Chapter 4 Level of Volatility in the Indian Stock Market

Transcription

1 Chapter 4 Level of Volatility in the Indian Stock Market Measurement of volatility is an important issue in financial econometrics. The main reason for the prominent role that volatility plays in financial markets is that volatility is associated with risk and risk & returns are the key elements involved in all financial decisions. They are key attributes in investing, option pricing, and risk management. All the asset pricing models (such as Portfolio theory, CAPM, Black Scholes model of option pricing etc.) require volatility as an input. Statistically, it is equivalent to dispersion and if we assume that it is time invariant (as it is generally assumed in classical regression models), it can be measured using standard deviation or variance. However, when we relax this assumption and realize that the variability of asset returns changes with time (sometimes market becomes highly volatile and sometimes it rests in the state of tranquility), then the measurement of volatility becomes little bit involving. The assumption of constant variance of errors term is called homoscedasticity and is an essential condition of traditional models. However this assumption is not consistent with the observed behaviour of the volatility of asset prices and many other economic variables, which is known to show clustering and high persistence. Time series data show certain stylized facts like heteroskedasticity, mean reversion & volatility clustering. The Autoregressive Conditional Heteroskedasticity (ARCH) family of models (Conditional volatility models) introduced by Engle (1982) and generalized by Bollerslev (1986) had the capability to capture these properties of volatility. The great workhorse of applied econometrics is the least squares model. This is natural because applied econometricians are typically called upon to determine how much one variable will change in response to a change in some other variable. Increasingly however, econometricians are being asked to forecast and analyze the size of the errors of the model. In this case the questions are about volatility and the standard tools have become the ARCH/GARCH models. The basic version of the least squares model assumes that, the expected value of all error terms when squared, is the same at any given point. This assumption is called homoskedasticity and it is this assumption that is the focus of 81

2 ARCH/GARCH models. Data in which the variances of the error terms are not equal, in which the error terms may reasonably be expected to be larger for some points or ranges of the data than for others, are said to suffer from heteroskedasticity. In the presence of heteroskedasticity, the regression coefficients for an ordinary least squares regression are still unbiased, but the standard errors and confidence intervals estimated by conventional procedures will be too narrow, giving a false sense of precision. Instead of considering this as a problem to be corrected, ARCH and GARCH models treat heteroskedasticity as a variance to be modeled. As a result, not only are the deficiencies of least squares corrected, but a prediction is created for the variance of each error term. The ARCH and GARCH models have been designed to deal with these set of issues. They have become widespread tools for dealing with time series heteroskedastic models. In this study GARCH model has been used to model the volatility to address the issue of impact of financial derivatives on stock market volatility in India. One of the objectives of the study is: To examine the level of volatility prevailing in the Indian stock market. In order to examine the level of volatility prevailing in the Indian stock market, we test the following null hypothesis: H o1 : there is no volatility in the Indian Stock Market. Daily closing prices of S&P CNX Nifty have been used as a proxy for Indian spot market as Nifty index is a broad based index representing 22 sectors of the economy. The daily closing price series has been converted into logarithmic returns by taking first differences. These closing prices have become stationary after they have been converted into log returns. The graph of the stationary nifty return series has been shown in fig. 1 in Annexure B. The stationarity of the series has also been confirmed using ADF test statistic testing the null of non stationarity. The low p value rejects the null of non stationarity. (result reported in table 1 in Annexure B). The descriptive statistics for nifty have been reported in figure 1 below. The descriptive statistics report that nifty return series is not normally distributed. It is negatively skewed and has excess kurtosis (greater than 3). Further, the Jarque Bera test statistic which assumes normality rejects the null, for p value 82

3 being low. The standard Deviation coefficient which is considered as a traditional measure of volatility has been reported as in fig 1. Fig. 1: Descriptive Statistics for Nifty returns Series: RETNIFTY Sample 4/01/1997 3/31/2010 Observations 3242 Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Standard Deviation as a measure of volatility has certain drawbacks. So we have employed the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model to detect changes in the level of volatility (unconditional variance) of the error terms. As a prior step for estimating GARCH equation, a mean equation needs to be formulated. The mean equation for GARCH (1,1) has been formulated as ARMA (1, 1) model using Box Jenkins methodology. The results for mean equation have been enumerated below: Table 1: ARMA(1,1) model Variable Coefficient Std. Error t-statistic Prob. C AR(1) MA(1) The mean equation has been selected on the basis of AIC criterion. The AIC criterion has been least for ARMA (1,1) model. So, the mean equation has been specified as ARMA (1,1). The coefficients of AR & MA terms are significant at 5% level. The constant term is 83

