Price Information of Options and the Construction of Volatility Index: An Empirical Evidence of Taiwan

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1 Price Information of Options and the Construction of Volatility Index: An Empirical Evidence of Taiwan David So-De Shyu Chih-Hsin Hung Yih Jeng Shyh-Weir Tzang Abstract VXO and VIX are difficult to measure with accuracy in an illiquid option market. The illiquid trading of the second and next to second nearby options will increase the measurement errors in constructing the volatility index of Taiwan. To account for the errors stemming from price discreteness and data unavailability, several rules to sample the option prices are proposed to construct volatility indexes. Based on the average of bid and ask quote prices on options, the constructed volatility index series are found to have less missing data by replacing nearby-options having 8 days to expiry with second-nearby options in volatility computation. The volatility indexes are further compared with the intraday high-low price range in their forecasting efficacy in Taiwan stock market. The high-low price range is found to be no less efficient than volatility indexes in volatility forecasting, which is consistent with recent literature. Keywords: implied volatility, realized volatility, VIX, VXO JEL classification code: C22; C53; G13; G14 1 Introduction The volatility indexes VXO and VIX are well-known indexes which are widely used to measure the perception of the option traders on stock market forward volatility. The VXO Shyu is a professor in the department of Finance, National Sun Yat-Sen University in Kaohsiung, Taiwan. Hung is an instructor of the Department of Finance, Fortune Institute of Technology, Taiwan. hunpeter65@yahoo.com.tw. Tel: Jeng is a professor in the department of Finance, National Sun Yat-Sen University in Kaohsiung, Taiwan. Tzang is an instructor of the Department of Finance, Yung Ta Institute of Technology & Commerce and Ph.D. candidate in the Department of Finance, National Sun Yat-Sen University in Kaohsiung, Taiwan. swtzang@gmail.com. Tel: The authors thank Taiwan Futures Exchange for their support for this study. All remaining errors are on the authors. 1

2 which CBOE has computed since 1993, originally called VIX, is the implied volatility of a hypothetical 30-calendar-day at-the-money S&P 100 index option. CBOE further introduced a new VIX volatility index in 2003 without using any particular option-pricing model. The new VIX incorporates information from the volatility skew by using a wider range of option series than just at-the-money option series. The so-called model-free implied volatilities provide exante risk neutral expectations of the future volatilities (Dupire, 1993; Carr and Madan, 1998; Britten-Jones and Neuberger, 2000; Lynch and Panigirtzoglou, 2003; Carr and Wu, 2005). Nonetheless, the existence of liquid options with strike prices covering the entire support of the return distribution is usually not available. Jiang and Tian (2005) find that VIX constructed by COBE is flawed as the measure introduces random noise as well as systematic errors into the index. One major flaw is the truncation errors due to the fact that very low and very high strikes are not available in practice. Moreover, Andersen and Bondarenko (2007) treats the VIX as a hybrid between a pure model-free implied volatility and a broad corridor implied volatility measure. 1 As the tradings of options required to compute volatility indexes are not as liquid as theorists have expected, the biases from the construction of volatilities merit further researches (Bandi and Russell, 2006; Andersen, Bollerslev, and Meddahi, 2006). Instead of getting involved in exploratory models, the paper tries to apply several sampling rules to option prices and to see the extent of which the discrepancies among the computed volatility indexes based on those sampling rules have been obtained. It is widely accepted that the true price processes are contaminated by market microstructure effects. The illiquid tradings and separate trading prices for buyers and sellers make observed prices diverge from their efficient values. By applying the construction measures of the volatility index proposed by CBOE to other emerging markets like Taiwan, the estimation biases from microstructure noises become ever serious when compared with the matured market like U.S. According to CBOE, the new VIX uses put and call options in the two nearest-term expiration months in order to bracket a 30-day calendar period. However, with 8 days left to expiration, the new VIX rolls to the second and third contract months in order to minimize pricing anomalies that might occur close to expiration. One major anomaly of Taiwan option market is the illiquid tradings for the second nearby options, not to mention the next to second nearby options. Only the active tradings are found mainly on nearby month options, and the tradings of the second nearby options will become active only when nearby month options come 1 Andersen and Bondarenko (2007) introduces the concept of corridor implied volatility (CIV) measure by regarding the VIX as one of its variant, with barriers being set to the lowest and highest strikes that CBOE uses a given day to compute the index. They also noted that liquidity of the options market and the barriers which change stochastically may induce systematic biases in the VIX. 2

