UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN BOOKSTACKS

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2 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN BOOKSTACKS

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5 FACULTY WORKING PAPER NO Volatility in Stock Index Futures and the Informational Content of Option Prices Hun Y. Park R. Stephen Sears JHEUBR^oE m JUL College of Commerce and Business Administration Bureau of Economic and Business Research University of Illinois, Urbana-Champaign

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7 BEBR FACULTY WORKING PAPER NO College of Commerce and Business Administration University of Illinois at Urbana- Champaign June 1984 Volatility in Stock Index Futures and the Informational Content of Option Prices Hun Y, Park, Assistant Professor Department of Finance R. Stephen Sears, Assistant Professor Department of Finance

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9 Abstract This paper empirically investigates the behavior of volatilities of stock index futures as reflected in the prices of option contracts on the futures. In contrast to previous studies measuring the volatilities of commodities and metals futures with limits on daily changes in futures prices on an ex-post basis, this paper analyzes volatility changes in prices of stock index futures contracts without such limits on daily changes in futures prices on an ex-ante basis. For the period examined, the empirical results reveal that the volatility of stock index futures which is implied in the values of options traded on those futures is changing in a systematic pattern over the lives of the options and futures contracts, with the characteristic being declining as the maturity date approaches. This result is contrary to the Samuelson hypothesis which states that the volatility of futures prices should increase as the contract approaches its maturity.

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11 Volatility in Stock Index Futures and the Informational Content of Option Prices A controversial issue concerning the behavior of futures prices is how and why the futures price volatility changes over the contract's life. At least two hypotheses have been developed as to why futures price volatility will change over time. One hypothesis, attributed to Samuelson (1965), postulates that futures price volatility should increase as the time to maturity decreases. However, this hypothesis is based on the assumptions that the futures price is an unbiased estimate of the expected spot price and that the spot price follows a first-order autoregressive process. Recently, research by Anderson and Danthine (1980), Richard and Sunderesan (1981) and Stein (1979) has begun to investigate the volatility-maturity issue within the context of demand/supply uncertainty, hypothesizing that the variance may change over time depending upon the distribution of certain underlying state variables. These analyses examine the volatility of futures prices within a framework of the simultaneous determination of cash and futures prices in market equilibrium. Although empirical analyses of these two hypotheses are mixed, the general consensus is that the volatility of futures prices is changing over the life of the contract (see Anderson (1982), Castelino and Francis (1982), Miller (1979), and Rutledge (1976)). The purpose of this paper is to empirically investigate the behavior of stock index futures volatility as reflected in the prices of option contracts on the New York Stock Exchange (NYSE) and Standard

12 -2- and Poors (S&P) 500 stock index futures. This paper differs from previous analyses of futures price variability in two respects. First, prior studies have examined the volatility for commodities and precious metals while this study examines the newly created stock index futures contracts. One of the factors which should not be ignored in an attempt to examine futures price volatility is the existence of limits on the daily changes in futures prices. However, unlike other commodities, the NYSE and S&P 500 stock index futures have no such,.. 2 limits. Second, prior research regarding futures price volatility has in one way or another measured the volatility on an ex-post basis. This analysis evaluates volatility changes on an ex-ante basis, as revealed in the prices of options on stock index futures contracts. For the period examined in this study, the results indicate that the volatility of stock index futures which is implied in option values is changing in a systematic pattern over the lives of the option and future contracts. Ex-ante evidence of futures price volatility should be important to both academicians as well as practitioners. First, since futures prices themselves are market expectations, the knowledge of changes in the volatility of futures prices would affect the design of statistical tools and the interpretation of the results in testing for market efficiency. Second, hedging activities are guided by price and volatility expectations; therefore, an ex-ante measure of volatility provides investors and portfolio managers with valuable information in establishing appropriate hedges, particularly anticipatory hedges (see

13 -3- Figlewski (1983), Figlewski and Kon (1982), and Moriarity, Phillips and Tosini (1981)). Finally, in light of Rutledge's (1978) finding that changes in trading volume respond to changes in price volatility, the ex-ante estimation of the variability will provide information on market activities in the future. Section I examines the informational content of options and describes the market for options on stock index futures along with a theoretical pricing model developed by Black (1976). Section II describes the data base and methodology employed while Section III presents the empirical results. A brief summary is contained in Section IV. I. Informational Content of Options on Stock Index Futures Several studies have examined the informational content of stock options for a variety of issues (see Chiras and Manaster (1978), Latane and Rendleman (1976), Manaster and Rendleman (1982), Patell and Wolf son (1979) and Schmallensee and Trippi (1978)). Of particular interest in this study is that modern option pricing theory (Black and Scholes (1972, 1973)) posits that an option contains an estimate of the expected volatility of the underlying asset. In the Latane and Rendleman study (1976), it was found that the ex-ante implied standard deviation (ISD) in an option was highly correlated with the ex-post future standard deviation of changes in the price of the underlying security. In light of the results which demonstrate instability in futures price volatility, an option on a futures becomes a useful tool in assessing the ex-ante volatility.

