The relationship between volatility , volume and open interest : some evidence from the South African futures market

Using the methodology devised by B~~sem~inder & Seguin, the relationships between volatility on the one hand and volume a_nd market d~pth in the South Afncan futures market are examined. Daily mark-to-market prices, trading volumes and open_int~rest ~,n six fu_tures contracts traded on SAFEX over the period 1990 to 1994 are utilized. The evidence suggests that linking pnce volat1hty to total ~olume does not capture all information. When total volume is divided into expected and u~expected components. the latter 1s show~ to have a more substantial effect on volatility. Furthermore, coefficients pertaining to open as wel~ as unexpected open t~terest tend to be negative, implying that lower volatility shocks are associated ~Ith a given v~lume .1~ deeper markets. It ts further shown that positive unexpected volume shocks are associated with ~tgher levels of volatthty and that asymmetry exists, insofar as positive shocks have larger effects on volatility than negative shocks.


Introduction
It has become conventional wisdom to state that a positive contemporaneous correlation exists between volume traded and price volatility in financial markets.Karpoff ( 1987) referenced 18 separate studies which had documented this relationship in a variety of financial markets.Some of these studies had also documented asymmetry in the relation, in the sense that positive price shocks are associated with larger volumes than negative price shocks.Karpoff ( 1987) argues that there are at least four reasons why the price-volume relationship requires better understanding.First, observed relationships between prices and volume can help discriminate between different hypotheses related to market structure.Second, in event studies the validity of tests can depend on the joint distribution of price changes and volume and third, the price-volume relation is critical in the debate about the distributions of speculative prices.In the fourth place, the nature of the price-volume relationship has significant implications for research into futures markets and the understanding of pricing behaviour in these markets.This research, using the methodology of Bessembinder & Seguin (1993).investigates two dimensions of the relationship between volatility and volume in the South African futures market.First, this article investigates whether the effect of volume on volatility is consistent by separating volume into an expected and unexpected element.Each element is allowed to have a separable effect on the observed price volatility.The effects of volume shocks on the volatility of prices are also examined to ascertain whether volatility responds asymmetrically to volume shocks depending on whether volume is above or below its expected value.
Second, the contribution of market depth is examined.According to Kyle ( 1985) market depth can be defined as the order flow innovation required to move prices by one unit.Market depth is proportional to the amount of noise trading and inversely proportional to the amount of private information not yet impounded into prices.It is hypothesized that depth varies with recent trading activity which is proxied by endogenously determined open interest.With large open interest, it is expected that volatility, conditional on contemporaneous volume, will be lower.
In the analysis empirical methods are employed that explicitly accommodate persistence in the volume-volatility relationship.
The research follows on the work of Smit & Nienaber (1996) who have demonstrated a positive relationship between equity volatility and expected and unexpected trading volumes in both the spot and futures markets.For the Gold and Industrial Indices it has been shown that the unexpected spot an<I futures-trading volumes have larger effects on volatility than does the corresponding expected trading volume.
Underlying the current research are theories that predict a positive contemporaneous relationship between price volatility and trading volume.The mixture of distributions hypothesis assumes that price variability is monotonically related to volume of the transaction.Price changes are sampled from a mixture of normal distributions with either the volume per transaction, number of transactions, or number of information arrivals per observation unit acting as the mixing variable (Clark, 1973;Epps & Epps. 1976;Tauchen & Pitts, 1983;Harris, 1986;and Lamoureax & Lastrapes, 1994 ).
With the sequential arrival of information models new information is distributed sequentially in time.This sequential arrival of new information generates both trading and volume price movements.which increase during periods characterized by frequent information shocks (Copeland, 1976(Copeland, . 1977;;Morse, 1981;Jennings, Starks & Fellingham, 1981;and Jennings & Barry, 1983).Admati & Pfleiderer (1988) show that traders with trade timing discretion choose to trade when recent volume is large.This has the effect that transactions and price movements are bunched in time, and the effect of volume on price movements depends on recent volume levels.Kyle ( 1985), defining depth as the volume of unanticipated order flows required to move prices by one unit, has developed a model which implies that larger volumes support more informed traders.According to this model depth varies with the level of noninformational trading activity.
It is neither the object of this article to choose between the alternative hypotheses, nor to expose the causal _structure between volume shocks and price changes.The pnm~ econometric objective, following Bessembinder & Se~um (1992), is to document partial relationships between pn~e-c~anges and shocks to volume and open interest while cond1t1onmg on the levels of recent activity.

