INDIA VIX AND FORECASTING ABILITY OF SYMMETRIC AND ASYMMETRIC GARCH MODELS

Maithili S. Naik¹ --- Y.V. Reddy²⁺

¹Assistant Professor in Commerce, Shree Damodar College of Commerce & Economics, Margao-Goa, India.
²Senior Professor, Goa Business School, Goa University, Goa, India.

Keywords: India VIX, Volatility forecasting, Implied volatility, Asymmetric GARCH, Volatility index, Symmetric GARCH.

JEL Classification: G1, G17, G130, C580.

ARTICLE HISTORY: Received: 9 December 2020, Revised: 12 February 2021, Accepted: 8 March 2021, Published: 17 March 2021.

Contribution/Originality: The study contributes to the existing body of literature on forecasting equity market volatility. It offers market participants a model that can assist them in reliably predicting the future direction of NSE’s volatility index (i.e., the India VIX).

1. INTRODUCTION

Uncertainty faced by investors is the starting point for every financial model and investigating the effects of uncertainty on investor sentiments and on share price movements is the aim of every financial model. Uncertainty is the foundation for the very existence of financial economics as a discipline. The study of risk and return trade-offs is critical for all investment decisions as inaccurate volatility forecasts can leave financial institutions deprived of capital for operations and investment. High volatility levels are perceived as indicators of market inefficiency and a potential threat to the very integrity of market mechanisms. The volatility in the stock market affects investment spending and investor confidence. Thus, risk-averse investors may shift their investments to less risky avenues. A volatile stock market may also lose out on important foreign investment. This, in turn, hampers economic progress.

Hedge funds, banks' proprietary and other professional option traders keenly observe the volatility implied by an option's market price as it supplements their research when making vital buying and selling decisions. An overpriced option is indicated by higher levels of implied volatility (IV) and vice versa. Furthermore, movements in implied volatility largely influence the earnings from volatile positions in complex options, such as straddles.

To gauge the market anxiety and to serve as a proxy for overall market uncertainty, the Chicago Board Options Exchange (CBOE) introduced the implied volatility-based Volatility Index (VIX) in 1993. It serves as a reference point for short-term volatility and allows trading of pure volatility. The US VIX was the first volatility index to be introduced and soon became popular as the benchmark for measuring volatility. Researchers globally have been interested in studying the accuracy and quality of estimates provided by the VIX. Giot (2002) attempted to ascertain if very high levels of volatility or the CBOE Nasdaq Volatility Index (VXN) imply oversold markets and could thus be viewed as a short-term or medium-term opportunity to buy stocks. He concluded that investors interested in entering oversold markets should wait until very high levels on these indices are witnessed but they should restrict their strategy strictly to the short term. Becker, Clements, & White (2007) compared VIX forecasts with the forecasts from the GARCH model, stochastic volatility model, realised volatility class model and models following the MIDAS (mixed data sampling) approach. They concluded that the VIX does not provide any additional information necessary for forecasting volatility. Hung, Ni, & Chang (2009) analyzed the S&P 500 VIX information content and range-based volatility by comparing their benefits to the forecasting performance of the GJR-GARCH volatility model. The author adopted a symmetric loss function to assess the efficiency and bias problem.

