On Stock Market Asymmetric Volatility and Trading Volume

Introduction

In the context of financial time series modeling, volatility remains to be a crucial issue since it is considered the most paramount characteristic of financial markets. The investment behavior of individuals and enterprises is largely affected by volatility owing to the existence of a direct relationship between volatility and the market uncertainty, see [1]. The risk and uncertainty of a stock market is increased by volatility which consequently is detrimental to the stock market transactions. It is, therefore, important to accurately measure the volatility of stock index returns so as to reduce this uncertainty. Volatility is measured by calculating the variance of the index returns, however, the estimation of volatility is not a perfect undertaking since the time series is auto-regressive and depends on past information, hence, the variance is non-constant (heteroskedastic). That is, stock market volatility is time-varying and exhibits volatility clustering, see [2].

The available literature reveals that modeling the correlation between stock price and its volatility has been the concern of researchers, for instance, in [3], the authors reveal a negative relationship between contemporaneous returns and return volatility. That is, positive (negative) returns are in general associated with downward (upward) changes of conditional volatility -a phenomenon referred to as volatility asymmetry. According to [4], asymmetric volatility is in most cases witnessed when stock markets experience a crash during which a large decline in stock price is associated with a significant increase in market volatility. Several empirical studies find asymmetric volatility to be a crucial factor in the On Stock Market Asymmetric Volatility and Trading Volume 91 understanding of trading volume-return volatility relationship. The asymmetry of volatility effect is largely associated with a greater rise in the volatility following an unexpected price fall compared to a price increase of the same magnitude, see [5,6]. Furthermore, this asymmetry of volatility effects is due to price fluctuations and these changes are in most cases negatively related with volatility changes. In [7], the authors argue that the cause for this asymmetric effect is due to leverage effect and a rise in the information flow following unfavorable news. Moreover, increase in information flow due to unfavorable news leads to relative rise of the rate of information flow across firms which in turn affects the co-variances across stock returns. In terms of the asymmetric issue, "bad (or unfavorable) news" refers to negative returns while during financial crises it refers to information with adverse effects across the integrated stock markets. In [8], the authors report that the effect of asymmetric volatility in the emerging market stock returns was lower compared to the developed stock market returns.

Moreover, another issue addressed in literature is on the relationship between trading volume and stock returns' volatility. In [9], the author investigates the relationship of the price-volume changes and the effect on volatility persistence after adding trade volume to the basic ARCH variance equation. The author reports that negative returns have a lower price-volume change slope than positive returns and also that volatility persistence reduces when trade volume is included in the GARCH variance equation as an exogenous variable. In [10], the authors evaluate the effects of trading volume as a proxy for the arrival of information on stock volatility, and the impact of adding trading volume into the conditional volatility equation on volatility persistence, using the EGARCH and TGARCH models. Their findings show a positive association between trading volume and stock returns, and that trading volume is a poor source of volatility on stock returns when used as a proxy for information flow. However, there is no observed change on volatility persistence when trading volume is added in the conditional variance equation. For the Johannesburg Stock Exchange (JSE) in South Africa, the authors in [11] look at the volume-volatility relationship using EGARGH and Granger causality models, as well as the volatility persistence before and after trade volume is included in the volatility model as an exogenous variable. The study reports a positive and contemporaneous association between trading volume and market volatility and that volatility persistence never died off after the explanatory variable was included in the volatility model. This study finds that most empirical studies in literature have mainly focused on modeling asymmetric volatility in developed markets than in emerging markets and a comparison of the two market situations is inadequate. As a consequence, this study aims at examining the relationship between stock returns and its volatility as well as asymmetric volatility in both developed and developing markets. Moreover, the effect of adding trading volume in the GARCH-type model's conditional variance equation on asymmetric volatility and volatility persistence is investigated.

The remainder of this paper is structured as follows: Section 2 describes the GARCH-type models utilized in the study and Section 3 presents the data, descriptive statistics and the discussion of the study findings. Section 4 concludes the paper.

