The Behavior of Option’s Implied Volatility Index: a Case of India VIX

The aim of this paper is to investigate the behavior of implied volatility in the form of day-of-the-week, year-of-the-month and surround the expiration of options. The persistence of volatility is modeled in ARCH/GARCH type framework. The empirical results have shown significant effects of the day-of-the-week, month-of-the-year and day of options expiration. The positive significant Monday effect explains that India VIX rises significantly on the initial days of the market opening, and the significant negative Wednesday effect shows that expected stock market volatility fall through Wednesday-Friday. Moreover, the study reveals the fact on options expiration, the evidence shows that India VIX fall significantly on the day of expiration of European call and put options. The March and December months have reported significant negative impact on the volatility index. Certainly, this kind of results holds practical implication for volatility traders, and helps to the market participant in hedging and pricing of options.


Introduction
Implied volatility is the inversion of the BlackScholes op tion pricing model, and it is the function of options' tra ded price; timetoexpiration; riskfreerateofinterest and dividend yield; strike price and spot price of underlying. Under the rational expectation and market efficiency, im plied volatility is the expectation of the future stock mar ket volatility. The market participant trade into options to hedge the market holdings and risk management, hence the expectation of the investors gauged into the price of options (call/put) and the same traded price used to calculate the implied volatility. Therefore, implied volatility is the best estimate of future realized return volatility (Christensen, Prabhala 1998;Hansen 2001;Shaikh, Padhi 2013, 2014a, 2014b for 30 days horizon (one month option).
The information content of implied volatility as the market's expected volatility has motivated to cons truct the volatility index, which is often referred as the the next 30 calendar days. India VIX (2007) uses the same methodology as developed by CBOE for VIX (2003) metho dology. The market participant, analyst and academician have been intrigued by the volatility index; the reason is that stock indices and implied volatility indices are negatively correla ted. The high level of implied volatility index signifies towards oversold market condition. The correlation between stock index and volatility index hovers in the rage of -0.70 to -0.90.
Unlike the previous studies on the seasonal anomalies in term of stock returns, exchange rate and fixed income securities, the aim of our study is to analyze the behavior of implied volatility in the emerging market like India. The seasonality of India VIX (herein after IVIX) has been asses sed like dayoftheweek, monthoftheyear and options expiration effects.
There are quite good number of attempts (e.g. Dzikevičius, Stabužytė 2012;MartinkuteKauliene 2013;Shaikh, Padhi 2013;Padhi, Shaikh 2014) that deals with the market efficiency, sensitivity of options and forecasting, investor sentiment and information content of option pri ces. However, we do not find any study that deals with the option volatility and stylized nature of implied volatility on Indian derivatives market; hence, this is an attempt in this direction to fillup the gap. Moreover, recent studies (e.g. Žilinskij, Rutkauskas 2012;Lukaševičius et al. 2013;Evrim Mandaci et al. 2013;Stádník 2013;Vilkancas 2014;Shaikh, Padhi 2014a, 2014b, 2014c deal on the firm's performance and potential return on investment; dynamics of stock price cycle; determinants of stock market dynamics in advanced and emerging economics; random walk and stock prices; portfolio optimization with respect to omega function; volatility index and forecasting performance of emerging market's volatility index. The studies are in association with the various issues on the stock market development and portfolio optimization, our study identifies the gap on the emerging market volatility index in terms of behavior of volatility index as the expected volatility of the future stock market realized volatility. The formal studies of seasonal anomalies are accessi ble in the works (e.g. Schwert 1989Schwert , 1990French 1980;Gibbons, Hess 1981;Keim, Stambaugh 1984;Fleming et al. 1995). These studies well document the seasonality of stock index returns and conclude the presence of seaso nal anomalies. Some of the earlier studies (e.g. Cross 1973;Jaffe, Westerfield 1985;Aggarwal, Rivoli 1989;Lakonishok, Levi 1982;Balaban et al. 2001) analyze the dayoftheweek effects and their empirical results has shown Monday and Friday effects. Particularly, these studies report significant negative returns on Monday, and Friday returns remain highest as compare to other days.