Of late, volatility forecasting has occupied the attention of academicians and practitioners as it plays a vital role in many areas of security analysis, portfolio management, economic policy formulation and risk evaluation. In the past, historical volatility methods were employed to estimate future volatility, but in recent times, more refined and advanced time series models are being advocated by experts for accurate evaluation of options. Balaban, Bayar, & Faff (2003) analyzed volatility prediction approaches across fourteen different Asian, Scandinavian and European countries along with the UK and US. The results of their research revealed that, based on symmetric error statistics, the volatility forecasts provided by exponential smoothing methods (ESMs) are the most superior and those by ARCH type models are the worst. Majumdar & Banerjee (2004) compared the forecasting ability of econometric models, namely GARCH, EGARCH, GJR-GARCH, APARCH & IGARCH in predicting volatility. They revealed that the asymmetric EGARCH model, which Nelson proposed in 1991, performed the best among the different models that were tested. Degiannakis (2008) employed a fractionally integrated ARMA model with the VIX as the dependent variable, and volatility measures based on interday and intraday forecasts as explanatory variables to investigate whether they provide incremental information in the forecasting of volatility. The results revealed that all the forecasting information was provided by the VIX. They realized volatility as well as conditional volatility does not provide significant predictability information and, therefore, do not provide any added value to the predictions.Fernandes, Medeiros, & Scharth (2014)examined the statistical properties of the CBOE daily market volatility index. The findings revealed that the VIX has a negative relationship with the S&P 500 index returns and it displays a positive relationship with the volume of the S&P 500 index. Credit spread has a negative long-term implication on the VIX. Kownatzki (2016) examines the precision of VIX and implied volatility in general as a proxy for risk. In normal market conditions the VIX overestimates volatility and underestimates it in times of financial crisis. Therefore, he concluded that the VIX cannot be considered a reliable tool for risk management. Miljkovic & SenGupta (2018) proposed a novel K-component type of regression model to analyze the S&P 500 market fluctuations through the VIX. They employed the scholastic Barndorff-Nielsen and Shephard models and their model gave a new dimension to building an indicator of non–Gaussian jumps using a mixture of Gaussian regression analyses for financial volatility data.Allen & Hooper (2018) captured the daily realized volatility of the S&P 500 to understand the casual relationship between the US implied volatility index and the S&P 500. Through the application of the artificial neural network (ANN) model they attempted to forecast the future daily values of the VIX. The results of their study suggest that the ANN method is successful in predicting the future direction of VIX using lagged daily realized volatility and the lagged daily S&P 500 index. They also concluded that causality runs between lagged realized volatility and lagged compounded daily returns. Siriopoulos & Fassas (2019) analyzed the information content of 50 different publically traded implied volatility indices. They examined their forecasting power and the return volatility relation and their findings showed that implied volatility has information about future volatility beyond that contained in past volatility. These findings are consistent across assets. They also found a significant contemporaneous relationship between implied volatility and underlying assets.Saha, Malkiel, & Rinaudo (2019) constructed a model with explanatory variables exogenous to the index and examined the model prediction errors. Their results show that daily changes in the VIX can be explained by market factors and are not manipulated. In the Indian context, Kumar (2010) investigated the behavior of the India VIX. His research concluded that the IVIX reflects all the stylized facts of volatility. He also made a proposition for the introduction of exchange traded volatility derivatives in India as it mirrors most of the empirical regularities. Fernandes et al. (2014)investigated the performance of the India VIX as a refined measure of market volatility compared to traditional approaches and explored the use of the India VIX as an appropriate tool for timely trading and risk management in the market. Their results demonstrated that the India VIX provides better estimates of market volatility compared to conditional volatility models, such as GARCH and EGARCH. Analysis of the timing strategy indicates that when the India VIX increases in markets the large cap equities help to maintain positive returns on investment portfolios, and when it decreases, the mid-cap equities perform the same function. Kownatzki (2016) attempted to determine the forecasting quality of the India VIX and his work revealed that it accurately predicts short term volatility and its prediction power is higher for upward market movements. He concluded that the VIX estimates are precise for low magnitude future price movements.

Though there have been numerous studies on the information content, efficiency and predictive ability of the India VIX, developing an appropriate model for predicting short-term volatility is an unexplored area in the Indian context. This is the first study that aims to evaluate the forecasting performance of symmetric/asymmetric GARCH models for the India VIX. Thus, through this paper an attempt has been made to model and forecast NSE’s implied volatility index (i.e., the India VIX). Over the years, academicians and econometricians have developed numerous models to forecast market volatility and applied them in practice. These range from the extremely simple random walk model to the relatively complex conditional heteroskedastic models of the GARCH family. The GARCH models are regarded to provide superior forecasts on volatility and have amassed a huge following in financial literature. The objective of the current study is to examine the forecasting ability of symmetric and asymmetric GARCH models, and the performance of the basic GARCH, EGARCH and TGARCH models is evaluated in forecasting the India VIX.

2. METHODOLOGY

Data: The daily closing levels of the India VIX are considered from March 2, 2009 to August 31, 2016. The India VIX data was extracted from the NSE India website. For the purpose of the study, the data was split into two parts – the in-sample and the out-sample data.
In-sample data: The period from March 2, 2009 to June 1, 2016 is considered as the in-sample period and consists of 1797 data points. The in-sample data was used to build a model that has a high degree of predictability.
Out-sample data: The period from June 1, 2016 to August 31, 2016 is considered as the out-sample period and consists of 64 data points. The model developed through in-sample data will be tested for forecasting performance on out-sample data. The forecasts provided by the models will be compared with the actual values of the India VIX.