Methodology

The GARCH model

The GARCH model is a basic conceptual structure given in [12] and it is a generalization of ARCH model. The model possesses some notable characteristics such as their capability to model volatility clustering as well as the ability to give account for the changing variance in time-series

is an independent and identically distributed (i.i.d.)

sequence of random variables with mean zero, i.e.,   0

and unit variance, i.e.,

is then a solution to the equations:

(2)

In equation 2,  is a constant variance corresponding to the long run average, 1  is the first-order ARCH term that broadcasts volatility information from a previous time, and 1  is the first-order GARCH term, which represents fresh information not available at the time of the prior forecast. The magnitudes of 1  and 1  determine the extent of volatility persistence, that is, the closer the sum of 1  and 1  to 1, the more the shocks to volatility do not die off.

The GARCH-M model

In finance, a stock's return may be influenced by its volatility and the GARCH-in-mean model, abbreviated as GARCH-M, is the best way to model this phenomenon. The following is a simple GARCH-M(1, 1) model:

(3) where t t r X , and 1  are the log return series, the mean-corrected log return series and the risk premium parameter, respectively. A positive value of 1 

implies that the stock return is positively correlated with its past volatility.

The GARCH-M model formulated in equation 3implies presence of serial correlations in the return series   t X which are caused by those in volatility process,  .

2 t  Therefore, another reason why stock returns have serial correlations is implied by the occurrence of the risk premium.

The exponential GARCH (EGARCH) model

This model has the ability to capture asymmetric responses of timevarying variance to shocks and leverage effects, that is, a negative relationship between stock returns and volatility shocks. The model ensures that the variance is always positive and utilizes

as the standardized

model is expressed as follows:

where i  is the asymmetric or leverage parameter that gives response to asymmetry. In most empirical cases, the value of i  is expected to be greater than 1, indicating that a negative shock can increase future volatility or uncertainty, whereas a positive shock decreases the effect on future uncertainty. A negative shock in financial market analysis usually means bad news, which leads to a more unpredictable future, whereas a positive shock means good news. As a result, investors, for example, would expect larger stock returns to compensate for the increased risk in their investment. This study employs EGARCH(1, 1) model defined as follows:

Results and Discussion

Empirical data

The data utilized here is the daily stock index and trading volume as reported in the FTSE100, S&P500 and Nairobi Securities Exchange for NSE20 share indices for the period: 1st Jan 2001 to 31st Dec 2017. The daily continuously compounded index returns and trading volume are calculated in terms of logarithmic change as follows:

where t S and

represent the daily closing indices at day t and day , 1  t respectively. Similarly, differenced trading volume,

where

are the trading volumes at day t and day , 1  t respectively. Table 1 presents the basic statistics for the FTSE100, S&P500 and NSE20 indices returns and log volume. The mean is positive and close to zero except for the FTSE100 log volume. A positive mean return shows that investors realized a positive return on the investment. The developed markets' indices (FTSE100 and S&P500) are negatively-skewed compared to the developing market index (NSE20) which has a positive skewness which is an indication that the distributions of both returns and volume are left-skewed and right-skewed for the developed and developing markets, respectively. The distributions of all the series are leptokurtic except for the log volume of the NSE20 which is platykurtic as depicted by the positive kurtosis. Moreover, the skewness and excess kurtosis are different from that of a normal distribution of zero and three which means that the series are not drawn from a normally distributed data. This claim of non-normality is further supported by the JB-statistic which is highly significant at 1% significance level as reported in Table 2. In order to analyze the data for desired results, the data set is differenced once to make it stationary. Moreover, the data set is tested for stationarity and serial correlation as well as for ARCH effects by use of Ljung-Box, Lagrange and ADF tests. The results presented in Table 2 report that the data is stationary, has no serial correlation and has ARCH effects.  which is a measure of volatility persistence, is large in both FTSE100 and S&P500 indices returns than in the NSE20 index returns. The results show that the returns have volatility clustering and clearly the developed markets are characterized by high clustering of volatility compared to low clustering of volatility in the emerging market. This means that shocks to conditional variance takes long to disappear in the developed market indices returns than in the NSE20 index returns. This claim is further confirmed by the value