The month of the year anomalies found in the works (e.g. Rozeff, Kinney 1976;Gultekin, M. N., Gultekin, N. B. 1983;Keim 1983;Lakonishok, Smidt 1984;Jones et al. 1987;Ariel 1987;Tong 1992;Pandey 2002), the January effect happens due to several reasons indentified like, it occurs due to tax motivated transaction, market participants intends to redu ce their tax expenses by closing their bad positions, returns realized on small and large firms. Moreover, the literature evidences on the dayoftheweek and options expiration effects are come up in the studies (e.g. Fleming et al. 1995;Dowling, Muthuswamy 2005, Frijns et al. 2010. More re cently, Fleming et al. (1995) describes how implied volatility index has been calculated, moreover they explain the beha viour of implied volatility over seven years of period in the form of dayofweek and on the options expiration. Their study strongly suggests the presence of seasonality and inter temporal relation between implied volatility and stock index returns. In particular, they find an inverse and asymmetric re lation among future stock market volatility and stock returns. Dowling and Muthuswamy (2005) examine the properties of Australian implied volatility index (AVIX) in the form of seasonality and the information content of AVIX as the predictor of future volatility. They find strong seasonal ano malies and contemporaneous asymmetric relation between AVIX and stock returns. Similarly, Frijns et al. (2010) revisits the study of Dowling and Muthuswamy (2005) and supports the previous work for the more recent period from 2002 to 2006. A great amount of literature we have been explored in the previous paragraph but very limited studies are based on the seasonal anomalies of implied volatility index, hence, this study is an attempt in this direction.
The aim of this paper is to explain the seasonal ano malies in the form of implied volatility index (India VIX). The statistical properties have shown the presence of sea sonal pattern like dayoftheweek and options expiration effects. Moreover, month of the year effects is also appa rent. The empirical model has been framed in the form of simple OLS and ARCH/GARCH type framework. The autocorrelation and ARCHtest reports significant presence of autocorrelation and heteroscedasticity in the residuals, consequently the regression models have been estimated using ARGARCH (1,1) specification. The empirical results show significant positive Monday effect on the expected market volatility, and significant negative impact of the day of the options expiration. In additions, there are some strong evidences on monthoftheyear effect like March, May and December effects.
The rest of the paper proceed as: Section 1 deals with the data and methodology and empirical model, Section 2 reports the empirical results and last section ends with our conclusions.

Data sources and empirical model
Our data sources consist of daily close of India VIX re trieved from the National Stock Exchange of India (NSE) website. The data points rages from November 1, 2007 to April 30, 2013, that has resulted into 1361 trading days. The returns on the volatility index has been obtained as one day contemporaneous continuously compounded logarithmic To isolate the effects of dayoftheweek, expiration of the options and monthoftheyear on expected stock mar ket volatility, a dummy ordinary least squares (DOLS) by allowing an AR term (i.e. ARDOLS) has been structured as follows.
The regression model based on the dayoftheweek anomalies is expressed: where: it D = 1, if Monday; = 0, otherwise i = 1: Monday, Tuesday, Wednesday, Thursday and Friday, and the behaviour of expected stock market volatility surround the cycle of one month options expira tion is written, where: jt D = 1, the day of option expiration (i.e. last Thursday of the month) = 0, otherwise j = -2, -1, 0, 1, 2. By combining the Eqs (1) and (2), Apart from the Eq. (3) we also develop the regression model that account for the monthoftheyear effects on the implied volatility, where: kt D = 1, January; = 0, otherwise k = January, February, ..., December. The Eqs (3) and (4) have been estimated by taking into accounts the problem of autocorrelation and heterosce dasticity. The Lagrange's Multiplier LMtest has shown significant presence of heteroscedasticity, hence the esti mation framework is expressed in ARCH/GARCH. The ARGARCH (1,1) model is structured as, where: 1 θ and 2 θ > 0 and 1 θ + 2 θ ≤ 1. The literature explains that the assumption of Gaussian (Normal) distribution for the errors may not be appropriate for the GARCH model. Thus, for GARCH models Generalized Error Distribution (GED) has been assumed.