Data and method
Daily mark-to-market prices, trading volumes, and open interest of six South African futures contracts are analysed over the period 2 May 1990 to 31 January 1994.The contracts investigated are those on the 3-month Bankers' Acceptance (LBA3); the Eskom 168 bond (EI68); the Dollar Gold Index (DGLD); the All Share Index (ALSI); the Industrial Share Index (INDI) and the All Gold Index (GLDI).Data were obtained from the South African Futures Exchange (SAFEX).
The percentage change on the daily mark-to-market price is calculated to obtain the daily return.Contracts closest to expiration are used, except within the delivery month in which the contract next closest to expiration is used.
Table I provides details of the contracts studied.Mean contract size and mean daily Rand volume are shown.The mean contract size is calculated by taking the total value of the contracts divided by the number of contracts.The All Gold Index contract (GLDI) has the smallest mean (R 16 063) as against the 3-Month Bankers' Acceptance contract (LBA3) which has the largest mean contract size of R96 I 462.
In calculating the mean daily Rand volume, the total Rand value of the contract for the period is used and divided by the number of days when trading actually took place.Here the All Share Index Contract has the largest mean daily volume (R54.7 million), against the Eskom EI68 contract which has a mean daily volume of R4.2 million and the Dollar Gold Index with a mean value of R2.3 million.
Table 2 provides a summary of the means, standard deviations and partial autocorrelations for returns, absolute returns, volume and open interest for each of the six contracts.The most volatile contract is that on the All Gold Index (GLDI) with a standard deviation of return equal to just more than 2.5% per day.In contrast, the daily standard deviation of the Eskom El68 future is 0.5%.Returns in general are not predictable, except in the case of the LBA3 contract.By analysing the autocorrelations of the absolute value of a time series, the time series properties of the variance of the original time series are determined (Bollerslev,J 988).Except for the GLDI contract all first order partial autocorrelation coefficients of absolute returns are positive and statistically significant, which implies that one day's absolute return is highly correlated with the previous day's absolute return.For higher order lags there is at least one positive and significant partial autocorrelation for each of these five contracts.This finding is similar to that of Bessembinder & Seguin (1993) in their analysis of the US market.In the case of the Gold Index Contract, there is no indication of autocorrelation in absolute returns, indicating a random walk.The fact that significant partial autocorrelations are present at higher lags in the case of the rest of the contracts implies that there is some persistence in volatility.
The most active contract on the basis of the mean open interest is the All Share contract, while the Dollar Gold futures market (DGLD) is least active.It is further clear from Table 2 that volume and open interest are highly autocorrelated.The first order partial autocorrelation coefficients for the volume series vary between 0.063 (LBA3) and 0.811 (ALSI).In the case of open interest, all the first order autocorrelations exceed 0.95.
The last column in Table 2 shows modified five-lag Dickey-Fuller test statistics for the presence of unit roots in the volume and open interest series.This unit root serves as an important first step in partitioning the series into its expected and unexpected components, and examines the s~tionarity of a time series.The existence of a unit root is rejected for all the volume series, but for only two of the open interest series.After taking first differences the existence of a unit root is rejected for all the series.
As can be seen in Table 3, four of the six contracts show statistical significant correlations between returns and absolute returns at the 5% level.Two of these correlation coefficients are significantly negative pointing towards negative skewness in returns.There are significant correlations between absolute returns and volumes in only two cases, which is in contrast to the US evidence provided by Bessembinder & Seguin (1993).They found absolute returns to be highly correlated with trading volumes in each of the eight markets analysed.
