99
CHAPTER IV
Asset pricing model should predict its price correctly as that of the
market and the residuals / error shoul systematic pattern with
any of its variables / parameters that ar
changes systematically with the change in variables then it is known as bias of
the model. The residuals should be distributed normally. This chapter explains
the predictability of the model and behaviour of th ption towards its
variables li ,
of returns of the stock, exercise price, and risk-free interest rate. The findings of
the study are explained in depth. Part research study was published in
the book “ Manage t Pra Polic Princ [105], by
The Allied Publishers Private Limited, New Delhi, after edited by a faculty of
Indian Inst agem dore xure
4.2 EARL DIE Bla stat mar s of tions “t differ in
certain sys ys” fr val by model tions with
less than hs to tion optio are ei eep in or
out of the rton ates est all pri r deep in
the money dee f th opt less t e market
prices. M and Mervil 4] op deep mone ons have
model prices those are lower than market prices, whereas deep out of money
options
PREDICTABILITY OF THE BS MODEL IN INDIAN OPTION MARKET
4.1 INTRODUCTION
d not have any
e used to predict the price. If the error
e call o prices
ke stock price and life of the option and the parameters like volatility
of this
Business men ctices, ies and iples”
itute of Man ent, In . (Anne I)
IER STU S
ck [21, 23] es that ket price call op end to
tematic wa om the ues given the BS for op
three mont expira and for ns that ther d
money. Me [98] st that BS imated c ces fo
as well as p out o e money ions are han th
acbeth le [9 ine that in the y opti
have model prices that are higher. These conflicts may perhaps be
100
reconciled by the fact that the studies examined market prices at different points
in time and these systematic biases vary with time (Rubinstein [41,118]).
4.3 DEFINITIONS
Based on the above studies, this research tries to find the truth about
the predictability of the BS Model and pricing biases, if any, in the Indian option
market. The Mean Absolute Errors (MAE) calculated with the following formula
for various moneyness and various lives of the options are tabulated in the
Table 4.1.
Σ ׀ PO – PT׀ Mean Absolute Error = ------------------- Σ PO
where PT is the call option price theoretically calculated using BS model and PO
is the observed call option prices in the market.
Moneyness is defined as S0 / X and the values ranging 0.99 to 1.01 are
taken as at-the-money options, values les ed as out-of-
the-money options and above 1.01 as in-the-money options. As the number is
increasing the mo
Mean percentage errors are calculated using the following formula.
[{(PO – PT) / P ] Mean Percentage Er -------------- n
where n is the sample
The percentage he formu O – PT) / PO} x100] in
each of the options taken in the sample. Then, the sum of the percentage errors
is calculated and result is divided by the number of data “n”.
s than 0.99 is consider
neyness is also increasing.
Σror = -----------------
O} x100
size.
error is found using t la [{(P
101
Though the mean percentage error is misrepresenting in the cases of
very low option prices, it gives irection of the err enlighten us whether,
the BS model is overpricing or under-pricing the options. Thus it is considered
the research studies abroad are not
considering the deep out-of-the-money options and deep in-the-money options,
ut, they have so little data that they
cannot predict the market price correctly. If the data volume is not sufficient
the d or. It
as an important measure and used in the research.
4.4 CLASSIFICATION OF DATA
First, the call options that are offered by NSE were collected, analyzed
and the traded options were segregated from the non-traded ones. Then, using
the ex-dividend dates and dates of board meetings related to dividend
decisions, the options related to cum-dividend stocks during the life of the
options were eliminated. Lastly, the data are classified into fourteen groups of
moneyness each having three consecutive classes of moneyness. The
definition of moneyness is So / X. The classifications started from 0.83 to 1.20.
