volatility from equity index options provides incremental information about firm-level conditional volatility. For the most of the firms, index implied volatility contains information beyond that in past returns shocks, suggesting an alternative method for modeling volatility for stocks without traded options. For a small part of firms with less actively-traded individual options, the index implied volatility provides incremental information beyond the own firm s implied volatility. Therefore, the equity index options appear to impound systematic volatility information that is not available from less liquid stock options.

19 15 Frijns et al. (2008) and Frijns et al. (2010) studied return volatility of Australian stock market of different period. Due to there is no implied volatility index published by Australian Stock Exchange, Frijns et al. (2010) computed the implied volatility index namely AVX on the basis of the European style index options traded on the Australian Securities Exchange. The approach of constructing AVX is similar to the way of computing VIX by CBOE. The distinctive feature is that the implied volatilities of eight near-the-money options were combined into a single at-the-money implied volatility index with a constant time to maturity of three months (Frijns et al. 2010: 31). Therefore, the computed AVX is considered to be the forecasted future return volatility of S&P/ASX 200 over the subsequent three months. The study results demonstrated that implied volatility outperforms RiskMetrics and GARCH and provides important information for forecasting future return volatility of Australian stock market. Furthermore, it is proposed that AVX could be valuable information to investors, corporations and financial institutions. To summarize, the empirical results of immediate studies favor the conclusion that implied volatility are more efficient and informative for forecasting future volatility of assets returns.

20 16 3 The CBOE Volatility Index-VIX 3.1 Implied Volatility Implied volatility is a prediction of process volatility rather than the estimate, and its horizon is given by the maturity of the option. In a constant volatility framework, implied volatility is the volatility of underlying asset price process that implicit in the market price of an option according to a particular model. If the process volatility is stochastic, implied volatility is considered to be the average volatility of the underlying asset price process that is implicit in the market price of an option (Alexander, 2001:22). The market price of options can be computed using various models. A simple model namely Black-Scholes model is widely used for European options pricing in practice. In practice, the theoretical market price and real price of option may differ from each other, whereas application of implied volatility can make these two prices equivalent (Alexander, 2001). A recognized fact is that different options on the same underlying asset can generate various implied volatilities. Furthermore, using different data can induce the irreconcilably different inferences of parameters value. Since implied volatilities are thought of the market s forecast of the volatility implied from the underlying asset of an option, the calculation of an implied volatility is closely associated with the option valuation model. Blair et al. (2001) argued that the inappropriate use of option valuation model can lead to mis-measurement in implied volatilities. For example, if implied volatilities of S&P 500 index option are calculated by an European model then error will be caused by the omission of the early exercise option due to is an American style option. In addition, Harvey and Whaley (1992) showed that if the option pricing model includes the early exercise option and the timing and level of dividends are assumed to be constant, then the option will be priced by error so that implied volatilities will be mis-measured.

21 The VIX Index The VIX index was introduced by the Chicago Board Options Exchange (CBOE) in By using the implied volatilities of various near-the-money options on the S&P 100 index, Whaley (1993) introduced the VIX index on the basis of a synthetic at-the-money option with a constant time to maturity of one-month, and demonstrated that the VIX index is not only an efficient index for market volatility, but also could be employed for hedging purpose by introducing options and futures on the VIX. The current calculation approach of VIX was changed since September 22, 2003, and it is now calculated from the bid and ask quotes of options on S&P 500 index rather than S&P 100 index. The S&P 500 index is the most popular underlying asset as well as the most widely used benchmark in the U.S market Before changing the calculation approach, the VIX index based on S&P 100 index is a weighted index of American implied volatilities derived from eight near-the-money, near-to-expiry, S&P 100 call and put options, and it was considered to be able to eliminate smile effects and most of problems of mis-measurement. It used the binominal valuation methods with trees that are adjusted to reflect the actual amount and timing of anticipated cash dividends. The midpoint of the most recent bid/ask quotes are used to calculate the option price and this way was considered to be able to avoid problems inducing by bid/ask bounce. Both call and put options were used in order to increase the amount of information and eliminate problems caused by mis-measurement of underlying index and put/call option clientele effects. VIX based on S&P 100 index represents a hypothetical option that is at-the-money and had a constant 22 trading days (30 calendar days) to expiry. It employed pairs of near-the-money exercise prices which are barely above and below the current index price. Otherwise, a pair of times to expiry was also used, one is at least eight calendar days to expiration and another one is the following contract month. Blair et al. (2001) showed that although VIX is robust to mis-measurement, it is still a biased predictor of subsequent volatility due to a trading time adjustment that typically multiplies conventional implied volatilities by approximately 1.2.

