2.1 Volatility Measures and Volatility Forecasting
Volatility is often defined as the (instantaneous) standard deviation (or „sigma‟) of the random Wiener-driven component in a continuous-time diffusion model.
Volatility is a major parameter in risk management, derivatives pricing, options trading, hedging and asset allocation, and has also been one of the most active and successful areas of research in time series econometrics and economic forecasting in recent decades. Blair, Poon & Taylor (2001) and Poon & Granger (2003) have summarized that volatility forecasting models can be classified in the following four categories: the historical volatility models (HISVOL), the GARCH family, the options implied standard deviation (ISD) model, and the stochastic volatility model (SV).2
Over the past decade, several researchers have focused on the univariate analysis of volatility, such as the estimation and properties of volatility (e.g., Engle 1982, Taylor 1986, Bollerslev 1986, Andersen and Bollerslev 1998) and forecasts of volatility (e.g., Fleming et al. 1995, Koopman et al. 2005, Poon and Granger 2005).
Other studies have focused on the multivariate analysis. Regardless of what categories of volatility are compared or composed, the main concerns of the forecasting model lie in investigating the possible indicators or properties which could improve the forecasting power and provide incremental information for application. The surveyed paper of Poon & Granger (2003, 2005) indicates that testing the effectiveness of a composite forecast is as important as testing the superiority of the individual models, but this has not been done more often or across different data sets. Multivariate forecasting models that consider the different categories of volatility models, such as the GARCH, historical volatility, stochastic volatility, and option implied volatility models, are constructed and compared hereafter (Engle & Gallo, 2006; Becker, Clements & White, 2007; Becker & Clements, 2008). In addition to the issue of the optimal combination of the multivariate volatility measures, there are other topics
2 Historical volatility models (HISVOL) include those related to the random walk, historical averages of squared returns or absolute returns. Also included in this category are time series models which are based on historical volatility using moving averages, exponential weights, autoregressive models or fractionally integrated autoregressive absolute returns, etc. All models in the HISVOL group model volatility directly by omitting the goodness of fit of the returns distribution or any other variable such as the options price (Poon & Granger, 2003).
examining the possible indicators which could improve the predictive power of forecasting and its application.
2.2 Volatility and Market Index
Whaley (1993, 2000) and Fleming et al. (1995), for example, find a negative correlation between volatility and the market index. In addition, Copeland and Copeland (1999) show that volatility is a leading indicator of market returns.
Latané and Rendleman (1976), Chiras and Manaster (1978) and Beckers (1981) indicate that, when compared with the earliest methods, volatility which is derived from the options pricing model can be regarded as a good predictor of future volatility.
There is also a growing volume of literature on the relationship between volatility and the market index. Wu (2001), Awartani and Corradi (2005) and Bollerslev et al. (2006) have recently claimed that causality between volatility and market index returns can be explained on the basis of the leverage effects (e.g., Black 1976) and volatility feedback (e.g., French et al. 1987).3 The nature of causality, which may be unidirectional or bi-directional, can be explained jointly by these two indistinguishable effects. Fleming et al. (1995) and Whaley (2000) point out that there is a highly negative correlation and asymmetric relationship between volatility and market index returns. In other words, losses lead to increases in volatility and gains result in decreases, but losses have a far greater impact on traded index volatility than gains. However, this is a direct violation of the predictions of classical finance theory.4 Montier (2002) claims that the asymmetric effect is just what the prospect theory, as proposed by Kahneman and Tversky (1979) in behavioral finance, would forecast.5
3 The leverage effect indicates that a drop in the value of equity increases financial leverage, and this makes the equity riskier and thus increases its volatility. Volatility feedback means that if volatility is priced, an anticipated increase in volatility raises the required return on equity. Hence, the leverage effect prescribes a causal nexus from returns to conditional volatility, while volatility feedback prescribes one from conditional volatility to returns.
4 Markowitz (1952) put forward the portfolio theory and assumed that risk was symmetric and could be expressed in terms of the standard deviation of asset returns.
5 The prospect theory, proposed by Kahneman and Tversky (1979) in behavioral finance, brings psychology into investors‟ decisions under uncertainty. It argues that investors have different risk tolerance in the face of gains and losses.
2.3 Investor Sentiment and Market Index
The causal relationships between sentiment indicators and stock market returns are mixed in previous studies. Clarke and Statman (1998) found that the sentiment of newsletter writers, whether bullish or bearish, does not forecast future returns, but that past returns and the volatility of those returns do affect sentiment. Causality would thus run from sentiment to market behavior if the noise trader explanation were to be accepted. However, Brown and Cliff (2004) and Solt and Statman (1988) documented that returns cause sentiment rather than the other way round. Brown and Cliff (2004) used a large number of sentiment indicators to investigate the relationship between sentiment and equity returns and found that returns cause sentiment rather than the opposite being the case. Brown (1999) supported the DSSW theory that irrational investors acting in concert and giving a noisy signal can influence asset prices and generate additional volatility. His tests used volatility instead of returns and his results indicated that deviations from the average level of sentiment are associated with increases in fund volatility only during trading hours. Lee, Jiang and Indro (2002) tested the impact of noise trader risk on the formation of conditional volatility and expected returns. Their empirical results show that sentiment is a systematic risk that is priced. Baker and Wurgler (2006) also indicated that investor sentiment affects the cross-section of stock returns. They found that when beginning-of-period proxies for sentiment are low, subsequent returns are relatively high for small stocks, young stocks, high volatility stocks, unprofitable stocks, non-dividend-paying stocks, extreme growth stocks and distressed stocks. Wang, Keswani and Taylor (2006) further tested the relationships between sentiment, returns and volatility. They also found strong and consistent evidence that sentiment measures, both in levels and first differences, are Granger-caused by returns. Banerjee, Doran and Peterson (2007) found that future returns are significantly related to both volatility index (VIX) levels and innovations for most portfolios, where the VIX is treated as a proxy variable for sentiment. While the causality test results presented above do not provide evidence of a consistent relationship between noise traders‟ sentiments and subsequent price movements, it might be possible that a relationship exists, but only in some special market scenarios.