The classification is made in a way that ATM options of 1.00 lies at the center of
the classification (0.99 to 1.01). Though,
they have also studied in Indian context. B
enough they are not considered for making conclusion. The details are given in
Table no. 4.1
102
TABLE 4.1
DETAILS OF CALL OPTION TAKEN AS SAMPLE AS PER MONEYNESS
Moneyness S0 / X No. of Data
< 0.83 188 0.84 – 0.86 377 0.87 – 0.89 1,028 0.90 – 0.92 3,204 0.93 – 0.95 8,760 0.96 51 – 0.98 17,20.99 – 1.01 2,1302 1.02 28 – 1.04 17,7 1.05 – 32 1.07 11,2 1.08 – 4 1.10 6,561.11 – 1 1.13 3,871.14 – 1.16 2,346 1.17 – 1.19 1,383
> 1 5 .20 51TOT 77AL 96,5
t o bov opti a 6 lier been na
the final sample size is 95,956. Box-Plot analysis is used to identify the outliers,
whi exp in detail at chapter 6. The call options with moneyness
between 0.93 and 1.10 are having a reason
related to deep in-the-money and deep ou the
very little number of traded options, for which the market may not correctly price
e options. The distribution of the call options looks like a normal curve, but
Ou f the a e call on dat 21 out s have elimi ted and
ch is lained
able number of data. The other data
t-of- -money options are having
th
having fat tail.
103
CHART 4.1
DISTRIBUTION OF CALL OPTION DATA FOR VARIOUS MONEYNESS
DISTRIBUTION OF CALL OPTIONS
0
5,000
10,000
15,000
20,000
25,000
< 0.
83
0.84
-0.8
6
0.87
-0.8
9
0.90
-0.9
2
0.93
-0.9
5
0.96
-0.9
8
0.99
-1.0
1
1.02
-1.0
4
1.05
-1.0
7
1.08
-1.1
0
1.11
-1.1
3
1.14
-1.1
6
1.17
-1.1
9
> 1.
20
MONEYNESS
NO
. OF
CA
LL O
PTIO
NS
4.5 PREDICTABILITY OF THE MODEL
The main objective of this empirical study is the predictability of the
model. The predictability of the model is verified through mean absolute errors,
ross various
determinants of the call option price.
4.5.1 M
mean percentage error and the distribution of these errors ac
EAN ABSOLUTE ERRORS
The options were classified with different categories of moneyness,
processed and the option prices were calculated using BS model. The actual
markets prices of call options taken from the NSE website [68] were compared
104
with the respective predicted prices by the model and the MAE thus calculated
are summarized and shown in the table no. 4.2.
TE ERRORS OF OPTIONS WITH VARIOUS MONEYNESS
TABLE 4.2 MEAN ABSOLU
Moneyness No. Of data
Total Observed
Price
Total Absolute
Error
Mean Absolute
Error
< 0.83 187 4,130 1,635 0.40
0.84-0.86 370 7,265 3,720 0.51
0.87-0.89 1,005 17,501 9,349 0.53
0.90-0.92 3,163 54,356 28,077 0.52
0.93-0.95 8,671 155,569 66,442 0.43
0.96-0.98 17,112 383,157 127,623 0.33
0.99-1.01 21,984 624,996 154,049 0.25
1.02-1.04 17,643 660,766 114,602 0.17
1.05-1.07 11,191 542,341 70,111 0.13
1.08-1.10 6,550 378,344 45,151 0.12
1.11-1.13 3,854 251,920 26,870 0.11
1.14-1.16 2,328 164,207 16,709 0.10
1.17-1.19 1,383 101,157 11,043 0.11
> 1.20 515 62,963 7,688 0.12
It may be observed from the above table, the MAE are as high as about
0.52 for the deep out-of-the-money options having moneyness 0.80 to 0.92.
Then it starts decreasing at a faster rate. For the moneyness of 0.93-0.95, it
dropped down by about 17% to 0.43 and for the next classification of 0.96-0.98,
it reduced by 23% to 0.33. Then, MAE are reduced by 24%, 32%, 23% and 7%
for next four moneynesses. At the end, it is almost flat.
105
The options that are having number of trades very less during the entire
period of five years and ten months (1716 working days) are illiquid and may
not reflect the correct price of the market. If we neglect the data related to
number of options less th ing the of study7; the MAE vary
from 0.12 to 0.43, with a m The clas tions of data with less than
5000 are shown in shades d not be considered for the conclusion
and are given only for the academic purpose. The mean predictability of the
model is around 76%. Mean is not a resistant summary of statistics and is
drastically influenced by th alues. Because of this, the predictability
of the model is low and the . For th ata without categories the
mean absolute error is 0.25 only. Let us also consider the resistant summary of
median based statistics. sorted in the ascending order. The
percentiles are calculated using SAS; values at t responding percentages
are given in the table no.4.3.
