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1、JOURNAL OF FINANCIAL AND OUANTITATIVE ANALYSIS VOL.25,NO.4.DECEMBER 1990The Dynamics of Stock Index and Stock Index FuturesRetumsHans R.Stoll and Robert E.Whaley*AbstractIn rational,efficiently functioning markets,the returns on stock index and stock indexfutures contracts should be perfectly,contem

2、poraneously correlated.This study investi-gates the time series properties of 5-minute,intraday retums of stock index and stockindex futures contracts,and finds that S&P 500 and MM index futures retums tend to leadstock market retums by about five minutes,on average,but occasionally as long as 10min

3、utes or more,even after stock index retums have been purged of infrequent tradingeffects;however,the effect is not completely unidirectional,with lagged stock index re-tums having a mild positive predictive impact on futures retums.I.IntroductionIn spite oftheir relatively short history,the stock in

4、dex futures contract mar-ket has become a controversial topic of discussion and debate.Beginning before,but particularly since the stock market crash of October 19,1987,stock indexfutures,index arbitrage,and program trading have been blamed for excessivestock market price swings.Many governmental an

5、d academic studies have ex-amined intraday pattems of index futures and stock price changes in the dayssurrounding and including October 19;however,little empirical analysis oftheintraday comovement of the prices of index futures and stocks in more nonnalperiods has appeared.2 Owen Graduate School o

6、f Management,Vanderbilt University,Nashville,TN 37203,andThe Fuqua School of Business,Duke University,Durham,NC 27706,respectively.This researchwas supported by a joint grant from the Chicago Board Options Exchange,the Chicago Board ofTrade,and the Chicago Mercantile Exchange.Helpful comments and su

7、ggestions by seminar partici-pants and colleagues at Duke University,The Ohio State University,Washington University,andYork University are gratefully acknowledged.In particular,comments by Kai Cohen,Doug Foster,Ken French,Larry Harris,Campbell Harvey,Michael Hemler,Naoki Kishimoto,John Long,JFQAMan

8、aging Editor Paul Malatesta,Tom Smith,Charles Whiteman,and an anonymous JFQA refereegreatly improved the paper.Very capable computer work was provided by Sunil Paremeswaran andJohn Sprow.Whaley also acknowledges the support of the Unisys Fund at The Fuqua School ofBusiness.1 Detailed plots of day-by

9、-day prices are contained in the Brady Commission Report(1988),theCFTC Report(1988),and the SEC Report(1988).A systematic examination ofthe relation betweenthe price movements of the S&P 500 and its futures contract on October 19 is contained in Harris(1988).2 Stoll and Whaley(1986),(1987)examine mi

10、nute-by-minute price change behavior of the S&P500 and Major Market indexes in the days surrounding the expirations of the S&P 500 and MMI441442 Journal of Financial and Quantitative AnalysisThe purpose of this paper is to model empirically the temporal relation be-tween the price movements of index

11、 futures contracts and stocks.In the process,the paper provides insights on the volatility of index futures versus stock indexesand the extent to which futures overshoot true values.The paper is distinguishedfrom prior work in several ways.First,a longer time intervalfive yearsanda finer retum gridf

12、ive minutesis examined than in other papers.Second,thedelay in the price change of a stock index due to the infrequent trading of thecomponent stocks is treated explicitly.A theoretical model of observed portfolioretums that incorporates the effects of infrequent trading and stock bid/askspreads is

13、developed and estimated,with the residuals of this model(i.e.,retuminnovations)then used to proxy for true stock index retums.To verify theresults,the retums of the most actively traded individual common stockIBMalso are used as a proxy for true index retums.Third,both the ChicagoMercantile Exchange

14、s S&P 500 and the Chicago Board of Trades Major Mar-ket Index futures contracts are considered.The paper is organized as follows.The theory underlying the covariation ofstock index futures and stock index retums is outlined in Section II,and the ef-fects of infrequent trading and transaction costs a

15、re identified.In Section III,thedata are described.All tests are based on 5-minute,intraday retums.The investi-gation period begins with the introduction of the S&P 500 futures contract by theChicago Mercantile Exchange in April 1982,extends through the introduction ofthe Major Market Index futures

16、contract by the Chicago Board of Trade on July23,1984,and ends on March 31,1987.Infrequent trading and bid/ask priceeffects are examined in Section IV.Serial correlations of retums in stock in-dexes,stock index futures,and IBM reveal the expected infrequent trading andbid/ask pattems.The effects of