3 close to expiry. Therefore, to analyze the pricing anomalies arising from the illiquid options market, the paper tries to adopt four sampling methods to select option prices to construct intraday volatility series. For the 1-minute trading interval, the intraday volatility index is computed with the option prices which are selected or computed according to the following criteria: (1) the last recorded trading price; (2) the average of all trading prices; (3) the mid of bid and ask price of the last quote on options; (4) the average of all the mids of bid and ask prices of all tradings. The 8-day roll-over rule for option contracts is also relaxed to allow less days to expiry, i.e., 1, 2, 3 and 7 days. By applying different days of roll-over rules to the selection methods mentioned in the above, the intraday 1-minute VXO and VIX index are computed respectively. The descriptive statistics of the volatility index series are further analyzed in detail to shed light on the characteristics of option trading in Taiwan market. Forecast quality of the volatility index based on these methods are further analyzed to provide investors in Taiwan with information on the application of the volatility index. Whaley (2000) has indicated that many investors will agree with the fact that when stock prices get higher, investors will raise their concern on stock prices. The level of concern which reflects the investors consensus view or sentiment on stock prices or market index can be represented by the forward volatility implied by traded options. Implied volatility has been gaining attention from institutions and academics for its ability in predicting future market volatility. For researchers and investors, the information content and forecast quality of implied volatility has become an important subject of research in recent years. Harvey and Whaley (1991) find that the implied volatility is able to forecast the forward volatility of the market. The forecast quality of implied volatility has been widely reviewed by a large amount of literature such as Latane and Rendleman (1976), Chiras and Manaster (1978), and Beckers (1981). They find that the implied volatility provides better estimates of future return volatility than standard deviations obtained from historical returns. Jorion (1995) also finds that the implied volatilities from options on foreign exchange offer better estimates than do the historical prices. Different results are also found by Day and Lewis (1988), Lamoureux and Lastrapes (1993) and Canina and Figlewski (1993). Their empirical studies show that the implied volatility is not qualified as a good predictor for future return volatilities. Christensen and Prabhala (1998) refuted their results by showing that the weakness of the implied volatility in future volatility prediction is mainly resulted from the methodological issues like overlapping sample and mismatched maturities. By using a longer time series and nonoverlapping sample data, Christensen and Prabhala (1998) find that implied volatility better predicts future realized volatility. 3

4 Christensen and Prabhala (1998), Christensen and Strunk-Hansen (2002) and Fleming (1998) find that the implied volatilities computed from option prices of S&P100 index tend to have an upward bias in forecasting realized volatility. The bias, however, is not significant in terms of statistical tests (Fleming, 1999). Blair, Poon, and Taylor (2001) further find that implied volatilities from option prices of S&P 100 index deliver better volatility forecasts than those obtained from either low- or high-frequency index returns. Corrado and Miller (2005) examine the forecast quality of three implied volatility indexes based on S&P100 (VXO), S&P500 (VIX) and Nasdaq 100 (VXN). Their empirical results reveal that the forecast quality of VXO and VIX has been improved since 1995 and VXN even provides better quality forecasts of future volatility. In view of the literature in the above, the implied volatility from option prices does have significant ability to forecast future volatility. However, the forecast quality highly depends on the depth and liquidity of the option market to convey full information through option prices. The implied volatilities from option prices in emerging market may be severely biased due to trading noise from market microstructure. Therefore, the paper tries to use intraday high-low price range as an alternative to forecast future volatility. The empirical studies of Lin and Rozeff (1994), Alizadeh, Brandt, and Diebold (2002), and Shu and Zhang (2006) have provided substantial support for the intraday high-low price range to ba a valid volatility forecast. Corrado and Truong (2007) further finds that the intraday high-low price range provides volatility forecast as efficient as high-quality implied volatility indexes published by CBOE on four stock indexes. The advantages of high-low price range as a volatility forecast prompts us to conduct a regression analysis to compare the forecast efficiency of price range with volatility indexes. The paper is structured as follows. Section 2 details the data and methods to select the option prices corresponding to different days of roll-over rule. The construction methods of volatility index by CBOE are briefly described. The estimation of realized volatility is reviewed and treated as a benchmark for comparison purpose only. Section 3 describes the forecasting models mainly based on GJR-GARCH. Section 4 reports the empirical results. The in-sample and out-of-sample performance of the various models to predict future volatility are explored. Section 5 provides the concluding remarks. 4