TABLE 4.3
an 5000 dur period
ean of 0.24. sifica
, which nee
e extreme v
error is high e full d
MAE are
he cor
PERCENTILES OF MEAN ABSOLUTE ERRORS
PERCENTILE ABSOLUTE ERRORS
100% 625.0199% 69.5195% 25.390% 15.62
Upper Quartile 75% 7.03Median 50% 3.12
Lower Quar 1.27tile 25%10% 0.475% 0.231% 0.050% 0
-------------------------------------------------------------------------------------------------------7Gurdip Bakshi, Charles Cao, and Zhiwu Chen in their study [10] considered data with
more than 4500 only during the period of 3 years.
106
A percentile8 is the value of a variable below which a certain percent of
observations fall. So the 20th percentile is the value (or score) below which 20
percent of the observations may be found. Fifty percentages of the options are
having MAE less than 3.12. The next 25% of the sample are having MAEs
within 3.12 to 7.03. The next 15%of the options are having errors from 7.03 to
15.62. It means, that the BS model predicted the call option prices with a
minimum accuracy of 84.38% for ninety percentages (86,360 options) of the
sample of 95,956. A very small portion of the options are having higher errors.
Thus, it may be inferred that the BS model is good as far as MAE are
considered. The median based statistics give better picture of the error
n the mean, which is influenced by the extreme values.
rs are considered to nullify the effect of positive and
egative errors canceling each other, thus showing smaller error values. But,
tion of the model errors. The
direction of the errors such as positive and negative errors will enlighten one to
entify
5.4% accuracy. If minutely
looked; the model predicts 50% of opt
and 89.19%. The next 30% options are ving predictability between 72% and
80.58% only. (10th percentile is -38.64 and 90th percentile is 39.76). The
accuracy of the prediction measured and summarized is given in table no. 4.4.
-------- ------- 8Source: (http://en.wikipedia.org/wiki/Percentile
distribution tha
4.5.2 MEAN PERCENTAGE ERRORS
The absolute erro
n
MAE has the disadvantage of not showing the direc
id , whether the model overprices or underprices the options.
The mean percentage errors are calculated and categorized into plus or
minus five percentages and exhibited in the table number 4.4. Let us consider
the resistance summary of median and quartiles. The median of the percentage
errors is 4.6, the lower quartile -10.81 and the upper quartile at 20.34. On an
average the model predicts the call option price by 9
ions with a predictability range 79.64%
ha
-------------------------------------------------------------------------------------------).
107
TABLE 4.4 PERCE E C DEL NTAGE WIS PREDI TABILITY OF THE BS MO
Accuracy Error
Number of
Negative Errors
Number of
Positive Errors
Total Number
of Errors
Cumu Total
Number of
Errors
% of Number
of Errors
Cumu % of
Number of
Errors
95% ± 1 18,305 .08 19.085 7,781 10,524 8,305 1990% ± 1 1 33,257 .58 34.660 5,659 9,293 4,952 1585% ± 1 45,207 .45 47.115 4,420 7,530 11,950 1280% ± 2 54,5410 3,338 5,996 9,334 9.73 56.8475% ± 2 62,1925 2,599 5,052 7,651 7.97 64.8170% ± 3 68,2650 2,026 4,047 6,073 6.33 71.1465% ± 35 9 73,265 .21 76.351,73 3,261 5,000 560% ± 40 77,230 .13 80.481,410 2,555 3,965 455% ± 45 80,474 .38 83.871,119 2,125 3,244 350% ± 5 7 83,065 .70 86.570 93 1,654 2,591 245% ± 5 794 1,330 85,189 .21 88.785 2,124 240% ± 60 713 1,056 1,769 86,958 1.84 90.62
Cumu - Cumulative
Out of 95,956 samples, the BS model has predicted the price with 95%
accuracy in 18,305 options; that is 19.08%. 54,541 options that are 56.84% are
predicted within 20% error. About 71.14% of options are having errors less than
30%. The BS model has predicted more number of options within a reasonable
accuracy and few have large errors. The same is depicted in the chart no.4.2.
