Implied Volatility Structure and Forecasting Efficiency: Evidence from Indian Option Market CHAPTER V FORECASTING EFFICIENCY OF IMPLIED VOLATILITY

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1 CHAPTER V FORECASTING EFFICIENCY OF IMPLIED VOLATILITY 5.1 INTRODUCTION The forecasting efficiency of implied volatility is the contemporary phenomenon in Indian option market. Market expectations are reflected in the option prices about the distribution of the future value of underlying assets. Under a rational expectations assumption, the market uses all the information available to form its expectations about future volatility. Therefore, the market option price displays the market s true volatility estimate. Furthermore, the implied volatility is the best possible forecast in case market is efficient. That is, implied volatility subsumed all necessary information generated by other explanatory variables to explain future realized volatility. In forecasting volatility, two approaches are followed. One is the historical variance of the future return and second is to elicit market expectations about the future volatility from observed option prices. Under the assumption that option pricing model correctly represents investors behaviour in deriving the implied volatility from observed option prices and other observable variables. When the market is informationally efficient, the implied volatility should reflect the information contained in historical volatility. Hence, forecasts based on historical volatility should not outperform forecasts based on implied volatilities. ~ Chapter 5 ~ 127

2 Latane and Rendleman (1976) are the first to examine 24 stock option traded on Chicago Board Option Exchange (CBOE) and report the predicative power of implied volatility. In the light of this paper, Chiras and Manaster (1978) and Beckers (1981) use a broader sample of individuals CBOE stock option to conduct static cross-sectional regressions. These earlier papers essentially document that implied volatility is superior to historical volatility in forecasting future volatility with cross-sectional data. It should be noted that these studies concentrate on static cross-sectional relation rather than on timeseries forecasts. With sufficient time-series data, later studies have focused on the information content of implied volatility in a dynamic setting. Day and Lewis (1992) examine option on the S&P 100 index between 1983 and 1989, and report that implied volatility does not contain more information content of future realized volatility than historical volatility. However, this study ignores the term structure of volatility in the measurement of realized volatility, which is not matched with the remaining life of options. Canina and figlewski (1993) conducted the regression of realized volatility over the remaining life of the option on the implied volatility of the S&P 100 index options 1983 to Surprisingly, they found that implied volatility has virtually no correlation with future return volatility. Lamoureux and Lastrapes (1993) examined options on individual stock from 1982 to 1984, and found that information contained in historical volatility about future ~ Chapter 5 ~ 128

3 realized volatility is more than that contained in implied volatility. These results are matched with the result of Day and Lewis (1992). Responding to the conclusions in the previous studies of individual stock options and index option, Jorion (1995) points out that there are two possible explanations: one is that the test procedure is faulty and the other is that the option market is inefficient. In contract with individual stock option and index options, he uses option on foreign currency future traded on Chicago Mercantile Exchange (CME), and concludes that implied volatility is an efficient but biased forecast of future realized volatility. Bates (1996) found that the implied volatility is biased and contains information about future volatility. Vasilellis and Meade (1996) examined 12 UK companies and suggested implied volatilities to produce better individual forecasts than time series. Fleming (1998) examined the S&P 100 index options from 1985 to He indicates that the implied volatility is an upward biased forecast, but also that it contains more information regarding future realized volatility than historical volatility. All of these studies above have used overlapping dataset and suffer from the serial correlation problem. To address this problem, Christensen and Prabhala (1998) have introduced a new sampling procedure which produces non-overlapping volatilities series. In simple term, exactly one implied volatility is responding to one realized volatility for each time period under consideration. With this sampling procedure and a longer volatility series from ~ Chapter 5 ~ 129

4 1983 to 1995, they found that implied volatility of S&P 100 index (OEX) options outperforms historical volatility in predicting future realized volatility. These conclusions are further enhanced by Christensen and Hansen (2002) where implied volatility is constructed as a trade weighted average of implied volatilities from both OEX calls and put. Following Christensen and Prabhala (1998) sampling procedure, few studies in other option market have been carried out, and view implied volatility as a better predictor of future volatility of underlying assets. Hansen (2001) analyses the information content of option in the Danish KFX share index. This option market is very illiquid with infrequent trade and low volume. However, he found that when the errors-in-variable problem is controlled by the instrumental variable technique, call implied volatility still contains more information about future volatility than historical volatility in such an illiquid option market. Shu and Zhang (2003) examined the options on S&P 500 index, and also report that implied volatility implied in the option prices outperforms the subsequently historical index return volatility. Szakmary. et al, (2003) examined 35 future option markets across eight separate exchanges and found that for a large majority of the commodities studies, implied volatility is a better predictor of future realized volatility than historical volatility. Jiang and Tian (2005) show that the at-the-money implied volatility correlated with future realized volatility and also report that model free volatility expectation is more highly correlated than implied volatility. Yu, ~ Chapter 5 ~ 130

