An Empirical Law of the Stock Option Market

Abel, U.; Boing, G. A icle An Empi ical Law o he S ock Op ion Ma ke Zei sch i ü Wi scha s- und Sozialwissenscha en (ZWS) - Vie eljah essch i de Gesellscha ü Wi scha s- und Sozialwissenscha en, Ve ein ü Socialpoli ik P o ided in Coope a ion wi h: Duncke & Humblo , Be lin Sugges ed Ci a ion: Abel, U.; Boing, G. (1986) : An Empi ical Law o he S ock Op ion Ma ke , Zei sch i ü Wi scha s- und Sozialwissenscha en (ZWS) - Vie eljah essch i de Gesellscha ü Wi scha s- und Sozialwissenscha en, Ve ein ü Socialpoli ik, ISSN 0342-1783, Duncke & Humblo , Be lin, Vol. 106, Iss. 1, pp. 15-24, h ps://doi.o g/10.3790/schm.106.1.15 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/291629 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen (insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en, gel en abweichend on diesen Nu zungsbedingungen die in de do genann en Lizenz gewäh en Nu zungs ech e. Te ms o use: Documen s in EconS o may be sa ed and copied o you pe sonal and schola ly pu poses. You a e no o copy documen s o public o comme cial pu poses, o exhibi he documen s publicly, o make hem publicly a ailable on he in e ne , o o dis ibu e o o he wise use he documen s in public. I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h ps://c ea i ecommons.o g/licenses/by/4.0/ Zei sch i ü Wi scha s- u. Sozialwissenscha en (ZWS) 106 (1986), S. 15 - 24 Duncke & Humblo , Be lin 41 An Empi ical Law o he S ock Op ion Ma ke By U. Abel and G. Boing The in es iga ion o he laws go e ning he p ices o s ock op ions is a p oblem o heo e ical and p ac ical in e es . The pape empi ically s udies he ela ionship be - ween he ma ke p ices o op ions ou o he money and he di e ence be ween s ock and exe cise p ices, he o me being ixed. We use linea eg ession wi h subsequen analysis o he esiduals. The esul s a e compa ed wi h hose ob ained by he Black- Scholes model. Se e al applica ions o he indings a e sugges ed. I. In oduc ion Op ion p ices depend on se e al pa ame e s, such as s ock p ice, s iking (exe cise) p ice, expi a ion ime, di idends paid on s ock be o e he expi a- ion o he op ion, in e es a es e c. Many a emp s ha e been made o heo e ically de i e op ion o wa an p ices unde mo e o less es ic i e model assump ions (e.g. Re e ences (1), (3), (4), (5), (7), (12), (14)). The mos amous alua ion o mula is undoub edly he one p oposed by Black and Scholes in 1973 o non di idend-paying s ocks (3). I has p omp ed an ex ensi e discussion o he model assump ions and empi ical s udies o he model i . Today he o mula (pe haps in an ex ended o m al- lowing o di idends (16)) appea s o be widely accep ed. I s g ea es sho coming is ha i assumes a cons an a iance a e (" ol- a ili y") 2 which is an explici model pa ame e . In eali y he ola ili y o a s ock can ha dly be ega ded as cons an o e he ime o ma u i y o he op ion, and, in any way, de e mining he a iance a e poses a p ac ical p oblem. Clea ly, his o ical es ima es o (2), (8), (9) can be un eliable and dange ous i money is a s ake. Implied es ima es a e mo e sa is ac o y as hey a e de i ed om he p esen ma ke . Thei a ionale is as ollows: Le k op ions o a s ock be in he ma ke p iced a Oi i = 1, . . . , k. Pu - whe e 6i (u) a e he p ices p edic ed by he Black/Scholes o mula o un- speci ied , es ima ed alues i . . . , k a e ob ained which can be weigh ed and a e aged o yield a weigh ed implied a iance a e o he s ock (6), (11), (13), (15). OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/schm.106.1.15 | Gene a ed on 2023-04-04 12:08:44 16 U. Abel and G. Boing Va ious wegh ing schemes ha e been used in he li e a u e, all mo e o less a bi a y o chosen on empi ical g ounds, so ha , in p inciple, he ap- plica ion o he Black/Scholes model sha es some ea u es wi h empi ical alua ion o mulas such as he one gi en by Kassou (10). II. An empi ical law o he op ion ma ke We ocus on he special p oblem o he ela ionship be ween he p ices o op ions which a e ou o he money and he di e ence be ween he s ock and exe cise p ices, he o me being ixed. We con end ha his ela ionships is app oxima ely loglinea . Fig. 1 a/b shows ha he hypo hesis o loglinea i y is no a e ched. o 0 20 40 60 |E-S| O O OCT X X JAN Fig. la The empi ical s udy was based on op ion p ices as published in he Wall S ee Jou nal. Only in o ma i e p ices we e aken in o conside a ion, ha is 1. Op ion p ices o Vie we e excluded because a his le el ( he lowes possi- ble) nume ous anomalies a ise. E.g., on Sep . 