108
CHART 4.2
PREDICTABILITY OF THE BS MODEL
0
5,000
10,000
PREDIC ABILITYT
15,000RO
20,000. O
F ER
RS
NO
± 5 ±10
±15
±20
±25
±30
±35
±40
±45
±50
±55
±60
PERCENTAGE ERRORS
4.6 BIASES OF THE MODEL ACROSS MONEYNESS
4.6.1 BIASES OF MEAN ABSOLUTE ERRORS
The plot of MAE should be of a straight line with only random variations.
Refer the table no 4.2 (pp 104); the errors are reducing from 0.43 to 0.12 as the
moneyness increases from 0.93 to 1.10. The MAE are coming down with
increase in moneyness and there is definite pattern against moneyness. That is,
MAE and moneyness is inversely related. The mean of the MAE is 0.25. The
rrors are not at random about the mean error. Hence, the pricing bias of the
model
e
over moneyness is proved beyond doubt.
The MAE corresponding to moneyness with more than 5000 data are
given in the chart no. 4.3
109
CHART 4.3
MEAN ABSOLUTE ERRORS FOR DIFFERENT MONEYNESS AFTER AN 5000 ELIMINATION OF DATA LESS TH
MEAN ABSOLUTE ERROR0.45
S
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.93-0.95 0.96-0.98 0.99-1.01 1.02-1.04 1.05-1.07 1.08-1.10MONEYNES
M A
E
The MAE are not at its minimum at moneyness of 1.00 but at its
minimum for the options at moneyness of 1.08. This is against the findings of
Black, Fisher [23] himself and o of the opinion that BS model
predicts correctly for options at-the-money. But our study is not coinciding with
this, a .08
– 1.10.
4.6.2 EXCE RORS OVER N OF EANS F AB UTE ERRO
If the errors are calculated around the mean of the above MAE, then the
mean error occurs at the moneyness of 1.00 and varies in opposite directions
thers. He is
s per our findings the minimum errors are only at the moneyness of 1
SS ER MEA M O SOLRS
110
as per mon results wn for a tte standing in
table no. 4.5 and char
E 4
EXCESS ERRORS OVER THE MEAN OF MEANS OF ABSOLUTE ERRORS FOR MONEY FOR DATA MORE THAN 5000
eyness. The are sho below be r under
t no. 4.4.
TABL .5
DIFFERENT NESS
Moneyness Excess Error
0.87 - 0.89 0.28
0.90 - 0.92 0.27
0.93 - 0.95 0.18
0.96 - 0.98 0.08
0.99 - 1.01 0.00
1.02 - 1.04 -0.08
1.05 - 1.07 -0.12
1.08 - 1.10 -0.13
1.11 - 1.13 -0.14
1.14 - 1.16 -0.15
1.17 - 1.20 -0.14
ed
areas of the above table. The rs around the
moneyness of t is ATM. ean error is positive over OTM
options and nega r ITM t is wn hart no. 4.4.
Excess errors are at zero around 0. .01 that is ATM options and
increases as moneyness deviates from moneyness of 1.00.
The MAE related to data more than 5,000 are shown in non-shad
mean of MAE of 0.25 occu
0.99 to 1.01 tha The m
sho
99 to 1
tive ove . Tha better in the c
111
CHART
EXCESS ERR THE M OF N LU RORS FOR DIFF NEYN OR A M TH 00
4.4
ORS OVER EAN MEA ABSO TE ERERENT MO ESS F DAT ORE AN 50
MEAN A LUT ROR
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.93-0.9 98 0 .01 1. 1.0 1.08-1.10
YNESS
M A
E
BSO E ER S
MONE
5 0.96-0. .99-1 1.02- 04 5-1.07
From the above chart, it may be noticed that the BS model predicts the
ITM options better than OTM options. The curve at deep ITM options and deep
OTM options remains flat. But, ITM options are having lesser errors than the
deep OTM options.
4.6.3 MEAN PERCENTAGE ERRORS (MPE)
For a better un s are also calculated
and studied. It shows the direction of errors, whether positive or negative. This
will enable one to understand whether the model is under-price or over-price
the options. The mean percentage error is calculated for the various
moneyness classifications and the results are tabulated in table no. 4.6. The
corresponding chart is given below in chart no. 4.5.