5 Lui and Wang (2009) investigated the efficiency of stock index options traded over-the-counter (OTC) in Hong Kong and Japan, and concluded that OTC market is more efficient than exchange in Japan. Taylor, et al. (2010) concluded that at-the-money implied volatilities generally outperform the model-free volatility expectations. In Indian contest, Rao (2005) found that option market inefficiency due to misspecification of the volatility process in the option valuation model. Mohan and Chaturvedual (2008) reported that implied volatility index (IMVI) is biased estimator of realized volatility. Panda. et al, (2008) examined the information content of call and put options during June 2001 and October 2004 and found implied volatility contains more information than historical volatility but slightly biased estimator of realized volatility. Kumar (2008) reports implied volatility is unbiased and informationally efficient predictor of future realized volatility. Devanadhen and Rajagopalan (2009) found at-the-money implied volatility is slightly biased and informationally efficient in predicting future realized volatility. The above Indian studies have used only the S&P CNX Nifty index option. So far no attempt has been made in stock option. In order to fill the gap in the Indian literature, the current study attempts to shed light on the selected individual stock option in India. ~ Chapter 5 ~ 131

6 5.2 DATA AND METHODOLOGY Data and Sample Procedure The study focuses on S&P CNX Nifty index option and selected five individual stock options (i.e. Infosys, ITC, Ranbaxy, Reliance and SBI) from January 1, 2002 to June 30, To obtain a relatively accurate measurement of implied volatility, the options must be chosen carefully. Options contracts are selected on the following criteria: 1. Non-overlapping near month option contract with a time to maturity of 30 calendar days was taken. 2. The option within the range of X/F from.95 to 1.05 (at-the-money option), where X is the exercise price of the option and F is the future price. 3. Options must be traded actively, i.e. a relatively highest trading volume. Criterion (1) is used to avoid the overlapping of data; this study follows the sampling procedure present by Christensen and Prabhala (1998). This approach results in that exactly one implied volatility is responding to one realized volatility for each time period under consideration. For example the first observation is for the January 2002 contract expiring on , so we move back 30 days from the expiry day i.e. on , observe the inputs to the option pricing model and extract the implied volatility. Similarly the next data point is the implied volatility from the February 2002 contract expiring on , so the prices on will be observed and they ~ Chapter 5 ~ 132

7 will be used to extract implied volatility. Criterion (2) is employed that option of different exercise prices have been known to produce different implied volatilities. A decision has to be made as to which of these implied volatilities should be used, or which weighing scheme should be adopted. The most common strategy is to choose the implied volatility derived from the at-themoney based on the argument that at-the-money option are the most liquid and hence at-the-money implied volatility is less likely to mispriced. Criterion (3) is employed to collect at-the-money option with highest trading volume in order to capture the investor expectation of future volatility. By applying the above sampling criteria and procedure, 612 (102 x 6) observations are obtained for each series of call and put options Methodology The methodology adopted for the measurement of time left to maturity and three volatility series namely, implied volatility, historical volatility and realized volatility will be discussed. After that, Unit root test, Ordinary Least Squared (OLS) method, Walt statistics and ARCH LM test, Hausman test and Instrument variable method will be described Measurement of time left to maturity In practice, the time for paying interest is based on calendar days, while the time of the life of an option is based on trading days. French (1994) suggests that, when calculating the option s price, time for paying interest and time for option s life should be measured separately. That is ~ Chapter 5 ~ 133

8 Hull (2003) suggests that, however, in practice, these two measurements do not result in a big difference except for options with very short life. The option s life in this study is about one month, so the above definition of time to maturity of option is used throughout the study. Time to maturity, T t is measured by the number of trading days between the day of trade and expiry day divided by the number of trading day per year which is taken as At-the-money option or weighted implied volatility? Under the framework of the Black-Scholas option pricing model, a single option price is sufficient to estimate the implicit parameter i.e. volatility of underlying assets return over the remaining life of the option. However, according to the fact that options with the same maturity but different exercise price yield different implied volatilities, a decision has to be made as to which of these implied volatilities should be used, or which weighing scheme should be adopted, to produce a single volatility assessment. It has been widely accepted in the literature that implied volatility computed from an at-the-money option is superior to volatilities obtained from out-of-the-money or in-the-money options. This is because at-the-money options are often most actively traded and hence volatility derived from these options should be least prone to measurement errors. Feinstein (1989) pointed ~ Chapter 5 ~ 134

9 out that for the stochastic volatility process described in Hull and White (1987), implied volatilities from at-the-money and near expiration options provide the closest approximation to the average volatility over the option s life, provided that volatility risk premium is either zero or a constant. This means that if volatility is stochastic, implied volatilities from at-the-money options are less likely to be biased compared to those from out-of-the-money and in-the-money options. On the other hand, many studies have focused on computing a composite implied volatility by placing different weights on at-the-money, outof-the-money and in-the-money options. Bates (1996) provide good summary of these weighing schemes, as shown in Table 5.1. Most assign the heaviest weight to at-the-money implied volatility. Other weighing schemes, such as equally weighted model by Schmalensee and Trippi (1978) and elasticity weighted model by Chiras and Manaster (1978) are less popular due to not emphasizing at-the-money implied volatility. Furthermore, Beckers (1981) shows that using only at-the-money options is preferable to various other weighing schemes, and hence questions the usefulness of weighting systems. Jorion (1995) also concentrates on at-themoney options, and suggests that using the arithmetic average of the implied volatilities from at-the-money call and put options alleviates some of the measurement problems. In chapter 4, the results indicate that the Black- Schloas-Merton model tends to overprice in-the-money call options and out-of- ~ Chapter 5 ~ 135