4 however, insignificant at 5% level. The specification of the mean equation in table 1 has been checked and tested for remaining serial correlation by looking at the correlogram of the standardised residuals with all Q statistics expected not to be significant for mean equation to capture all characteristics of the data. The Ljung-Box Q-statistic test with 12 lags of autocorrelations has been utilized to test if the errors showed any correlation after the equation has been specified. This requires an inspection of the correlogram (autocorrelations and partial autocorrelations) of the standardised residuals & the squared standardised residuals (reported in Annexure B in Table 2.1 & 2.2) The correlogram of the residuals do not reveal any significant correlation between the error terms as Q statistics being insignificant. However, the squared residuals revealed significant correlation among the error terms with all Q statistics being significant as is evident from low p values reported in the last column of the table. These errors have been further tested using ARCH LM test at lag 12 (reported in Annexure B in Table 3) The ARCH LM Test has been used to check for heteroskedasticity testing the null hypothesis of no heteroskedasticity between the error terms. The results of ARCH LM Test have been found to be significant being p value for chi square distribution reported as zero. Thus, sufficient evidence for using GARCH model has been generated. Further to model the variance, GARCH (1, 1) equation has been estimated. GARCH (1,1) equation is given by: (1) The result of the GARCH (1,1) model have been reported in table 2 below. Table 2: GARCH (1, 1) model: Coefficient Std. Error z-statistic Prob. C 8.80E E RESID(-1)^ GARCH(-1) GED PARAMETER

5 To examine the level of volatility prevailing in the Indian stock market, GARCH (1,1) equation has generated the values for different parameters. These parameter values have been found to be significant as p value is zero for the constant, the ARCH term & the GARCH term. The level of volatility in the Indian stock market has been examined using unconditional variance using the formula:... (2) where, α 0 = intercept α 1 = ARCH term = GARCH Thus various values generated using GARCH (1,1) has been put into equation (2) and the level of volatility has been estimated. The result derived is Level of volatility prevailing in the stock market has been found to be: So, we Reject the Null Hypothesis that there is no volatility in the Indian stock market. Our next objective is to examine if volatility is static or has changed over time. To examine if volatility is static the following null hypothesis has been tested: H o2 : there is no change in level of stock market volatility after introduction of derivatives. To examine if the volatility has changed over time, unconditional variance has been estimated. Any change in the unconditional variance of an asset price after derivatives has been detected by augmenting equation ( 1) to include a dummy variable as in equation (3) below. The dummy variable is equal to 0 for all pre-event periods and to 1 afterwards (i.e. from 9 th Nov., 2001).. (3) where D stands for dummy variable. The results for GARCH (1,1) model with a dummy variable has been given in Table 3: 85

6 Table 3: GARCH (1,1) model with a dummy variable Variable Coefficient Std. Error t-statistic Prob. C 3.26E E RESID(-1)^ GARCH(-1) DUMMY -1.80E E GED PARAMETER All the parameter values for GARCH (1,1) have been found to be significant for p values being low. After the parameter values have been estimated using augmented GARCH (1,1) equation, the unconditional variance has been estimated. The unconditional variance after event has been calculated as in equation (4):.(4) A significant positive (negative) coefficient of the dummy variable points to an increase (decrease) in the volatility as a result of futures trading (Samanta & Samanta 2007). The results from table 3 report a dummy variable coefficient to be with a p-value of The value of the coefficient of dummy variable is significant & negative. Hence we Reject the Null Hypothesis that there is no change in level of stock market volatility after introduction of derivatives. The significant & negative coefficient of dummy variable indicates that long term volatility has changed with time and has shown a decline. The level of volatility in the post derivatives period as estimated from equation (4) comes out to be