derstanding, mean percentage error
112
TABLE 4.6
MEAN PERCENTAGE ERRORS FOR DIFFERENT MONEYNESS
Moneyness No. of Data
Total Observed
Price
Mean Percentage
Error
< 0.83 187 4130.05 32.63
0.84 - 0.86 370 7,265 29.78
0.87 - 0.89 1,005 17,501 27.41
0.90 - 0.92 3,163 54,356 18.75
0.93 - 0.95 8,671 155,569 7.10
0.96 - 0.98 17,112 383,157 -1.26
0.99 -1.01 21,984 624,996 -3.51
1.02 -1.04 17,643 660,766 -1.66
1.05 -1.07 11,191 542,341 0.03
1.08 -1.10 6,550 378,344 0.59
1.11 - 1.13 3,854 251,920 0.20
1.14 - 1.16 2,328 164,207 0.72
1.17-1.19 1,383 101,157 -0.98
> 1.20 515 62,963 -3.81
The mean percentage errors are positive and very high for the deep
OTM options, which are related the moneyness 0.83 to 0.92. If one looks at the
number of data in each of the above classifications, they are less than 5000 for
the entire period of five years and ten months. These options can be considered
as illiquid and the results may not be reliable. For the moneyness of 0.93-0.95,
the mean percentage error wor 10 and mean percentage error
changes its direction around moneyness of 0.96 and 0.98 from positive error to
negative er
from 0.96 to percentage
ks out to 7.
rors. The mean percentage errors are negative for the moneyness
1.04. From the moneyness of 1.05, again the Mean
113
errors change its direction from negative to positive but they are at the
minimum.
The Mean percentage errors vary from -3.51 to 7.10. That is the BS
model predicts the call option price with a minimum accuracy of 92.90% and a
maximum accuracy of 96.49%. This should be a wonderful result.
CHART 4.5
MEAN PERCENTAGE ERRORS FOR DIFFERENT MONEYNESS FOR DATA MORE THAN 5000
MEAN PERCENTAGE ERROR
-4
-2
0
2
4
6
8
0.93-0.95 0.96-0.98 0.99-1.01 1.02-1.04 1.05-1.07 1.08-1.10
MONEYNESS
M P
E
-6
From the chart, it is easily inferred that in-the-money options are better
firmed by the
strike biases for the 1975-1976 periods, but found that it was the reverse of the
biases reported by Black, Fisher. The research paper of Bakshi, Gurdip, Cao,
Cha ion
priced than the OTM options. The model systematically underpriced at the deep
OTM options and the deep ITM options. The near OTM options and at-the-
money options are overpriced. This is against the prediction of Black, Fisher
[23] that the model systematically underpriced deep OTM options and
overprices the deep ITM options. MacBeth and Merville [94], con
rles and Chen, Zhiwu [10], “Empirical Performance of Alternative Opt
114
pricing Models” where, call optio a pe 88
to May 1991, the Mean percentage pricing errors vary from 2.49% to - 65.78%
for th eyne 0 o e
moneyness. For te h gs t rr
from 10 to 100%.
e conclu of Fisher Black, MacBeth and Merville, and Bakshi,
Gurdip that the model correctly prices the A optio e not c t in I
option market. This may be due to the fact that from 5-05-2003 to 31-10-2007,
the stock market was having a bull market, wherein, the NIFTY galloped from
9 0.65. Th igh n upw as in the market.
4.7 BIASES OVER TIME TO EXPIRATION
greater than 61 days. This is in line with the NSE offering
urrent month, near month and far off month expirations. The
38,749 ns (S&P 500) for riod of June 19
e mon ss range o
tune, Pe
f 1.06 to
r [53], in
.94. It als
is findin
opined t
inferred
hat biases
hat the e
xist over
ors vary
But, th sions
TM ns ar orrec ndian
45.40 to 590 ere m t be a ard bi
4.7.1 MEAN ABSOLUTE ERRORS
Next, the study is made to understand whether there exist biases over
the time to expiration (life of the option). The time to expiration is divided into
three categories; life with less than or equal to 30 days, life between 31 days to
60 days and life
options with c
options are segregated accordingly and the summary is given in the table
no.4.7. Around 78.01% of options are with life less than or equal to 30 days,
options with life between 31 days to 60 days are 21.77% and the options with
life more than 61 days are only 210 with a percentage of 0.22. Though for the
options with more than 61 days the results are tabulated, we can omit the
findings as it may not represent a correct sample size and illiquid.
