Intraday relationships between volatility in S&P 500 futures prices and volatility in the S&P 500 index

Journal

of Banking

and

Finance

14 (1990) 373-397.

North-Holland

INTRADAY RELATIONSHIPS BETWEEN VOLATILITY IN S&P 500 FUTURES PRICES AND VOLATILITY IN THE S&P 500 INDEX*

Ira G. KAWALLER Chicago

Mercantile

Exchange,

Chicago,

IL,

USA

Paul D. KOCH Unicersity

of Kansas,

Lawrence,

KS 66045,

USA

Timothy W. KOCH Unioersity

of South

Carolina,

Columbia,

SC 29208,

USA

This paper examines the intraday relationships between the volatility of S&P 500 futures prices and the volatility of the S&P 500 index. We calculate variance measures for minute-to-minute price changes on a daily basis and across 30-minute intervals. The empirical results indicate that: (i) futures volatility is greater than index volatility, (ii) volatility increased for both futures prices and the index in absolute terms from 1984 through 1986, (iii) both futures and index volatility increased directly with futures trading volume, and (iv) index volatility was systematically greater during the lirst 30 minutes of trading each day than at other times. Granger tests, however, reveal no systematic pattern of futures volatility leading index volatility, or index volatility leading futures volatility.

1. Introduction

Market observers generally agree that the introduction of stock index futures has altered both the nature of equity management strategies and the structure of U.S. equity markets. Institutional investors, in particular, employ index arbitrage, portfolio insurance, and tactical asset allocation programs that affect the price adjustment dynamics between futures and the associated indexes. Despite the benefits of futures trading from lower transactions costs and improved trading efficiency, critics claim that implementation of futuresbased strategies exacerbates volatility in underlying equity prices. Increased volatility, in turn, presumably reduces investor confidence which leads to lesser trading activity, lower liquidity and ultimately a higher cost of owning *We wish to thank Steve Hillmer, Paul Potvin, an anonymous reviewer, and seminar participants at the University of Kansas and the University of South Carolina for helpful comments. Financial support from University of Kansas grant GRF-39634038 is gratefully acknowledged. 0378-4266/9O/‘rSO3.50

Q

1990-Elsevier

Science

Publishers

B.V. (North-Holland)

equities. This issue has become increasingly important with the stock market crashes of 1987 and 1989 and concern that individuals are withdrawing from direct participation in the market. While the volatility relationship between futures and equity prices has yet to be tested rigorously, a variety of evidence suggests that futures-induced volatility is plausible. First, a substantial portion of volume on the New York Stock Exchange (NYSE) can be attributed to program trading strategies involving stock index futures. Such trading is often associated with sharp swings in stock prices within short intervals. Second. many analysts claim that futures activity exacerbated the stock market crash of October 19, 1987. Third, recent research by Harris (1989), Herbst, McCormick and West (1987), and Kawaller, Koch and Koch (1987) indicates that movements in the ievel of S&P 500 futures prices sytematically lead movements in the S&P 500 index. We thus might reasonably expect the same relationship with the volatility of index futures prices leading volatility in the index. The purpose of this research is to examine intraday relationships between the volatility of S&P 500 futures prices and volatility of the S&P 500 index. We contend that, because of modern communications systems and improved technology, traditional volatility measures based on day-to-day observations ignore critical information concerning intraday price patterns and linkages.’ We overcome this problem by constructing a variety of intraday volatility measures using minute-to-minute data for S&P 500 futures prices and the S&P 500 index throughout the fourth quarters of 1984, 1985, and 1986. We then conduct Granger tests that relate futures price volatility to subsequent index volatility, and vice versa, to examine the lead/lag relationships between volatility changes in the futures and equity markets.* The empirical results indicate that the minute-to-minute price variance for futures is at least five times greater than the minute-to-minute variance of the index. We attribute this to infrequent trading in index stocks relative to index futures and to the role of the bid ask spread. Furthermore, while there has been a trend toward increased intraday voIatility of both futures and index prices in absolute terms, no robust systematic Iead/iag relationship is evidenced between futures and index price volatility in either direction. During some periods S&P 500 futures price volatility leads S&P 500 index ‘The debate over whether futures market trading influences cash market prices appeared with the develooment of the first commoditv futures contract. Working (1960) and Turnovsky (1983) summarize key research in the commodity markets and generally conclude that the introduction of futures trading does not increase price volatility in the associated cash markets. This and later research on financial futures focuses on closing prices and thus generally ignores intraday price volatility. 21deally, we would like to conduct the same tests with data from the fourth quarters or 1987 and 1989. During the respective market crashes, however, trading on both the futures and equity markets stopped temporarily. Price quotes for some equities were subsequently not provided for long periods of time and the index did not reflect current transactions opportunities. lntraday Granger tests would not be meaningful with such breaks in the data.

l.G.

Kanaller

et al., Intmday

relationships

375

volatility, while in other periods the lead runs from the index to futures, or no lead appears. Thus, any existing intermarket lead/lag relationship in volatility is unstable, and the contention that volatility in stock index futures is systematically responsible for subsequent volatility in the equities market appears to be unfounded. The paper is organized as follows. In section 2 we describe the data, construct separate intraday volatility measures over one-day and 30-minute time intervals, and compare the behavior of these measures across different contracts and time intervals. Section 3 introduces the model that characterizes the dynamic interrelationship between S&P 500 futures and index intraday price volatility. We describe the estimation procedure, formalize specific hypothesis tests and address econometric issues. Section 4 presents the empirical results for the price volatility tests. The final section summarizes the results and offers conclusions. 2. The construction and comparison of volatility measures 2.1. Data

Transactions data on all values of nearby S&P 500 futures contracts (Fr ) and all values of the S&P 500 index (I,) were obtained from the Chicago Mercantile Exchange (CME) for every business day in the fourth quarters of 1984-1986.3 These data are used to construct time series reflecting volatility movements in futures and index prices over the life of a nearby futures contract. The comparison of volatility is complicated, however, by the fact that futures prices do not coincide with the index calculation. While the index is computed regularly each minute, futures prices reflect all trades throughout each trading day. On most days there are several futures prices available each minute, while no quotes appear for many minutes on a few days. In order to statistically investigate leads and lags connecting movements in the volatility of the index and futures prices, we must first have time series in the prices themselves that represent matched observations taken at equispaced time intervals. We resolve this problem by matching the last index quote each minute with the last futures quote each minute. Thus over the course of a 6: hour trading day our effective sample includes 390 pairs of ‘The price series, (F.r), reflects all S&P 500 futures transactions quoted each day. Prior to June 13, 1986, the index (I,) was available once each minute. Since then the index has been quoted approximately every fifteen seconds. Furthermore, before September 30, IYYS both markets opened at 9~00 a.m. Central Standard Time. Since then the markets have opened at 8:30 a.m. Finally, the stock market has traditionally closed at 3:00 p.m. C.S.T.. while the S&P 500 futures market has remained open until 3:15 p.m. In our analysis all futures quotes after 3:oO p.m. are ignored, so that volatility movements in the two markets are matched over the same time frame. Thus there were six hours of synchronous trading activity each day of 198-t and six and one-half hours throughout the fourth quarters of 1985 and 1986.