Descriptive statistics

Empirical findings and discussion

which is close to one and is high in developed markets than in emerging market index returns. Furthermore, the coefficient 1  is a measure of the extent to which the present time volatility shock feeds through into the volatility occurring in the next period. This value is large in the emerging market index returns than in the indices returns of the developed market stock, which means that, in comparison to developed stock market returns' volatility, the volatility of emerging stock market returns is influenced more by previous volatility than by comparable news from the previous period. The parameter estimates of EGARCH(1, 1) model is presented in Table 5. It is evident that the conditional mean, , explains volatility persistence and a high value is reported in the developed market indices returns than in the emerging market index returns. This implies that volatility persists for a long time in developed markets than in emerging market. The asymmetric or leverage parameter 1  is positive and high in the NSE20 index returns compared to the FTSE100 and S&P500

indices returns. In addition, the parameter 1  is positive and significant in all indices returns which means there is non-existence of leverage effects but asymmetric volatility is present among the indices returns and thus the impact of negative news does not outweigh positive news, that is, good news increases volatility more than bad news. It is also noted that the asymmetry parameter 1  is big in NSE20 index returns than in FTSE100 and S&P500

indices returns which shows that volatility asymmetry is more in emerging market than in established markets. This means positive shocks affects volatility more than negative shocks in developing markets compared to developed markets. On the other hand, the ARCH effect coefficient, , 1  is negative except in the NSE20 index returns for student-t distribution which

is an indication that the variance goes up more after negative returns than after positive returns. The results further imply a positive and significant relationship between the stock returns and conditional volatility since the value of 1  is positive and significant. In order to check the effect on volatility persistence in the three market indices after inclusion of lagged trading volume into the GARCH model, the

reported for GARCH(1, 1) model is compared with that of GARCH(1, 1) model with log volume included. As displayed in Table 6, the volatility persistence decreased in both emerging and developed markets even though the normal distribution and student-t distributions reported contrary findings for the FTSE100 and NSE20 indices returns, respectively.  is positive and decreases when trading volume is included into the conditional variance equation for developed markets but increases for the emerging market. This is an indication that indices returns and volatility have a positive association and that the risk of holding asset returns from developed markets is less compared with that of holding asset returns from emerging market.  which is the measure for volatility persistence, increases with volume addition into the equation of conditional variance.

Conclusion

This study investigated the asymmetric volatility in both developed and developing markets in addition to examining the effect of including trading volume into the conditional variance equation on volatility persistence by utilizing data from both developed and emerging markets.

The result of the study reveals that developed markets are described by high volatility clustering and volatility persistence as compared with the emerging markets. Furthermore, volatility asymmetry is high in developed markets than in emerging markets and this is an indication that volatility is increased by positive shocks than by negative shocks in emerging markets than in developed markets. The volatility asymmetry decreases when trading volume is added to the conditional variance equation and this shows trading volume affects the flow of information into the markets. Thus, it should be noted that bad news has more impact on conditional volatility than good news which indeed is a further confirmation that the markets are characterized by asymmetric volatility.

Moreover, when trading volume is added to the conditional variance equation of result reports an increased volatility persistence. Also, the risk parameter 1  is greater than zero and decreases when trading volume is included into the conditional variance equation for developed markets but increases for the emerging market. This is an indication that indices returns and volatility have a positive association and that the risk of holding asset returns from developed markets is less compared with that of holding asset returns from emerging market. Finally, we suggest that this research can be extended in future by using more empirical data set to see whether similar results will be achieved and especially with EGARCH(1, 1) model.