10 the-money put options and underprice in-the-money put option and out-of-themoney call options. At-the-money options are often most actively traded and hence they are less likely to be mispriced. Therefore, implied volatilities from at-the-money options were taken instead of weighted implied volatility. Table 5.1 Methods for Computing Weighted Implied Volatility Model Formula Comments Schmalensee and Trippi (1978) Latane and ˆ 1 N i, where i is the implied volatility from the i th option price O i w i i ˆ, 2 Oi w i Rendleman (1976) wi i Modified Latane and Rendleman Whaley (1982) Beckers (1981) ˆ w w i i, 2 i i 2 i i Oi wi ˆ arg min O O w Oi, wi w w Oi Oi ˆ arg min w w Oi, wi w 3 i i 3 i i i i i i i 2 Chiras and Manaster wi i Oi ˆ, wi (1978) wi Oi i At-the-money ˆ ATM Equal weights. Typically implemented on a restricted set of options (e.g., excluding deep out-of-themoney options) Weights don t sum to one, creating biased volatility estimates Heaviest weight on near-themoney options. In-themoney and out-of-themoney options weighted symmetrically Even heavier weight on near-the-money options than the Modified Latane and Rendleman. Typically implemented on transactions data, which affects the relative weights. 2 Even heavier weight on near-the-money options than Whaley (1982) Elasticity-weighted, with heaviest weight on lowpriced, deep out-of-themoney options. Increasingly standard. A readily replicable benchmark based on actively traded ~ Chapter 5 ~ 136

11 options Measurement of Implied volatility The Merton (1973) model generalized the Black-Scholas model by relaxing the assumption of no dividend paid during the life of option is employed to derive at-the-money implied volatilities from the observed market prices. To obtain the dividend yield estimation, the future contracts are utilized, which have exactly the same expiry cycle, expiry date and underlying asset with the option contract Realized volatility The realized volatility can be measured by standard deviation of the daily return over the remaining life of an option. Let n be the number of trading days before expiration, P i be the closing price on the day of the remaining life of the option and R i denotes daily continuously compounded return on the i th day. Then R i Pi ln Pi 1 for i = 1,2,3... n. Thus, the annualized realized volatility can be expressed as ~ Chapter 5 ~ 137

12 r, t 252 n 1 i 1 n Ri, t Rt 2 where Rt denotes the mean of daily continuously compounded return during the period t Historical volatility The historical volatility ( h, t 1 ) is calculated as annualized standard deviation of daily continuously compounded return of the period going back T days (i.e.,30 calendar days) from time. h, t T 1 i 1 T R t Rt i, Where R t 1 denotes the mean of daily continuously compounded return during the period t Unit root test The study first tests the stationarity of the volatilites series of implied volatility, historical volatility and realized volatility. Engle and Granger (1982) have shown that many time series variables are non-stationary or different order of integration i.e. I(1) series. Since most of time series have unit roots and are non-stationary as indicated by Nelson and Plosser (1982), and as proved by Stock and Watson (1988), that conventional regression techniques on non-stationary time series may produce spurious regression. Hence, the ~ Chapter 5 ~ 138

13 Augmented Dickey Fuller (ADF) test and Phillips-Perron (PP) test are employed to infer the stationarity of the series Augmented Dickey Fuller (ADF) test Augmented Dickey Fuller (1979) implicitly assumes that the estimated errors are statistically independent and homoscedastic. Heteroskedasticity does not affect a wide range of unit root test statistics. However, a problem will occur if the estimated residual ε t is not free from autocorrelation since this invalidates the test. The well-known example of unit root non-stationary is the random walk model. There might be three possibilites for any time sereis. The time series might be a random walk, a random walk with drift, or random walk with drift and time trend. The three possible forms of the ADF test are given by the following equation: Y y y p t 1 t 1 i t 1 t i 1 Y y y t 0 1 t 1 i t 1 t i 1 p Y y t y t 0 1 t 1 2 i t 1 t i 1 p ~ Chapter 5 ~ 139

14 Where ε t is white noise. The additional lagged difference terms are being determined by minimum number of residuals free from autocorrelation. This could be tested for in the standard way such as Akaike Information Criterion (AIC) or Schwartz Bayesian Criterion (SIC), or more usefully by the lag length criteria of the white noise series. The tests are based on the null hypothesis (H 0 ): Y t is not I (0). If the calculated ADF test statistics are less than their critical values from table, then the null hypothesis (H 0 ) is accepted and the series are non-stationary or integrated to zero order Phillips-Peron (PP) test The distribution theory supporting the Dickey-fuller tests is based on the assumption that the error terms are statistically independent and have a constant variance. Thus, while using the ADF methodology one has to make sure that the error terms are uncorrelated and that they really have a constant variance. Phillips and Perron (1988) have developed a generalization of the ADF test procedure that allows for fairly mild assumptions concerning the distribution of errors. The PP regression equations are as follows: Y t 1 0 yt 1 t where the ADF test corrects for higher order serial correlation by adding lagged differenced terms on the right-hand side, the PP test makes a correction to the t statistics of the coefficient γ from AR (1) regression to account for the serial correlation in ε t. The statistics are all used to test the hypothesis γ = 0, ~ Chapter 5 ~ 140

