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 FACULTY WORKING 
 PAPER NO. 1011 
 
 Stochastic interest Rates, Changing 
 Volatility and the Pricing of 
 Options on Stock Index Futures 
 
 Hun Y. Park 
 
 ft Stephen Sears 
 
 College o1 Commerce and Business Adminisiral 
 Bureau ..>- ; Economic and Business Research 
 cis. Urbana-Cbam r -3ian 
 
BEBR 
 
 FACULTY WORKING PAPER NO. 1011 
 College of Commerce and Business Administration 
 University of Illinois at Urbana- Champaign 
 February 1984 
 
 Stochastic Interest Rates, Changing Volatility and the 
 Pricing of Options on Stock Index Futures 
 
 Hun Y. Park, Professor 
 Department of Finance 
 
 R. Stephen Sears, Professor 
 Department of Finance 
 
 This research is supported in part by the Investors in 
 Business Education and the University Research Board at the 
 University of Illinois. We gratefully acknowledge the compu- 
 tational assistance of Messrs. Chen-Chin Chu and Prabir Datta. 
 This is a preliminary draft and is not for quotation. Comments 
 are welcome. 
 
Digitized by the Internet Archive 
 
 in 2011 with funding from 
 
 University of Illinois Urbana-Champaign 
 
 http://www.archive.org/details/stochasticintere1011park 
 
Abstract 
 
 Black [2] has recently formulated a model for the pricing of options 
 on futures contracts under the assumption that futures contracts are 
 equivalent to forward contracts. Since futures and forward contracts 
 do differ because of the effects of changing interest rates, the vola- 
 tility which is implied in the value of the option may be changing over 
 the life of the option. This paper investigates the impact of changing 
 volatilities on the pricing of options on futures by comparing the pric- 
 ing performance of the Black, model under the two alternative assumptions 
 of constant and changing implied volatility. The empirical results, 
 using options on New York Stock Exchange futures contracts, provide 
 motivation for the development of a theoretical pricing model based on 
 variable interest rates. 
 
Stochastic Interest Rates, Changing 
 Volatility and the Pricing of Options on 
 Stock Index Futures 
 
 The original Black-Scholes [4] stock option pricing model assumes 
 that the interest rate is constant over the life of the option. Merton 
 [11] relaxed this assumption by introducing a variable interest rate 
 option pricing model in which the capital market perceives a continuous 
 sample path for equilibrium stock and default-free discount bond prices 
 so that the option price is a linear homogeneous monotonic function 
 of the ratio of the price of the underlying stock to the price of a 
 default-free discount bond. Merton [11] has shown the Black-Scholes 
 model to be a special case of his model in which the instantaneous bond 
 price variance and the covariance between the stock and the bond price 
 are both zero. 
 
 Using the same assumptions as Black and Scholes, Black [2] has 
 developed a formula for pricing call options on futures contracts under 
 the assumption that futures contracts are equivalent to forward con- 
 tracts. This model differs from the original Black-Scholes model only 
 in that the current stock price is replaced by the present value of the 
 futures price. This transformation comes from the intuition that an 
 investment in a futures contract requires no commitment of funds (see 
 also [1], [12]). 
 
 Recent research ([5], [8] and [15]), however, has demonstrated that 
 a futures contract will differ from a forward contract because of the 
 "marking-to-market" effect, which in turn is a function of changing 
 interest rates. Empirically, as Figure 1 illustrates, the volatility 
 
-2- 
 
 o 
 o 
 
 oo_ 
 
 o 
 o 
 
 CD _| 
 
 O 
 
 o 
 
 o 
 a 
 
 r\j J 
 
 a 
 
 c. 
 
 a 
 
 a 
 
 oo 
 
 o 
 
 a 
 
 CO 
 
 c 
 a 
 
 a 
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 "\j. 
 
 0. 00 
 
 October 6, 1979 
 
 « 
 
 
 
 40. CO SO. 00 
 
 120. 00 
 
 160. 00 
 
 I ME 
 
 200.00 240.00 280.00 
 
 Figure 1. 
 
 Weekly Averages of Six-Month 
 Treasury Bill Yields for the Period 
 January, 1978 - Sepr^ber. 1981. 
 