5 2 Data and Method Description 2.1 Data With the increasing trading volume of index options in Taiwan since December 2001, the implied volatility of this market have drawn attentions from practitioners and researchers. Table 1 shows that the average daily trading volume in 2005 is almost twice as large as that in Therefore, the sample period spans from 2004/1/2 to 2007/12/31, and the option data before 2004 are excluded as the trading of options is still thin in comparison with the trading in subsequent years. The intraday prices of index options (with ticker symbol TXO) are obtained from Taiwan Futures Exchange (TAIFEX). The underlying cash index is obtained from Taiwan Stock Exchange (TAIEX). As there is a 15-minute difference between the close of TAIFEX and TAIEX, the nonsychronicity biases between closing prices of the options and the level of stock index may arise. Therefore, the sampling time period of options for a day is set to be the same as the stock index, starting from 9:00 to 13:30. The records are further filtered on a minute-by-minute basis. By taking into account of the intraday bid and ask quote on option prices of TXO, two construction methods of CBOE, VXO and VIX, are used to find the implied volatility fittest for Taiwan market. The construction methods of VIX and VXO are briefly reviewed in the following. 2.2 Measures of Implied Volatility Index By the white paper published by CBOE, VXO is constructed by using the implied volatilities of eight nearby and second-nearby options series with the nearest at-the-money strikes. The eight option series are equally divided into two groups: nearby and second-nearby options. For the nearest at-the-money options, there are two pairs of call/put option series with strike prices just bracketing the underlying stock index respectively. To compute the VXO, the term structure of volatility is assumed to be linear, and the implied volatility with 22 trading days (30 calendar days) to expire can be interpolated. Let N t1 and N t2 be the days to expiry for nearby and second nearby options. VXO will be obtained as follows: ( ) ( ) Nt Nt1 VXO = σ 1 + σ 2 N t2 N t1 N t2 N t1 σ 1 and σ 2 are implied volatilities of the nearby and second nearby options, respectively. The computation of VIX by CBOE is described in the following formula, in which C(K, T) and P(K, T) denote prices of call and put options with strike price K and days to maturity T. By 5 (1)

6 assuming strike prices to be order such that K j+1 > K j, the restriction is further imposed on two maturities such that T 2 22 T 1 8. VIX = 2 h=1 ( 1) ht h 22 T 2 T 1 N j=1 K j+1 K j 1 K 2 j min(c(k j, T h ), P(K j, T h )) (2) To reduce biases in computing the volatility indexes, CBOE uses the prices of second nearby options to replace nearby options with 8 days to expire. However, for options trading in Taiwan market, the second-nearby options and the next to second-nearby options have rather thin tradings. In panel A of Table 2, the second nearby and next to second nearby options have an average daily trading volume of less than 16% and 1% of the nearby options when nearby options have eight days to expiry. When nearby options have three days to expiry, the average trading volume of the second nearby and next to second nearby options just grows to barely 50% and 2% of the nearby options. The sparse trading records in the next to second nearby options will inevitably bias the computation of the volatility index series. To measure the effects of the illiquid trading and mitigate the biases from prices, the paper proposes to adopt four different methods to select the option prices from the trading interval: (1) the last recorded trading price in the interval; (2) the average of all trading prices within the trading interval; (3) the mid of bid and ask price of the last quote on options in the trading interval; (4) the average of all the mids of bid and ask prices for all tradings within the trading interval; The trading interval is set up to be 1-minute to compute intradaily volatility series. In other words, there will be 270 intradaily price series generated from one trading day. For comparison purpose, the 8-day roll-over rule set by CBOE is modified to allow five different days to expiry: 1 day, 2 days, 3 days, 7 days and 8 days. That means volatility index is computed by using option prices of second nearby and next to second nearby month when nearby options have 1 day, 2 days, 3 days, 7 days and 8 days to expiry, respectively. 2 The five kinds of roll-over rules are used respectively to compute the volatility index. Based on the four methods of data selection and five roll-over rules, VXO and VIX are computed and the descriptive statistics are listed in Table 3 and 6. Additionally, for comparison purpose, the volatility index of the first 2 The options in TAIFEX are the type of open-expiry with expiry on a third Thursday of a contract month. The 4th and 5th calendar days before expiry fall on Sunday and Saturday and thus are not considered. The 6th day before expiry falls on Friday and is not considered due to the possible weekend effect. 6

7 and the last trading minute within a day are also reported in Table 4, 5, 7 and 8. Each table has four panels (A-D) representing the price sampling methods of (1) to (4). In Table 3-5, the average VXO indexes in panel A and B are smaller than those in panel C and D by roughly 1%. The mean and standard deviations of VXO indexes tend to increase slightly as nearby options approach expiry. Despite the negligible differences in statistics summary, substantial changes are found in the number of sample across the panels. The missing data arises from the unavailability of option prices which are required in computing the volatility index. In other words, the second nearby option prices might be unavailable due to thin trading which thus makes the computation of volatility index infeasible. According to CBOE, when options prices are not available, the most recent available option prices will be used instead. Instead of following the method of CBOE, the paper treats the volatility index as being missing when the required second nearby option prices are not available. Therefore, the difference between the number of sample and the actual trading days is the number of missing data during computation. In Table 3-5, the number of sample in panel C and D is consistently larger than in panel A and B. The number of sample also decreases as the days to expiry for nearby options increase from one to eight days in panel A and B. However, panel C and D show that the number of sample is relatively stable. The analysis of Table 6-8 for VIX index has shown results similar to VXO. The changes in the number of sample are consistent with the fact that active trading is limited to nearby options in Taiwan causes insufficient trading data in volatility computation. As a result, to reduce the effects of the missing data on volatility computation due to illiquid trading of the second nearby options, option prices based on the bid and ask quote prices seem to be more appropriate than those based on trading prices. 2.3 Estimation of Realized Volatilities The recent development in estimating realized volatilities have seen a large amount of literature, especially after the thriving high frequency data. The accuracy of the realized volatility measure will improve as the sampling frequency increases and its nature is well documented in references like Barndorff-Nielsen and Shephard (2002) and Andersen, Bollerslev, Diebold, and Labys (2001). Due to the limited space, however, the high-frequency data is not adopted as a measure of the realized volatility. One simple measure of realized volatility is used in the paper for comparison purpose: the squared daily returns which is computed from natural logarithms 7