15 i.e., there exists a unit root. So, the PP statistics are just modifications of the ADF t statistics that take into account the less restrictive nature of the error process Ordinary Least Squares (OLS) Method The OLS regressions are used to examine the relationship between implied volatility and historical volatility. To investigate the information content of implied volatility by OLS estimate, the following regressions are conducted for both call and put series: (5-1) (5-2) To explore the information content of historical volatility, the regression is given below: (5-3) To examine the implied volatility of call and put option includes all information to predict future realized volatility; the following regressions are given below: ~ Chapter 5 ~ 141

16 (5-4) (5-5) where,, and denotes realized volatility, implied call volatility, implied put volatility and historical volatility respectively. There are few hypotheses to be examined (Christensen and Prabhala (1998); Szakmary, et al. (2003)) 1. If volatility contains some information about future realized volatility, then the co-efficient of the volatility (, and ) in regression (5-1) to (5-3) should be non-zero. 2. If volatility is an unbiased forecast of future realized, then the coefficient of volatility (, and ) should be equal to one and the coefficient of intercept ( ) should be equal to zero in regression (5-1) to (5-3) 3. If implied volatility includes more information than historical volatility, higher Adjusted R 2 would be expected from regression (5-1) and (5-2) than (5-3) 4. If implied volatility is an informationally efficient predictor of future realized volatility, i.e., implied volatility efficiently includes all information to predict future volatility, then the co-efficient of historical ~ Chapter 5 ~ 142

17 volatility ( ) in both (5-4) and (5-5) should be zero, and the error term should be white noise and thereby uncorrelated with any explanatory variable in the market s information set. In addition, to examine the relative information content of implied call and put volatility, the following regression can be conducted: (5-6) (5-7) where,, and denote realized volatility, implied call volatility, implied put volatility and historical volatility respectively. Several hypotheses can be tested within regression (5-6) and (5-7) (Hansen (2001), Christensen and Hansen (2002)). 1. The issue of which of the two implied volatilities has more information about the future realized volatility can be investigated by comparing with in regression (5-6). In fact, it could help to find out an optimal weighting of these two implied volatilities to forecast future volatility. 2. If implied call and put volatility is an informationally efficient predictor of future realized volatility, the co-efficient of historical volatility ( ) is excepted to be zero in regression (5-7) ~ Chapter 5 ~ 143

18 Wald test The wald test computes a test statistics based on the unrestricted regression. It is also called co-efficient restrictions test. The wald statistic measures how close the unrestricted estimates come to satisfying the restrictions under the null hypothesis. If the restrictions are in fact true, then the unrestricted estimates should come close to satisfying the restrictions. Wald test is used to examine the unbiasedness of implied volatility ARCH LM Test The ARCH LM test is called Lagrange multiplier (LM) test for autoregressive conditional heteroskedasticity (ARCH) in the residuals. The ARCH LM test statistics is computed from an auxiliary test regression. To test the null hypothesis, there is no ARCH up to q order in the residuals (residual are white noise up to q order). The regression is q 2 2, t 0 s t s, t s 1 where,, t is the residual. This is a regression of the squared residuals on a constant and lagged squared residual up to q order. The ARCH LM test is applied on the residual of the OLS regression Hausman (1978) test ~ Chapter 5 ~ 144

19 Hausman (1978) test is to verify the presence of the Error-in-variable (EIV) problem in this study. If the EIV problem exists, then the OLS estimates can be not only biased but also inconsistent. Thus, the OLS estimates of regression (5-1), (5-2), (5-4) and (5-5) may yield misleading results. In this case, the remedy is the alternative method, Instrument variable method. However, the Instrument variable estimates are less efficient than OLS estimates if the EIV problem does not exist. Thus, the Hausman (1978) test is needed to test the presence of EIV problem. The basic idea of Hausman (1978) test is to construct a chi-square test statistics based on the difference between OLS estimator and IV estimator. But as suggested by Davidson and Mackinnon (1989, 1993), one never need to construct the difference between two estimators to compute the statistics. Hausman (1978) test can be illustrated by a simple version in which an auxiliary regression is utilised. To carry out to Hausman test, two OLS regressions are run. The first regression, implied volatility ( ) is regressed on all exogenous variable and instrument variable. In this case, the first regression is given as (5-8) The second regression is to re-estimate regression (5-1) by including the residual from the first regression (5-8) as an additional regressor (5-9) ~ Chapter 5 ~ 145

20 If the OLS estimates are consistent, then the co-efficient on the first stage residuals ( ) should not be significantly different from zero Instrument variable (IV) method The IV method can be achieved by the two stage least squares procedure (2SLS). In the first stage, implied volatility is regressed on the lagged implied volatility and historical volatility as instrument variable by OLS method: (5-10) In the second stage, the regressions (5-1), (5-2), (5-4) and (5-5) are reexamined by replacing respective implied volatility with the fitted implied volatility from the first stage regression (5-10) (5-11) and (5-12) 5.3 EMPIRICAL RESULTS This section presents the empirical results on the relationship between implied and realized volatility of S&P CNX Nifty index option and selected five individual stock options. Firstly, the descriptive statistics for implied call volatility, implied put volatility, historical volatility and realized volatility are ~ Chapter 5 ~ 146