To establish the expected and unexpected components in volume and open interest, univariate forecasting methods are used.ARIMA models are fitted assuring that the partial autocorrelations and autocorrelations of the residuals are well within the 95% limits.Seasonal lengths of five days are u~ in all calculations.The expected volume or open interest ts • the given by the model forecast, and the unexpected value ts difference between the actual and the model values.Therefore the sum of the expected and unexpected values equal ~e actual (observed) value.The ARIMA models fitted to obtaI~ the expected and unexpected components for volume an open interest are presented in Table 4.
The method used by Bessembinder & Seguin (1993) is utilized to iterate between a conditional mean and a conditional volatility equation.The conditional mean equation is of the form:  then E(lxl) = ../2/rr)cr.Since x in this case is a vector of OLS residuals, the assumption that the mean of the distribution is zero is not a problem.However. the distributional assumption of conditional normality must be maintained.The presence of skewness or kurtosis could impart a bias in mean absolute deviation-based estimates of volatility.They further state that the effects of changes in higher moments on inferences made using this class of volatility estimate are negligible for equity returns.
In the conventional regression analysis the following assumptions regarding residuals are usually included: -residuals are identically distributed; and -residuals are independently distributed.Financial models do not always satisfy these assumptions and information contained in residuals is lost.The model utilizes the information contained in the residuals by following an iterative procedure in which residuals from a first equation are fed into a second equation.
Fitted values from Equation ( l) estimate conditional expected returns while o, is, assuming conditional normality, an unbiased estimate of the daily return standard deviation.Daily dummies are needed to capture differing mean daily returns, while lagged returns allow for short-term shifts in expected returns.Following Bessembinder & Seguin ( 1992).ten lags are used in the analysis.In Equation (2) conditional standard deviations are estimated by regressing those standard deviation estimates on daily dummies, lagged standard deviation estimates and lagged raw residuals obtained from Equation ( I ).Lagged standard deviation estimates are included to allow for the persistence of volatility shocks.Past unexpected returns are included because past studies indicated that these lags have explanatory power; and including both signed forecast errors and the lagged o., allows for the relation between unexpected return and volatility to vary depending on the sign of the unexpected return.Daily dummies make provision for day-of-the week differences in mean volatilities.All regression analysis is done in TSP using the least squares (LS) method.S.Afr.J.Bus.Manage.1996 27(41 In brief, equations (I) and ( 2) are determined in the following way: Equation (I) is estimated without lagged volatilities.The transformation ~I = Ju,JJrt/2 is then applied tO the residuals, and Equation (2) is estimated.Fitted values of Equation ( 2) are used as regressors in the re-estimation of Equation (I).Lastly Equation ( 2) is re-estimated by using the residuals from Equation Lastly the residuals from the final Equation (I), (U,), are used to obtain the final conditional volatility Equation (2), of which the coefficients are presented in Table 6.
The Second, market depth depends on the willingness and ability of traders to risk capital and position themselves in response to deviatior.sbetween spot price and perceived fair value.Willingness is dependent on the trader's risk aversion and the ability to trade is a function of wealth.If these lags.This provides mixed evidence on the existence of a pos-