14 -4- In February 1982, the Commodity Futures Trading Commission (CFTC) approved the trading of futures contracts on the Value Line Index at the Kansas City Board of Trade. This action was quickly followed by the introduction of futures contracts on the S&P 500 Index (Chicago Mercantile Exchange) and the NYSE Index (New York Futures Exchange) in April and May of 1982, respectively. These stock index futures contracts differ from other physical commodity futures contracts because of their cash settlement procedure. These new futures contracts have maturity dates in the months of March, June, September, and December. In 1983, the CFTC approved the trading of options on stock index futures contracts. Options on the S&P 500 futures are traded on the Chicago Mercantile Exchange while the NYSE futures are traded on the New York Futures Exchange. These options share the same maturity months as the corresponding futures contracts. A call (put) option on a futures contract conveys the right to go long (short) in a futures contract at a specific price (called exercise or striking price) during a specified time period. A pricing model for call options on futures contracts was developed by Black (1976) under the same assumptions as the original Black-Scholes (1973) model and is given in equation (1): C = FN(d.) - XN(d 2 ) (1) where: C = the value of a call option on a futures contract F = the present value of the futures price

15 -5- X = the present value of the exercise price = the price of a default-free discount bond which pays the exercise price on the expiration date d = ln(f/x)/a /t~ +.5a /F d 2 " d l " 'J* 2 a = instantaneous variance of percentage changes in futures prices t = time to expiration of the option N(*) = cumulative normal distribution Because all of the variables in the model are directly observable, 2 except a, equation (1) can be used to derive the implicit volatility present in the call option value, C. As such a time series of option prices can reveal the anticipated changes in futures price variability. II. The Data and Methodology Daily closing call option and underlying futures price data for both the NYSE and S&P stock indexes were gathered from the Wall Street Journal for the period: January 28, 1983-June 24, 1983 for options maturing in March, June, September, and December of January 28, 1983 marks the first trading day for options on the stock index futures. This period provides a total of 2156 observations for the NYSE options and 1444 observations for the S&P 500 options. Interest rates on United States Treasury Bills were gathered from the Wall Street Journal and updated daily. An average of the bid and asked discount rate for the Treasury Bill having a maturity closest to the

16 -6- expiration date of the option was calculated and converted to an equivalent bond yield. An important issue in previous empirical studies of stock option pricing models is the technique which is used in the estimation of the expected volatility on the underlying security. Previous approaches include, among others, calculating: 1) a historical estimate from ex-post price changes (Black and Scholes (1973)) 2) using a weighted ISD (Schmallensee and Trippi (1978), Latane and Rendleman (1976), Chiras and Manaster (1978), Patell and Wolf son (1979) and MacBeth and Merville (1979)) Schmallensee and Trippi and Patell and Wolfson calculate the implied standard deviation which would make the model price equal to the observed price for each option and then employ a simple equallyweighted arithmetic average of the ISD regardless of maturity. Latane and Rendleman weight each individual volatility estimate by the partial derivative of the model option price with respect to the ISD. Chiras and Manaster and MacBeth and Merville use a relative weighting technique following somewhat the same logic. This paper employs the 3 technique introduced by Whaley (1982) which: 1) computes an implicit weighting scheme that yields an estimate of the volatility which minimizes the sum of squared errors and 2) calculates the implied volatility at time t-1 to circumvent the selection bias problem pointed out by Phillips and Smith (1980), By segregating options according to contract maturity date, the Whaley technique can be employed to yield a separate ISD for each maturity option. Repeating this procedure on each day yields a time series of ISD's for each contract maturity which can be used for the