21 given in Section ADF and PP Unit root test results are exhibited in section Regression results of OLS methods are reported in section Wald test, ARCH LM test and Hausman test result are present in section 5.3.4, and respectively. Lastly, Instrument variable method is adopted to overcome Error-in-Variable problem and the results are reported in Descriptive statistics Table 5.2 exhibits the descriptive statistics of implied volatility. Panel A and Panel B present the descriptive statistics of implied call and put volatility of S&P CNX Nifty and selected five individual stock options respectively. Panel A in Table 5.2 shows mean call volatility of S&P CNX Nifty option is , the maximum and minimum call volatility during the sample period is and respectively. The standard deviation of S&P CNX Nifty call volatility is The skewness and kurtosis presents the normal distribution of the series. For a normal distributed variable, skewness =0 and kurtosis =3. As per the table value, the skewness and kurtosis of S&P CNX Nifty call volatility are and respectively. ~ Chapter 5 ~ 147

22 TABLE 5.2 Descriptive Statistics of Implied Volatility This table reports descriptive statistics of implied volatility of S&P CNX Nifty and five selected individual stock options. Here, Black-Scholas-Merton implied volatility is computed on the at-the-money option which has highest trading volume, measured at time t. Statistics are based on non-overlapping monthly observation over the period from January 2002 to June Panel A contains implied call volatility Mean, Maximum, Minimum, Standard Deviation, Skewness and Kurtosis of S&P CNX Nifty and selected five individual stock options. Panel B contains implied put volatility Mean, Maximum, Minimum, Standard Deviation, Skewness and Kurtosis of S&P CNX Nifty and selected five individual stock options. Jarque-bera statistics test the normal distribution of the volatility series. a denotes significant at 1% level. Panel A: IMPLIED CALL VOLATILITY Particular S&P CNX NIFTY INFOSYS ITC RANBAXY RELIANCE SBI Mean Maximum Minimum Standard Deviation Skewness Kurtosis Jarque-Bera a a a a Panel B: IMPLIED PUT VOLATILITY Particular S&P CNX NIFTY INFOSYS ITC RANBAXY RELIANCE SBI Mean Maximum Minimum Standard Deviation Skewness Kurtosis Jarque-Bera a a a a a ~ Chapter 5 ~ 148

23 Panel B in Table 5.2 indicates mean put volatility of S&P CNX Nifty option is , the maximum and minimum put volatility during the sample period are and respectively. The standard deviation of S&P CNX Nifty put volatility is The skewness and kurtosis of S&P CNX Nifty put volatility are and respectively. In the both cases, the right tail of the skewness is observed with extreme and reveals positively skewed effect. The kurtoses indicate leptokurtic; it is a representation of volatility clustering, fat tail, etc. From the Jarque-bera statistics, implied call and put volatility of S&P CNX Nifty is not normally distributed. The S&P CNX Nifty mean put volatility is higher than call volatility. The standard deviation of implied put volatility is slightly more volatile than the call volatility. Panel A in Table 5.2 presents the mean implied call volatilities of Infosys, ITC, Ranbaxy, Reliance and SBI are , , , and respectively and , , , and are their respective standard deviations. From the Jarque-bera statistics, Infosys and ITC stock options are normally distributed, other options reveal positively skewed effect and kurtosis indicates leptokurtic, it is a representation of volatility clustering, fat tail, etc. In panel B, the mean implied put volatilities of Infosys, ITC, Ranbaxy, Reliance and SBI are , , , and respectively and , , , and are their respective standard deviations. From the ~ Chapter 5 ~ 149

24 Jarque-bera statistic, only ITC stock is normally distributed; other options reveal positively skewed effect and kurtosis indicates leptokurtic. It is a representation of volatility clustering, fat tail, etc. The standard deviation of individual stock options put volatility is slightly more volatile than call volatility. The individual stock option mean put volatility is higher than call volatility except SBI. The mean put volatility is higher than call volatility and standard deviation of put volatility is slightly more volatile than call volatility. According to Harvey and Whaley (1991, 1992), buying pressure on put option is larger than on call option, because put option is relatively inexpensive and convenient way for hedging. Consequently, implied put volatility is higher than implied call volatility on average and standard deviations of put volatility are slightly more volatile. ~ Chapter 5 ~ 150

25 ~ Chapter 5 ~ 151

26 TABLE 5.3 Descriptive Statistics of Historical and Realized Volatility This table reports descriptive statistics of historical and realized volatility of S&P CNX Nifty and five selected individual stocks. Realized volatility is calculated as annualized standard deviation of the daily log-return over the remaining life of an option. Historical volatility is calculated as annualized standard deviation of daily log-return during the period before time t and with same length as that of the realized volatility (i.e. 30 calendar days). Statistics are based on monthly observation over the period from January 2002 to June Panel A contains historical volatility mean, Maximum, Minimum, Standard Deviation, Skewness and Kurtosis of S&P CNX Nifty and selected five individual stocks. Panel B contains realized volatility mean, Maximum, Minimum, Standard Deviation, Skewness and Kurtosis of S&P CNX Nifty and selected five individual stocks. Jarque-bera statistics test the normal distribution of the volatility series. b and a denotes significant at 5% and 1% level respectively. Panel A: HISTORICAL VOLATILITY Particular S&P CNX NIFTY INFOSYS ITC RANBAXY RELIANCE SBI Mean Maximum Minimum Standard Deviation Skewness Kurtosis Jarque-Bera a a a a a 7.56 b Panel B: REALIZED VOLATILITY Particular S&P CNX NIFTY INFOSYS ITC RANBAXY RELIANCE SBI Mean Maximum Minimum Standard Deviation Skewness Kurtosis Jarque-Bera a a a a a 6.78 b ~ Chapter 5 ~ 152