Hypotheses of the models:
(i) Dayoftheweek effects: if dayoftheweek anomalies holds in the Indian capital market than the slopes i δ should be statistically different from zero. Previous studies on stock returns anomalies found out significant negative Monday effect, hence in our model (the volatility index) the slope of Monday should appear positive and statistically significant.
(ii) Optionsexpirationeffects: Fleming et al. (1995) analyzes the behavior of expected market volatility surround options expiration. It is expected that on the day of options expiration, market positions are cleared and ambiguity re garding the market also get resolved, and investors become certain about the market condition. Hence, the slope j ϕ on the day of options expiration should appear negative and statistically significant.
(iii) Monthoftheyear effects: if India VIX behaves systematically and hold an important pattern based on the months, than the slopes k π should be different from zero and remain significant. Generally speaking at the end of the year investors clear their market positions, hence it is expected that December and January months should have significant impact on the expected stock market volatility. Moreover, the corporate results are due in the month of July (Q1), October (Q2), January (Q3) and April (Q4), hence the ambiguity of corporate performance get resolved in these months, con sequently the slope for these months should appear negative as on these months VIX reaches its normal level.

Statistical properties of India VIX
In this section some of the statistical properties of implied volatility index have been presented based on the seaso nality and options expiration cycle. Table 1 reports the summary statistics of daily closing of India VIX and its returns. The summary statistics shows the mean, maximum, minimum and standard deviation of the India VIX. In additions, the measures of autocor relation and autoregressive conditional heteroscedasticity are calculated up to three lags. The analysis has been repor ted for full sample, calendar year wise and normal period. Now starting with the Panel A, the average close of IVIX for the whole sample period found to be 27.74%, while for the normal period it is calculated 21.11%. The average range of IVIX for the calendar years appears between 21.82% (2010) to 39.34% (2008). The measure of central tendency speaks that implied volatility was remain quite normal during the year 2010-13. Generally it is believed that the reading of VIX between 15 to 30% is good for the market performan ce of the future realized volatility. The average close of the years 2008-09 violates the normal range of VIX, it is due to the extreme nature of stock market happened on the counter part of global financial crises took place during 2008-09. The maximum and minimum values of IVIX to be observed 85.13% and 13.04% for the full sample, while for normal period it is 37.19% and 13.04%. The standard deviation of the entire sample (11.14) is more than the normal (4.70) period, and the range of volatility of volati lity for the calendar years appears between 4.20 (2012) to 11.43 (2008). The empirical results are reported for the entire sample and low volatility (normal) period based on these statistical properties.
Panel B of the Table 1 shows the statistical measures of returns (IVIX). The stock index and volatility index are ne gatively correlated, hence the average returns on VIX expec ted to be negative. The means score of India VIX returns for full and normal period appears respectively -0.0621% and 0-0.560%. The average return of the calendar years ranges between -0.2510% (2009) to 0.2250% (2008). The volatility of the returns is found to be 7.39% for the entire data points and 5.16% for the low volatility period. Once again, the vo latility of the volatility index remains higher for the calendar years 2008 and 2009, which appears more than 10%. There are some evidences of autocorrelation and heteroscedastici ty in the returns of India VIX, hence in the empirical model an AR term has been added to resolve the autocorrelation problem, and the resulted residuals are modelled in ARCH/ GARCH framework to control the heteroscedasticity. Table 2 summarizes the behavior of India VIX based on the dayoftheweek and options expiration. The average close of the India VIX on Monday (28.18%) is remain higher as compare to other days, and the corresponding IVIX re turn on Monday also observed to be positive (2.06%), that implies implied volatility becomes more volatile on the mar ket opening (see Fig. 1). The standard deviation of the IVIX close and returns appears respectively 11.76 and 8.48, which is higher than the other day's volatility. The descriptive sta tistic on the options expiration and surround the expiration cycle (i.e. Thursday, the last week of the respective months) the average close and returns to be recorded respectively 27.47% and -2.76%, this numbers explains that India VIX on the day of options expiration remains more normal and falls significantly. The patterns surround the options expi ration show that VIX increases prior to the expiration and keeps on falling after the scheduled expiration. The volatility of the volatility also confirms these patterns. Table 3 explains the changes in the India VIX based on the monthoftheyear (also see Fig. 1). The Panel A and B shows the descriptive statistics for VIX close and correspon ding returns. The average highest VIX close calculated for  the month of May and November are respectively 29.94% and 31.02%, and the average corresponding returns found to be respectively 0.36% and 0.04%. This signifies that for the whole sample the May and November months appears to be more volatile for the investors. The SD's of the respective months also appear to be high as compare to other mont hs. More particularly, the months July (Q1), October (Q2), January (Q3) and April (Q4) are the months in which the quarterly corporate results are scheduled to be announced, we can observe that the average return on VIX recorded negative for the April and July. This signifies that VIX index is an efficient investor's fear gauge index that reflects the corporate announcements. On the other hand, we record positive returns on the India VIX for other months, plau sible reason could be, more uncertainty about the future corporate results and other domestic economic factors.