376

1.G. Kawaller

et al.. Intraday

relationships

minute-to-minute observations on F, and I,. We then take first differences to generate time series reflecting minute-to-minute changes in each price: fr=Fr-FT-r and ir=Ir-Ir_,.4 We construct volatility measures by calculating sample variances of these time series (jr and ir) over two different time intervals. Initially, we construct daily volatility measures using all observations throughout each day. We then construct volatility measures for 30-minute intervals. Over each day or 30-minute time interval, r, the sample variance of each price series is recorded as that interval’s observation of the volatility of futures prices.(c,,) and the index ( uit):

where n= the number of observations available within each interval; f and T indexes the i equal the means for fr and i, within each interval; minute-to-minute observations on fT and i,,. and r indexes the time intervals chosen to measure volatility.5 This procedure implicitly assumes that the variances are constant during the intervals used to contruct c’/, and rir. but time dependent beyond that point. It is unlikely that if the variances are changing across these time intervals, they are not also changing within the intervals. We have also investigated volatility relationships computing variances every 10 minutes with results that are virtually identical to those for 30-minute intervals. This suggests that any changes within 30-minute intervals are small. 2.2. Characteristics

of the volatility

measures

The observations for each variance measure, series that characterizes movements in intraday 30-minute time intervals. We initially examine of these futures and index variance measures

ul, and rir, represent a time price volatility across days or the time series characteristics to investigate whether price

“This procedure ignores any price variation that might occur within minutes. If there are no futures quotes during one or more minutes, then the most recent futures quote is retained as the value of the futures contract to be matched with the index quote each such subsequent minute. Because we use tirst differences of these minute-to-minute observations to compute the volatility measures, periods without any futures quotes do not increase the measure of futures price volatility. If the levels of F, and I, were used instead, such periods would increase futures price volatility. Using first differences reduces the impact of rising price levels on variance measures. sThe underlying time series on the minute-to-minute first differences in the price levels and are therefore ergodic. Fuller (1976) typically display only first order autocorrelation. establishes that the sample variance (s’) is an appropriate measure of the variance of an ergodic time series. The use of 30-minute segments seems reasonable given that Kawaller, Koch. and Koch (1987) determine that, on average, S&P 500 futures prices lead the S&P 500 index by 20 to 40 minutes.

1.G. Kawaller

et al., Intrada!

Table Summary

377

relarionships

1

statistics for volatility measures of S&P 500 futures price changes the S&P 500 index (r,,): Fourth quarters of 1984, 1985 and Daily YLII

1984: Quarter

T Lir

30-minute SC&

F

volatility s,,/s;;;

measures

T Lif

,s,c tit

49.5 41.5 43.3 32.2

0.37 0.19 0.27 -

5.9 6.4 8.4 7.7

0.40 0.38 0.30 -

49.8 41.7 43.4 32.3

0.67 0.52 0.58 0.58

5.1 5.6 7.0 7.1

0.90 0.87 0.96 1.02

45.7 58.5 69.7 61.2

0.35 0.37 0.43 -

7.0 10.8 II.5 9.1

0.48 0.31 0.39 -

46.0 58.5 69.5 62.0

0.67 0.69 0.78 0.94

6.3 9.1 10.5 8.2

0.95 0.83 0.75 0.53

134.5 134.8 120.0 143.8

0.45 0.33 0.27 -

19.3 28.0 26.0 23.1

0.56 0.42 0.43 -

135.0 135.3 121.3 146.2

0.94 0.81 0.86 1.17

17.1 25.1 24.3 21.5

1.69 1.59 2.05 I .02

IV

All 64 business days Last 15 business days Last 5 business days Expiration day 1986: Quarter

sr,,‘C

measures

in

IV

All 64 business days Last I5 business days Last 5 business days Expiration day 1985: Quarter

volatility

(cl,) and changes 1986.”

IV

All 64 business days Last I5 business days Last 5 business days Expiration day

‘Mean values. G and G, and coeflicients of variation [standard deviations (s,, and s,J divided by means] are provided for each volatility measure. computed over one-day intervals and over 30-minute intervals.

volatility changed notably from the fourth quarter of 1984 to the fourth quarter of 1986. Sample means and coefftcients of variation for the time series, L’/, and vi,, are presented in table 1 for the entire fourth quarters of 1984, 1985 and 1986, as well as for selected subsamples, consisting of the last 15 business days, the last five business days and expiration day of each futures contract period. Separate summary statistics are provided for measures calculated over one day and 30-minute intervals. Consider initially the means of c’,-, and tiit. Regardless of the quarter and subsample considered or time interval for calculating variance, the mean futures volatility measure typically exceeds the comparable mean index volatility measure by a factor of live to ten. This robust relationship largely reflects inertia in index prices relative to futures prices due to infrequent trading of stocks. When new information becomes available, participants can take a position in index futures quickly with low transactions costs. The same position in the cash market involves buying or selling all 500 stocks that comprise the S&P 500 index. Not only are transactions costs higher, there is considerable uncertainty over the exact price at which each stock will trade. Thus minute-to-minute changes in the index exhibit greater inertia than comparable changes in futures prices, and

37s

I.G.

Kawtrlirr

er cd., Inrrutlu~

reltrtionships

g will generally be less than c. In addition, bid ask spread differences will affect the comparative variances. Our futures price series represents actual transactions data. As orders reach the market some quotes will reflect trades at the bid price and others the ask price, such that rfr will be higher than if just one side of the spread was used. 6 While stock trades will similarly reflect the bid or ask price alternatively, any movement between the bid and ask will be averaged across 500 trades and thus uir should again be lower than G. Consider next the comparison of mean variances across time. Three results stand out. First. both mean variances are remarkably stable across the two time intervals for calculating variances. Second, the mean variances exhibit no consistent pattern within each contract period. In 1984 cfl was highest during the early part of trading and lowest on expiration day. In 1985 it was highest during the last five business days and lowest early in the contract and in 1986 it was the largest on expiration day and lowest during the preceding four business days. Over the same periods C’iroften displayed a different pattern, such as in 1984 when it increased as expiration approached. Third, the data exhibit a strong trend toward increased volatility from 1984 to 1986. Specifically both mean variance measures more than doubled from 1985 to 1986 regardless of the time interval for comparative volatility. This is due in large part to the genera1 rise in both S&P 500 futures prices and the index, and is consistent with public perceptions that volatility measured in absolute terms has increased. The coefficients of variation generally support these conclusions. For daily volatility measures the degree of dispersion about the mean variance is quite stable within each contract period, and for 1984 compared with 1985 and 1986. For 30-minute measures, however, the relative dispersion about the mean variance rises sharply in 1986 from previous levels. Clearly, intraday price volatility for both the index and futures increased in 1986 relative to prior years.

3. The dynamic interrelationship

between volatility

in S&P 500 futures prices

and the S&P 500 index

The availability of stock index futures and technological improvements with communications and order execution now enable highly-capitalized institutions to trade stocks comprising the S&P 500 virtually simultaneously with S&P 500 index futures in order to take advantage of perceived price discrepancies. Arbitrageurs essentially buy the cheaper instrument and sell

6Unfortunately. bid and ask data are not availabir

for futures

transactions.