The frame dependence theory, proposed by Shefrin (2000) in behavioral finance, argues that investors‟ decisions are sensitive to different market scenarios. This
motivates us to investigate whether there are dynamic causal relationships between sentiments and returns. Besides considering both positive and negative market scenarios, we infer that investors may exhibit dissimilar behaviors depending on the level of sentiment, and therefore different dynamic relationships may exist between stock market returns and sentiment indicators. Giot (2005) found that for very high (low) levels of the VIX, future returns are always positive (negative). His findings suggested that extremely high levels of the VIX might signal attractive buying opportunities. Banerjee et al. (2007) examined the relationship between returns and the VIX, the proxy variable for sentiment, for different levels of market performance and relatively high or low levels of volatility. Banerjee et al. (2007) defined those returns above and those below the sample median as constituting a „bull market‟ and a
„bear market‟, respectively. Volatilities above the median level of the VIX are said to be in a „high volatility‟ period and those below the median in a „low volatility period‟.
They provided two analyses, one of the „bull and bear market‟ and the other of „high and low volatility‟. Their findings suggested that the market states based on directional movements (positive and negative returns) or volatility levels (above or below the average) do not make a difference. On the contrary, we believe that the results will be misunderstood if the separation of the different market states is defined subjectively.
2.4 Volatility Forecasting and Investor Sentiment
From the behavioral finance point of view, the investors‟ behavior could be influenced by psychology or by bullish/bearish sentiment proxies (Montier, 2002;
Shefrin, 2007). De Long, Shleifer, Summers, & Waldmann (DSSW (1990) hereafter) point out that investors are subject to sentiment and model the influence of noise trading on equilibrium prices. Their study motivates empirical attempts to substantiate the proposition that noise traders‟ risks indexed by sentiment influence either the mean or variance of asset returns. Sentiments are therefore proposed as one of the indicators which could enhance the incremental explanation of the future volatility.
A large body of literature focuses on the relationship and information content between returns and sentiment (Solt & Statman, 1988; DSSW, 1990; Clarke &
Statman, 1998; Fisher & Statman, 2000; Wang, 2001; Simon & Wiggins, 2001;
Brown & Cliff, 2004; Baker & Wurgler, 2006; Baker & Wurgler, 2007, Han, 2008).
While less attention is given to the impact of sentiments on the realized volatility or vice versa (Brown, 1999; Lee, Jiang & Indro, 2002; Low, 2004; Wang et al. 2006;
Banerjee, Doran & Peterson, 2007; Verma & Verma, 2007), the exact role of sentiment in the price formation process is still a topic worth looking into.
To sum up, the information content of sentiment may be useful for volatility forecasting. However, the precise form in which sentiment will affect or predict volatility is not clear ex ante. For this reason, in our empirical analysis the possible sentiment indicators in the Taiwan stock market are constructed by referring to the previous literature, the predictive ability of sentiment to volatility is examined, the forecasting performance of the competitive models is compared, and finally effective option trading strategies are proposed based on the volatility forecasting.
2.5 Related Studies in Taiwan Stock Market
There are some related studies that focus on the Taiwan derivatives market. Lee, Lu and Chiang (2005) compare the characteristics and construction methodology of the volatility indexes across different countries. They find that the volatility index for the TAIEX (VXT) is a good estimator of future volatility. Besides, the VXT has negative and asymmetric relationship with the TAIEX and may be a contrarian trading signal when the market plunges. In contrast to Lee et al. (2005), the contribution of our study lies in the econometric analysis of the relationship between the information content of the volatility index and the TAIEX. Lee and Yuan (2005) investigate whether the traders‟ risk preference in the Taiwan stock market can be perceived by the volatility index. They find that investors in the Taiwan stock market tend to hedge the risk perception by put option contracts and the tendency is only remarkable in the bear market. Hsieh, Lee and Yuan (2006) separately construct the call and put implied volatility in the Taiwan Stock Market. Their empirical results show that put implied volatility is more closely linked to the spot index and is more sensitive to the change in the spot index than the call volatility. The strategy based on the information content of the put volatility index also outperforms the benchmark buy-and-hold strategy. In contrast to their study that the put volatility reveals more information content, our study indicates that the implied index derived from call options takes precedence over the underlying TAIEX.