 
-3- 
 
 of short-term interest rates has changed dramatically over the past few 
 years. This is due in part to a change in the Federal Reserve's policy 
 (October, 1979) of monitoring monetary aggregates rather than interest 
 rates. As such, it no longer seems reasonable to assume interest rates 
 are constant and thus independent of futures price movements. Because 
 the difference between futures and forward prices is affected by the 
 covariability between futures and bond prices, it is apparent that the 
 Black model may be misspecified because of this effect. 
 
 The purpose of this paper is to examine the efficiency of the 
 Black model in the pricing of options on stock index futures and the 
 impact of changing implied volatility, which is possibly due to a 
 stochastic interest rate, on the pricing of such options. To facili- 
 tate this test, the Black model is examined under two alternative 
 scenarios: constant and varying implied volatilities. The results 
 indicate that the volatility which is implied in the value of an op- 
 tion on the NYSE stock index is changing over the life of the option. 
 Recognizing this in the pricing process produces option prices signifi- 
 cantly closer to their actual market values than is the case under the 
 assumption of constant volatilities. These empirical results provide 
 motivation for the development of a pricing model for options on 
 futures along the lines of Merton's variable interest rate stock option 
 model. 
 
 Section I describes the market for options on stock index futures 
 and how the changing volatility can affect the pricing of such options. 
 Section II describes the data base and methodology while Section III 
 presents the results. A brief summary is contained in Section IV. 
 
-4- 
 
 I. The Pricing of Options on Stock Index Futures 
 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 and 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 con- 
 tracts 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 these new futures 
 contracts. Cptions on the Value-Line futures are traded on the Kansas 
 City Board of Trade, and options on the S and P 500 futures and the 
 NYSE futures are traded on the Chicago Mercantile Exchange and the New 
 York Futures Exchange, respectively. 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 [2] under the same assumptions as the original Black-Scholes 
 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 
 
-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 it 
 
 d 2 = i x - O f /I 
 
 2 
 a = instantaneous variance of percentage changes in futures 
 
 prices 
 t = time to expiration of the option 
 N(*) = cumulative normal distribution 
 The impact that stochastic interest rates can have upon the pricing 
 of options on futures can be examined within the context of Merton's 
 variable interest rate stock option model. When interest rates are 
 stochastic, the variance implied in the value of an option on a futures 
 can be specified as: 
 
 x = / S 2 (t)dt (2) 
 
 
 
 where 
 
 2 2 2 
 S - <J f + a b - 2p Vb 
 
 2 
 a = instantaneous variance of percentage changes in bond prices 
 
 p = correlation between changes in bond prices and futures prices 
 
 Note that the variance in Black's model is a special case of this spe- 
 cification in which a = p = and a. is constant. 
 
 b r 
 
 The variable x, called a "time" variable by Merton himself, is 
 worthy of special notice, since all of the variables are directly 
 observable in the market except the variance variable. Conceivably, 
 the "time" variable can be split into two components: a pure time 
 
-6- 
 
 component and a volatility component. The former can be expressed as 
 a fraction of a year and the latter as an annualized cumulative variate 
 rate. In this manner, Black's model can be interpreted to be derived 
 under the assumption that the volatility component is constant. In 
 extending the Black-Scholes stock option model to include the possi- 
 bility of stochastic interest rates, Merton [11] demonstrates that 
 the correct maturity to form perfect hedge positions is the one which 
 matches the maturity date of the options. (Note that in the Black- 
 Scholes model, it does not matter whether hedgers borrow or lend long 
 or short maturities since the model assumes a constant interest rate.) 
 Given the empirical evidence of stochastic interest rates (see Figure 
 1) and thus non-constant instantaneous volatilities, the volatility 
 component of x is expected to be different for options on futures con- 
 tracts of different maturities. Therefore, by splitting the variable 
 t into two components, a time component and a volatility component, the 
 impact of changing volatilities upon the pricing of options on futures 
 can be examined. Empirically, the explanatory power of Black's model 
 under the assumption of constant volatility can be examined by esti- 
 mating the single implied standard deviation (ISD) which best fits 
 groups of options of different maturities. On the other hand, the 
 impact of changing volatility can be examined by estimating ISD's for 
 options segregated by time to maturity. If the change in the under- 
 lying volatility is not important in the pricing of these options, 
 then there should be no difference between the values determined by 
 these two approaches. 
 
