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
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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).
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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.
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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
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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
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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*
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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 ---
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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.
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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.
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