115
TABLE 4.7
DETAILS OF DATA OVER LIFE OF OPTIONS AND MONEYNESS
MONEYNESS < = 30 DAYS
31- 60 DAYS
> 61 DAYS TOTAL
< 0.83 113 70 4 187
0.84 - 0.86 202 163 5 370
0.87 - 0.89 612 390 3.00 1,005
0.90 - 0.92 1,923 1,227 13 3,163
0.93 - 0.95 5,728 2,916 27 8,671
0.96 - 0.98 12,149 4,916 47 17,112
0.99 - 1.01 16,882 5,063 39 21,984
1.02 - 1.04 14,568 3,034 41 17,643
1.05 - 1.07 9,632 1,542 17 11,191
1.08 - 1.10 5,771 771 8 6,550
1.11 - 1.13 3,425 427 2 3,854
1.14 - 1.16 2,099 226 3 2,328
1.17 - 1.19 1,274 108 1 1,383
> 1.20 479 36 0 515
TOTAL 74,857 20,889 210 95,956
se definite patterns
of residuals / errors show the bias over life of the options also.
The MAE as per life of the options were calculated and tabulated in
Table no. 4.8. The MAE were not equal for the three different periods of
expirations, they change significantly. For options ITM; the MAE were less for
options with life less than 30 days compared to options with life between 31 to
60 days. For OTM options; options with lives less than 30 days have more
errors when compared to life in between 31 to 60 days. The
116
TABLE 4.8
MEAN ABSOLUTE ERRORS FOR VARIOUS LIVES OF OPTIONS
MEAN ABSOLUTE ERRORS MONEYNESS
S0 / X ALL
Up to 30
DAYS 31- 60 DAYS
> 61 DAYS
<= 0.83 0.40 0.45 0.34 0.66
0.84 - 0.86 0.56 0.70 0.46 0.54
0.87 - 0.89 0.60 0.74 0.44 0.28
0.90 - 0.92 0.58 0.62 0.52 0.79
0.93 - 0.95 0 0.58 8.58 0.5 0.42
0.96 - 0.98 0 0.51 7.50 0.4 0.50
0.99 - 1.01 0.37 0.37 0.39 0.29
1.02 - 1.04 0 0.23 0.24 0.3 0.18
1.05 - 1.07 0.1 0.16 16 0.2 0.11
1.08 - 1.10 0.1 0.13 43 0.1 0.10
1.11 - 1.13 0.11 0.11 0.13 0.06
1.14 - 1.16 0.11 0.11 0.11 0.18
1.17 - 1.19 0.12 0.12 0.13 0.31
> 1.20 0.19 0.18 0.37 Nil
erstood from the chart no. 4.6 shown.
From the table, it may be observed that the ranges of MAE are almost
the same for different categories of life of options. For lives with less than 30
days, it varies from 0.58 to 0.13 and for the lives in between 31 to 60 days it
ranges from 0.58 to 0.14. The patterns of the distribution of MAE are clearly
viewed and und
117
CHART 4.6
MEAN ABSOLUTE ERRORS OVER ARIOUS LIVES OF THE OPTIONS AND MON L DATA)
VEYNESS (FUL
MEAN ABSOLUTE ERRORS (MAE)0.90
0.000.100.20
0.300.400.500.60
0.700.80
0.84 -0.86
0.87 -0.89
0.90 -0.92
0.93 -0.95
0.96 -0.98
0.99 -1.01
1.02 -1.04
1.05 -1.07
1.08 -1.10
1.11 -1.13
1.14 -1.16
1.17 -1.20
>1.20
MONEYNESS (S / X)
M A
E
Total "=< 30 DAYS" 31-60 DAYS "> 61 DAYS"
In the deep OTM, ranging from 0.84 to 0.92 the number of options are
ery less (Please refer Table Nos. 4.7 & 4.8 above) and hence, the pattern of
Moreover, the number of data for the life of the option more than
1 days is as low as 210. Thus, the predictions based on life more than 61 days
can be
v
errors are erratic and not uniform. Also, for the deep ITM options above the
moneyness of 1.10, the errors are having varying patterns as the number of
data is small.
6
neglected.
If we consider the options across 0.93 to 1.10, the MAE are reducing as
the moneyness increases or the options tend to move from deep-out–of money
to deep-in-the-money. The chart restricted to this range is given below in chart
No. 4.7.