I.G. Kwaller

rr al.. Inrrcday

relurion.ships

379

the more expensive one. locking in a minimum rate of return as the two prices must converge at expiration, if not sooner.’ The fundamental impact of arbitrage is that the S&P 500 futures price (Fr) and the S&P index (I,) are linked throughout the course of a typical trading day and throughout an entire contract period. Harris (1989), Herbst McCormick and West (1987), and Kawaller, Koch and Koch (1987) document that F, and I, display a substantive and robust intraday lead’lag relationship. While futures and index prices move largely in unison, futures price adjustments also generally lead index price moves by 20 to 40 minutes. The lead from index price moves to futures, in contrast, rarely extends beyond one minute. These results largely reflect greater inertia in the index compared with futures prices, as all stocks do not trade coincidentally at new prices consistent with new futures values each minute. This robust empirical relationship in which futures prices systematically lead index prices might lead some to expect that futures volatility also leads index volatility. Such a relationship does not necessarily follow, however, because the behavior of the second moments of two time series does not necessarily follow the behavior of their first moments. Appendix A presents a plausible model of price formation to demonstrate that the level of one price series can lead another while the variance in the latter leads the variance in the former. Thus the nature of any relationship between volatility in two price series is an empirical issue.

Because index arbitrage and portfolio insurance link stock index futures and underlying equity prices, volatility in S&P 500 futures and index price movements may each provide predictive information about subsequent price volatility in their own and the other’s market. That is, movements in the volatility of the S&P 500 index (vi,) may be dependent upon past behavior of volatility in the index, the history of volatility in the S&P 500 futures price (~~0, and other factors reflective of market conditions or structures. Likewise, volatility in the futures price depends upon its own past volatility, past volatility in the index, and other similar market factors. This interaction can be expressed in the following two-equation model of the determination of S&P 500 futures and stock price volatility:

‘Considerable research has been devoted to explaining the basis, or price differential, between stock index futures and the underlying indexes. See, for example, Cornell and French (1983). Modest and Sundaresan (1983) and Kawaller (1987). Arbitrageurs realize their minimum rate of return by offsetting their initial positions at expiration when the basis equals zero. If the basis converges sufficiently prior to expiration, participants may earn even greater returns. Brennan and Schwartz (1988) offer a model that addresses the issue of pre-expiration arbitrage reversals.

I.G.

350

L’~,=Q+

Kwallrr

‘2

akCir_k+

k=i

c

bkrf,-k+e,,,

y

rrlatiunships

(1)

k=l

.U2

L’f,=do+

et al.. Inrradu~

M

ckt’ir-k+

L

,L, dkuft-k+errq

(2)

k=l

where e,, and ezt reflect other relevant information that affects price volatility in the stock and index futures markets, respectively. Because the ‘other relevant information embodied in each error term could conceivably affect volatility in both markets, the error terms may be contemporaneously correlated. This suggests that eqs. (1) and (2) may be rewritten as the following joint system of seemingly unrelated regressions.

(3)

where the distributed lags specified in (1) and (2) appear as polynomials in the lag operator, L (LkL’it= Uil-k), and e,, and e,, may be contemporaneously correlated. Again, this formulation implicitly assumes that variances are constant during the intervals indexed by t.

3.2. Estimation

procedure

and hypothesis

tests

Note that the system (3) is specified in terms of levels of volatility. Consistent with the relationships analyzed by Wiggins (1987), we employ a natural log transformation of vjl and air in the regression analysis. As described below, this corrects for possible hereoskedasticity in the error terms. Before system (3) can be estimated, we must select a finite lag parameterization. While longer lag lengths lessen the chance of misspecification, they also consume more degrees of freedom. Hence we should choose the minimum lag length that specifies the relationship accurately. Geweke (1978) argues that the number of lags on the dependent variable in each equation (M,) should be kept generous to minimize the chance of serially correlated errors, while the number of lags on the other variables (M2) should be set lower to retain power in the hypothesis tests. To investigate the robustness of the results, we examine the results for several different lag lengths. When the time interval for volatility is one day, lag lengths are specified to be ill, = fifteen days on the lagged dependent variable in each equation and ,V,= three, five, and ten days on the right-hand side volatility variable. Alternatively, when the volatility interval is 30 minutes the lag lengths applied are

I&. Kauallrr

et al., 1nrrada.r

relationships

381

M, = 24 half-hour periods (two days) and hfcI,=6, 12, 18, and 24 periods (one-half day to two days).8 The model in (3) is estimated using the standard Zellner-Aitken technique. If the ‘other relevant information’ embodied in each error term affects the volatility in both markets, this would imply contemporaneous correlation between the errors, and Ordinary Least Squares would yield inefficient estimates.’ Volatility movements in the futures market are said to Granger-cause volatility movements in the S&P 500 index (L’fr+L’il) if index volatility can be predicted more accurately by including past movements in futures and index volatiliy, than by including just past movements in index volatility alone.” Absence of such Granger-causality (c,,+u~,) can be investigated in the context of model (3) by using the Wald F-statistic to test zero restrictions on the lagged ulr coefficients in the first equation:

Ht: u/r%L’it

(B(L)=O).

If H, is rejected by the data, then evidence suggests that volatility in the futures maret is systematically followed by volatility in the stock market. The analogous hypothesis that volatility in the stock market does not lead to subsequent volatility in the futures market is tested as:

H,: Uir%Cft

(C(L)=O).

Infrequent trading of index stocks should induce positive serial correlation into cash price movements that is not observed in futures prices. This has been documented and used as an explanation for the intraday lead of futures price levels over index levels [Herbst, McCormick and West (1987); and Kawaller, Koch and Koch (1987)]. But what does this imply about the behavior of the variances, uir and u,.,? The same infrequent trading phenomenon might also be expected to induce inertia into the cash index variances that may not be observed in the futures variances, although such behavior of second moments does not necessarily follow the behavior of first moments. To investigate, we have computed the autocorrelation and partial autocorrelation functions of both sGeweke and Meese (1981), Judge et al. (1985, ch. 21) and Thornton and Batten (1985) discuss the tradeotT between bias and efliciency in the selection of lag lengths for distributed lag models such as (3). and propose model selection criteria based on weighting schemes for this tradeoff. The problem is important only if the estimation results and hypothesis tests are not robust with respect to the lag length selection. We explore several lag lengths in the analysis of section 4 and show that the &anger test results are generally robust with respect to this choice. ‘The models analyzed in section 4 have also been estimated with ordinary least squares, and the resulting parameter estimates and hypothesis test results are generally robust. “‘Granger causality diNers from a traditional, philosophical concept in that Granger’s definition simply refers to whether lagged movements in one variable provide statistically significant explanatory power regarding subsequent movements in another [Granger (1969)].