Results
itive relationship between rates of return and expected volatility.Estimates of Equations (l) and ( 2) are prepared for each of Table 6 documents the regression results of the conditional the six contracts.The results of the estimation of the convolatility equation.Significant lagged volatilities are found in ditional mean equations are shown in Table 5.
the following cases: ALSI at lag 1; DGLD at lags 4 and 9; The highest R 2 -value is equal to 4.7% in the case of the LBA3 at lags l, 2, 3 and 4; El68 at lags 2, 3, 9 and IO; and LBA3 contract.It is therefore clear that there is very little pre-INDI at lags I. 4 and l 0. dictive power in the model.Day-of-the-week dummies are Generally speaking, the lagged volatilities tend not to be not significant except in the case of the Eskom EI68 contract, where the Monday dummy is significant at the 5% level.Lag-significant seeing that only 14 out of the 60 coefficients are ged returns for Eskom E 168 and the LBA3 contract are sig-significant.However, persistence is clearly present insofar as 50 out of the 60 estimated coefficients are positive and signiftrial Index, two are significant.For the Eskom 168.only one ,cant volatilittes are present in 5 of the 6 contracts studied.coefficient is significant.In general the lagged unexpected re-In analysmg the lagged unexpected returns, the Eskom turns are not significant, only six out of 60 coefficients being E 168 coefficients are negative in seven out of the ten lags, significant, although 36 are negative, the latter implying that while the Industrial Index and the All Share Index each has in most cases the unexpected return has a negative impact on nine negative coefficients and the Gold Index has eight.This volatility.This is in contrast to the findings in the US market means that unexpected return shocks tend to be negatively re- (Bessembinder & Seguin, 1993).lated to conditional volatilities.In the case of the Gold Index, All 12 coefficients for expected and unexpected volumes none of the coefficients are significant.while for the Indusare positive, while ten are significant at the 5% level.All the coefficients are significant at the I 0% level.The coefficients Except for the Dollar Gold contract all the coefficients for for unexpected volume are higher than those for expected expected open interest are negative.Only two of these are s1gvolume.except for the Eskom E 168 and LBA3 contracts.
nificant, namely for the Gold Index and the LBA3 contracts.with the highest ratio between unexpected and expected coef-This is consistent with the joint hypothesis that: (i) the exficients that of the Gold Index contract at 3.4.A one unit pected open interest is related to the number of trades or change in unexpected volume has roughly twice the effect on amount of capital affiliated with the market; (ii) that these volatility than a unit change in expected volume, which is factors enhance market depth; and (iii) there are lower volatilmuch less than in the American market where it is approxiity shocks associated with a given volume in deeper markets mately seven times, according to Bessembinder & Seguin and is supportive of the findings in the US market (Bessem-(1993).
binder & Seguin, 1993).Estimated coefficients relating unexpected open interest to volatility for five of the six contracts are also negative, of which only one, the Eskom 168, is significant at the 5% level.This means that an increase in open interest during the trading day lessens the impact of a volume shock on volatility.This is also supportive of the findings in the US market (Bessembinder & Seguin, 1993).
Table 7 shows the results when unexpected changes in volume and open interest on volatility are allowed to vary with the sign of the shock to investigate the existence of asymmetries in volume and open interest shocks.Dummy variables are defined and set equal to O for a negative shock (activity below expectation) and I for a positive shock (activity above the expected level).Thereafter the product of the indicator variable and unexpected activity series is created.The coefficient associated with the unexpected series represents the marginal impact of a negative volatility shock, while the marginal effect of a positive shock can be determined by adding the coefficients associated with the unexpected series and the cross product.The coefficients for the unexpected volume shocks are all positive.All these coefficients are significant except in the case of the Eskom E168 contract.This means that negative volume shocks are associated with lower levels of volatility.All of the cross-product terms are positive, four of them significantly so.This reinforces the previous finding that positive shocks are associated with higher levels of volatility, and further indicates that positive shocks have a larger effect than negative shocks.Once more these findings support those of Bessembinder & Seguin ( 1993).
In the case of open interest, the coefficients associated with unexpected open interest are all positive, although not significantly so.Three of the coefficients associated with the crossproduct term are positive and three are negative, none of them being significant.Here the results are inconclusive and are not supportive of the Bessembinder & Seguin findings.
According to Kyle ( 1985), market depth can be defined as the order flow necessary to move prices by one unit.Table 8 shows Rand trading volumes required to move prices by I% and the capital required to move prices by the average absolute return.To illustrate, the Rand amount for LBA3 (R769.98 million) when divided by the average value of the contract from Table I, yields a number of contracts (800), and when multiplied by the estimated coefficient, which links volatility to unexpected positive volume shocks per contract from Table 7. (0.00004 + 0.0012) yields 1%.The probability that the Gold Index will move by I% per day is much larger than a similar probability for the LBA3 contract.To accommodate this. the Rand volume needed to move the futures prices by their average absolute return, was calculated.This Rand amount (e.g. for LBA3 = R299.52 million) when divided by the average value of the contract from Table I yields a number of contracts (311) and when multiplied by the estimated coefficient linking volatility to unexpected volume shocks from Table 7 (0.00004+0.00l2), yields the mean of the absolme return series for that asset (0.389 for LBA3).The rankings do not differ following this approach and it is clear that the depth of the LBA3 contract is the highest while that of the Gold Index is the lowest for unexpected positive volume shocks.The rankings also follow the rankings of S.Afr.J .Bus.Manage.1996 2 714 ) comparable contracts in the US market (Bessembinder & Seguin, 1993).