27 ~ Chapter 5 ~ 153

28 Table 5.3 exhibits the descriptive statistics of historical and realized volatility. Panel A and Panel B present the descriptive statistics of historical and realized volatility of S&P CNX Nifty and selected five individual stocks respectively. Panel A in Table 5.3 indicates the mean historical volatility of S&P CNX Nifty option is , the maximum and minimum historical volatility during the sample period are and respectively. The standard deviation of S&P CNX Nifty historical volatility is The skewness and kurtosis presents the normal distribution of the series. For a normal distributed variable, skewness =0 and kurtosis =3. As per the table value, the skewness and kurtosis of S&P CNX Nifty historical volatility are and respectively. Panel B in Table 5.3 shows mean realized volatility of S&P CNX Nifty option is , the maximum and minimum realized volatility during the sample period are and respectively. The standard deviation of S&P CNX Nifty realized volatility is The skewness and kurtosis of S&P CNX Nifty realized volatility are and respectively. From the Jarque-bera statistics, Historical and realized volatility of S&P CNX Nifty is not normally distributed. The right tail of the skewness observed with extreme and reveals positively skewed effect. The kurtoses indicate leptokurtic; it is a representation of volatility clustering, fat tail, etc. The S&P CNX Nifty mean historical volatility is higher than realized volatility. The ~ Chapter 5 ~ 154

29 standard deviation of historical volatility is slightly more volatile than the realized volatility. Panel A in Table 5.3 presents the mean historical volatilities of Infosys, ITC, Ranbaxy, Reliance and SBI are , , , and respectively and , , , and are their respective standard deviations. From the Jarque-bera statistics, historical volatility of individual stocks is not normally distributed. The right tail of the skewness observed with extreme and reveals positively skewed effect and kurtosis indicates leptokurtic; it is a representation of volatility clustering, fat tail, etc. In panel B, the mean realized volatilities of Infosys, ITC, Ranbaxy, Reliance and SBI are , , , and respectively and , , , , are their respective standard deviations. From the Jarque-bera statistic, the individual stocks are not normally distributed; the right tail of the skewness observed with extreme and reveals positively skewed effect and kurtosis indicate with leptokurtic, it is a representation of volatility clustering, fat tail, etc. Historical volatility of individual stocks is higher than realized volatility. Historical volatility of individual stock is slightly more volatile than realized volatility. The historical volatility is higher than realized volatility, and the standard deviation of historical volatility is slightly more volatile than realized volatility. This may be due to the excess volatility on the expiration days, ~ Chapter 5 ~ 155

30 since the expiration day is excluded in the measurement of realized volatility, while it is included in the measurement of historical volatility. Tables 5.2 and 5.3 indicate that the average implied call and put volatility is larger than the average realized and historical volatility for S&P CNX Nifty and selected four stock options (except SBI). This indicates that Black-Scholes-Merton Model tends to overprice both at-the-money call and put option on average. A similar finding has been demonstrated by Christensen and Prabhala (1998), Hansen (2001), Kumar (2008), Li and Yang (2008), Devanadhen and Rajagopalan (2009) Unit root results One of the main concerns of the time series is to investigate whether the volatilities series contain unit root. Unit root data will lead to spurious regression. To avoid this, the study conducts a unit root test in order to examine whether the series are stationary or not. If the series are nonstationary at level, they have to be differentiated either once or more until they are stationary. In order to check the stationary of the series, both Augmented Dickey-Fuller (ADF) test and Phillips-Perron (PP) test were conducted and their results are presented in Table 5.4. The unit root results of implied call volatility, implied put volatility, historical volatility and realized volatility are exhibited in Panel A, B, C and D respectively. According to Augmented Dickey-Fuller (ADF) test and Phillips-Perron (PP) test under ~ Chapter 5 ~ 156

31 constant, the null hypothesis (H 0 ) constitutes that unit root exists. The alternative hypothesis (H 1 ) is stationary. The critical values at 1% and 5% level of significance for both unit root test under constant are and respectively. The null hypothesis is rejected and denotes that all volatility series are stationary. According to Augmented Dickey-Fuller (ADF) test and Phillips-Perron (PP) test under constant and trend, the null hypothesis (H 0 ) constitutes that trend unit root exists. The alternative hypothesis (H 1 ) is trend stationary. The critical values at 1% and 5% level of significance for both unit root test under constant and trend are and respectively. The null hypothesis is rejected and denotes that all volatility series are trend stationary. ~ Chapter 5 ~ 157