8 of closing levels of TAIEX as a benchmark for forecasting, R 2 t = ln2 ( c t c t 1 ) (3) R t represents the index return for day t based on index levels at the close of c t and c t 1. Based on Parkinson (1980), the second measure is the intraday high-low range volatility based on the squared range of natural logarithms of intraday high-low levels for TAIEX. W 2 t = ln2 (h t /l t ) 4 ln 2 (4) W t represents the intraday high-low range volatility on day t. h t and l t denote the highest and lowest index levels observed during trading on day t. To be consistent with other daily volatility measures, all implied volatility indexes are squared and divided by 252, the assumed number of trading days in a calendar year. The summary statistics of four measures and their correlation are listed in Table 9 and 10. Figure 1 is the graphical display of the volatility series. 3 Forecasting Models The paper uses the generalized autoregressive heteroskedasticity model, or GJR-GARCH model developed by Glosten, Jagannathan, and Runkle (1993), to account for the effect of good news and bad news on conditional volatility. The implied volatility indexes and high-low range volatility are included in the model for further examination. The augmented GJR-GARCH model is denoted as the following: where: R t = µ + ε t h t = α 0 + α 1 ε 2 t 1 + α 2 s t 1 ε 2 t 1 + βh t 1 + γ IVOL 2 t 1 + δw 2 t 1 (5) R t = return on day t h t = conditional volatility on day t IVOL t 1 = implied volatility at the end of option trading on day t 1, and s t 1 = 1 if ε t 1 < 0, and 0 otherwise. W t 1 = intraday high-low range volatility on day t 1. s t is an indicator function to account for effect of the good news (ε t > 0) and bad news (ε t < 0) on conditional variance. Therefore, the effect of good news is measured by α 1, and 8

9 the effect of bad news is measured by (α 1 + α 2 ). The effects of implied volatility and high-low range volatility on conditional variance are measured by coefficients γ and δ. To assess the incremental information provided by implied volatilities and high-low price range, four models are examined as follows: Model 1: h t = α 0 + α 1 ε 2 t 1 + α 2s t 1 ε 2 t 1 + βh t 1 Model 2: h t = α 0 + α 1 ε 2 t 1 + α 2 s t 1 ε 2 t 1 + βh t 1 + γ IVOL 2 t 1 Model 3: h t = α 0 + α 1 ε 2 t 1 + α 2s t 1 ε 2 t 1 + βh t 1 + δw 2 t 1 Model 4: h t = α 0 + α 1 ε 2 t 1 + α 2 s t 1 ε 2 t 1 + βh t 1 + γ IVOL 2 t 1 + δw 2 t 1 In model 2 and model 4, the IVOL can be substituted for by VIX or VXO to evaluate their forecasting quality. Model 2-1 and 4-1 denote the IVOL replaced by VIX, and model 2-2 and 4-2 denote the IVOL replaced by VXO. To mitigate the potential bias from large sample sizes leading to Type II errors, the critical test statistic values of t and F are also adjusted Out-of-Sample Forecasts and Efficiency Evaluations As the sample size is limited to 992 days, the first 500-day data are used as in-sample data to estimate the required parameters. To ensure the stability of the estimated parameters, 250- day in-sample data are also used to estimate the parameters for comparison purpose. Based on the initial estimated parameters, the out-of-sample volatility forecasts are obtained for the next N = 1, 10, 20 days after the estimation period. One day forecast is generated from each model specification. 10- and 20-day forecasts are obtained by multiplying 1-day volatility forecast by 10 and 20, respectively. the corresponding realized volatilities are computed as the sum of squared daily returns over each forecast period. The parameters are recalibrated by moving the sample window based on 250- and 500-day in-sample data. Four evaluation criteria are listed in the following: P-statistic: Suggested by Blair et al. (2001), P-statistic measures the proportion of the 3 As shown by Lindley s paradox, the t-and F-statistics are adjusted when their absolute values are greater than the value of the following: t = T k (T 1/T 1), F = T 1 K 1 (T (k 1)/T 1) where T is the sample size and k is the number of degrees of freedom lost in the regression. 9