Empirical results and discussion
This section presents the empirical results obtained on the behavior of India VIX in the form of dayoftheweek, options expiration and monthoftheyear effects. The em pirical results are presented using Eqs (3) and (4) based on AR-DOLS and AR-GARCH framework. The results are organized for entire data points and normal/low volatile period. Table 4 reports the AR-DOLS/AR-GARCH estimates for the seasonal anomalies of India VIX, in which we test the dayoftheweek and options expiration effects. The results are presented in four columns, the column (1) and (2) show the output on AR-DOLS and AR-GARCH for the full sample and column (3) and (4) for low volatile normal period. The column (1) shows significant positi ve Monday effect and negative options expiration effect. The slope of Monday effect appears 0.0235 (2.35%) and remains statistically significant at 1% level of significan ce. We do not find significant changes on India VIX on other trading days. When we fit AR-GARCH (1,1) the slope of Monday calculated positive 2.44%, this signifies that on the market opening the IVIX increases by 2.44%, hence investors can plan their profitable strategy through volatility trading. The practical implication of this phe nomenon explains that (i) a rise in the VIX level implies the "fear" among the investors on Monday, hence the op tions seller can make profit by selling the options at high rate of premium (ii) the market uncertainty experienced by the investors in holidays (i.e. SAT-SUN) get reflected on the Monday, and due to market uncertainty investors bid higher premium for the call/put options, ultimately it results into rises of volatility of Black-Scholes model.
The AR-GARCH (1,1) for Tuesday shows significant negative impact on India VIX, the estimated slope (-0.129) explain that after Monday, once the market continuous its business VIX keeps on falling as the market uncertainty re solved , and keeps on falling through Tuesday-Wednesday-Friday, but not significantly. At this point we can conclude that India VIX hold dayoftheweek effects pattern unlike previous studies have shown Monday significant negati ve effect on stock indices. More particularly, the markets' expected volatility rises significantly on the market opening and falls significantly on the other days. The similar kinds of results are also obtained for the normal period as shown in columns (3) and (4). Table 4 also reports the behaviour of expected stock market volatility surround the expiration of one-month European options, which are cash, settled. Generally it is believed that market participant buys options to hedge their market holding, market players buy one month options to protect their portfolio in near term. The European options can be exercised only on the day of scheduled expiration; hence, investor exercises their right to buy/sell of the un derlying if the trade is profitable. The estimate of the slope 0 ϕ = -0.032, which is negative and statistically significant at 1% level. The result signifies that on the day of options expiration India VIX falls significantly by 3.2%. This hap pen due to positions cleared by the investors and they take new market positions on the next trading day. The slope of AR-GARCH (1,1) also appears statistically significant, and the results on the normal period are also identical with the full sample. We do not find any significant movement in the India VIX before and after, from the day of options expira tion. The AR-GARCH parameters appears highly statically significant that implies that volatility persist in the returns of expected stock market volatility index. The LB-Q(12) statistic speaks that results are not suffering from autocor relation. Finally, at this stage we can conclude that India VIX holds some seasonal anomalies like dayoftheweek and options expiration effects. This kind of predictive elements can helpful to the volatility traders for the risk management and profitable trade from the trading of options. Where jt D = 1, the day of option expiration, otherwise zero; j = -2, -1, -0, 1, 2. The value with bold letter signifies statistically significant at 1%, 5% and 10% level of significance. The LB-Q(12) and LB-Q 2 (12) explain that residual are free from autocorrelation. Where VIX Table 5 shows the results on monthoftheyear anoma lies, the evidences show significant March, May, June and December effects. The column (1) reports the AR-DOLS re sults in which only December months appears with negative significant slope. The column (2) reveals the slope of March -0.0085 (0.85%) and for the December it is -0.0087 (0.87%), both the slopes are statistically significant at 5% level. While the slope of May appear with positive value 0.0092(0.92%), and significant at 5% level. These are the prima facie evi dences of seasonal anomalies in the form of monthofthe year effects on the expected stock market volatility. One can explain that India VIX rises significantly in the month of May and fall significantly during March and December.