-7- 
 
 II. The Data and Methodology 
 Daily closing call option and underlying futures price data for 
 the NYSE stock index were gathered from the Wall Street Journal for 
 
 the period: January 28, 1983-June 24, 1983 for options maturing in 
 
 2 
 March, June, September and December of 1983. January 28, 1983 marks 
 
 the first trading day for options on NYSE index futures. This period 
 
 provides a total of 2156 observations. 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 expiration date of the option was 
 
 calculated and converted to an equivalent bond yield. 
 
 An important issue in previous empirical studies of option pricing 
 
 models is the estimation of the expected volatility on the underlying 
 
 security since, as Black and Scholes suggested, the usefulness of the 
 
 model will depend on the market's ability to make an accurate estimate 
 
 of this parameter. Previous approaches include, among others, 
 
 calculating: 
 
 1) a historical estimate from ex-post price changes (Black and 
 Scholes [3], and Galai [7]) 
 
 2) using a weighted ISD (Schmallensee and Trippi [16], Latane 
 and Rendleman [9] , Chiras and Manaster [6] , Patell and Wolfson 
 [13] and MacBeth and Merville [10]) 
 
 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 equally- 
 weighted 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 
 
-8- 
 
 anci Manas ter and MacBeth and Merville use a relative weighting tech- 
 nique following somewhat the same logic. While the discussion of 
 ail the pros and cons of these techniques is beyond the scope of this 
 paper, prior research results tend to support the superiority of the 
 
 ISD measure, regardless of the weighting scheme. This paper employs 
 
 3 
 
 the technique introduced by Whaley [18] 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 [14] 
 
 The changing volatility issue is examined in the following manner. 
 
 On each day during the period examined, Whaley' s technique is used to 
 
 compute ISD's in two ways: 
 
 1) all options with different exercise prices are lumped together, 
 regardless of time to maturity, and a single ISD is computed 
 
 2) options are segregated according to time to maturity and a 
 different ISD is computed for each time to maturity category 
 
 As such, two model prices are generated for each option in the sample 
 
 where method 1) assumes volatilities are constant and thus prices all 
 
 options with different times to maturity on a given day with the same 
 
 ISD and method 2) recognizes that volatilities may change and prices 
 
 options with different ISD's depending upon the time to maturity. We 
 
 now examine the differences in these two pricing methods. 
 
 III. Empirical Results 
 
 For each futures, implied standard deviations are computed daily 
 using a tolerance criterion of K = .0001 (see footnote 3). Figure 2 
 illustrates the daily behavior of the implied standard deviations for 
 
 * 
 
 each of the four maturity cycles during the period examined. The first 
 
07900+ 
 
 C720O- 
 
 -9- 
 
 orc- 
 
 65C 
 
 <* 
 
 oseccj 
 
 Tfe 
 
 m 
 
 5 i OJ,- 
 
 □a / ^f 
 
 h 
 
 4 4 G 0- 
 
 IP 
 
 . 037004- 
 
 
 X 
 
 m 
 
 k A 
 
 
 3000- 
 
 17 
 
 ?? 
 
 49 
 
 65 
 
 TIME 
 
 Legend: 
 
 Figure 2. Implied Standard Deviations (ISD) 
 Across Time (measured in days) 
 Mar 33, □ June 83, + Sept 83, X Dec 83 
 
 8i 
 
 y / 
 
-10- 
 
 trading date (January 28, 1983) for options on NYSE index futures is 
 designated as time 1. In general, the ISD estimates are declining over 
 the life of each option. Equally importantly, the graph shows a wide 
 divergence in the implied volatilities of the different contracts. In 
 particular, the longer the time to maturity, the higher Che variability 
 implied in the option value. It appears that investors are uncertain 
 about the true future variability of the market (e.g., NYSE futures) 
 and thus translate alternative variability estimates into the prices 
 of options written on futures of different maturities. In light of 
 Merton's time variable, the declining ISD over the life of each option 
 is consistent with the notion that as option approaches maturity, the 
 market perceives less chance for bond price variability and hence co- 
 variability with futures prices. 
 