2.1. Methods of Volatility Forecasting

The most commonly applied class of time-varying models are the generalized autoregressive conditional heteroskedasticity (GARCH) models. Ever since their introduction by Engle (1982) and subsequent generalization by Bollerslev (1986) these advance models have been modified, refined and stretched in several ways to make them best suited for exploring the volatility process. ARCH class models help in capturing the volatility clusteringphenomenon in financial time series. Their first two moments -the mean and the variance – define these models. The mean equation describes the behavior of the dependent variable over time. The functional forms of different models have been expressed below.

The ARCH (q) model is expressed as:

Tim Bollerslev (1986) developed the GARCH model. In this model the conditional variance is allowed to depend on its own past lags. The GARCH model is expressed as:

The GARCH(1,1) model can be extended to GARCH(p,q) where the current conditional variance is modeled as a function of q lags of the squared error term and p lags of the conditional variance. GARCH models enforce a symmetric response of volatility to positive and negative shocks, but in financial time series it is argued that a negative shock is likely to have a higher impact on volatility than a positive shock. The threshold GARCH (TGARCH) or the GJR-GARCH and the exponential GARCH (EGARCH) models are the popular extensions of the GARCH models that capture the asymmetries in financial data.

The TGARCH model was developed by Zakoian (1994), and the GJR-GARCH model was named after its developers, Glosten, Jagannathan, & Runkle (1993). The model includes an additional term to capture the asymmetries in data sets and is an extension of the GARCH vanilla model. The conditional variance in this case is modelled as:

 2.2. Forecast Performance Evaluation

An important element of an exhaustive forecasting exercise encompasses comparing the forecast performance of competing models. We evaluated the predictive ability of the competing models using popular evaluation measures used in previous studies – root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE) and Theil’s inequality coefficient. These are termed as symmetric forecast error statistics as they treat over-predictions and under-predictions equally. These are defined as follows:

The evaluation techniques discussed above were employed in this study and the model with the fewest forecast errors was ranked as the best model. The out-sample forecasts given by these three models were compared with the actual India VIX and the one producing the fewest deviations was selected.

3. EMPIRICAL RESULTS AND DISCUSSION

Before developing an appropriate GARCH model it is pertinent to discuss the properties of the India VIX. The descriptive statistics of the daily closing price of the India VIX index for the sample period are reported in Table 1. The mean value of the daily India VIX was found to be 21.37, and the maximum and minimum values using the study period seem to be 56.07 and 11.56, respectively. Figure 1 shows the daily India VIX closing price during the sample period and it seems to have an upward and downward trend. The standard deviation was found to be higher (7.33) as it largely deviated from the mean value of 21.37. The India VIX closing price series is positively skewed (1.68) and the Kurtosis value is more than 3 (6.33), i.e., excess kurtosis, thus they are leptokurtic. Likewise, the non-normalcy of the distribution is confirmed by a highly significant and large Jarque Bera statistic. It was found to be statistically significant at the 1% level, rejecting the null hypothesis that the India VIX closing price series is normally distributed, confirming the non-normal distribution of data series.

Given the time series nature of the data, the initial step in the analysis was to test whether the series is stationary or not. The paper employed the augmented Dickey–Fuller (ADF) unit root test for the daily closing price of the India VIX series and the results are reported in Table 2. The ADF statistics result indicates that the price series is stationary. The ADF test statistics reject the hypothesis of a unit root at the 1% level of significance in the price series, implying that the VIX series is stationary.

To test whether there is an ARCH effect in the Indian VIX price series, the Ljung Box Q2 statistical test was used on squared residual series of the mean model and the results are presented in the Table 3. Q2[1], Q2[4], Q2[12] and Q2[24] represent the Ljung Box Q2 statistics for the squared standardized residuals using 1, 4, 12 and 24 lags, respectively. The Ljung Box Q2 statistics tested the null hypothesis of no ARCH effect against the alternative hypothesis of the existence of an ARCH effect. From the test of all lags, it was observed that the null hypothesis was rejected at the 1% level of significance, indicating the existence of an ARCH effect in the India VIX during the sample period. Furthermore, to test the null hypothesis of no ARCH effect on the India VIX series, the Engle (1982) ARCH-LM test was conducted. The test results are shown in Table 3. The ARCH-LM test statistics were highly significant at the 1% level, confirming the existence of significant ARCH effects on the India VIX data series during the study period. Hence, the results of both the tests confirmed the presence of ARCH effects in the residuals of time series models in the VIX series and hence the results warrant the estimation of GARCH family models.

Figure 2 exhibits the residual series of the India VIX. From the figure, it appears that there are stretches of time where the volatility is relatively high and relatively less, which suggests an apparent volatility clustering, or ARCH effect, in some periods. Statistically, a strong autocorrelation in squared returns is implied by volatility clustering.

