The objective of this study was to analyze and model periodic behavior observed in India's Nifty Implied Volatility (VIX) Index and to seek the origins of these previously unreported calendar variations. Implied volatility and its variations are important to understand as the pricing of many financial assets and derivatives depend upon this variable. A novel modeling approach to this task uses Fourier analysis for the first time in the literature of implied volatility and calendar effect modeling. For this purpose, daily closing levels for the Nifty VIX Index were gathered covering trading days from 2010 through January 2014. Time series of VIX levels after removing a trend line were tested for normality, stationarity, and autocorrelation. Nifty index and Nifty VIX data were also tested for two-way Granger causality. Detrended Nifty VIX levels were Fourier analyzed to determine primary Fourier frequencies contributing to VIX periodic behavior. A Fourier model of VIX movements was constructed using four primary frequencies from the Fourier power spectrum. This model showed surprising accuracy in identifying the temporal location of VIX peaks and troughs. The probable origin of one recurring VIX calendar frequency was traced to India's earnings release cycle.

1. Ahoniemi, K. (2006). Modeling and forecasting implied volatility - An econometric analysis of the VIX index (Discussion Paper). Helsinki: Helsinki Center of Economic Research.
2. Ariel, R. A. (1990). High stock returns before holidays: Existence and evidence on possible causes. The Journal of Finance, 45(5), 1611-1626. DOI: 10.1111/j.1540-6261.1990.tb03731.x
3. Bollerslev, T., Chou, R. Y., & Kroner, K. F. (1992). ARCH modeling in finance: A review of the theory and empirical evidence. Journal of Econometrics, 52 (1-2), 5-59. DOI: 10.1016/0304-4076(92)90064-X
4. Bollerslev, T., Engle, R., & Nelson, D. (1994). ARCH models. In R. Engle, & D. McFadden, Handbook of econometrics (pp. 2959-3038, Vol 4). Amsterdam: North Holland Press.
5. Brooks, C., & Oozeer, M. (2002). Modeling the implied volatility of options on long gilt futures. Journal of Business Finance & Accounting, 29 (1 - 2), 111-137. DOI: 10.1111/1468-5957.00426
6. Chandra, A. (2011). Stock market anomalies: A test of calendar effect in the Bombay Stock Exchange. Indian Journal of Finance, 5 (5), 23-31.
7. Choudhary, K. (2008). Calendar anomalies In Indian stock market. Indian Journal of Finance, 1 (5), 3-10.
8. Danielson, G. C., & Lanczos, C. (1942). Some improvements in practical fourier analysis and their application to X-ray scattering from liquids. Journal of the Franklin Institute, 233 (4), 365 - 380. DOI: 10.1016/S0016-0032(42)90767-1
9. French, K. R. (1980). Stock returns and the weekend effect. Journal of Financial Economics,8 (1), 55-69. DOI: 10.1016/0304-405X(80)90021-5
10. Harvey, C., & Whaley, R. E. (1992). Market volatility prediction and the efficiency of the S&P 100 index option market. Journal of Financial Economics, 31(1), 43-73.
11. Haugen, R., & Lakonishok, J. (1988). The incredible January effect. Homewood, IL: Dow Jones-Irwin.
12. Jacobsen, B., & Zhang, C. Y. (2012). The halloween indicator: Everywhere and all the time. DOI : <a href=http://dx.doi.org/10.2139/ssrn.2154873 target="_blank">http://dx.doi.org/10.2139/ssrn.2154873</a>
13. Kumar, G. P., & Saravanan, S. (2013). Prediction of stock option prices using volatility (GARCH(1,1)) adjusted black scholes option pricing model. Indian Journal of Finance, 7(9), 36-43.
14. Lakonishok, J., & Smidt, S. (1988). Are seasonal anomalies real? A ninety-year perspective. Review of Financial Studies, 1 (4), 403-425.
15. Malkiel, B. (2003). The efficient market hypothesis and its critics. Journal of Economic Perspectives, 17(1), 59-82.
16. McMillan, L. (2012, November 6). $VIX Seasonality in election years. Retrieved from <a href=http://www.optionstrategist.com/blog/2012/11/vix-seasonality-election-years target="_blank">http://www.optionstrategist.com/blog/2012/11/vix-seasonality-election-years</a>
17. Mixon, S. (2002). Factors explaining movements in the implied volatility surface. Journal of Futures Markets, 22 (10), 915-937.
18. Narang, S., & Vij, M. (2013). Long-term effects of expiration of derivatives on Indian spot volatility. ISRN Economics, Article 718538. DOI:<a href=http://dx.doi.org/10.1155/2013/718538 target="_blank">http://dx.doi.org/10.1155/2013/718538</a>
19. National Stock Exchange (n.d.). Nifty VIX with linear trend line. Retrieved from <a href=http://www.nseindia.com/products/content/equities/indices/historical_vix.htm target="_blank">http://www.nseindia.com/products/content/equities/indices/historical_vix.htm</a>
20. NYU Stern. (2014, April 17). V-Lab Beta. Retrieved from <a href=http://vlab.stern.nyu.edu/ target="_blank">http://vlab.stern.nyu.edu/</a>
21. Rajan, M. (2011). Volatility estimation in the Indian stock market using heteroscedastic models. Indian Journal of Finance, 5 (6), 26-32.
22. Rogalski, R. J. (1984). New findings regarding day-of-the-week returns over trading and non - trading periods: A note. The Journal of Finance, 39 (5), 1603-1614. DOI: 10.1111/j.1540-6261.1984.tb04927.x
23. Shaikh, I., & Padhi, P. (2014). The forecasting performance of implied volatility index: Evidence from India VIX. Economic Change and Restructuring, May 2014. DOI : 10.1007/s10644-014-9149-z
24. Vipul, V. (2005). Futures and options expiration-day effects: The Indian evidence. The Journal of Futures Markets, 25(11), 1045-1065. DOI: 10.1002/fut.20178
25. Zhang, C. Y., & Jacobsen, B. (2012, September 6). Are monthly seasonals real? A three century perspective. DOI: <a href=http://dx.doi.org/10.2139/ssrn.1697861 target="_blank">http://dx.doi.org/10.2139/ssrn.1697861</a>