An Empirical Law of the Stock Option Market

Abel, U.; Boing, G. Article An Empirical Law of the Stock Option Market Zeitschrift für Wirtschaftsund Sozialwissenschaften (ZWS) - Vierteljahresschrift der Gesellschaft für Wirtschaftsund Sozialwissenschaften, Verein für Socialpolitik Provided in Cooperation with: Duncker & Humblot, Berlin Suggested Citation: Abel, U.; Boing, G. (1986) : An Empirical Law of the Stock Option Market, Zeitschrift für Wirtschaftsund Sozialwissenschaften (ZWS) - Vierteljahresschrift der Gesellschaft für Wirtschaftsund Sozialwissenschaften, Verein für Socialpolitik, ISSN 0342-1783, Duncker & Humblot, Berlin, Vol. 106, Iss. 1, pp. 15-24, https://doi.org/10.3790/schm.106.1.15 This Version is available at: https://hdl.handle.net/10419/291629 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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Sozialwissenschaften (ZWS) 106 (1986), S. 15 - 24 Duncker & Humblot, Berlin 41 An Empirical Law of the Stock Option Market By U. Abel and G. Boing The investigation of the laws governing the prices of stock options is a problem of theoretical and practical interest. The paper empirically studies the relationship between the market prices of options out of the money and the difference between stock and exercise prices, the former being fixed. We use linear regression with subsequent analysis of the residuals. The results are compared with those obtained by the BlackScholes model. Several applications of the findings are suggested. I. Introduction Option prices depend on several parameters, such as stock price, striking (exercise) price, expiration time, dividends paid on stock before the expiration of the option, interest rates etc. Many attempts have been made to theoretically derive option or warrant prices under more or less restrictive model assumptions (e.g. References (1), (3), (4), (5), (7), (12), (14)). The most famous valuation formula is undoubtedly the one proposed by Black and Scholes in 1973 for non dividend-paying stocks (3). It has prompted an extensive discussion of the model assumptions and empirical studies of the model fit. Today the formula (perhaps in an extended form allowing for dividends (16)) appears to be widely accepted. Its greatest shortcoming is that it assumes a constant variance rate ("volatility") v2 which is an explicit model parameter. In reality the volatility of a stock can hardly be regarded as constant over the time to maturity of the option, and, in any way, determining the variance rate poses a practical problem. Clearly, historical estimates of v (2), (8), (9) can be unreliable and dangerous if money is at stake. Implied estimates are more satisfactory as they are derived from the present market. Their rationale is as follows: Let k options of a stock be in the market priced at Oif i = 1, . . . , k. Putwhere 6i (u) are the prices predicted by the Black/Scholes formula for unspecified v, estimated values vif . . . ,vk are obtained which can be weighted and averaged to yield a weighted implied variance rate of the stock (6), (11), (13), (15). OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.106.1.15 | Generated on 2023-04-04 12:08:44 16 U. Abel and G. Boing Various weghting schemes have been used in the literature, all more or less arbitrary or chosen on empirical grounds, so that, in principle, the application of the Black/Scholes model shares some features with empirical valuation formulas such as the one given by Kassouf (10). II. An empirical law of the option market We focus on the special problem of the relationship between the prices of options which are out of the money and the difference between the stock and exercise prices, the former being fixed. We contend that this relationships is approximately loglinear. Fig. 1 a/b shows that the hypothesis of loglinearity is not farfetched. o 0 20 40 60 |E-S| O O OCT X X JAN Fig. la The empirical study was based on option prices as published in the Wall Street Journal. Only informative prices were taken into consideration, that is 1. Option prices of Vie were excluded because at this level (the lowest possible) numerous anomalies arise. E.g., on Sept. 21st, the October $ 15, 20 and 25 puts of Homestake were all priced at Vi6. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.106.1.15 | Generated on 2023-04-04 12:08:44 An Empirical Law of the Stock Option Market 17 A X A X o —I—r 1 I 1 1 20 IE-SI n—I—I—r n I I r O O OCT X X JAN A A APR 10 Fig. lb 30 2. Stocks with less than three prices above Vis of options out of the money were excluded since they carried no information as to our hypothesis. We were slightly more restrictive in that three option prices > Vi6 for successive exercise prices had to be available in order to qualify the stock for the analysis. For puts and calls different days had to be chosen for we failed to find one single day where the criteria of selection were met by a satisfactorily large sample of both calls and puts. The data bases for the analysis were the following: Calls expiration in October 1982 prices of June 23rd, 1982 20 eligible stocks Puts expiration in January 1983 prices of September 22nd, 1982 22 eligible stocks. For each selected stock a simple linear regression of log (O/S) vs A = 100-\S-E\/S 2 Zeitschrift für Wirtschaftsund Sozialwissenschaften 1986/1 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.106.1.15 | Generated on 2023-04-04 12:08:44 18 U. Abel and G. Boing was performed (O, E, 5 denoting the option, exercise and stock prices respectively). Of course, log (O/S) and A are linearly related if and only if log (O) and E — S are. The standardization was introduced in the hope that the result would prove independent of S and possibly even of the stock. Unfortunately, no general test for linearity of a regression exists unless there are more degrees of freedoms than abscissa values, and this is not the case, here. It is, however, reasonable to assume that any alternative to linearity is either concavity or convexity. In both cases a systematic effect must show in the successive differences dj = r{ +1 - r{ of the residuals belonging to increasing abscissa values for each stock. In case of convexity the di, ¿=1,2,... should increase, in case of concavity they should decrease. Tables 1 a/b and 2 a/b show the results. The d* are very small compared with the change in the log (0/S)-values as predicted by the regression. This indicates that the linear model fits well. In two cases the log (O/S) lie even on straight lines. The Hodges-Lehmann estimators of the median differences of the diy i = 1, 2, 3 were -0.001 and -0.11 for calls and 0.009 and 0.19 for puts. While there is no perceptible monotonic trend in the d, of calls, such a trend, though very slight, can be ascertained in the samples of puts (Jonckheere test against ordered alternatives, p < 0.01). However, there were only 10 out of 22 stocks with a strictly montonic increase. On the other hand, there were 4 stocks with a strictly monotonic decrease and 2 stocks with equality of the d{. Summarizing, deviations from linearity, if any, were small and not systematic in the majority of the stocks. Table 1 a/b shows that the intercepts of the regression lines were densely packed, while the differences in the slopes were rather marked. Since calls and puts had about the same time to maturiy, it makes sense to compare their regression parameters, especially for those stocks appearing in both analyses. Slopes and intercepts were smaller in puts than in calls (the arithmetic means were - 2.39 and - 6.99 in puts versus - 2.27 and - 6.24 in calls). The parameters of puts of a stock were strikingly similar to those of the calls of the same stock, and, as far as were differences, there seems to be no rule for their sign. We have seen that the regression lines for different stocks were not equal. It might still, however, be true that, for a given stock, they do not depend on the stock price S. An empirical check of this hypothesis is difficult and must rely on few data because large changes of stock prices require some time to OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.106.1.15 | Generated on 2023-04-04 12:08:44 An Empirical Law of the Stock Option Market 19 Table 1 a STOCK INTERC. SLOPE FEDEXP - 1.944107 - 7.94420 FLUOR - 2.531504 - 2.95640 HALBTN - 2.366208 - 4.91840 HOMESTK - 2.027928 - 2.89740 MERCK - 2.868246 - 8.36761 MONSAN - 2.497658 - 10.60607 PENNZ - 2.221898 - 3.03217 STORTEC - 2.240172 - 3.81818 TELDYN - 2.052711 - 8.53185 AMEXP - 2.368962 - 9.64208 DIGEQ - 2.251492 - 9.15036 HUTTON - 1.815444 - 3.78057 LILLY - 2.540796 - 11.78133 MERRIL - 2.077980 - 5.36468 COMSAT - 2.177308 - 9.29057 OAK - 2.164429 - 2.73604 WESTUN - 2.208561 - 6.73506 BALDUN - 2.388497 - 4.81473 DSHAM - 2.546658 - 3.36863 WANGB - 2.023090 - 5.05841 Table 1 b STOCK INTERC. SLOPE BURLN - 2.493704 - 6.51496 EASTKD - 2.626853 - 12.17279 FEDEXP - 2.203536 - 7.34584 HOMESTK - 2.031629 - 4.37037 IBM - 2.900605 - 6.53806 JOHNJ - 2.761150 - 9.10545 MMM - 2.575750 - 8.86155 MONSAN - 2.894036 - 9.44603 PEPSI - 3.237197 - 8.24209 TELDYN - 2.019670 - 6.77224 TEXIN - 2.251436 - 5.65140 AMEXP - 1.981937 - 10.68347 DIGEQ - 2.248021 - 8.94600 DUPONT - 2.821536 - 8.84224 HUTTON - 1.941320 - 4.62676 MERRIL - 1.840691 - 5.77109 MOTORLA - 2.365029 - 9.87831 PROCG - 2.953498 - 8.93818 WESTNG - 2.585875 - 7.68488 WESTUN - 2.081289 - 10.57049 BALDUN - 1.844625 - 6.32225 WANGB - 2.111259 - 6.38418 2* OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.106.1.15 | Generated on 2023-04-04 12:08:44 20 U. Abel and G. Boing Table 2 a Successive Differences of Residuals NO D1 D2 D3 D4 FEDEXP - .235175 - .006917 .244398 _ FLUOR .202733 - .202733 - - HALBTN .020637 .017681 .002938 - HOMESTK .160299 .077412 .263515 - MERCK .026622 .026622 - - MONSAN .095310 .095310 - - PENNZ .071381 .053032 - .120991 .0305587 STORTEC .127967 .127967 -- TELDYN - .062600 .065234 - .024378 - AMEXP - .052680 .052680 - - DIGEQ - .113763 .048756 .048756 - HUTTON .220916 .220916 -- LILLY .309520 .309520 - - MERRIL .314304 - .314304 -- COMSAT .038481 - .038481 - - OAK .111572 - .111572 - - WESTUN - .235002 .235002 - - BALDUN .009475 .343481 - .467449 - DSHAM .412088 - .412088 -- WANGB - 235002 .235002 - - Table 2 b Successive Differences of Residuals NO D1 D2 D3 D4 D5 BURLN - .018870 .018870 _ _ _ EASTKD - .105532 .042684 .048619 - - FEDEXP - .078327 .078327 - - - HOMESTK - .022607 - .059962 .102557 -- IBM - .155301 .020430 .128061 -- JOHNJ - .111572 .111572 --- MMM - .008687 .072049 - .087379 - - MONSAN .426039 - .380436 .377249 - .421258 - PEPSI - .020411 .020411 --- TELDYN - .142430 - .291465 .253466 .194043 - .157934 TEXIN - .221645 - .012083 - .237756 -- AMEXP - .011236 - .011236 -- - DIGEQ .033605 - .257272 .161438 .110145 - DUPONT - .053815 .053815 - - - HUTTON - .000000 .000000 -- - MERRIL - .031143 .006598 .022346 - - MOTORLA - .020469 - .168685 - .184189 .549780 - PROCG .159227 - .159227 -- - WESTNG .235002 .235002 -- - WESTUN .000000 .000000 --- BALDUN - .154809 .154809 --- WANGB .018184 - .018184 --- OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.106.1.15 | Generated on 2023-04-04 12:08:44 An Empirical Law of the Stock Option Market 21 occur so that the change in time to maturity exerts influence on the option prices. The examples given in Table 3 show that the parameters of the linear regression remain constant even after pronounced short-term changes of the stock price. This stability is remarkable in view of the extreme sensitivity of the parameters to small variations of the option prices. Table 3: Changes of the parameters of the linear regression of log (O/S) vs. |E — S| after sharp movements of stock prices. Motorola St. Oil Ohio Coleco maturity Apr. 1984 maturity March 1984 maturity Apr. 1984 datete) Dec. 2nd Dec. 14th Dec. 2nd Dec. 14th Dec. 16th Dec. 23rd stock price 142 Vi 1353/4 453/4 413/4 253/s 205/s calls- *ntercePt slope (b) - 2.56 - 0.055 - 2.61 - 0.154 - 2.86 - 0.165 - 1.86 - 0.115 - 1.88 - 0.1 intercept puts: , ^ slope - 2.88 - 0.065 - 2.74 - 0.065 - 2.94 - 0.254 (b) - 1.92 - 0.159 - 1.64 - 0.16 (a) Year 1983. - (b) No calculation possible. Without presenting a detailed analysis we finally note that for options in the money a loglinear relation between the premia and A holds, too, though outliers are more frequent. m. Discussion The empirical investigation has shown that for fixed stock prices S a loglinear relationship between the prices O of options out of the money and the difference between 5 and the exercise price E very approximately holds. Observe that this is not a consequence of the interval E — S being small (so that it necessarily follows by Taylor expansion approximation). For a stock priced at about $ 100, a $ 40-interval (such as in Fig. 1) is not small as any investor knows, and, moreover, linearity between S - E and some other function of O (say OorO2) definitely does not hold. The relationship put forward in this article is not incompatible with Black/Scholes's formula as Fig. 2 shows. However, we have been unable, so far, to deduce approximate loglinearity from this formula. Our finding gives a strong support to a special estimator for implied volatility of a stock, viz. the value v which minimizes 2 (log (Oi) - log (6i))2. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.106.1.15 | Generated on 2023-04-04 12:08:44 22 o.o — -2.5 — -5.0 -7.5 — U. Abel and G. Boing —I I I I | I I I I | I I I I | I I I I | I I I I | 0 10 20 30 40 50 E-S O O OCT XX JAN Fig. 2 Apart from this, it can be applied in several ways. First, it can be used for predictions of option prices for future stock prices. If changes in stock prices do not occur rapidly, prognosis will require interpolation between the prices predicted for the available times to maturity. Second, it allows the easy detection of options which are underor overvalued with respect to other options of the same stock. Third, it can be exploited for practical investment. We suggest the following (winning?) strategy: Determine the intercept and slope for the puts and calls of the same stock. Calculate the value 5min such that a given spraddle, i. e. a combination of one put and one call of the stock with •Ecall > s > Eput , assumes its minimum1. Invest when S = Smin. Take profit on any movement of the stock price. 1 Given the regression parameters, Smin can be determined analytically by standard calculus. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.106.1.15 | Generated on 2023-04-04 12:08:44