After volatility clustering was confirmed with the daily closing price of the India VIX series, stationarity using the ADF test, heteroskedasticity effect using Ljung and Box (1978) Q2 statistics and the ARCH-LM test, the study suggests that the GARCH-type models are capable and deemed fit for modeling the VIX of the Indian market, as it sufficiently captures volatility clustering and heteroskedastic effects. Therefore, the GARCH-type family models were used to examine the predictive ability of the Indian VIX series during the sample period.

The symmetric GARCH(1,1) and asymmetric EGARCH(1,1) and TGARCH(1,1) estimations for the daily closing price of the India VIX are shown in Table 4. The results reveal a strong support for ARCH and GARCH effects in the estimation, which is suggested by the positive and significant ARCH and GARCH terms in conditional variance equations at the 1% level in all estimations. In addition, the results reveal that the equations of conditional variance of the symmetric GARCH(1,1) and asymmetric EGARCH(1,1) and TGARCH(1,1) models were all close to one. This implies that the innovation in conditional variance takes a longer time to die down and is exceedingly persistent in nature. The results are also suggestive of a model that is covariance stationary with long memory and a high degree of persistence in the conditional variance. The α1 and δ1 coefficients appear to be statistically significant in the EGARCH(1,1) model, which confirms the dependency of volatility on its past behavior. The asymmetric coefficient γ1 (0.202547) shows that the VIX exhibits statistically significant asymmetric effects at the 1% level. This indicates that positive shocks (good news) have greater impacts on this VIX than the negative shocks (bad news). This implies that during market upward movement, the impact of volatility is larger compared to market downward movement of the same magnitude, i.e., the volatility caused by positive news is higher than that caused by negative news. This could be attributed to the large impact on volatility caused by the activity of noise traders in the Indian equity market. The TGARCH(1,1) model results also support the finding that the impact of positive news is higher compared to that of negative news as the parameter estimate δ (-0.214376), which captures the asymmetric effect, is less than zero.

The robustness of the estimated mean equations of the GARCH-type models and the presence of the remaining ARCH effects in the standardized residuals was tested with the Lagrange multiplier (ARCH-LM) test. The ARCH-LM(1,1) test results revealed no ARCH effects in the standardized residuals of variance equation of the model, thus indicating that the mean and the variance equations of the GARCH models are appropriately defined.

The Akaike Information Criteria (AIC) and Schwarz Information Criteria (SIC) were employed for the selection of the most fitted GARCH model for predicting the India VIX price series. Overall, the GARCH(1,1) was the most preferred based on the minimum AIC and SIC. Despite the presence of an asymmetric response to news, as suggested by the minimum AIC and SIC, the GARCH(1,1) produced an unbiased estimate leading to consistent and precise inferences pertaining to the modeling and forecasting of the India VIX price series.

The forecasting ability of the models in predicting the future VIX series was evaluated. The standard (symmetric) loss functions, such as the root mean square error (RMSE), the mean absolute error (MAE), the mean absolute percentage error (MAPE) and Theil’s inequality coefficient were employed to evaluate the performance of the competing models. In Table 5, the results of the out-sample forecast performance of the estimated models are shown for the last 64 observations from June 1, 2016 to August 31, 2016. The model displaying the fewest error measurement values is considered to be the best one and the results show that the GARCH(1,1) model is the most fitted and provides superior forecasts of the India VIX price series compared to other models. The TGARCH(1,1) model is considered second best in forecasting the India VIX price series. To summarize, the empirical results revealed that, despite a leverage effect being present in the data series, the performance of the symmetric GARCH model in forecasting the India VIX price series is better than the asymmetric GARCH models.

4. CONCLUSION

Volatility forecasting plays an important role in investment decision, security valuation, risk management and monetary policy formulation. There is extensive literature available on volatility forecasting for implied volatility indices of developed markets, but very limited work has been done in terms of emerging economies. Moreover, Kumar (2010) and Shaikh & Padhi (2015) examined the behavior and forecasting efficiency of the India volatility index, respectively, but this is the first study that attempts to test the predictive ability of the symmetric GARCH(1,1) and asymmetric TGARCH(1,1) and EGARCH(1,1) models in forecasting the India VIX. The estimated results from the models confirmed the dependency of volatility on its past behavior and revealed that innovation to the conditional variance is highly persistent and takes longer to dissipate.

The asymmetric coefficients of the EGARCH(1,1) model show that at that 1% level of significance the VIX series displays statistically significant asymmetric effects. According to the results, positive innovation (good news) has a greater impact on the India VIX than negative innovation (bad news). This implies that the volatility is larger during market upward movement as compared to market downward movement of the same magnitude. This may be attributed to  the activity of noise traders in the Indian equity market.