17、infrequent trading and bid/ask spreads on theobserved structure of retums are tnodeled and estimated,and then retum innova-tions are generated.In Section V,the temporal relation between the futures andstock retums is estimated in a multiple regression framework.Generally speak-ing,the retums in the

18、futures market lead those in the stock market,even afteradjusting for the infrequent trading of stocks.The evidence also shows that thelead has diminished through time.The paper concludes in Section VI with a sum-mary.II.TheoryThe theoretical relation between the price of an index futures contract a

19、ndthe price level ofthe underlying index is,(1)F,=s/futures contracts,but do not examine nonexpiration days.Kawaller,Koch,and Koch(1987a),(1987b)use intraday data to examine price changes of the S&P 500 index and index futures;how-ever,they examine only three quarters ofthe data and do not correct f

20、or infrequent trading.MacKin-lay and Ramaswamy(1988)also use intraday data,but they focus on deviations from the cost of carryequilibrium.Ng(1987)uses interday data to investigate the price behavior of S&P 500 index futuresprices and its ability to predict the S&P 500 index level.Chan,Chan,and Karoi

21、yi(1990)use intradayprice data to examine the transmission of volatility between the stock and index futures markets.stoll and Whaley 443where F,is the index futures price at time t,S,is the index price at time t,r-d isthe net cost of carrying the underlying stocks in the index,that is,the rate ofin

22、terest cost r less the rate at which dividend yield accrues to the stock indexportfolio holder d.T is the expiration date of the futures contract,so T-tis thetime remaining in the futures contract life.Note that in this formulation the risk-less rate of interest and the dividend yield on the underly

23、ing stock index are as-sumed to be known,constant,continuous rates.The market force driving the cost-of-carry relation(1)is the never-endingsearch for a free lunch.When the futures price is above the level implied bythe right-hand side of(1),a riskless arbitrage profit equal to the difference be-twe

24、en the futures price and the index price plus the cost of carry,a long arbitrageprofit ofF,-S,e-F,can be eamed by buying the futures andselling the portfolio of stocks,investing the proceeds of the sale of stock at theriskless rate of interest.The use of a single,computer-generated order to buy orse

25、ll an entire portfolio of stocks is known as program trading.In perfectly efficient and continuous futures and stock markets absent trans-action costs,riskless arbitrage profit opportunities should not appear so the cost-of-carry relation(1)should be satisfied at every instant t during the futures c

26、on-tract life.If such is the case,the instantaneous rate of price appreciation in thestock index equals the net cost of carry of the stock portfolio plus the instantane-ous relative price change of the futures contract;that is,(2)Rs,t=ir-d)+R,whereas,=Ln(5,/5,_,)and/?,=Ln(F,/F,_,).Several implicatio

27、ns follow from(2)under the assumptions that the short-term interest rate and the dividend yield rate of the stock index are constant andthat the index futures and stock markets are efficient and continuous:(a)The expected rate of price appreciation on the stock index portfolio EiR,)equals the net co

28、st of carry ir-d)plus the expected rate of retum on thefutures contract EiRp,).(b)The standard deviation of the rate of retum on the futures contract equalsthe standard deviation ofthe rate of retum ofthe underlying stock index.(c)The contemporaneous rates of retum of the futures contract and the un

29、der-lying stock portfolio are perfectly positively correlated.(d)The rates of retum ofthe futures contract and ofthe underlying stock indexportfolio are serially uncorrelated.(e)The noncontemporaneous rates of retum of the futures contract and theunderlying stock portfolio are uncorrelated.3 Technic

30、ally speaking,more than an assumption of market efficiency is needed to ensure seri-ally uncorrelated rates of retum.It must also be the case that the expected rates of retum ofthe futuresand stock index are constant(see Fama(1976),pp.149-151).An assumption of constant expectedretums is reasonable h

31、ere since intraday rate of retum series are examined.444 Journal of Financial and Quantitative AnalysisNaturally,all ofthe above implications are based on the assumption that thecost-of-carry relation(1)holds at all points in time.It has been shown,however,that(1)does not hold exactly;indeed one of

32、the puzzles in stock index futures isthe frequency with which deviations from(1)are observed.Stoll and Whaley(1986),Table 23A),for example,report frequent violations ofthe cost-of-carryrelation in excess of transaction costs using hourly S&P 500 index and indexfutures data during the period April 19

