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2. Literature Review
A large literature has argued about the specification of disturbances. To further analyze time series data, Engle (1982) develops the AutoRegressive Conditional Hetereskedasticity (ARCH) model, in which the conditional variance is a function of past residuals. Bollerslev (1986) then extends Engle’s ARCH model to Generalized ARCH (GARCH) model by allowing the conditional variance to be a function of the lagged variance. However, there are some drawbacks existing in such models. Diebold (1986) and Lamoureux and Lastrapes (1990) propose that the high persistence in the GARCH model may reflect structural change in the variance process.
Following this line of thought, Hamilton (1989) introduces the Markov-switching autoregressive model and it becomes the widely employed method in describing time series data. Later on, Turner et al. (1989) examine a variety of models in which the variance of a portfolio’s excess return depends on a state variable generated by a first-order Markov process. Hamilton and Susmel (1994) apply Markov-switching ARCH (SWARCH) model, which incorporates the features of both Hamilton’s (1988, 1989) switching-regime model and Engle’s (1982) ARCH model, to model the high persistence of variance. Cai (1994) parameterizes a similar model to analyze the volatility of US Treasury-bill yield. Ramchond and Susmel (1998) also estabalish the bivariate SWARCH model to study the relations among major stock market in the
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world.
While Markov-switching models have been successfully used to model level changes for many economic and financial time series data, many researchers apply the similar models to describe other variables, such as business cycles (Wang, 2007), aggregate output (Huang et al., 1998), exchange rates (Dueker and Neely, 2007), interest rate (Smith, 2002), crude oil market (Zou and Chen, 2013) and stock market (Turner, Startz, and Nelson, 1989).
There are some indices and theories concerning the volatility for an equity index option. The option pricing theory, Black-Scholes (1973), indicates that volatility plays an important role in determining the fair value for an option, or any derivative instrument with option features. The Chicago Board Options Exchange (CBOE) proposes the concept of Volatility Index, VXO in 1993 and VIX in 2003. Following the same concept, the TAIEX Options Volatility Index, which applies the CBOE’s methodology to trading activity in Taiwan option market, reflects current price volatility in the market.
Empirical studies discussing volatility forecasting for equity index options are as follows. Figlewski (1997) investigates how best to obtain future volatility forecasts from historical data and from implied volatility in pricing options, and suggests a hypothesis that implied volatilities from different option markets contain relatively more
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or less information depending on whether the arbitrage trade in that market is easy or hard. Zhuang, Chang, and Wang (2003) compare the performance of predicting realized volatilities for TAIEX Options between three most commonly adopted estimation models, Historical Volatility (HV), GARCH model, and Implied Volatility (IV). They use TAIEX Options data, including contracts of expiration within one month and contracts of expiration within two months, for both call and put. The sample period is from 2002/3/1 to 2003/2/28 with a total of 6,723 observations. The empirical results report that first, the predicting performance of Implied Volatility for TAIEX Options is better than that of Historical Volatility and GARCH model, especially for current-month option contracts. Second, the contained information contents in both Implied Volatility and Historical Volatility are independent in explaining Realized Volatility, therefore, adding Historical Volatility into implied volatility regression model as another regressor improves the model performance. However, the contained information contents in GARCH model could be explained by Historical Volatility and Implied Volatility, which means adding GARCH(1,1) into implied volatility regression model is useless in enhancing the model performance. Lastly, they found that generally, taking trade volume into consideration does not improve the estimation models’ predicting performances for predicting Realized Volatility.
There are other researches concerning the information content implied by option
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volatility. For example, Mayhew and Stivers (2003) examine 50 firms with the highest option volume on the CBOE, and indicate that compared to a time-series method, the implied volatility of equity index options provides reliable incremental information about future firm-level volatility. Kuo, Chen, and Chiu (2010) explore whether there are information content in TAIEX VIX and VXO for the future volatility and the return of TAIEX, and the results show that VIX can best describe the future volatility effectively with a positive correlation. Backus, Chernov, and Martin (2011) further quantify the distributions of consumption growth disasters by using prices of equity index options, S&P 500. First, they compare pricing kernels constructed from macro-finance and option-pricing models. Second, they compare option prices derived from a macro-based model to those we observe. Lastly, they compare the distribution of consumption growth estimated from international macroeconomic data with one derived from option prices.
The empirical results show that option prices are a reasonably good indicator of the likelihood of disasters in consumption growth, and the probabilities of large negative realizations of consumption growth implied by option prices are smaller than we see in international macroeconomic data.
Besides, equity index options are also analyzed in portfolios. Chan, Shih (2005) investigate the correlations among the spot, futures, and options of TAIEX, and find that financial derivatives are more efficient than the underlying asset in conveying
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information. Chan, Cheng, and Lung (2005) analyze the impact of option trading activity on implied volatility changes to returns in the S&P 500 Index futures option market. Chang (2006) investigates the lead-lag relationships among the spot, futures, and option markets in TAIEX. Su et al.(2006) apply the TAIEX Options to test the two-state volatility model with Markov process, and show that the two-state volatility model outperforms Black-Scholes and CRR as used for pricing, whether the price conditions are deep in-the-money or out-of-the money, and outperforms Black-Scholes and CRR as used for hedging, no matter what volatility state of the time periods (high or low). Wu, Liao, and Lin (2009) also use TAIEX options to analyze option pricing under GARCH-Lévy processes. Kuo, Chen, and Chen (2013) test whether there exists a common volatility factor and a long-run stable relationship between the electronic sector index options and the TAIEX options.
However, recent studies concerning option time value are limited. In this paper, we concentrate on examining the volatilities of TAIEX Call Options and analyzing the relationship between option premium and time value.