382

I.G. Kawallrr

er al., lntruduy

relationships

series, ri, and c/,, for all contract periods and both time intervals. The results indicate that there is generally only first order autocorrelation observed in both vi, and cl, over both the daily and 30-minute intervals. Furthermore, the partial autocorrelation for uJ, at lag one is generally at least as large as that for ttit, suggesting that there is at least as much serial correlation observed in futures variances as in index varances. This result is consistent with that of MacKinlay and Ramaswamy (1988) who show that autocorrelation in S&P 500 futures prices is high when autocorrelation in the index is high. Implicitly, non-synchronous data do not account for all the observed autocorrelation. In any case, the relative inertia in ci, and cl1 has little bearing on the Granger test results. In our tests of system (3) such inertia is washed out by incorporating lagged values of the dependent variable in each equation so that the error terms are not autocorrelated. While there is some inertia observed in ri, and ~‘r*,there is no autocorrelation in the estimated residuals of the model as evidenced by the Ljung-Box (1978) test provided in tables 3 and 5 and discussed later.

3.3. Nonprice fuctors

that affect

volutilit>

Stock index futures and index price volatility can be expected to vary with other market factors. It is widely accepted, for example, that stock price volatility varies with general marker movements. Black (1976) specifically determined that volatility tends to increase as stock prices decline and decrease as stock prices rise. Recent research also indicates that price volatility for index component stocks increases dramatically during the last to hour of trading on futures expiration days. ” This rise may be attributed index arbitrageurs offsetting their initial positions with market-on-close orders when the basis equals zero. Aside from timing considerations, Merrick (1987) finds evidence that truding volume and price volatility are directly related. This logic suggests that if intermarket activity increases the volume of futures and stock transactions significantly, it may directly increase volatility. Samuelson’s (1965) theory of futures price volatility also suggests that the variance of futures price movements systematically increases the closer it is to expiration of a futures contract. This implies an upward trend in our futures volatility series throughout the life of a nearby contract. Extending the argument, if index futures prices represent unbiased estimates of forthcoming “Stall and Whaley (1986) document this effect along with greater stock market volume on expiration Fridays. Prior to the June 1986 futures contract, the CME’s S&P 500 futures. the CBOE’s S&P I00 options, and individual stock options all expired on the same day. The last hour of trading on those expiration Fridays was thus referred to as the triple witching hour.

I.G. Kwuller

er al.. Intruda~ relurionships

383

equity prices and equity prices adhere to a stationary first-order autoregressive process, the impact of new information on futures prices would be greater near expiration. Implicitly. more information is disclosed as time elapses, so variance would rise. Anderson (1985) analyzes this issue and a competing state variable hypothesis which asserts that the variance of futures price movements may be highly variable throughout a contract’s life depending on the degree of uncertainty surrounding market transactions. Empirical evidence using daily data generally supports the increasing volatility hypothesis, while evidence from monthly data does not show any pattern. When the time interval for volatility is one day, we capture the effects of general market price movements, volume changes, and any trend component of volatility by extending system (3) to incorporate market returns, transactions volume measures, and a linear trend. General market movements are captured by including lagged returns based on the S&P 500 index. Specititally, daily returns are measured as the natural log of the ratio of the last index quote each day to the first index quote for the day, with lagged values for the previous five business days added as explanatory variables in each equation of (3). Three volume variables are also added separately, including the daily volume of NYSE listed stocks, the number of daily block trades (greater than 10,000 shares) on the NYSE, and daily volume for all S&P 500 futures contracts traded on the CME. We also employ a time trend and dayof-the-week dummy variables. If volatility increases (decreases) when the market falls (rises), volatility should vary inversely with the returns variables. The inclusion of transactions volume data tests whether there is any consistent relationship between volume and price volatility as suggested by Merrick. The time trend allows us to investigate the possibility that volatility systematically increases as contract expiration approaches, as Samuelson proposes. Finally, day-of-theweek dummy variables will indicate whether any days exhibit consistently high or low volatility. Because reliable volume data are not available at 30-minute intervals, tests on system (3) covering volatility movements over 30-minute intervals incorporate only returns measures, a trend variable, and dummy variables for the different intraday periods, with the last half-hour the omitted group. Returns are again measured as the natural log of the ratio of the last index quote to the first index quote during each 30-minute interval, with two days of lagged 30-minute returns added in each equation. The use of a separate dummy variable for each half hour interval should capture intraday patterns in price volatility. In particular, volatility during the first and last 30 minutes of each day may differ substantially from that for other intervals. Although trading in U.S. markets is closed overnight, new information continues to accumulate and influence trading particularly when markets open and close [Ahimud and Mendelson (1987); Wood, McInish and Ord (1985)]. This incorporation

I.G.

38‘l

Kawailer

et ul., intraday

relationships

of systematic timing patterns over different intraday time intervals allows the estimation of lag effects beyond one day in system (3). 3.4. Econometric issues If we assume that the minute-to-minute price change series (fr and ir) are normally distributed, then the time series observations on their sample variances, v/, and rir, are random variables with known asymptotic distributions: vfr 5

N [CT&, k4,,‘n-

t]. (4)

l’ir 5

N[cr&Zaf/n - I],

where n is the sample size, and a$ and CY~are the true variances of fr and i,, respectively, for the time interval, t. We thus employ proxies for the true variances over each time interval. This implies three complications to the model in eqs. (1) and (2). The first problem is potential heteroskedasticity in e,, and ezr Each interval’s observation on ufr and Uir is computed from underlying observations on fr and ir, From (41, the precision of each observation depends on the true afl and qi”,during that time interval. Therefore the precision of our data on L’,., and Ci, varies as the true volatility does. If the errors are heteroskedastic, we cannot rely on the F tests of H, and H,. Given the disapnormality assumption underlying i f and jr, this heteroskedasticity pears with the natural log transformation:

After applying the natural log transformation to u,-~ and Vi,%there is no heteroskedasticity from this source in (1) and (2).12 The second problem lies in potentiai autocorrelation in e,, and ez2. Since ln(uJ and In(c,J are still random (measured with error), the lagged values of these variables that appear on the right-hand side of eqs. (1) and (2) have ‘lagged errors’ in them. This may introduce inertia into the error structure embodied in e,, and e,,. If the errors are autocorrelated, then we cannot rely on the F tests of H, and H,. For Granger tests to be valid, we must include “The rule provided in Judge et al. (198.5, p. 160, eq. 5.320) suggests that we consider functions of the volatility measures that will stabilize the error variances. The natural log transformation of rIr and L‘~,is such a function to correct for heteroskedasticity that might be transferred to the regression error terms.

I.G.