21s , he Oc obe $ 15, 20 and 25 pu s o Homes ake we e all p iced a Vi6. OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/schm.106.1.15 | Gene a ed on 2023-04-04 12:08:44 An Empi ical Law o he S ock Op ion Ma ke 17 A X A X o —I— 1 I 1 1 20 IE-SI n—I—I— n I I O O OCT X X JAN A A APR 10 Fig. lb 30 2. S ocks wi h less han h ee p ices abo e Vis o op ions ou o he money we e excluded since hey ca ied no in o ma ion as o ou hypo hesis. We we e sligh ly mo e es ic i e in ha h ee op ion p ices > Vi6 o suc- cessi e exe cise p ices had o be a ailable in o de o quali y he s ock o he analysis. Fo pu s and calls di e en days had o be chosen o we ailed o ind one single day whe e he c i e ia o selec ion we e me by a sa is ac o ily la ge sample o bo h calls and pu s. The da a bases o he analysis we e he ol- lowing: Calls expi a ion in Oc obe 1982 p ices o June 23 d, 1982 20 eligible s ocks Pu s expi a ion in Janua y 1983 p ices o Sep embe 22nd, 1982 22 eligible s ocks. Fo each selec ed s ock a simple linea eg ession o log (O/S) s A = 100- S-E /S 2 Zei sch i ü Wi scha s- und Sozialwissenscha en 1986/1 OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/schm.106.1.15 | Gene a ed on 2023-04-04 12:08:44 18 U. Abel and G. Boing was pe o med (O, E, 5 deno ing he op ion, exe cise and s ock p ices e- spec i ely). O cou se, log (O/S) and A a e linea ly ela ed i and only i log (O) and E — S a e. The s anda diza ion was in oduced in he hope ha he esul would p o e independen o S and possibly e en o he s ock. Un o una ely, no gene al es o linea i y o a eg ession exis s unless he e a e mo e deg ees o eedoms han abscissa alues, and his is no he case, he e. I is, howe e , easonable o assume ha any al e na i e o linea i y is ei he conca i y o con exi y. In bo h cases a sys ema ic e ec mus show in he successi e di e ences dj = { +1 - { o he esiduals be- longing o inc easing abscissa alues o each s ock. In case o con exi y he di, ¿=1,2,... should inc ease, in case o conca i y hey should dec ease. Tables 1 a/b and 2 a/b show he esul s. The d* a e e y small compa ed wi h he change in he log (0/S)- alues as p edic ed by he eg ession. This indica es ha he linea model i s well. In wo cases he log (O/S) lie e en on s aigh lines. The Hodges-Lehmann es ima o s o he median di e ences o he diy i = 1, 2, 3 we e -0.001 and -0.11 o calls and 0.009 and 0.19 o pu s. While he e is no pe cep ible mono onic end in he d, o calls, such a end, hough e y sligh , can be asce ained in he samples o pu s (Jonckhee e es agains o de ed al e na i es, p < 0.01). Howe e , he e we e only 10 ou o 22 s ocks wi h a s ic ly mon onic inc ease. On he o he hand, he e we e 4 s ocks wi h a s ic ly mono onic dec ease and 2 s ocks wi h equali y o he d{. Summa izing, de ia ions om linea i y, i any, we e small and no sys ema ic in he majo i y o he s ocks. Table 1 a/b shows ha he in e cep s o he eg ession lines we e densely packed, while he di e ences in he slopes we e a he ma ked. Since calls and pu s had abou he same ime o ma u iy, i makes sense o compa e hei eg ession pa ame e s, especially o hose s ocks appea ing in bo h analyses. Slopes and in e cep s we e smalle in pu s han in calls ( he a i h- me ic means we e - 2.39 and - 6.99 in pu s e sus - 2.27 and - 6.24 in calls). The pa ame e s o pu s o a s ock we e s ikingly simila o hose o he calls o he same s ock, and, as a as we e di e ences, he e seems o be no ule o hei