 Table I summarizes the differences between market prices and model 
 prices of options calculated by the two aforementioned methods. Even 
 though the models tend to misprice options in the same direction, dif- 
 ferences between the two procedures are striking. When a single ISD is 
 used, the average overpricing is 9.26 percent (t = 12.1112). However, 
 even though the segregated (according to time to maturity) approach 
 produces statistically significant mispricings (t = 3.6054), the magni- 
 tude of the average differential is only 1.14 percent. 
 
 We also examine the impacts of exercise price (X/F) and time to 
 maturity (t) on differences between market prices and model prices. With 
 the exception of in-the-money options (X/F < .95), prices based upon 
 segregated maturities (differing volatilities) are significantly closer 
 to actual market values than are prices based upon constant implied 
 
-11- 
 
 Table I 
 
 Differences between Market Prices and Model Prices 
 
 or, ..--r . fcU fc fc i .. • -■ • _ B(raodel) - A(actual) ,^„ 
 
 Average % difference with constant volatility = r - — r - x 100 
 
 A(actual) 
 
 . . i-cc -«.u • i ^ • i • ». M(model) - A(actual) „ ..,.- 
 Average A difference with varying volatility = -~ ^— ^ X 100 
 
 
 
 
 
 Number of 
 
 
 (B-A)/A 
 
 (M-A)/A 
 
 (M-B)/A C 
 
 Observations 
 
 All options 
 
 9.26% 
 
 1.14% 
 
 -7.99% 
 
 2156 
 
 
 (12.1112) 
 
 (3.6054) 
 
 (-9.7780) 
 
 X/F < .90 
 
 -1.55 
 
 -1.58 
 
 -.03 b 
 
 231 
 
 
 (-17.1255) 
 
 (-18.8868) 
 
 (-.2434) 
 
 .90 _<_ X/F < .95 
 
 -.93 
 (-9.9891) 
 
 -1.22 
 (-17.3851) 
 
 -.29 
 (-2.4604) 
 
 516 
 
 .95 <_ X/F < 1.05 
 
 13.12 
 (12.5890) 
 
 2.68 
 (7.1041) 
 
 -10.44 
 (-9.4167) 
 
 1398 
 
 X/F 2. 1.05 
 
 31.42 
 (4.3147) 
 
 -.30 b 
 (-.0924) 
 
 -31.72 
 (-3.1525) 
 
 111 
 
 t < 3 months 
 
 24.80 
 
 1.69 b 
 
 -23.11 
 
 759 
 
 
 (12.47564) 
 
 (1.9784) 
 
 (-10.6745) 
 
 3 months < t 
 < 6 months 
 
 2.53 
 (7.5196) 
 
 1.21 
 (4.9733) 
 
 -1.32 
 (-3.1445) 
 
 743 
 
 t > 6 months 
 
 -1.57 
 (-12.3579) 
 
 .42 
 (3.4036) 
 
 1.99 
 (11.2378) 
 
 654 
 
 Numbers in parentheses are t values 
 
 Not significantly different from zero at the 1% level 
 'Difference between (M-A)/A and (B-A)/A 
 
-12- 
 
 volatilities. (Note, however, that for in-the-money options, the dif- 
 ferences in mispricing between the two alternatives are only .03 percent 
 and .29 percent when X/F < .90 and ,90__X/F < .95, respectively.) 
 Differences between market prices and model prices for different X/F 
 values under the assumption of constant volatility and varying volatility 
 are graphically displayed in Figures 3A and 3B , respectively. As the 
 scales on these figures indicate, the constant volatility (3A) approach 
 can produce large percentage deviations (several are in excess of 1.00 
 (100%)). 
 
 With regard to the time to maturity effect, differences between the 
 two pricing formulations are also significant, most notably in options 
 with maturities less than 3 months. The magnitude of mispricing under 
 the assumption of constant volatility varies much more across the alter- 
 native times to maturity categories than does the one under the varying 
 volatility assumption. For example, the average mispricings across 
 maturities ranges from 24.80% to -1.57% for the constant volatility 
 column. On the other hand, with varying volatilities, the mean mis- 
 pricing levels are quite similar (1.69% to .42%). Since the ISD mea- 
 sured in the first column is an average of the maturity classes' ISD's, 
 it is not surprising that the least amount of relative mispricing 
 (statistically) occurs for intermediate term options (3 months < t < 6 
 months) when all options are pooled to compute the ISD. From Figure 2 
 it can be seen that the intermediate term ISD most nearly approximates 
 the overall ISD. Because the three-month ISD differs more from the 
 other maturity ISD's, a pooled estimate will most seriously misprice 
 these options. This overpricing effect is consistent with the failure 
 