The results of the TGARCH(1,1) model confirm that positive shocks have a greater impact than negative shocks, as suggested by the asymmetric coefficient. The models were evaluated in terms of their forecasting ability of future volatility by employing standard (symmetric) loss functions. The results showed that the GARCH(1,1) model gives a superior performance in forecasting the Indian VIX price series compared to the other models.

Finally, Table 6  shows the comparison of actual VIX along with the forecasted values of the Indian VIX, which were predicted through the GARCH(1,1), EGARCH(1,1) and TGARCH models for the last 64 observations from June 1, 2016 to August 31, 2016. The results reveal that forecasted series of VIX from the GARCH(1,1) model has the lesser deviation from the realized VIX.

REFERENCES

Allen, D. E., & Hooper, V. (2018). Generalized correlation measures of causality and forecasts of the VIX using non-linear models. Sustainability, 10(8), 2695.Available at: https://doi.org/10.3390/su10082695.

Balaban, E., Bayar, A., & Faff, R. (2003). Forecasting stock market volatility: Evidence from fourteen countries. Retrieved from: http://dx.doi.org/10.2139/ssrn.339520 .

Becker, R., Clements, A. E., & White, S. I. (2007). Does implied volatility provide any information beyond that captured in model-based volatility forecasts? Journal of Banking & Finance, 31(8), 2535-2549.Available at: https://doi.org/10.1016/j.jbankfin.2006.11.013.

Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307-327.Available at: https://doi.org/10.1016/0304-4076(86)90063-1.

Degianakis, S. A. (2008). Forecasting VIX. Journal of Money, Investment and Banking, 4, 5-19.

Engle, R. F. (1982). Autoregressive conditional Heteroskedasticity with estimates of the variance of United Kingbom inflation. Econometrica, 50(4), 987-1008.

Fernandes, M., Medeiros, M. C., & Scharth, M. (2014). Modeling and predicting the CBOE market volatility index. Journal of Banking & Finance, 40, 1-10.Available at: https://doi.org/10.1016/j.jbankfin.2013.11.004.

Giot, P. (2002). Implied volatility indices as leading indicators of stock index returns? CORE Discussion Papers 2003027.

Glosten, l. R., Jagannathan, R., & Runkle, D. E. (1993). On the relation between the expected value and the volatility of the nominal excess return on stock. The Journal of Finance, 48(5), 1779-1801.Available at: https://doi.org/10.1111/j.1540-6261.1993.tb05128.x.

Hung, J. C., Ni, R. X., & Chang, M. C. (2009). The information contents of VIX index and range-based volatility on volatility forecasting performance of S &P 500. Economics Bulletin, 29(4), 2592-2604.

Kownatzki, C. (2016). How good is the VIX as a predictor of market risk? Journal of Accounting and Finance, 16(6), 39-60.Available at: 10.13140/RG.2.1.2486.4887.

Kumar, S. (2010). The behaviour of India’s volatility index. Indore Management Journal, 2(2), 27-33.

Ljung, G. M., & Box, G. E. (1978). On a measure of fit in time series model. Biometrika, 65(2), 297-303.Available at: https://doi.org/10.1093/biomet/65.2.297.

Miljkovic, T., & SenGupta, I. (2018). A new analysis of VIX using mixture of regressions: Examination and short-term forecasting for the S & P 500 market. High Frequency, 1(1), 53-65.Available at: https://doi.org/10.1002/hf2.10009.

Majumdar, U., & Banerjee, A. (2004). VIX forecasting. Social Science Research Network e-library, Working Paper id:533583.

Saha, A., Malkiel, B. G., & Rinaudo, A. (2019). Has the VIX index been manipulated? Journal of Asset Management, 20(1), 1-14.Available at: https://doi.org/10.1057/s41260-018-00102-4.

Shaikh, I., & Padhi, P. (2015). The implied volatility index: Is ‘investor fear gauge’or ‘forward-looking’? Borsa Istanbul Review, 15(1), 44-52.Available at: https://doi.org/10.1016/j.bir.2014.10.001.

Siriopoulos, C., & Fassas, A. P. (2019). Implied volatility indices – a review. SSRN Online Library. Retrieved from https://ssrn.com/abstract=1421202.

Zakoian, J.-M. (1994). Threshold heteroskedastic models. Journal of Economic Dynamics and Control, 18(5), 931-955.Available at: https://doi.org/10.1016/0165-1889(94)90039-6.