sign. We ha e seen ha he eg ession lines o di e en s ocks we e no equal. I migh s ill, howe e , be ue ha , o a gi en s ock, hey do no depend on he s ock p ice S. An empi ical check o his hypo hesis is di icul and mus ely on ew da a because la ge changes o s ock p ices equi e some ime o OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/schm.106.1.15 | Gene a ed on 2023-04-04 12:08:44 An Empi ical Law o he S ock Op ion Ma ke 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 unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/schm.106.1.15 | Gene a ed on 2023-04-04 12:08:44 20 U. Abel and G. Boing Table 2 a Successi e Di e ences o 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 Successi e Di e ences o 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 unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/schm.106.1.15 | Gene a ed on 2023-04-04 12:08:44 An Empi ical Law o he S ock Op ion Ma ke 21 occu so ha he change in ime o ma u i y exe s in luence on he op ion p ices. The examples gi en in Table 3 show ha he pa ame e s o he linea eg ession emain cons an e en a e p onounced sho - e m changes o he s ock p ice. This s abili y is ema kable in iew o he ex eme sensi i i y o he pa ame e s o small a ia ions o he op ion p ices. Table 3: Changes o he pa ame e s o he linea eg ession o log (O/S) s. |E — S| a e sha p mo emen s o s ock p ices. Mo o ola S . Oil Ohio Coleco ma u i y Ap . 1984 ma u i y Ma ch 1984 ma u i y Ap . 1984 da e e) Dec. 2nd Dec. 14 h Dec. 2nd Dec. 14 h Dec. 16 h Dec. 23 d s ock p ice 142 Vi 1353/4 453/4 413/4 253/s 205/s calls- *n e ceP slope (b) - 2.56 - 0.055 - 2.61 - 0.154 - 2.86 - 0.165 - 1.86 - 0.115 - 1.88 - 0.1 in e cep pu s: , ^ slope - 2.88 - 0.065 - 2.74 - 0.065 - 2.94 - 0.254 (b) - 1.92 - 0.159 - 1.64 - 0.16 (a) Yea 1983. - (b) No calcula ion possible. Wi hou p esen ing a de ailed analysis we inally no e ha o op ions in he money a loglinea ela ion be ween he p emia and A holds, oo, hough ou lie s a e mo e equen . m. Discussion The empi ical in es iga ion has shown ha o ixed s ock p ices S a log- linea ela ionship be ween he p ices O o op ions ou o he money and he di e ence be ween 5 and he exe cise p ice E e y app oxima ely holds. Ob- se e ha his is no a consequence o he in e al E — S being small (so ha i necessa ily ollows by Taylo expansion app oxima ion). Fo a s ock p iced a abou $ 100, a $ 40-in e al (such as in Fig. 1) is no small as any in es o knows, and, mo eo e , linea i y be ween S - E and some o he unc ion o O (say Oo O2) de ini ely does no hold. The ela ionship pu o wa d in his a icle is no incompa ible wi h Black/Scholes's o mula as Fig. 2 shows. Howe e , we ha e been unable, so a , o deduce app oxima e loglinea i y om his o mula. Ou inding gi es a s ong suppo o a special es ima o o implied ol- a ili y o a s ock, iz. he alue which minimizes 2 (log (Oi) - log (6i))2. OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/schm.106.1.15 | Gene a ed 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 Apa om his, i can be applied in se e al ways. Fi s , i can be used o p edic ions o op ion p ices o u u e s ock p ices. I changes in s ock p ices do no occu apidly, p ognosis will equi e in e - pola ion be ween he p ices p edic ed o he a ailable imes o ma u i y. Second, i allows he easy de ec ion o op ions which a e unde - o o e - alued wi h espec o o he op ions o he same s ock. Thi d, i can be exploi ed o p ac ical in es men . We sugges he ollowing (winning?) s a egy: De e mine he in e cep and slope o he pu s and calls o he same s ock. Calcula e he alue 5min such ha a gi en sp addle, i. e. a combina ion o one pu and one call o he s ock wi h •Ecall > s > Epu , assumes i s minimum1. In es when S = Smin. Take p o i on any mo e- men o he s ock p ice. 1 Gi en he eg ession pa ame e s, Smin can be de e mined analy ically by s anda d calculus. OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/schm.106.1.15 | Gene a ed on 2023-04-04 12:08:44