10 variances of realized volatilities explained by volatility forecasts. (T S)/N (y S+N(i 1),N ŷ S+N(i 1),N ) 2 P = 1 i=1 (T S)/N i=1 ( ) 2 (6) y S+N(i 1),N ȳ y t,n denotes realized volatility measured as the sum of squared daily returns over an N-day forecast period beginning on day t, and ŷ t,n denotes the corresponding volatility forecast for the same forecast period. For the data of TAIEX used in the paper, T = 992 and S=250 and 500 for the two sample periods. N is the forecast horizon of 1, 10 and 20 days. Root mean square error (RMSE): RMSE = Mean absolute error (MAE): MAE = 1 (T S)/N 1 (T S)/N R 2 from the following regression: (T S)/N i=1 (T S)/N i=1 ( y S+N(i 1),N ŷ S+N(i 1),N ) 2 (7) y S+N(i 1),N ŷ S+N(i 1),N (8) y t,n = α + β ŷ t,n + e t (9) R 2 measures the ability of volatility forecast ŷ t,n to explain the corresponding N-day realized volatility y t,n. By following Blair et al. (2001) to compare forecasts from models 2-1, 2-2 and 3, the forecast errors from each model are regressed on forecasts from model 2 and 3 specified in the following: y t,n ŷ M 2 t,n = a 1 + a 2 ŷ M 2 t,n + a 3 ŷ M 3 t,n + ε t (10) y t,n ŷ M 3 t,n = b 1 + b 2 ŷ M 2 t,n + b 3 ŷ M 3 t,n + ε t (11) ŷ M 2 t,n and ŷm 3 t,n denotes an N-day forecast from model 2 and model 3. The forecasts ŷm 2 t,n and ŷm 3 t,n are regarded as efficient when R 2 values from the above two regressions are not significantly different from zero. 10

11 4 Empirical Results 4.1 In-Sample Model Results Table 11 shows the results of GJR-GARCH parameter estimates and related statistics across all models and different sample sizes. To account for effects of sample size on estimated parameters, 250-day sample sizes are considered in panel B. VIX and VXO replacing the variable IVOL in equation (5) are denoted as model 2-1 and 2-2. Similar notation applies to model 4. Columns 2 through 7 state parameter estimates with robust t-statistics in parentheses below each coefficient estimate. Model 4-2 in panel A yields the largest log-likelihood value and the model 2-2 and 4-1 ranked 2nd and 3rd. Model 4-2 in panel B still has the largest log-likelihood value but with a reverse ranking for model 2-2 and 4-1. In panel A and B, coefficients α 1 are estimated to be negative and significantly different from zero for model 2, 3 and 4. Coefficients α 2 measuring the effect of good news are all positive and significantly different from zero in both panels. For models 2, 3 and 4, the sums of coefficients α 1 and α 2 are all positive, which is consistent with result of Corrado and Truong (2007) that asymmetric news effects on conditional volatility exist for stock index in Taiwan. The coefficients γ and δ are significantly different from zero across all models in panel A. However, coefficient γ is not significantly different from zero in model 4-1 and 4-2 of panel B. In the general model 4-1 and 4-2, the forecasting ability of volatility index tends to increases with in-sample size, but the absolute values of γ are smaller than those of δ, suggesting that the intraday high-low range volatility dominates implied volatility index in predicting the conditional variance Out-of-Sample Forecast Evaluations The rankings of models based on the three measures of forecasting, P-statistics, RMSE and MAE, are listed in Table 12. The underlined number is the highest ranking in its respective column. For the 500-day in-sample data, model 3 ranked highest in terms of overall rankings, a result different from the intuition that volatility index should be effective in volatility prediction. For 1- and 10-day forecast in panel A, model 3 ranked highest. For 20-day forecast, model 1 and 3 ranked the first and the second. For 250-day in-sample data, model 4-1 ranked highest 4 The paper has also used 125-day in-sample data to estimate the γ and δ in model 4. Both estimated coefficients are not significantly different from zero. In consideration of possible biases from small sample size and the limited space of the paper, the estimated results are not reported in the paper. 11

12 in terms of overall rankings. Model 4-1 also ranked highest for 1-, 10- and 20-day forecasting periods. As in-sample size increases, the high-low range volatility seems to subsume VIX as a dominating indicator in volatility prediction. However, the result is still consistent with the Corrado and Truong (2007) that the high low range volatility is no less efficient than VIX in volatility prediction. Table 13 reported the estimated coefficients and R 2 values by regressing the realized volatility on out-of-sample volatility forecasts across models and periods based on two in-sample sizes. In comparison with Table 12, the results of Table 13 are mixed. In panel A, model 3 has the highest R 2 of for 10-day forecast. For 1-day forecast, model 4-1 and 4-2 have higher R 2 values of and than other models. However, their efficacy decreases as forecast period increases to 10 and 20 days. For 20-day forecast, all models have insignificant R 2. In panel B, model 4-1 consistently delivered better results than other models. In general, for 500-day in-sample data, model 4-1 and 4-2 yielded better results for 1-day forecast; model 3 is better in 10-day forecast and has the highest R 2 value among all models. For 250-day in-sample data, model 4-1 consistently performs better than others across all forecast periods. Table 14 and?? reported the regression results by regressing the volatility forecast errors on out-of-sample model 2 and model 3 forecasts. In panel A of Table 14, the estimated coefficients of forecast errors from model 2-1 represent a 1, a 2 and a 3 of equation (10). The estimated coefficients of forecast errors from model 3 represent b 1, b 2 and b 3 of equation (11). The slope coefficients a 1, a 2 from equation (10) and b 1, b 2 from equation (11) are expected not to differ significantly from zero, if the forecasts are efficient. The F-test and its corresponding P values are shown in each panel to test the hypotheses a 1 = a 2 = 0 and b 1 = b 2 = 0. In panel A of 500-day in-sample data, all R 2 values are not significantly different from zero at critical p level of In panel B of 250-day in-sample data, model 2-1 and 3 delivered efficient forecasts only for 20-day forecast period. In general, the forecasting efficiency of model 2-1, 2-2 and 3 is found to be better as the sample size increase from 250 to 500. The forecasting efficiency for 20-day period is better than 1- and 10-day forecast in 250 sample size. 5 Conclusion The volatility index has been widely accepted as an indicator to measure the sentiments of investors. In emerging market like Taiwan, however, the volatility index is still new to many investors as the option market is not as well-developed as in US. According to CBOE, the 12