Generally, the market participants are uncertain about the corporate earning declared quarterly (i.e. during July (Q1), October (Q2), January (Q3) and April (Q4)). Hence, the slope of the months (July, October, January and April) should appear negative and for the rest of the months it should be positive. The slopes with negative sign are April, July and October (for normal period) but not statistically significant, this indicates corporate scheduled announce ments determine the expected level of implied volatility. In particular, the slope with negative sign are (March, April, June, September, October, November, and December) but only March and December appear statistically significant, this implies before declaration of corporate earnings, results for the quarter 3 and 4 matter for the investment decisi ons. The market participants take into account the Q3 and Q4 results in their assets valuation that is reflected in the expected market volatility. The rest of the months (January, February, May, July and August) appears with positive slo pes, only May shows significant positive impact on IVIX, the plausible reason could be investors remain more uncer tain about their investment during the May month, hence they buy more and more hedge funds (options) to protect their portfolio, consequently VIX level increases. We do not find any significant results for the low volatility period only except to the October month.
Finally, the empirical results have supported the presence of seasonal anomalies in the expected stock market volatility. , where: kt D = 1, January, otherwise zero; k = January, February, ..., December.
The value with bold letter signifies statistically significant at 1%, 5% and 10% level of significance. The LB-Q(12) and LB-Q 2 (12) explain that residual are free from autocorrelation. Where: Unlike the previous studies, volatility index also holds the seasonal component in the form of dayoftheweek, options expiration and monthoftheyear effects. Our empirical evi dences have shown significant impact of seasonal anomalies on the India VIX. This kind of predictive pattern can help to the volatility traders, policy makers and financial institutions for potential investment and financing decisions.

Conclusions
This study demonstrates the seasonal anomalies of the emerging market's volatility index in the form of dayof theweek, options expiration and monthoftheyear effects based on India VIX. To the best of our knowledge, this is the first attempt in the emerging markets like India that ana lyzes the behavior of volatility index based on seasonality.
The results have been presented based on simple dummy OLS and conditional volatility GARCH framework.
The important finding of the study has shown signifi cant positive Monday effect on the expected stock market volatility. The average VIX close of the Monday is recorded 28.18% with positive return 2.06%. The slope of the Monday appears positive 2.44%, which signifies on the initial market opening VIX rises significantly by 2.44%, and it fall signi ficantly on Wednesday. Unlike the previous studies, India VIX also shows the positive Monday effects. Moreover, our findings reports significant negative impact of the day of options expiration, the India VIX falls by 2.64% on the Thursday (the last week of the month). Most interesting evidence on the monthoftheyear effect reveals that March and December have significant negative impact on the India VIX, while the month May reports positive impact. There are some evidences of the effects of quarterly announcement of corporate earnings on the India VIX. The practical im plications of the empirical evidence are definitely helpful to the volatility traders who trade in the options. The seasonal anomalies of the India VIX provide an insight for the pricing of future options. We strongly believe that the India VIX is the gauge of the investors' expectation about the future market volatility, hence the NSE can introduce some more volatility products like futures and options on India VIX and this will allow more liquidity in the derivative market.