17 -7-4 ex-ante investigation of changing volatility of futures prices. We now examine the time series behavior of the implied volatilities. III. Empirical Results For each futures contract maturity, implied standard deviations are computed daily using a tolerance criterion of K =.0001 (see footnote 3). Figures 1 and 2 illustrate the daily behavior of the implied standard deviations for each maturity cycle in the NYSE and S&P 500 options, respectively. The first trading date (January 28, 1983) for options on the stock index futures is designated as time 1. In general, the graphs show a wide divergence in the implied volatilities of the different contracts on a given day. More importantly, the longer time to maturity of a given contract, the higher the variability implied in the option value. It appears that investors are uncertain about the true future variability of the market (e.g., NYSE and S&P 500 futures) and thus translate alternative variability estimates into the prices of options written on futures of different maturities. In the absence of limits on the daily changes in futures prices of both markets (NYSE and S&P 500), these two figures provide ex-ante market perceptions on changes in the volatility of stock index futures prices, with a characteristic being declining over the lives of each of the contracts examined during the period. Statistically, the relationship between time to maturity and the ex-ante estimation of stock index futures price volatility is significantly positive as shown in Table I. Table I summarizes the regression results of the relationship between the implied standard deviation and the time to maturity. The regression results have been corrected for

18 the positive serial correlation present in the residuals. As illustrated in Figures 1 and 2 and tested in Table I, the relationship between 1SD and maturity is positive for the sample as a whole and for each option contract maturity. It is particularly strong in the two June contracts, where time to maturity explains about two-thirds of the futures price volatility. Contrary to the Samuelson hypothesis, these results are consistent with the view that the implied volatility would be revised as anticipated information is released (see Patell and Wolf son (1979) and Whaley (1982)). As there might be many anticipated information arrivals, a variety of frequently changing implied volatilities would be anticipated with the characteristic being a higher implied volatility for longer-maturity options. The purpose of this paper is not to claim that the time to maturity is necessarily the only determinant of the stock index futures price volatility. As recent research has shown, there are several possible economic reasons why futures price volatility may change over the life of the contract. Our interest is in the examination of how the market perceives (ex-ante) changes in stock index futures price variability as revealed in the values of options traded on such contracts. Since options on stock index futures are relatively new securities, more general conclusions on the volatilitymaturity issue can be drawn as the trading history for these instruments becomes longer. Nevertheless, this paper provides strong evidence that the volatility of futures price changes decreases as the time to maturity increases, at least on ex-ante basis.

19 -9- IV. Summary This paper empirically investigates the behavior of volatilities of stock index futures as reflected in the prices of option contracts on the futures. In contrast to previous studies measuring the volatilities of commodities and metals futures with limits on daily changes in futures prices on an ex-post basis, this paper analyzes volatility changes in prices of stock index futures contracts without such limits on daily changes in futures prices on an ex- ante basis. For the period examined, the empirical results reveal that the volatility of stock index futures which is implied in the values of options traded on those futures is changing in a systematic pattern over the lives of the options and futures contracts, with the characteristic being declining as the maturity date approaches. This result is contrary to the Samuelson hypothesis which states that the volatility of futures prices should increase as the contract approaches its maturity.

20 -10- Footnotes It has been shown that futures prices are not necessarily unbiased estimates of expected spot prices even in an efficient market (see Richard and Sunderesan (1981)).? Prior to the introduction of options on the S&P 500 futures, limits were imposed on the daily changes in S&P 500 stock index futures. However, at the time options on the S&P 500 stock index futures were introduced, the limit was removed. 3 Whaley's procedure to estimate the ISD can be summarized in the following manner. First, options written on the same security (at a given point in time) can be expressed as: C. = C(a). + e. (a) J J J where: C. = the market price of option j C(a). = the model price J e. = the residual 3 An estimate of o" is determined by minimizing the sum of the squared observed residuals,.. An iterative (non-linear) technique is used J to minimize the sum of the squared residuals by first obtaining an initial estimate of a by using a Taylor series approximation: C. = C(0. + 8C./3a_(a-o\J higher order terms + e. (b) J j j 3 where a n = initial value of volatility a = true volatility Assuming that the higher order terms are trivial, (b) can be written as: C. - C(a n ). = 3C./3a n (a-o n ) + e. (c) J j j j An estimate of the volatility, a, is found by applying OLS repeatedly until the estimate satisfies an accepted tolerance K: (a-a )/a < K where K = We use the same criterion as Whaley's in this paper.

21 -11-4 In the derivation of equation (1), Black (along the same lines as 2 Black and Scholes) assumes that the variance in futures prices a, is constant over the life of the options. However, as discussed in Patell and Wolf son (1979) and Whaley (1982), one should expect empirical analyses of ISD's to demonstrate variation because of such factors as the market anticipation of new information arrival.