-IS- 
 
 C' 
 
 + 
 
 ) 
 
 
 \ C 
 
 + 
 
 + 
 
 t 
 
 + 
 + 
 
 
 + 
 
 -rr r 
 
 . + 
 
 ^r^- rru ~+* 
 
 + 
 
 ?f 
 
 
 •+ 
 
 O 
 
 ■+W'-i 
 
 I 
 
 C. 80 
 
 Q. 84 
 
 •G. 88 
 
 0. 92 
 
 -F 
 
 Q. 96 
 F 
 
 00 
 
 04 
 
 i. 08 
 
 Figure 3A. Differences between Market Prices and Model Prices 
 for Different X/F Values Under the Assumption of 
 Constant Volatility 
 
-14- 
 
 
 
 I 
 
 + 
 
 + 
 
 -It 
 
 c 
 
 cr 
 i 
 
 
 
 -rt- 
 
 -r +r 
 
 I 
 
 0. 80 
 
 0. 84 
 
 C 
 
 88 
 
 C. 92 
 
 0. 96 
 
 1. DO 
 
 04 
 
 i. 08 
 
 Figure 3B. Differences between Market Prices and Model Prices 
 for Different X/F Values with Varying Volatility 
 
-15- 
 
 of the model to correctly measure a downward changing implied volatility, 
 The mispricing versus time dimension (as measured by months to maturity) 
 is displayed in Figures 4A and 4B under the two pricing formulations. 
 As a final test, all of the observations are pooled and the rela- 
 tionships between mispricing and X/F and time to maturity are examined 
 and presented in Table II. Segregating according to maturity in order 
 to capture the changing volatility reduces significantly the pricing 
 bias associated with the exercise price/futures price ratio and for the 
 sample as a whole eliminates any bias due to time to maturity. 
 
 IV. Summary 
 Interest rate variability has increased dramatically in the last 
 few years. Theoretically, this gives rise to non-constant volatilities 
 which are implied in the values of options traded on futures contracts. 
 Empirical evidence of this non-stationarity can be obtained through an 
 examination of the implied volatilities on option contracts of dif- 
 fering maturities. Correcting for this problem by pricing options with 
 ISD's derived from segregating maturities provides a more accurate 
 assessment of market values. These empirical results provide motiva- 
 tion for the development of a more precise option on futures pricing 
 model — one which incorporates the impact of changing volatility on 
 value estimation. 
 
c 
 
 -16- 
 
 
 + 
 
 ex 
 
 
 \ 
 
 a 
 
 n 
 
 c 
 
 ci 
 
 p~ 
 
 en 
 
 
 LJ 
 
 
 
 C 
 
 
 C 
 
 + 
 
 + 
 
 -nj. 
 
 + + 
 
 t -pr 
 
 + 
 
 + 
 
 T t 
 
 ,1" .-r 
 
 + 
 
 
 ,-f^ 
 
 C 
 
 I- ■"•" i -Jr-^r *r . -r- 
 
 _^r 
 
 
 G. GO 
 
 i. 25 
 
 2. 5Q 
 
 Figure 4A. Differences between Market Prices and Model Prices 
 
 for Different Times to Maturity Under the Assumption 
 of Constant Volatility. Time is measured as the 
 number of months to maturity. 
 
 
-17- 
 
 o 
 
 CD 
 
 O 
 
 O 
 
 03 
 
 \ 
 
 a 
 
 R- CD 
 
 C 
 
 a 
 
 a 
 
 a 
 
 ^> 
 
 a. 
 
 a 
 
 00 
 
 o. 
 
 i 
 
 a 
 
 + 
 
 + 
 
 +H 
 
 1- 
 
 + 
 
 + + 
 
 + C 
 
 + 
 
 + 
 
 + 
 
 
 + 
 
 -r 
 T + 
 
 + 
 
 + 
 
 .^ 
 
 4. 
 