13 nearby and second nearby options prices will be used in the construction of the volatility series. However, the illiquid trading of the second nearby options in Taiwan may create missing data in computing the volatility index. To gain more insight into the options market in constructing the volatility indexes, different sampling methods are adopted to obtain a better measure of implied volatility series. Among the sampling methods, the method of using the average of all the mids of bid and ask prices on options is found to be able to provide the most complete price information on options, thus delivering more accurate measures of VXO and VIX for Taiwan. The paper further compares volatility indexes with daily high-low range volatility in their forecasting efficiency. The general regression formula takes the form of GJR-GARCH model with additional explanatory variables of implied volatility index and high-low range volatility. The results show that the high-low range volatility is at least as efficient as the volatility indexes across different forecasting periods. For large 500-day in-sample data, the high-low range volatility outperforms the volatility indexes in 1-, 10- and 20-day volatility forecasts. However, for 250-day in-sample data, the volatility index seem to be able to provide incremental information on volatility forecast. The general GJR-GARCH model with VIX index consistently performs better than other models in three forecast periods. The regression of forecasting errors on various model specifications also confirms, though partially, the aforementioned results. The general GJR-GARCH model with VIX index is found to be able to deliver better forecasts in small in-sample data across all forecast periods. With a large in-sample data, the high-low range volatility seems to be more effective in forecasting than volatility indexes. Due to the limit of the sample size used in the paper, a more robust conclusion remains to be obtained when larger sample data is available. 13

14 Table 1: Summary of TXO trading information Year Volume Trading Average daily (contracts) days trading volume ,824, , ,096, , ,929, , ,585, ,841 Table 2: Average daily trading volume of option contracts (2004/1/2-2007/12/31) * Types Option series Time to maturity 1 day a 2 days 3 days 6 days 7 days 8 days Panel A: by moneyness In-the-money nearby 35,975 32,100 33,913 30,379 23,949 23,650 second nearby 5,279 3,919 3,036 1,897 1,208 1,420 next to second nearby At-the-money nearby 162, , , , , ,508 second nearby 30,753 21,756 15,100 11,408 7,367 5,621 next to second nearby Out-of-the-money nearby 70,228 85, , , , ,846 second nearby 101,729 74,716 52,693 42,026 30,058 24,364 next to second nearby 2,349 1,852 1,555 1,417 1,343 1,320 Panel B: by calls/puts calls nearby 143, , , , , ,338 second nearby 75,526 54,275 38,994 31,219 21,698 17,326 next to second nearby 1,702 1,302 1, puts nearby 125, , , , , ,665 second nearby 62,236 46,115 31,835 24,112 16,935 14,080 next to second nearby 1,265 1, * For in-the-money option series, the table lists the average daily trading volume of three option series when nearby option series have 1 day, 2 days, 3 days, 6 days, 7 days and 8 days left to expiry. a It means there is 1 day left to expiry for nearby options. Similar reason applies to the rest. 14

15 Table 3: Descriptive statistics of intradaily VXO based on four sampling methods (2004/1/2-2007/12/31) * Time to maturity a Mean Panel A: last recorded trading price Standard # of Skewness Kurtosis Max Min deviation sample b 1 day ,533 2 days ,642 3 days ,271 7 days ,417 8 days ,203 Panel B: average of all trading prices 1 day ,605 2 days ,684 3 days ,294 7 days ,428 8 days ,213 Panel C: last mid of bid and ask prices 1 day ,513 2 days ,060 3 days ,691 7 days ,239 8 days ,299 Panel D: average of all mids of bid and ask prices 1 day ,536 2 days ,074 3 days ,690 7 days ,245 8 days ,304 * Panel A to D represent VXO indexes based on method 1 to 4. The intradaily trading interval is 1 minute. a When nearby options have 1, 2, 3, 7 and 8 days left to expire, the volatility index is computed by rolling to the second and third contract month options, respectively. b The number of TAIFEX trading days from 2004/1/2 to 2007/12/31 is 992. For price per minute, there will be a total of 267,840 recorded prices for 992 days. However, due to insufficient data from the thin trading for some options, the number of sample will be less than expected. 15