Kawaller

et al., Intraday

relationships

385

enough lagged dependent variables in each equation to eliminate any autocorrelation in the errors. We can then test the hypothesis that the errors are not autocorrelated for every model we estimate. If we do not reject the null hypothesis that e,, and ezt are not autocorrelated, then any autocorrelation in the residuals is effectively eliminated. Our tests follow this procedure. This leads to a related problem regarding how to test whether e, is autocorrelated. The Durbin-Watson test cannot be used since there are lagged endogenous variables on the right-hand side of each equation. We could use the Durbin h test, but it lacks power and is a restricted test only for first order autocorrelation. Instead, we use the Ljung-Box (1978) test on the autocorrelation function of the residuals of each equation because it is a more general test for autocorrelation of any order (1978). The results which are presented in tables 3 and 5 almost uniformly do not reject the hypothesis that e, is not autocorrelated. This lends credence to the validity of the F tests. The final problem rests in the fact that In(u,.,) and In(v,J are still measured with error, as shown in (5). If right-hand side variables are measured with error, then all coefftcient estimates are biased and inconsistent, and the F tests are invalid. However, the extent of the bias depends on the magnitude of the measurement error variance relative to the residual variance [Maddala (1977)]. Fortunately, in this context we know the variance of our measurement error. Recall from eq. (5) that the asymptotic variance of In(o,,) and ln(u,J is 2/n. Thus we need to compare 2/n with the estimated residual variances for eqs. (1) and (2). For the daily analysis, n=360; var(tl/,)= var(t+,) =2/360=0.0056; and the residual variances for (1) and (2) are seen to be approximately S: =0.055, and 8: =0.027, respectively. For the 30-minute analysis, n = 30; var (u,J = var( uir)= 2/30 = 0.067; and the residual variances for (1) and (2) are approximately 3: =0.294, and 6: -0.192, respectively. These reflect typical residual variances for the fourth quarters of 1984 and 1985. For 1986 the residual variances are approximately twice these values. Thus, the variance of the measurement error is only 10% to 25% of the residual error variances over the 1984 and 1985 samples, and 5% to 15% of the residual error variances over the 1986 sample. This suggests that the extent of the bias is reasonably small [McCallum (1972) Wickins (1972)].

4. Empirical results: Granger causality tests 4.1. Volatility between S&P 500 futures prices and the index: Dail}* variance intervals

The calculation of ult and uit over each entire business day yields time series reflecting daily movements in intraday volatility of S&P 500 futures prices and the index. After taking the natural logs of clr and Vi,, we conduct

J.B.F.

F

386

1.G. Kawaller

et al.. Inrrudar

rrlutionships

the tests associated with system (3) to obtain evidence concerning hypotheses H, and H, for the fourth quarters of 1984-1986. While our primary concern is whether futures volatility systematically leads index volatility, the underlying regression estimates of the nonprice factors yield interesting results. These estimates and the subsequent F-tests are discussed in turn. 4.1.1. Regression estimates of nonprice factors The inclusion of nonprice factors in the system (3) regression analysis is intended to enhance the model’s structure, and thus its ability to reveal any systematic lead/lag relationships between volatility in the two markets. The results, however, suggest several important conclusions.r3 First, lagged daily market returns generally display no significant relationship with volatility in either futures prices or the index. Their coefficients typically vary in sign and have large standard errors. The one exception is for the fourth quarter of 1984, where the five lagged return coefficients in the index volatility equation are all positive, with two significantly different from zero at the 5% level. In contrast to the effect suggested by Black (1976) who considers much longer sample intervals, positive runs in the market were associated with subsequent higher index volatility during this period. Second, the variance of both futures prices and index does not exhibit any systematic positive trend as suggested by Samuelson. Third, there is no evidence of any systematic day-ofthe-week impact for either volatility measure. Finally, two of the three daily volume measures exhibit some significant relationships with volatility of futures prices and the index. While the number of block trades on the NYSE is never significantly associated with any volatility measure, volume on the NYSE and in S&P 500 futures display some significant relationships. The coefftcient estimates for daily volume on the NYSE (VOLNYSE) and all S&P 500 futures contracts (VOLFUT) are presented in table 2. Each row provides estimates with a different assumed lag length (MJ on the right-hand side price volatility variable in eqs. (1) and (2). Results show that VOLNYSE never significantly affects futures volatility, but does display a significant positive relationship with index volatility for certain lag lengths and sample periods, consistent with the work of Merrick (1987). This result is not robust, however. In contrast, VOLFUT always varies directly with the variance of futures prices. This at least partly reflects the impact of bid ask spread. On days when futures volume is high minuteto-minute futures volatility will be high because trades alternate at a higher frequency between the bid and ask prices relative to days when volume is low. Futures volume also varies directly with the variance of the index in 1986, suggesting that when aggregate futures volume is higher underlying stock prices are more volatile during this sample period. This potentially “These

results are available

on request.

1.C. Kawaller

et al., Intrada!

387

relationships

Table 2 Regression estimates of the impacts of daily volume measures on daily volatility ci, equation _ VOLNYSE

r,, equation M,b

VOLNYSE

VOLFUT

measures.’

VOLFUT

1984.1 v 3 5 10

0.0014 0.0002 -0.ooO8

(0.0105) (0.0111) (0.0131)

0.0216 0.0200 0.0213

(0.0044)’ (0.0057)* (0.0079)*

(0.0074) (0.0079) (0.009 1)

0.0208 0.0223 0.0222

(O.G042)* (0.0047)’ (0.0056,*

0.0’40 0.0133 0.0’05

(0.0117)’ (0.0136) (0.0213)

0.0175 0.0104 0.0142

(0.0081)* (0.0086) (0.011 I)

(0.01 I I) (0.01 17) (0.0145)

0.0044 0.0061 0.0025

(0.007 I) (0.0077) (0.0103)

(0.0077) (0.0082) (0.0086)*

0.0287 0.0282 0.0292

(0.0055)* (0.0060)* (0.0071)*

198S.IV 3 5 IO

0.0014 0.0013 0.0046

-0.0038 -0.0037 -0.0019

1986.1V 3 5

- 0.0009 0.0003

(0.0048) (0.0053)

0.0205 0.0197

(0.0041)* (0.0044)*

10

0.00 1 I

(0.0058)

0.0226

(0.0051)*

0.0127 0.0131 0.0168

“The volume variables are labelled as follows: VOLNYSE= total daily volume on the NYSE, and VOLFUT= total daily volume on all S&P 500 futures contracts. Coefticient estimates are presented, and standard errors appear in parentheses. Signilicance at the 0.10 level is indicated with an asterisk. ‘This column presents the number of daily lags (.\Iz) on the right-hand side variable in each equation of the regression model.

results from greater futures volume inducing greater volume in index stocks from program trading. It is consistent with greater futures activity producing greater stock price volatility and/or higher stock price volatility requiring a greater reliance on futures for risk management. 4.1.2. F-tests associated with Granger causality Table 3 presents the F-statistics for the hypothesis tests associated with H, and H, and the Ljung-Box (1978) test statistics for white noise residuals. Importantly, the null hypothesis that the errors of each equation are white noise is rejected in only one instance at the 0.10 level, which lends credence to the validity of the F-tests. According to the F-tests, we reject H, at the 0.10 level of significance just once, out of all lag lengths and fourth quarter contracts examined. H, is never rejected at the 0.10 significance level. These results suggest no systematic lead or lag relationship between day-to-day movements in cl, and vi,. This result is not surprising given that prior tests of the dynamic interrelationship between the levels of futures and index price movements show a complete adjustment within one hour. Implicitly, any linkages that might systematically spread volatility in one market to volatility in the other presumably arise within one day. This leads to the consideration of volatility movements over 30-minute time intervals.

388

I.G. Kawallrr

ef al.. introda_v relarivnships

Table 3 Granger

tests on intraday

data for the fourth quarters of 1984-1986: variance computed over one-day intervals.