33、82 through December 1985.The fre-quency of violation is nearly 80 percent for the June 1982 futures contract;how-ever,for more recent contract maturities,the frequency falls below 15 percent.MacKinlay and Ramaswamy(1988),Table 6)report similar results for the S&P500 futures contracts expiring in Sep

34、tember 1983 through June 1987.Using 15-minute price data,they find that the cost-of-carry relation is violated 14.4 percentofthe time,on average.Violations of the cost-of-carry relation may appear for a variety of reasons.Some are purely technical.An important one is the infrequent trading of stocks

35、within the index.Markets for individual stocks are not perfectly continuous.Consequently,stock index prices,which are averages of the last transactionprices of component stocks,lag actual developments in the stock market.Fisher(1966)describes this phenomenon.Cohen,et al.(1986),Ch.6)give a moregenera

36、l discussion of serial correlation of stock index retums in terms of delaysin the price adjustment of securities.Lo and MacKinlay(1988)model the effectsof infrequent trading on index retums under restrictive assumptions.Assumingthat the index futures prices instantaneously refiect new information,ob

37、servedfutures retums should be expected to lead observed stock index retums becauseof infrequent trading,even though there is no economic significance to this be-havior whatsoever.A second reason for violation of relation(2)is that transaction costs tend toinduce noise in the relation(2).The prices

38、used in the computation of retums aretransaction prices,and these transaction prices tend to fiuctuate randomly be-tween bid and ask levels.This random price movement between bid and askprices in successive transactions induces negative serial correlation in observedretums even though the true retum

39、s are serially independent.t At the individualsecurity level,the negative serial correlation due to the bid/ask price effect isunderstandable,but the effect seems less likely when one considers a stock indexportfolio for which movements between the bid and ask for some stocks could beoffset by oppos

40、ite movements from the ask to bid for other stocks;however,tothe extent that the rates of retum of the stocks in the index are positively corre-lated and/or that the index is narrowly based,negative serial correlation in indi-vidual stock retums attributable to the bid/ask price effect also might ap

41、pear inthe stock index retums.A third reason for violation of the cost-of-carry relation has to do with timedelays in the computation and reporting of the stock index value.Once a transac-tion in the stock market takes place,the transaction information is entered into acomputer and transmitted to th

42、e particular service that updates and transmits the t Roll(1984)develops a simple model to show the relation between the bid/ask spread and theserial covariance in stock retums.Stoll(1989)and others have extended Rolls work.stoll and Whaley 445index level.5 Three time delays are therefore possible:(

43、a)the delay in enteringthe stock transaction into the computer;(b)the delay in computing and transmit-ting the new index value;and,(c)the delay in recording the stock index value atthe futures exchange.Assuming that new information arrives in the stock andfutures markets simultaneously and that pric

44、e changes in the futures market arerecorded instantaneously,such delays would tend to show the futures marketretums leading stock index retums.Finally,lead/lag behavior of stock index and stock index futures retumsmay reflect the greater speed with which investors views are reflected in futuresmarke

45、ts.Investors with strong beliefs about the direction of the market as awhole(as opposed to a trend in the price of an individual stock)may trade indexfutures rather than individual stocks because transaction costs are lower and thedegree of leverage attainable is higher.Such trading moves futures pr

46、ices first,and then pulls stock prices when index arbitrage responds to the deviations fromthe cost-of-carry relation(1).III.DataA.Data Description and SourcesThe data used in this study were obtained from three separate sourcestheChicago Mercantile Exchange(CME),the Chicago Board of Trade(CBOT),and

47、Francis Emory Fitch,Inc.(Fitch).The CME provided the S&P 500 index andindex futures price data for the period April 21,1982,through March 31,1987.These data,referred to as Quote Capture information,contain the time(to thenearest 10 seconds)and price(to the nearest 0.05 index points)of every futurest

48、ransaction in which the price has changed from the previously recorded transac-tion,as well as the S&P 500 index level(to the nearest 0.01 index points)eachtime it is computed and transmitted to Chicago.Prior to June 13,1986,the stockindex was computed approximately once a minute;but,since that time

49、,it hasbeen computed and reported approximately four times per minute.The S&P 500futures are on a quarterly expiration cycle(i.e.,March,June,September,andDecember),and it is always the nearby contract that is the most active in terms oftrading volume.Since the time series tests require the most freq

50、uent retum obser-vations possible,only the data for the nearby futures contract are used.The CBOT provided the Major Market Index(MMI)and MMI futures pricedata for the period July 23,1984,through March 31,1987.Like the Quote Cap-ture information from the CME,the data file contains the time(to the ne