 + + + . + 
 
 w 
 
 
 I J 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 + 
 
 I li in* i 
 
 0. 00 
 
 1. 25 
 
 2. 50 
 
 75 
 
 5. 00 
 
 TIME 
 
 6. 25 
 
 7. 50 
 
 8. 75 
 
 Figure 4B. Differences between Market Prices and Model Prices 
 
 for Different Times to Maturity with Varying Volatility, 
 Time is measured as the number of months to maturity. 
 
-18- 
 
 Table II 
 
 Regression Results' 
 
 Test 
 
 '0 
 
 a. 
 
 R 2 
 
 SSE 
 
 1. (B-A)/A = a Q + a 1 (X/F) 
 
 -2.0643 2.2191 .1077 .1093 
 (-15.4532) (16.1593) 
 
 (M-A)/A = a Q + a 1 (X/F) 
 
 -.4475 .47241 .0275 .0209 
 (-7.6677) (7.8737) 
 
 2. (B-A)/A * a + at 
 
 .2737 -.0439 .0980 .1105 
 (19.7077) (-15.3925) 
 
 (M-A)/A = a Q + o^t 
 
 .0177 -.0001 -.0005 .0215 
 (1.9066) (-.0570) 
 
 numbers in parenthesis are t values 
 not significant at the 1% level 
 
 sum of squared errors 
 
-19- 
 
 Footnotes 
 
 This effect can be seen through an examination of the time 
 variable in the Merton model. Conceivably as the bond approaches 
 maturity, its price converges to the face value; thus, its variance 
 rate and the covariance term approach zero for the same reason. Thus, 
 even though the futures price variance were constant through time, the 
 instantaneous variance of the price ratio need not be constant over 
 time. Another reason for possible differences in volatility across 
 maturities, as noted by Pat tell and Wolf son [13] and cited by Whaley 
 [18], is the effect of anticipated information arrival. For instance, 
 consider the information effect on options for different maturities 
 when the information set is limited to news about inflation. One would 
 not be surprised to see a higher implied standard deviation in options 
 which expire after the anticipated announcement date of news about 
 inflation. As there might be many such anticipated information arri- 
 vals, a variety of frequently changing implied volatilities might be 
 anticipated with the characteristic being a higher implied volatility 
 for longer-maturity options (see [13]). This effect would be stronger 
 
 the greater the uncertainty about inflation (see figure 1). 
 
 2 
 We also obtained daily closing call option and underlying futures 
 
 price data for the S & P 500 index from the Wall Street Journal for the 
 
 same time period. The results for the S & P 500 options do not deviate 
 
 significantly from those for the NYSE options and are not reported here 
 
 to conserve space. They are available from the authors upon request. 
 
-20- 
 
 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(cr). = the model price 
 J 
 
 e. = the residual 
 J 
 
 An estimate of a is determined by minimizing the sum of the squared 
 
 observed residuals, £.. An iterative (non-linear) technique is used to 
 
 J 
 
 minimize the sum of the squared residuals by first obtaining an initial 
 estimate of a by using a Taylor series approximation: 
 
 C. = C(o ). + 3C./8a (a-a ) + ... + higher order terms + e. (b) 
 
 where a = initial value of volatility 
 
 a = true volatility 
 Assuming that the higher order terms are trivial, (b) can be written 
 as: 
 
 A A 
 
 C. - C(a_). = 3C./3a n (a-o A ) + e. (c) 
 
 3 j j j 
 
 An estimate of the volatility, a, is found by applying OLS repeatedly 
 until the estimate satisfies an accepted tolerance K: 
 
 \Co-o Q )/a \ < K 
 
 where K = .0001. We use the same criterion as Whaley's in this paper. 
 
-21- 
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-22- 
 
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 18. R. Whaley. "Valuation of American Options on Dividend-Paying 
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 (1983), pp. 29-58. 
 
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-2- 
 
 PERCEN'T 
 20 
 
 MONEY MARKET HAILS 
 
 rfONTHLT *ve«»8SI OF OAILT FISURES 
 
 PERCENT 
 20 
 
 1978 1979 1930 193 1 1982 1983 
 
 LATEST DATA N.OTTED. SEfTEySER 
 
 fREfARED Br FEDERAL RESERVE BANK OF ST. LOUIS 
 
 Figure 1. Weekly Averages of Six-Month Treasury 
 Bill Yields for the Period: 
 January, 1978 - September, 1983