Granger test F statistics”

~__ N,: rir+rr,

measures

Ljung-Box test chi square statistics” ---N,: ea not N,: et, not autocorrelated autocorrelated

DF

n,: ny,jWir

1984.1V 3 5 10

(3, 34) (5, 30) (10. 20)

1.25 (0.306) 1.69 (0.167) 0.78 (0.650)

0.31 (0.819) 0.33 (0.889) 0.43 (0.914)

7.3 (0.838) 14.3 (0.285) 13.3 (0.350)

13.2 (0.354) 11.0 (0.530) 16.9 (0.154)

1985.1v 3 5 10

(3, 32) (5, 281 (14 18)

0.31 (0.818) 0.33 (0.892) 0.40 (0.928)

0.32 (0.808) 0.39 (0.853) 0.58 (0.807)

12.6 (0.402) 10.1 (0.607) 11.1 (0.517)

16.3 (0.177) 14.3 (0.283) 15.6 (0.212)

(3. 34) (5, 30) (10, 20)

2.66 (0.064)’ 1.47 (0.230) 1.72 (0.146)

0.55 (0.649) 0.41 (0.839) 0.99 (0.486)

7.9 (0.796) 7.9 (0.790) 12.7 (0.391)

16.2 (0.181) 17.7 (0.124) 23.5 (0.024)*

1986.1 v

3 5 10

“W, and W, test the null hypotheses of no Granger-causality in each direction. if, and H, test the null hypotheses that the residuals of eqs. (1) and (2) are white noise. These latter test statistics each incorporate twelve residual-autocorrelation coefftcients, and have an approximate asymptotic chi square distribution with twelve degrees of freedom. Failure to reject H, and HA suggests that the F tests of H, and H, are valid. The approximate marginal significance level appears in parentheses beside each F statistic and chi square statistic presented. Signiticance at the 0.10 level is marked with an asterisk. bThis column presents the number of daily lags (Ma) on the right-hand side variable in each equation of the regression model. ‘This column presents the degrees of freedom on the F-statistics testing the null hypotheses, H, and Hz, in each row.

4.2. Volatility between S&P $00 futures prices and the index: 30-minute variance intervals When variance is measured over 30-minute intervals, the test results reveal substantive volatility relationships during some periods but not others, with leads running from either futures to the index or the index to futures, or both. The fact that there is no consistent systematic relationship indicates that the volatility dynamics are unstable. There are, however, several important results that appear repeatedly. 4.2.1. Regression estimates of nonprice factors The regression estimates for the lagged returns reveal no consistent relationships. In generaf, lagged returns caicufated over 30-minute intervals are not sign~~~antly associated with either futures or index volatility.

I.G.

Kawaller

et al.. Inrradav

relarionships

389

Table 4 Regression estimates for trend and dummy variables for the first 30 minutes of trading: Variance measures computed over 30-minute invervals.” rir equation

u,, equation M,

DI’

Trendb

DI’

Trend

1984.1 v 6 12 18 24

- 0.00007 - 0.00007 -0.00007 - 0.00007

(O.OOGO8) (0.00008) (0.00008) (0.00008)

-0.078 -0.087 - 0.099 - 0.030

(0.087) (0.092) (0.093) (0.094)

O.C0016 0.00016 0.00015 0.ooo15

(0.001IlO) (O.ooOlO) (O.OOOlO) (0.OcNll0)

0.385 0.383 0.371 0.380

(0.116)* (0.117); (0.120); (0.121)*

1985.1V 7 13 20 26

-0.00014 -0.00015 -0.00014 -0.00014

(0.00008) (O.oooo9) (0.00010) (0.00010)

-0.100 -0.105 -0.114 -0.104

(0.075) (0.077) (0.079) (0.080)

o.OcO31 0.00031 0.00033 0.00033

(0.00012)* (0.00012)’ (0.00012)* (o.OOOl2)*

0.340 0.351 0.340 0.354

(0.094)* (0.097)1 (0.098)’ (o.lOO)*

(0.102)* (O.lOS)* (0.109)* (0.114);

0.00063 0.00065 0.00068 0.00066

(0.00017)* (0.00018)* (0.00019)’ (0.00019)*

0.880 0.921 0.925 0.875

(0.148)* (0.152); (0.157)* (0.162)*

1986.1 v 7 -0.00017 13 -0.00021 20 -0.00021 26 -0.00022

(0.00011) (0.00012) (O.OcO13) (0.00014)

0.264 0.300 0.313 0.265

‘Standard errors appear in parentheses beside each parameter estimate. An asterisk beside the standard error indicates signiticance at the 0.05 level. bThis column presents the trend estimates in each of the regressions. ‘DI takes a value of 1 for the variance during the first 30 minutes of trading each day, and zero elsewhere. Dummy variables for all but the last 30 minutes of trading, the omitted group, are also included in the regression models.

Typically, very few (approximately two of the 24) lagged return coefficient estimates in each equation are significantly different from zero. It is important, however, to incorporate the lagged returns in the model because their inclusion affects the test results of hyotheses H, and H, for some quarterly samples and subsamples. Most notably, the last four weeks subsample of fourth quarter 1986 displays sensitivity to this specification. When lagged returns are omitted, volatility in each market significantly leads volatility in the other. When they are incorporated for this subsample, several lagged return coefficients are signiticant, and almost all Granger tests become insignificant (see table 6). Estimates for the trend and first half-hour dummy variable for each day are reported in table 4. Dl equals one for the interval encompassing the first 30 minutes of trading each day, and zero elsewhere. Similar dummy variables are included for the other 30-minute intervals, except the last one in each day which represents the omitted group. As with the tests using daily data, futures volatility displays no systematic trend as contract expiration approaches. In contrast, index volatility increases in 1985 and 1986 as expiration day draws closer. While no obvious explanation exists, this may be due to program trading comprising an increasingly higher fraction of total

390

i.G. Kawnllrr

et al., Intrudu~

rrlarionships

trading as futures expiration approaches. The more frequent trading of index stocks in turn might add to the variance estimate. The estimates for Dl further indicate that, during the fourth quarter of 1986, futures volatility was systematically greater during the first half hour of trading than during the last half hour. Results for index volatility were more robust. Index volatility was greater during the first hatf hour of trading in all contract periods investigated over the three years, a result consistent with previous studies [Ahimud and Mendelson (1987); Wood, McInish and Ord (1985)]. New information typically arises between the previous day’s market close and the opening of business the following day and generally influences trading in overseas markets. This information and market activity is presumably at least partly responsible for the higher volatility in the U.S. near the opening of trading. Ahimud and Mendelson attribute timing influences to different trading mechanisms; specificatiy, call trading at the open compared with price setting by market makers at other times. In general, volatility from the second 30-minute interval until the last interval is significantly lower than the last 30-minutes of trading. The interruption of trading has the apparent effect of fostering higher volatility immediately before U.S. markets close, and higher volatility still immediately after U.S. markets open. These effects likely arise from the greater uncertainty associated with breaks in information flows when no trading is allowed.” 42.2. F-tests assoriate~ with ~~ang~~ ~a~sa~it~ Results of the Granger tests for 30-minute intervals over the entire fourth quarter samples are presented in table 5. The different rows again represent the F statistics and Ljung-Box (1978) statistics based on various lag lengths of the respective volatility measures. In all cases the estimated residuals are not autocorrelated. The Granger test results suggest that any association between futures and index price volatility is unstable. During the fourth quarter of 1984, none of the F-statistics are significant at the 0.10 level. Thus there is no evidence that volatility in either the index or futures price SystematicaIly leads to subsequent votatility in the other price series once this contract reached nearby status. This conclusion holds for 30-minute lags ranging from one-half day to two days. In contrast to results for 1984, significant leads do appear in later contract periods. During the fourth quarter of 1985, for example, index volatility leads futures volatility for all the lag lengths. In sharp contrast, volatility never runs from futures to the index. The empirical evidence during the fourth quarter of 1986 differs stiI1. Regardless of lag length, index volatility never leads futures volatility, as it does with the 1985 data. During this period, ‘*Admati and Pfleiderer (1988) and Ahimud and Mendelson rationale for the observed empirical pattern of higher volume the beginning and end of each trading day.