22 18 The new calculation approach makes the VIX index to be much closer to the real financial practices and become the practical standard for trading and hedging volatility. It is widely accepted and considered to be the market s expected volatility of the S&P 500 index. Since the computation augments a wide range of exercise prices, the VIX index based on S&P 500 index become more robust. In addition, VIX is computed directly from option prices rather than seeking it by the use of the Black-Scholes option pricing model (Ahoniemi 2006). The popularity of VIX are developing rapidly and it has become the main index for the U.S stock market volatility. So far, VIX has been a tradable asset for both option and futures with 6-year history. In terms of CBOE proprietary information (2009), VIX is computed by the at-the-money and out-of-the-money call and put option prices using the formula where σ denotes VIX divided by 100, T is time to maturity, r is the risk free interest rate, F is the forward index level computed by the index option prices, denotes the first strike below the forward index level (F), is the strike price of i th out-of-the-money option (a call if ; a put if ; both call and put if ), stands for the midpoint of the bid-ask spread for each option with strike, is the interval between strike prices and it is calculated by the difference between the strike on either side of divided by two, /2. Since VIX forecasts 30-day volatility of S&P 500 index, the near-term and next-term put and call options of the first two contract months are used to compute VIX. For near-term options, the time to maturity should equal one week at least so that can minimize the potential pricing anomalies which could happen near the time to maturity. If the expiration date of the near-term options is less than one week, then must roll to the next two contract months (CBOE proprietary information 2009).

23 19 4 Time Series Models for Volatility Forecasting 4.1 Random Walk Model Perhaps the random walk model is the simplest one for modeling volatility of a time series. Under the efficient market hypothesis, the stock price indices are virtually random. The standard model for estimating the volatility of stock returns using ordinary least square method is the random walk model based on the historical price: 2 where denotes the stock index return at time t; μ is the average return under the efficient market hypothesis, and it is expected to be equal to zero; is the error term at time t, and its auto-covariance should equal to zero over time. 4.2 The ARCH (q) Model Engle (1982) introduced the autoregressive conditional heteroskedasticity ARCH (q) model and documented that the serial autocorrelated squared returns (conditional heteroskedasticity) can be modeled using an ARCH (q) model. The framework of the ARCH (q) model is: where denotes the conditional mean given information set available at time

24 20 1; represents a sequence of iid random variables with mean equals zero and unit variance. The constraints of parameters that 0 and 0 1,, ensure the conditional variance is non-negative. The equation (5) for can be expressed as an AR (q) process for the squared residuals: 6 where is a martingale difference sequence (MDS) since 0 and it is assumed that (Zivot 2008:4). The condition for to be covariance stationary is that the sum of all parameters of past residuals 1,, should be smaller than unity. The measurements of persistence of and are and 1, respectively. 4.3 The GARCH (p,q) Model The generalized ARCH (GARCH) model, proposed by Bollerslev (1986), is the extension of ARCH model. It is based on the assumption that the conditional variance to be dependent upon previous own lags, and it replaces the AR (q) representation in equation (5) with an ARMA (p,q) process: 7 where the parameter constraints 0 0,1,, and 0 1,, assure that σ 0. The equation (7) together with equation (3) and (4) is known as the basic GARCH (p,q) model. If 0, the GARCH (p,q) model became an ARCH(q) model. In the interest of the coefficient estimates of the GARCH term to be identified at least one of parameters 1,, must be significant from zero. For the basic GARCH (p,q) model, the squared residuals behave like an ARMA process. It

25 21 is required that 1 for the covariance stationarity. The unconditional variance of is computed as : 1 8 In practice, the GARCH (1, 1) model comprising only three parameters in the conditional variance equation is sufficient to capture the volatility clustering in the data. The conditional variance equation of GARCH (1,1) model is 9 Due to, the equation (9) can be rewritten as 10 The equation (10) is an ARMA (1,1) process for, and it is followed by many properties of GARCH (1,1) model. For instance, the persistence of the conditional volatility is captured by, and the constraints 1 assures the covariance stationarity. The covariance stationary GARCH (1,1) model has an ARCH representation with, and the unconditional variance of is 1. (Zivot, 2008:6) The Stylized Facts of Volatility The stylized facts about the volatility of economic and financial time series have been studied extensively. The most important stylized facts are known as volatility clustering, leptokurtosis, volatility mean reversion and leverage effect. The volatility clustering can be interpreted by GARCH (1,1) model of equation (9). For many daily or weekly financial time series, a distinctive feature is that the