32 TABLE 5.4 Augmented Dickey-Fuller and Philips-Perron Unit Root Test This table displays unit root test of implied volatilities, historical volatility and realized volatility series of S&P CNX Nifty and selected five individual stock options over the period from January 2002 to June The Augmented Dickey-Fuller and the Phillips- Perron unit root test were employed to check the stationary of volatility series. The null hypothesis of Augmented Dickey-fuller and Philips-Perron unit root test under constant and constant & trend is non-stationary. The alternative hypothesis is stationary. Unit root results of volatilities series are exhibited in separate panel; Panel A contains implied call volatility, Panel B contains implied put volatility, Panel C contains historical volatility and Panel D contains realized volatility. b and a denote significant at 5% and 1% level respectively. Augmented Dickey-Fuller Phillips-Perron (PP) Unit (ADF) Root Test Root Test Particular Constant Constant Constant Constant and Trend and Trend PANEL A: IMPLIED CALL VOLATILITY S&P CNX NIFTY b a b a INFOSYS b b a a ITC a a a a RANBAXY a a a a RELIANCE a a a a SBI b a a a PANEL B: IMPLIED PUT VOLATILITY S&P CNX NIFTY b a a a INFOSYS a a a a ITC a a a a RANBAXY b a a a RELIANCE b a a a SBI b a a a PANEL C: HISTORICAL VOLATILITY S&P CNX NIFTY a a a a INFOSYS a a a a ITC a a a a RANBAXY b b a a RELIANCE a a a a SBI a a a a PANEL D: REALIZED VOLATILITY S&P CNX NIFTY a a a a INFOSYS a a a a ITC a a a a RANBAXY a a a a RELIANCE a a a a ~ Chapter 5 ~ 158

33 SBI a a a a ~ Chapter 5 ~ 159

34 5.3.3 OLS estimates Table 5.5 displays the OLS estimate of regression. Realized volatility ( ) is regressed on implied call volatility ( ). The slope co-efficients ( ) of implied call volatility of S&P CNX Nifty, Infosys, ITC, Ranbaxy, Reliance and SBI is , , , , and respectively. The slope co-efficient ( ) of implied call volatility is found to be significant at 1% level. The intercepts ( ) of S&P CNX Nifty, Infosys, ITC, Ranbaxy, Reliance and SBI are , , , , and respectively. The intercept ( ) of SBI is found to be significant at 1% level and remaining intercepts are insignificant. The adjusted R 2 of S&P CNX Nifty, Infosys, ITC, Ranbaxy, Reliance and SBI are 57%, 38%, 21%, 37%, 35% and 11% respectively. If implied call volatility contains information about future realized, then the co-efficient of the implied call volatility ( ) should be non-zero or significant. From the Table 5.5, it is clearly shown that implied call volatility ( ) co-efficients of S&P CNX Nifty, Infosys, ITC, Ranbaxy, Reliance and SBI are significant and contain some information about realized volatility. In addition, the F-test is used to check the overall significance of the regression (5-1). According to the F-statistics, it is found that regression (5-1) of S&P CNX, Infosys, ITC, Ranbaxy, Reliance and SBI is significant at 1% level. This result appears to indicate that implied call volatility contains some ~ Chapter 5 ~ 160

35 information about future realized volatility. ~ Chapter 5 ~ 161

36 ~ Chapter 5 ~ 162

37 TABLE 5.5 Realized Volatility Regressed on Implied Call Volatility Estimator This table presents the OLS estimate of regression for implied call volatility series:... (5-1) Where denotes the realized volatility (annualized standard deviation) of daily log-return over the remaining life of option; denotes the Black- Scholas-Merton implied call volatility for at-the-money options with highest trading volume, measured at time t; denotes error term. The data consist of non-overlapping monthly observation of the S&P CNX Nifty and selected five individual stock options for the period January 2002 to June denotes the intercept, denotes the co-efficient of implied call volatility and the numbers in parentheses are t-statistics. a denotes significant at 1% level. Particular Adj R 2 F-statistics S&P CNX NIFTY INFOSYS ITC RANBAXY RELIANCE SBI a a (0.8646) ( ) a a (1.3022) (7.6214) a a (3.7021) (4.8806) a a (0.7171) (7.0461) a a (0.9259) (7.1635) a a a (2.8493) (3.4589) ~ Chapter 5 ~ 163

38 TABLE 5.6 Realized Volatility Regressed on Implied Put Volatility Estimator This table presents the OLS estimate of regression for implied put volatility series:...(5-2) Where denotes the realized volatility (annualized standard deviation) of daily log-return over the remaining life of option; denotes the Black- Scholas-Merton implied put volatility for at-the-money options with highest trading volume, measured at time t; denotes error term. The data consist of non-overlapping monthly observation of the S&P CNX Nifty and selected five individual stock options for the period January 2002 to June denotes the intercept, denotes the co-efficient of implied put volatility and the numbers in parentheses are t-statistics. b and a denotes significant at 5% and 1% level respectively. Particular Adj R 2 F-statistics S&P CNX NIFTY INFOSYS ITC RANBAXY a a (1.1424) ( ) a a (3.8433) (4.8894) a a (1.0800) (6.0713) a a (1.2851) (5.2609) RELIANCE a a (1.5830) (7.5363) SBI a a b ~ Chapter 5 ~ 164