Conclusions
It has been shown that the most volatile future contract is that on the All Gold Index (GLDI), while the Eskom 168 (El68) has proved to be the least volatile contract.The All Share (ALSI) contract is most actively traded.as against the Dollar Gold (DOLD) contract, which has been the least active.
Generally speaking, returns in the South African futures market are not predictable from their past histories.Absolute returns, however, show significant autocorrelations which means there is some persistence in volatility.Volume and open interest are highly autocorrelated, while there appears to be no significant correlation between absolute return and volume as in the US market.
Looking at the conditional mean equation, it is clear that there is little predictive power in the models and that day-ofthe-week effects are not significant.Regarding the lagged volatilities, no clear pattern is visible in the data.
In the conditional volatility equation the volatilities show no clear pattern.The lagged unexpected returns are mostly negative, although in general not significantly so.Expected and unexpected volumes both are positively, and in general, significantly related to volatility and a unit change in unexpected volume has roughly twice the effect on volatility as a unit change in expected volume.
The coefficients pertaining to expected open interest as well as unexpected open interest tend to be negative implying that lower volatility shocks are associated with a given volume in deeper markets.When allowing the relationship between unexpected changes in volume and open interest and volatility to vary with the sign of the shock, it is demonstrated that positive shocks are associated with higher levels of volatility and that positive shocks have larger effects than negative shocks.In the case of open interest, this analysis has proved to be inconclusive.
0 + L m/1,_j + L Tl;d; + L µ;Ak + L P;~r-1 + e,return on day t; the residual from Equation I representing unex-pc1:tcd returns; IU,! lrr/2 is the estimated conditional return standard deviation on dav t: d, = a dummy variable representing days of the week; and A. = an activity variable (volume traded and open interest).Bessemhinder & Seguin (1993:23) state that if x-N(O,cr 2 ) Share returns are regressed over ten lagged returns to estimate the short-term movements in conditional returns.Dummy variables d, representing the day of the week are included to capture differences in mean returns.The residuals from this regression, JO 4 = ~ + "" yR .+ "" p.d. return on day t, where the hat '"' denotes an estimate.The absolute residual IU,I obtained is then multiplied by the factor (1t/2)Y2 to estimate the standard deviation of the futures return in period t.According to Schwert (l 990) this estimator is unbiased if the conditional distribution of returns is normal.Next, the conditional volatility equation is estimated by Equation (2).The trading activity variables, Ak, are the expected and unexpected volume and open interest which have been derived by means of the Box-Jenkins technique as described earlier.Fitted values from Equation (2) are then used for the final reestimation of Equation (I).These fitted values are represented by & in Equation (I).From this estimation process the coefficients in Table 5 are obtained.
effects of trading activity on conditional volatilities are captured by the activity variables, Ak, representing volume traded (expected and unexpected) and open interest (expected and unexpected).The expected portion of open interest variable reflects open interest at the start of a trading day.while the unexpected component captures unanticipated changes in net contract formation.As a result, expected open interest is approximately equal to yesterday's level, while the unexpected component approximately equals the change in open interest during the day.Bessembinder & Seguin (l 993) argue that open interest measures are relevant for two reasons.First, in using open interest together with volume, insights may be gained into the effects of market activity which is generated by informed versus uninformed traders.They argue that many speculators are day traders who dot not hold open positions overnight and that the open interest at close of trading primarily reflects hedging activity which they associate with uninformed trading.

Table 1
Mean contract size and mean daily Rand volumes of the South African futures contracts for the period 2 May 1990-31 January 1994

Table 2
Summary statistics of futures returns, risk and trading volumes (daily observations 2 May 1990-31 January 1994) Asterisks indicate partial autocorrelations coefficients which are significantly larger than zero at the 5% level.Limits are computed in Statgraphics and are equal to 2/'fu.In the Dickey-Fuller column • denotes rejection of the null hypothesis of a unit root at the 5% level.n 4 n R,=a+Ly.R, +'pd.+'nrs,+U, {})

Table 3
Correlation coefficient between returns, absolute returns and volume

Table 4
ARIMA models used for establishing expected and unexpected returns and open interest

Table 5
Model estimates for daily futures returns

Table 6
Model estimates of daily return standard deviation

Table 7
Model estimates of daily return standard deviations on trading activity, allowing for asymmetries

Table 8
Estimates of market depth