16 Table 4: Descriptive statistics of VXO at the opening minute based on four sampling methods (2004/1/2-2007/12/31) * Time to maturity a Mean Panel A: last recorded trading price Standard # of Skewness Kurtosis Max Min deviation sample b 1 day days days days days Panel B: average of all trading prices 1 day days days days days Panel C: last mid of bid and ask prices 1 day days days days days Panel D: average of all mids of bid and ask prices 1 day days days days days * Panel A to D represent VXO indexes based on method 1 to 4. The intradaily trading interval is 1 minute. a When there are 1, 2, 3, 7 and 8 days to expiration for nearby options, the volatility index is computed, respectively, by rolling to the second and third contract month options. b The number of trading days from 2004/1/2 to 2007/12/31 is 992. However, due to insufficient data from the thin trading for some options, the number of sample will be less than the expected. 16

17 Table 5: Descriptive statistics of VXO at the closing minute based on four sampling methods (2004/1/2-2007/12/31) * Time to maturity a Mean Standard # of Skewness Kurtosis Max Min deviation sample b Panel A: last recorded trading price 1 day days days days days Panel B: average of all trading prices 1 day days days days days Panel C: last mid of bid and ask prices 1 day days days days days Panel D: average of all mids of bid and ask prices 1 day days days days days * Panel A to D represent VXO indexes based on method 1 to 4. The intradaily trading interval is 1 minute. a When there are 1, 2, 3, 7 and 8 days to expiration for nearby options, the volatility index is computed, respectively, by rolling to the second and third contract month options. b The number of trading days from 2004/1/2 to 2007/12/31 is 992. However, due to insufficient data from the thin trading for some options, the number of sample will be less than the expected. 17

18 Table 6: Descriptive statistics of intradaily VIX based on four sampling methods (2004/1/2-2007/12/31) * Time to maturity a Mean Panel A: last recorded trading price Standard # of Skewness Kurtosis Max Min deviation sample b 1 day ,100 2 days ,196 3 days ,782 7 days ,555 8 days ,299 Panel B: average of all trading prices 1 day ,086 2 days ,183 3 days ,768 7 days ,541 8 days ,285 Panel C: last mid of bid and ask prices 1 day ,526 2 days ,527 3 days ,630 7 days ,630 8 days ,630 Panel D: average of all mids of bid and ask prices 1 day ,531 2 days ,532 3 days ,636 7 days ,636 8 days ,636 * Panel A to D represent VIX indexes based on method 1 to 4. The intradaily trading interval is 1 minute. a When nearby options have 1, 2, 3, 7 and 8 days left to expire, the volatility index is computed by rolling to the second and third contract month options, respectively. b The number of TAIFEX trading days from 2004/1/2 to 2007/12/31 is 992. For price per minute, there will be a total of 267,840 recorded prices for 992 days. However, due to insufficient data from the thin trading for some options, the number of sample will be less than the expected. 18

19 Table 7: Descriptive statistics of VIX at the opening minute based on four sampling methods (2004/1/2-2007/12/31) * Time to maturity a Mean Panel A: last recorded trading price Standard # of Skewness Kurtosis Max Min deviation sample b 1 day days days days days Panel B: average of all trading prices 1 day days days days days Panel C: last mid of bid and ask prices 1 day days days days days Panel D: average of all mids of bid and ask prices 1 day days days days days * Panel A to D represent VIX indexes based on method 1 to 4. The intradaily trading interval is 1 minute. a When there are 1, 2, 3, 7 and 8 days to expiration for nearby options, the volatility index is computed, respectively, by rolling to the second and third contract month options. b The number of trading days from 2004/1/2 to 2007/12/31 is 992. However, due to insufficient data from the thin trading for some options, the number of sample will be less than the expected. 19

20 Table 8: Descriptive statistics of VIX at the closing minute based on four sampling methods (2004/1/2-2007/12/31) * Time to maturity a Mean Standard # of Skewness Kurtosis Max Min deviation sample b Panel A: last recorded trading price 1 day days days days days Panel B: average of all trading prices 1 day days days days days Panel C: last mid of bid and ask prices 1 day days days days days Panel D: average of all mids of bid and ask prices 1 day days days days days * Panel A to D represent VIX indexes based on method 1 to 4. The intradaily trading interval is 1 minute. a When there are 1, 2, 3, 7 and 8 days to expiration for nearby options, the volatility index is computed, respectively, by rolling to the second and third contract month options. b The number of trading days from 2004/1/2 to 2007/12/31 is 992. However, due to insufficient data from the thin trading for some options, the number of sample will be less than the expected. 20