39 (2.9238) (3.0570) ~ Chapter 5 ~ 165

40 ~ Chapter 5 ~ 166

41 Table 5.6 displays the OLS estimate of regression. Realized volatility ( ) is regressed on implied put volatility ( ). The slope co-efficient ( ) of implied put volatilities of S&P CNX Nifty, Infosys, ITC, Ranbaxy, Reliance and SBI are , , , , and respectively. The slope coefficient ( ) of implied put volatility is found to be significant at 1% level. The intercepts ( ) of S&P CNX Nifty, Infosys, ITC, Ranbaxy, Reliance and SBI are , , , , and respectively. The intercept ( ) of SBI is found to be significant at 1% level, and the remaining intercepts are insignificant. The adjusted R 2 of S&P CNX Nifty, Infosys, ITC, Ranbaxy, Reliance and SBI are 56%, 27%, 39%, 33%, 37% and 7% respectively. If implied put volatility contains information about future realized, then the co-efficient of the implied put volatility ( ) should be nonzero or significant. From the Table 5.6, it is clearly shown that the implied put volatility co-efficients ( ) of S&P CNX Nifty, Infosys, ITC, Ranbaxy, Reliance and SBI are significant and contain some information about realized volatility. In addition, the F-test is used to check the overall significance of the regression (5-2). According to the F-statistics, it is found that regression (5-2) of S&P CNX, Infosys, ITC, Ranbaxy and Reliance is significant at 1% level and SBI is significant at 5% level. This result appears to indicate that implied put volatility contains some information about future realized volatility. ~ Chapter 5 ~ 167

42 ~ Chapter 5 ~ 168

43 TABLE 5.7 Realized Volatility Regressed on Historical Volatility Estimator This table presents the OLS estimate of regression for historical volatility series:... (5-3) Where denotes the realized volatility (annualized standard deviation) of daily log-return over the remaining life of option; denotes historical volatility (as annualized standard deviation) of daily log-return during the period before time t and with same length as that of the realized volatility (i.e. 30 calendar days); denotes error term. The data consist of non-overlapping monthly observation of the S&P CNX Nifty and selected five individual stock options for the period January 2002 to June denotes the intercept, denotes the co-efficient of historical volatility and the numbers in parentheses are t-statistics. a denotes significant at 1% level. Particular Adj R 2 F-statistics S&P CNX NIFTY INFOSYS ITC RANBAXY RELIANCE SBI a a (3.9232) (9.0743) a a a (4.9875) (5.7436) a a a (5.6093) (4.3492) a a a (4.5308) (5.0515) a a a (4.6601) (6.2680) a a a (4.5247) (7.9860) a ~ Chapter 5 ~ 169

44 Table 5.7 displays the OLS estimate of regression. Realized volatility ( ) is regressed on historical volatility ( ). The slope co-efficients ( ) of historical volatility of S&P CNX Nifty, Infosys, ITC, Ranbaxy, Reliance and SBI are , , , , and respectively. The slope co-efficient ( ) of historical volatility is found to be significant at 1% level. The intercepts ( ) of S&P CNX Nifty, Infosys, ITC, Ranbaxy, Reliance and SBI are , , , , and respectively. The intercepts ( ) of S&P CNX Nifty and selected five individual stock are found to be significant at 1% level. The adjusted R 2 of S&P CNX Nifty, Infosys, ITC, Ranbaxy, Reliance and SBI are 46%, 26%, 18%, 23%, 29% and 41% respectively. If historical volatility contains information about future realized, then the co-efficient of the historical volatility ( ) should be non-zero or significant. From the Table 5.7, it is clearly shown that historical volatility coefficients ( ) of S&P CNX Nifty, Infosys, ITC, Ranbaxy, Reliance and SBI are significant and contain some information about realized volatility. In addition, the F-test is used to check the overall significance of the regression (5-3). According to the F-statistics, it is found that regression (5-3) of S&P CNX, Infosys, ITC, Ranbaxy and Reliance and SBI is significant at 1% level. This result appears to indicate that historical volatility contains some information about future realized volatility. ~ Chapter 5 ~ 170

45 TABLE 5.8 Comparison of Adjusted R 2 of Realized Volatility Regressed with Different Volatilities This table reports adjusted R 2 of regression (5-1) (5-2) Where ICV denotes the adjusted R 2 of realized volatility ( (5-3) ) regressed with implied call volatility ( ); IPV denotes the adjusted R 2 of realized volatility ( ) regressed with implied put volatility ( ); HV denotes the adjusted R 2 of realized volatility ( ) regressed with historical volatility ( ). The data consist of non-overlapping monthly observation of the S&P CNX Nifty and selected five individual stock options for the period January 2002 to June Particular ICV IPV HV S&P CNX NIFTY INFOSYS ITC RANBAXY RELIANCE SBI Table 5.8 presents the comparison of adjusted R 2 of realized volatility regressed with implied call volatility (ICA), implied put volatility (IPA) and historical volatility respectively. For S&P CNX Nifty, the adjusted R 2 of regression suggests that 57% and 56% variations in future realized volatility ( ) are explained by implied call volatility ( ) and implied put volatility ( ) respectively, while the historical volatility ( ) ~ Chapter 5 ~ 171