(1987) also provide a theoretical and greater return variability at

I.G. Kawallrr

et al.. Imraduy

Table &anger

tests

on

intraday

5

data for the fourth quarters of 1984-1986: computed over 30-minute intervals.

DF’

Variance

measures

Ljung-Box test chi square statistics”

Granger test F statistics”

MLb

391

relarionships

H,: cr,+~t,

H,: e,, not autocorrelated

H,: e2, not autocorrelated

1984.1V 6 12 I8 24

(6, (12, (18, (24.

1354) 1342) 1330) 1318)

1.44 0.98 0.95 0.88

(0.194) (0.464) (0.515) (0.629)

0.19 0.27 0.71 0.90

(0.979) (0.994) (0.802) (0.599)

0.46 0.55 0.53 0.62

(0.999) (0.999) (0.999) (0.999)

0.3 I 0.37 0.23 0.43

(0.999) (0.999) (0.999) (0.999)

(7, (13, (20, (26,

1434) 1422) 1408) 1396)

1.58 1.08 1.13 0.96

(0.136) (0.374) (0.311) (0.526)

4.29 2.60 1.88 I.91

(0.001); (0.002)* (0.01 I)* (0.004)*

0.29 0.5 I 0.56 0.60

(0.999) (0.999) (0.999) (0.999)

0.58 0.56 0.7 I 0.74

(0.999) (0.999) (0.999) (0.999)

(7, (13, (20, (26,

1468) 1456) 1442) 1430)

3.15 1.91 1.39 I.17

(0.003)* (0.026)* (0.119) (0.252)

0.67 0.55 0.48 0.59

(0.696) (0.897) (0.976) (0.950)

0.85 0.95 I.10 1.00

(0.999) (0.999) (0.999) (0.999)

0.16 0.17 0.22 0.30

(0.999) (0.999) (0.999) (0.999)

19SS.I v 7 I3 20 26 1986.1 V 7 I3 20 26

‘H, and Hz test the null hypotheses of no Granger-causality in each direction. H, and H, test the null hypotheses that the residuals of eqs. (I) and (2) are white noise. These latter test statistics each incorporate twelve residual-autocorrelation coeflicients, and have an approximate asymptotic chi square distribution with twelve degrees of freedom. Failure to reject H, and H, suggests that the F tests of H, and H2 are valid. The approximate marginal signilicance level appears in parentheses beside each F statistic and chi square statistic presented. Any marginal significance level below 0.10 (or statistical significance at the 0.10 level) is marked with an asterisk. ‘This column presents the number of 30-minute lags (,LfJ on the right-hand side variable in each equation of the regression model. Each day in 1984.IV had six (twelve 30-minute intervals) of trading, whereas almost every day in 1985.IV and 1986.1V has six and one half hours (thirteen 30-minute intervals) of trading. Thus, the lag lengths chosen reflect lags of f day, I day. I+ days. and 2 days for each sample. ‘This column presents the degrees of freedom on the F-statistics testing the null hypotheses, H, and H,, in each row.

however, futures volatility leads index volatility for lags of both one-half day and one day. No lead is detected with longer lags. The fundamental implication is that the volatility relationships are not robust across different contract periods. To examine whether these results are stable Gthin each contract period, we divide each fourth quarter contract considered into three subsamples comprising the first live weeks, the next four weeks, and the last four weeks. We then repeat the estimation and hypothesis tests of system (3) for each

392

I.G. h’awaller

et al.. Intraduy

relationships

subsample. l5 Table 6 presents the results for these subsamples of the fourth quarter contracts of 1984, 1985, and 1986. The results differ substantially from those reported in table 5, for the entire fourth quarter samples. In 1984, for example, futures volatility leads index volatility during all but the middle four weeks, while index volatility also leads futures volatility during the last four weeks. This differs substantially from no volatility relationships suggested when the entire sample is employed. During the fourth quarters of 1985 and 1986, there is little evidence that significant leads appear in either direction during any of the subperiods. Results for the entire fourth quarter contracts, in contrast, indicate that index volatility systematically leads futures volatility in 1985, while the opposite occurs in 1986. Clearly the estimated volatility leads and lags are sensitive to the sample period selected. Evidently, the interaction of volatility in stock index futures and equity markets varies with market conditions over a time frame much shorter than an entire nearby futures contract period. This contrasts sharply with the robust relationship between intraday movements in the levels of S&P 500 futures prices and the S&P 500 index, documented by Herbst, McCormick and West (1987) and Kawaller, Koch and Koch (1987). It is possible that interaction between volatility movements in the two markets also adheres to a robust relationship over shorter time periods such as one day or one week. However, the problems of constructing time series reflecting volatility movements in the two markets inhibits any investigation of this market interaction over such a short time horizon. Importantly, the lack of any systematic relationships indicates that futures activity does not predictably increase index volatility in any consistent fashion, nor does index volatility predictability influence futures volatility.

5. Summary

Many critics of futures markets argue that S&P 500 futures-based trading strategies increase the volatility of the stocks underlying the index. The October stock market crashes of 1987 and 1989, in particular, have raised concerns that index arbitrage and portfolio insurance activity accelerated equity price moves. This research examines whether intraday S&P 500 index futures and S&P 500 index price volatility has changed notably in recent years, and whether intraday volatility in futures prices has systematically led to intraday volatility in the index. We address these issues by calculating variance measures for minute-to-minute futures and index price changes on a daily basis and across 30-minute intervals for the fourth quarters of 1984, 1985, and 1986. These measures indicate that average intraday volatility for IsParameter estimates for the trend and dummy so that the implications are identical.

variables

are consistent

with those in table 4

I.G. Kawaller et al.. lntraday relationships

Table

6

Granger tests on intraday data; 30-minute variance weeks l-5, weeks 6-9. and weeks l&13 within F statistics

393

intevals for’subsamples each contract period.

of

(probability) H,: c,,+Q,

H,: uir+u,,

(23, 382)

1.89 1.48 1.68 1.62

(0.082)* (0.127j (0.041)* (0.033)*

0.34 0.55 0.91 0.89

(0.917) (0.885j (0.564) (0.612)

(6. (12, (18, (24,

322) 310) 298) 286)

0.28 0.44 0.46 0.60

(0.946) (0.949) (0.971) (0.930)

1.89 1.25 1.34 1.31

(0.082)* (0.250) (0.163) (0.156)

6 12 18 24

(6, (12, (18, (24,

346) 334) 322) 310)

3.18 1.67 1.26 1.17

(o.OOs)* (0.073); (0.212) (0.269)

1.89 2.07 1.63 1.45

(0.082)* (0.019); (0.052)* (0.083);

weeks l-5:

7 13 20 26

(7, ( 13, (20, (26,

420) 408) 394) 382)

0.91 0.86 0.68 0.61

(0.501) (0.595) (0.849) (0.934)

1.41 1.05 0.89 1.48

(0.200) (0.400) (0.596) (0.063)*

weeks 6-9:

7 13 20 26

(7, (13, (20, (26,

376) 364) 350) 338)

1.19 1.23 0.99 0.93

(0.307) (0.253) (0.476) (0.571)

1.46 0.87 0.79 0.81

(0.178) (0.586) (0.722) (0.741)

1985.IV weeks IO-13:

7 13 20 26

(7, (13, (20, (26.