26 22 coefficient estimate of the GARCH term approximates 0.9. This implies that the large (small) value of the conditional variance will be followed by the large (small) value. The same discursion can be derived by the ARMA representation of GARCH models in equation (10), i.e. the large changes in will be followed by the large changes, and small changes in will be still followed by small changes. (Zivot, 2008) Compared to the normal distribution, the distributions of the high frequency data usually have fatter tails and excess peakedness at the mean. This fact is known as leptokurtosis, and it suggests the frequent presence of the extreme values. The kurtosis is a statistic for measuring the peak of a distribution of time series compared to a normal distributed random variables with constant mean and variance, and it is calculated by a function of residuals and their variance : kurtosis = (11) The kurtosis of a normal distribution is three and the excess kurtosis which equals to kurtosis minus three is zero. The normal distribution with zero excess kurtosis is known as mesokurtic. A distribution with the excess kurtosis larger than three is referred to as leptokurtic, and the distribution is said to be platykurtic if the excess kurtosis is smaller than three. Sometimes financial markets experience excessive volatility, however, it seems that the volatility can ultimately go back to its mean level. The unconditional variance of the residuals of the standard GARCH (1,1) model is computed by 1. In order to clarify that the volatility can be finally driven back to the long run level, we consider the interpretation by rewriting the ARMA representation in equation (10): 12 by successively iterating k times,

27 23 13 where γ is a moving average process. Due to 1 is required for a covariance stationary GARCH (1,1) model, approach zero as k increase infinitely. Although may deviate from the long run level at time t, will approach zero as k becomes larger, and this implies that the volatility will eventually go back its long run level σ. The half-life of a volatility shock suggests the average time for to decrease by one half, and it is measured by 0.5. Therefore, the speed of mean reversion is dominated by, i.e. if the value of 1, the half-life of a volatility shock will be very long; if 1, the GARCH model is non-stationary and the volatility will ultimately explode to infinity as k increases infinitely (Zivot 2008:8). The standard GARCH (p,q) model enforce a symmetric response of volatility to positive and negative shocks because the conditional variance equation of the standard GARCH (p,q) model is a function of the lagged residuals but not their signs, i.e. the sign will be lost if the lagged residuals are squared (Brooks, 2008). Therefore, the standard GARCH (p,q) model cannot capture the asymmetric effect which is also known as the leverage effect in the distribution of returns. One alternative is modeling the conditional variance equation augmented with the asymmetry. Another approach is allowing the residuals to have an asymmetric distribution (Zivot 2008). In order to overcome this limitation of the standard GARCH (p,q) model, a number of extensions have been built such GJR and the exponential GARCH (EGARCH) models. 4.4 The GJR (p,q) Model The GJR (p,q) model is built with the assumption that the unexpected changes in the market returns have different effects on the conditional variance of returns. Compared to the basic GARCH (p,q) model, the GJR (p,q) model augments with an

28 24 additional term which is used to account for the possible asymmetries. The function form of the conditional variance is given by: (14) where I (.) represents the dummy variable that takes value one if 0, otherwise zero. If γ 0, the leverage effect exhibits and suggests that the negative shocks will have a larger impact on conditional variance than positive shocks; if γ0, the news impact is asymmetric. Since the conditional variance should be positive, therefore, the constraints of parameters are 0, 0, 0 and 0. When 0, the model is still admissible even if γ 0. The model is stationary if γ The EGARCH (p,q) Model The exponential GARCH (EGARCH) model introduced by Nelson (1991) incorporates the leverage effect and specifies the conditional variance in the logarithmic form. The conditional variance equation of the EGARCH model is expressed as: 15 If 0 or there is arrival of good news, the total effect of is 1 ; if 0 or there is arrival of bad news, the total effect of is 1. The EGARCH model has three advantages over the basic GARCH model. First, since the conditional variance is modeled in the logarithmic form, the variance will always be positive even if the parameters are negative. With appropriate condition of the parameters, this specification captures the fact that a negative shock leads to a higher conditional variance in the next period than a positive shock. Second,