22 -12- Bilbiography Anderson, R. (1982): "The Determinants of the Volatility of Futures Prices," Columbia University CSFM Working Paper #33. and Danthine, J. (1980): "The Time Pattern of Hedging and the Volatility of Futures Prices," Columbia University CSFM Working Paper #7. Bigraan, D., Goldfarb, D., and Schechtraan, E. (1983): "Futures Market Efficiency and the Time Content of the Information Sets," The Journal of Futures Markets, 3: Black, F. (1976): "The Pricing of Commodity Contracts," Journal of Financial Economics, 3: and Scholes, M. (1972): "The Valuation of Option Contracts and a Test of Market Efficiency," Journal of Finance, 27: (1973): "The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81: Castelino, M. and Francis, J. (1982): "Basis Speculation in Commodity Futures: The Maturity Effect," The Journal of Futures Markets, 2: Chiras, D. and Manaster, S. (1978): "The Informational Content of Option Prices and a Test of Market Efficiency," Journal of Financial Economics, 6: Cornell, B. (1980): "The Relationship Between Volume and Price Variability in Futures Markets," The Journal of Futures Markets, 1: Figlewski, S. (1983): "Hedging With Stock Index Futures: Theory and Application in a New Market," Columbia University CSFM Working Paper #62. and S. Kon (1982): "Portfolio Management with Stock Index Futures," Financial Analysts Journal, 20: Latane, H. and Rendleman, R. (1976): "Standard Deviations of Stock Price Ratios Implied by Option Preraia," Journal of Finance, 31: MacBeth, J. and Merville, L. (1979): "An Empirical Examination of the Black-Scholes Call Option Pricing Model," Journal of Finance, 34:

23 -13- Manaster, S. and Rendleraan, R. (1982): "Option Prices as Predictors of Equilibrium Stock Prices," Journal of Finance, 37: Miller, K. (1979): "The Relation Between Volatility and Maturity in Futures Contracts," in R. Leuthold (ed.) Commodity Markets and Futures Prices, Chicago Mercantile Exchange, Modest, D. and Sundaresan, M. (1983): "The Relationship Between Spot and Futures Prices in Stock Index Futures Markets: Some Preliminary Evidence," The Journal of Futures Markets, 3: Moriarity, E., Phillips, S., and Tosini, P. (1981): "A Comparison of Options and Futures in the Management of Portfolio Risk," Financial Analysts Journal, 19: O'Brien, T. and Schwartz, P. (1982): "Ex-Ante Evidence of Backwardation/Contango in Commodity Futures Markets," The Journal of Futures Markets, 2: Patell, J. and Wolf son, M. (1979): "Anticipated Information Releases Reflected in Call Option Prices," Journal of Accounting and Economics, 1: Phillips, S. and Smith, C. (1980): "Trading Costs for Listed Options: The Implications for Market Efficiency," Journal of Financial Economics, 8: Richard, S. and Sundaresan, M. (1981): "A Continuous Time Equilibrium Model of Forward Prices and Futures Prices in a Multigood Economy," Journal of Financial Economics, 9: Rutledge, D. (1976): "A Note on the Variability of Futures Prices," Review of Economics and Statistics, 58: (1978): "Trading Volume and Price Volatility: New Evidence on the Price Effects of Speculation," International Futures Trading Seminar, Vol. 5, Chicago Board of Trade: Samuelson, P. (1976): "Is Real-World Price a Tale Told by the Idiot of Chance," Review of Economics and Statistics, 58: (1965): "Proof that Properly Anticipated Prices Fluctuate Randomly," Sloan Management Review, 6: Schmallensee, R. and Trippi, R. (1978): "Common Stock Volatility Estimates Implied by Option Premia," Journal of Finance, 33: Stein, J. (1979): "Spot, Forward and Futures," Research in Finance, 1:

24 -14- Whaley, R. (1982): "Valuation of American Options on Dividend-Paying Stocks: Empirical Tests," Journal of Financial Economics, 10: 29-58, Wilson, J. (1982): "Comment on Ex-Ante Evidence of Backwardation/ Contango in Commodities Futures Markets," The Journal of Futures Markets, 2: D/230

25 1 1 CY C720C ^" O CD 5 1 O1, P", 1- if' fc rti, n": &J 1 fi* " aftop x i X h k QE x^ I i4\ Vj feci l TIME Figure 1 Implied Standard Deviations (ISD) Across Time (measured in days) for Options on NYSE Futures Legend: Mar 83, June 83, f Sept 83, K Dec 83

26 . w L/ w ^ _/ i C7900r C72 - c c; n pi 030CO TIME Figure 2 Implied Standard Deviations (ISD) Across Time (measured in days) for Options on S&P 500 Futures legend: C Mar 83, G June 83, -f- Sept 83, X Dec 83

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