350) 338) 324) 312)

1.83 1.11 1.07 0.91

(0.079)* (0.349) (0.376) (0.589)

2.36 1.44 1.17 1.06

(0.023)* (0.137) (0.274) (0.388)

1986.IV weeks I-5:

7 13 20 26

(7, (13, (20. (26,

454) 442) 428) 416)

0.82 1.14 0.99 0.82

(0.572) (0.323) (0.467) (0.728)

0.53 0.41 0.63 0.60

(0.817) (0.965) (0.887) (0.944)

1986.IV Ireeks 6-9:

7 13 20 26

(7, (13, (20. (26,

376) 364) 350) 338)

I .02 1.03 1.1 I 1.22

(0.420) (0.422) (0.341) (0.216)

0.75 0.66 0.84 1.03

(0.634) (0.805) (0.666) (0.432)

1986.IV weeks 10-13:

7 13 20 26

(7, (13, (20. (26,

350) 338) 324) 312)

1.92 1.31 1.05 0.97

(0.065); (0.203) (0.404) (0.511)

0.95 1.61 1.29 1.03

(0.470) (O.OSO)* (0.183) (0.421)

‘M,b

DF’

6 12 18 24

( 12, 406) ( 18, 394)

1984.IV weeks 6-9:

6 12 18 24

1984.1V weeks IO-131

I98S.IV

1985SV

1984.IV weeks I-5:

-

(6, 418)

“Approximate marginal signilicance levels appear in parentheses beside each F-statistic presented. Any marginal significance level below 0.10 (or statistical significance at the 0.10 level) is marked with an asterisk. Ljung-Box white noise tests indicate that we never reject the null hypothesis that the residuals of each equation are white noise. and are omitted for brevity. ‘This column presents the number of 30-minute lags (M,) on the right-hand side variable in each equation of the regression model. Each day in 1984.W had six hours (twelve 30-minute intervals) of trading, whereas almost every day in 1985.N and 1986.1V had six and one half hours (thirteen 30-minute intervals) of trading. Thus, the lag lengths chosen reflect lags of i day, I day, li days, and 2 days for each subsample. ‘This column presents the degrees of freedom on the F-statistics testing the null hypotheses in each row.

394

I.G. Kudlrr

et cd.. Inrruday

rehtionships

both S&P 500 futures and index prices increased from 1984 through 1986. Given these measures, we then conduct Granger tests by relating movements in the intraday variance of futures prices to movements in the intraday variance of index prices. Nonprice actors are included in each equation of the model to enhance the ability of the Granger tests to reveal any systematic lead/lag relationships between volatility movements in the two markets. When daily intervals are employed, these factors include lagged index returns, daily volume data for both the cash equities market and the futures market, a time trend, and dayof-the-week dummy variables. The parameter estimates indicate that daily volume on the NYSE sometimes affects daily movements in index volatility but never affects futures volatility. In contrast, on days when there is more volume in the futures market, futures volatility is systematically higher for all three sample periods and index volatility is higher in 1986. This result is consistent either with greater futures activity producing greater stock price volatility or higher stock price volatility requiring a greater reliance on futures for risk management. Index returns, the time trend and day-of-week dummies reveal no systematic relationships with either index or futures volatility. When 30-minute intervals are employed, coefficient estimates for the nonprice factors indicate no substantial trend in futures volatility, but significant positive trend in index volatility throughout the fourth quarters of 1985 and 1986. The intraday 30-minute dummy variables also reveal a robust daily pattern in which both index and futures price volatility increase substantially just prior to closing each day, and increase further just after opening the next day. The interruption of trading has the apparent effect of fostering higher volatility immediately before U.S. markets close, and still higher volatility immediately after U.S. markets open. Finally, Granger tests reveal no systematic pattern of futures volatility leading index volatility. Depending on the time interval considered, intraday volatility leads run in one or the other direction, in both directions, or in neither direction. The fundamental implication is that volatility relationships are not robust within or across the different nearby futures contract periods examined. Any systematic linkages that might occur apparently arise within a very short time frame.

Appendix A

The following model is designed to show that, even if the level of one nonstationary price series (x,) leads the level of another (y,) movements in the variance of y, may still lead movements in the variance of x,. In this case, let .Y, be the futures price series and y, be the associated cash price series.

!.G. Kawaller

rt al.. Inrraday

395

relationships

Assume x,=s,+e,,

and

(A.1) (A.2)

where s,, e,,, and ezt are random variables, with s, independent Let var (s,) = oz, where 0: is constant over time (s, is stationary). Then assume

of e,, and e,,.

e,,=xe,,_,+~‘,,

with e 2,hN(0,a:,),

(A.3) al-N(0,

var (e,,) = a$,,

ai), and with e,,_,

while

and U, independent.

Then (A.4) (A.%

Note that the variances of et, and e,, are time dependent, and therefore indexed over I, while 0: is constant. Because e,, and e2*‘are nonstationary, x, and y, are also nonstationary. From the above we have the following results:

yt=x,-1

+Et,

where

s,=e,,-e,,_,.

(2) var(x,)=af+x,o:,_,+a,2. (3) var (yJ = 0: + a:,,

(A.7) such that

OS,= var (y,) - ~7:. (4) Substituting

(A.61

(A.8)

(A.8) into (A.7),

=(l-r2)o,2+a,Z+a2var(y,_,).

(A.9)

Eq. (A.6) shows that x, leads y, in this model through the impact of s, in (A.l) and (A.2). Eq. (A.9) shows that movements in the variance of yr lead

396

I.G. Kawullrr

rt al., Intraday

relarionships

movements in the variance of e,,, from (A.5). Hence, the important assumption for this result appears in eq. (A.3). It is easy to imagine a relationship between e,, and e2, such as (A.3); lagged noise in the cash market could influence the level of the futures price and therefore the variance of the futures price, even though the level of the futures price leads the level of the cash price as in (A.2). There is no analytical reason why the assumption in (A.3) cannot hold, and hence no reason why this result might not appear in practice. An important implication of this model is that, whether or not the level of one variable leads movements in the level of another, the question of the nature of any dynamic relationship between their variances is an empirical issue. not a theoretical one.

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