21 .7 VXO.5 VIX Squared Returns.6 High-low range Figure 1: Volatility series from 2004/1/2-2007/12/31 21

22 Table 9: Summary Statistics for Volatility Series (2004/1/2-2007/12/31) * VXO VIX Ret2 HLR Mean Median Maximum Minimum Std. Dev Skewness Kurtosis * Ret2 and HLR are the annualized volatility series based on daily squared returns and high low range. Table 10: Correlations of the Volatility Series (2004/1/2-2007/12/31) * VXO VIX Ret2 HLR VXO 1 VIX Ret HLR * Ret2 and HLR are the annualized volatility series based on daily squared returns and high low range. 22

23 Table 11: GJR-GARCH In-Sample Regressions for Daily TAIEX Index Volatility Model a α 0 α 1 α 2 β γ δ Log-L Panel A: 500-day sample period Model (1.8706) (2.0318) (2.3955) ( ) Model ( ) ( ) (4.3011) (4.5720) (3.5805) Model (0.5902) ( ) (4.5079) (4.7302) (3.8593) Model (1.5655) ( ) (3.4612) ( ) (3.9489) Model (0.3017) ( ) (4.6316) (7.8358) (2.4578) (1.8434) Model (0.1384) ( ) (4.6625) (9.0213) (2.8611) (1.9644) Panel B: 250-day sample period Model (1.7368) (0.8609) (1.8331) (9.2914) Model ( ) ( ) (1.4332) (1.3237) (3.3477) Model ( ) ( ) (1.4740) (0.6389) (4.6826) Model (1.1707) ( ) (2.3994) (7.2680) (2.5227) Model (0.1874) ( ) (2.3156) (5.5250) (0.8216) (1.8855) Model ( ) ( ) (2.3898) (5.0021) (1.1169) (1.7463) a The general model, or model 4, denotes the augmented GJR-GARCH model and are specified as follows: h t = α 0 + α 1 ε 2 t 1 + α 2s t 1 ε 2 t 1 + βh t 1 + γ IVOL 2 t 1 + δw 2 t 1 where s t 1 = 1 if ε t 1 < 0, and 0 otherwise. Log-L is the maximum likelihood values. Robust t-statistics are reported in parentheses below estimated coefficients. Model 2-1 and 2-2 use implied volatilities based on VIX and VXO, respectively. The same notation applies to model 4. * The notations of *, ** and *** represent 0.1, 0.05 and 0.01 significance level, respectively. 23

24 Table 12: Forecast rankings for the TAIEX Index * 1-Day Forecast 10-Day Forecast 20-Day Forecast P R M sum a P R M sum P R M sum total b Panel A: 500-day sample period Model Model Model Model Model Model Panel B: 250-day sample period Model Model Model Model Model Model * P, R and M represent P-statistic, RMSE and MAE values, respectively. a Measures of P-statistics, RMSE and MAE are applied to models and their respective rankings are summed up. The underlined number denotes the highest ranking among the models. b Total denotes the sum of each sum column. 24

25 Table 13: Regression of Realized Volatility Against Out-of-Sample Volatility Forecasts * Panel A: 500-day sample period 1-Day forecast 10-Day forecast 20-Day forecast α β R 2 α β R 2 α β R 2 Model (0.5057) (7.9549) (0.9141) (4.1507) (1.7826) (0.5478) Model 2-1 a (0.6187) (8.1836) (0.9733) (4.0011) (2.2614) (0.5836) Model 2-2 b (0.1704) (8.2055) (1.2683) (3.7535) (2.3928) (0.5159) Model (0.3039) (8.2449) (0.6580) (4.7077) (1.9021) (0.5947) Model 4-1 a (0.6911) (8.3879) (1.0491) (3.9089) (2.2813) (0.5375) Model 4-2 b (0.5562) (8.4681) (1.3542) (3.6993) (2.4351) (0.4997) Panel B: 250-day sample period Model (3.6308) (7.8698) (3.5345) (3.3331) (5.7509) (0.2698) Model (3.6987) (7.9711) (3.0288) (3.9119) (1.4412) (3.1560) Model (2.9615) (8.6235) (3.4744) (3.8336) (1.6119) (2.9295) Model (3.3390) (8.1384) (3.4195) (3.3538) (1.5099) (2.5037) Model (3.4804) (8.1966) (3.0950) (4.0534) (1.3915) (3.2705) Model (4.0201) (7.8703) (3.3868) (3.8477) (1.3907) (3.2441) * The estimated regression is as follows: y t,n = α + β ŷ t,n + e t, y t,n denotes N-day realized volatility and ŷ t,n is the volatility forecast for the same period. Robust t-statistics are reported in parentheses below estimated coefficients. a Model 2-1 and Model 4-1 use implied volatility VIX for IVOL. b Model 2-2 and Model 4-2 use implied volatility VXO for IVOL. 25