Volatility Measures

6 By referring to John C. Hull (2006), this study assumes that there are 252 trading days in each year.

7 A multiple indicators volatility forecasting model jointly considers absolute daily returns (|R|), daily high-low range (HL) and daily realized volatility (RV) as proposed by Engle & Gallo (2006). The three variables have different features relative to one another, the main difference being that the daily return uses information regarding the closing price of the previous trading day, while the high-low spread and the realized volatility are measured on the basis of what is observed during the day. The former takes all trade information into account, and the latter is built on the basis of quotes sampled at discrete intervals.

8 By taking the price limits in the Taiwan stock market into consideration⁸, we transfer the high-low range to the degree of fluctuation relative to the price variation limits for each day. The daily price

the RV is calculated by summing the corresponding five-minute interval squared returns⁹ (e.g., Andersen & Bollerslev, 1998; Barndorff-Nielsen & Shephard, 2002, among others), and the variable is expressed in terms of percentage annual terms. The calculations can be expressed as follows: variation at time t, RV_t is the daily realized volatility at time t, S_t is the closing price on trading date t, S_t-1 is the closing price on the previous trading day, H_t is the highest price on date t, L_t is the lowest price on date t, S_t+i is the intraday index level of the

i-th interval on trading day t, S

n is the number of time intervals in each day.

3.1.3 Volatility Index

In 1993, the Chicago Board Options Exchange (CBOE) introduced the Volatility Index (VIX) based on the S&P 100 index options, which can be defined as the magnitude of price variations for the next 30 days. In 2003, the CBOE published the new VIX, which is based on the S&P 500 index options prices.¹⁰ The construction of the CBOE‟s new volatility index incorporates information from the skewness of volatility by using a wider range of strike prices including the out-of-the-money call and put option contracts rather than just the at-the-money series.¹¹ The new VIX is

limits on day t in the Taiwan stock market are -7% and +7% of the previous day‟s closing price. Thus, the maximum price variation on day t would be 14% based on the previous day‟s closing price.

9 The latest observations available before the five-minute marks from 09:00 until 13:30 are used to calculate the five-minute returns. We sum the 54 squared intra-day five-minute returns and the previous squared overnight returns to construct the daily realized volatility.

10 In March 2004, the CBOE futures exchange (CFE) introduced volatility futures, and volatility options were launched in February 2006. The underlying index is just the VIX published in 2003. The volatility index comprises tradable derivatives. The CBOE new VIX takes into account a wide range of strike prices for the same 30-day maturity, thus freeing its calculation from any specific option pricing model.

11 For details of the index‟s construction, the interested reader may refer to the white book publish by the CBOE in 2003. http://www.cboe.com/micro/vix/vixwhite.pdf

not calculated from the Black-Scholes-Merton option pricing model which implies that the calculation is independent of any model. However, the fundamental features of the volatility index between the old and new versions remain the same. Since the new VIX is more precise and robust than the original version, we construct a volatility index for the Taiwan stock market based on the CBOE‟s last revision of the volatility index.

In the construction of the Taiwan stock market VIX, the interest rate has been adjusted accordingly. The risk-free rate is calculated from the monthly average one-year deposit rates at the Bank of Taiwan, Taiwan Cooperative Bank, First Bank, Hua Nan Bank and Chang Hwa Bank. The CBOE‟s volatility index (VIX) uses put and call options in the two nearest-term expiration months in order to bracket a 30-day calendar period. With 8 days left to expiration, CBOE‟s VIX „rolls‟ to the second and third contract months in order to minimize pricing anomalies that might occur close to expiration. However, the nearest-term expiration contract usually has high trading volume and the next nearest-term contract usually has low trading volume in the Taiwan options market even if the nearest-term contract is traded on the last trading day. In considering the market structure of liquidity and trading volume for the second and third contract months, we have revised the rollover rule from 8 days to 1 day prior to expiration in constructing the volatility index in Taiwan.

Options market-based implied volatility can reflect the expectations with respect to price changes in the future, and it can be treated as an indicator of sentiment. Olsen (1998) indicated that the volatility index has been viewed as a „sentiment indicator‟ in the recent behavioral finance literature and can be regarded as a market indicator of rises and falls in the underlying index. Whaley (2000) and research conducted by the Chicago Board Options Exchange (CBOE) have indicated that the greater the fear, the higher the VIX level is. Therefore, the volatility index is commonly referred to as the

„investor fear gauge‟. Baker and Wurgler (2007) also treated option-implied volatility as one of the sentiment measures in investigating the investor sentiment approach.

Therefore, the Taiwan stock market volatility index (TVIX) could be one of the volatility measures and one of the sentiment proxy variables in the Taiwan options market.

The hypothesis that volatility could reflect the expectations of future price changes and be treated as an indicator of sentiment is well documented (Whaley 2000, Baker and Wurgler 2006). Research by the Chicago Board of Options Exchange

(CBOE) indicates that, the greater the fear, the higher the Volatility Index (VIX) level is. The volatility index is therefore referred to as the “investor fear gauge”. Olsen (1998) indicates that the volatility index has been viewed as the “sentiment indicator”

in the recent behavioral finance literature and can be a market indicator of rises and falls in index returns in the future.