118
CHART 4.7
MEAN ABSOLUTE ERRORS OVER VARIOUS LIVES OF THE OPTIONS AND MONEYNESS FOR DATA MORE THAN 5000
MEAN ABSOLUTE ERRORS
0.35
0.50
0.9 6 -0.9 .99 -1.0 2 -1.04 -1.07 -1.10
ONEY
MEA
N A
BS
OLU
TE E
RR
OR
S
0.400.45
0.000.050.100.150.200.250.30
0.93 - 5 0.9 8 0 1 1.0 1.05 1.08
M NESS
ALL DATA <= 3 S 0 DAY 3 DAYS1 - 60
AE c s con approximately at the moneyness of 0.99 to
1.01. For the life less than or equal to 30 days, the slope of the curve upto 1.04
is stee
g
bias over the moneyness and life. The meaning is that BS model predicts the
ITM options better than OTM options. This is in accordance with the findings of
the Black [23], and against the findings of Macbeth and Merville [94], and
The M urve verge
per and becomes flatter after this point. The curve of 31-60 days is also
steeper earlier and slightly less steeper afterwards. It is a straight line from
moneyness of 0.99 up to 1.10; which is related to Deep ITM options. Options
with current month expiration, i.e. up to 30 days life are having less MAEs
compared to options with life 31-60 days, especially ITM options. They are
higher for OTM options. For an unbiased model, the line should be more or less
horizontal. The chart reveals the systematic biases of the model across various
moneyness. The errors are decreasing as the moneyness increased, showin
119
Rubinstein [41]. The reason being the sample timings played a major role.
Some aggressive bull conditions and crash involved recessions may be the
reasons that can be explained for the said differences in the findings of different
studies.
4.7.2
Black opined that ATM options are correctly priced and the errors are
found ITM and OTM options. But, so far in our findings the MAE are not zero for
ATM o
MEAN OF MEANS OF ABSOLUTE ERRORS
ptions. Hence it is a curiosity where the mean errors are at the minimum
for ATM options. Accordingly, the errors are analyzed as a deviation from the
mean of means of absolute errors and given in the table no. 4.9.
TABLE 4.9
EXCESS ERROR OVER MEAN OF MEANS OF ABSOLUTE ERRORS
ALL Up to 30 DAYS 31- 60 DAYS Moneyness
S0 / X Absolute
Errors Excess Errors
Absolute Errors
Excess Errors
Absolute Errors
Excess Errors
0.84 - 0.86 0.56 0.22 0.70 0.34 0.46 0.140.87 - 0.89 0.60 0.26 0.74 0.38 0.44 0.120.90 - 0.92 0.58 0.24 0.62 0.26 0.52 0.200.93 - 0.95 0.58 0.24 0.58 0.22 0.58 0.260.96 - 0.98 0.50 0.16 0.51 0.15 0.47 0.150.99 - 1.01 0.37 0.03 0.37 0.01 0.39 0.071.02 - 1.04 0.24 -0.10 0.23 -0.13 0.30 -0.021.05 - 1.07 0.16 -0.18 0.16 -0.20 0.21 -0.111.08 - 1.10 0.13 -0.21 0.13 -0.23 0.14 -0.181.11 - 1.13 0.11 -0.23 0.11 -0.25 0.13 -0.191.14 - 1.16 0.11 -0.23 0.11 -0.25 0.11 -0.211.17 - 1.20 0.12 -0.22 0.12 -0.24 0.13 -0.19
120
The mean of MAE for the life less than or equal to 30 days is 0.36 and
he MAE of the full data is 0.34 and
error for less than or equal to 30 days life is higher than it and the error is lower
ss than or equal
to 30 days and at 1.02 to 1.04 for options with lives between 31 to 60 days.
CHART 4.8
EXCESS ERROR OVER MEAN OF MEANS OF ABSOLUTE ERRORS
for lives in between 31 to 60 days is 0.32. T
to it for 31 to 60 days life. As it has been argued earlier, the data for life greater
than 61 days are not considered for the analysis. The excess errors are plotted
in chart no 4.8.
From this table, it is observed that the mean of MAE are at minimum and
almost zero for options ATM (moneyness of 1.00). ITM options are having less
excess errors when compared to OTM options. Options with lives less than or
equal to 30 days are having less excess errors that of options with lives 31 to
60 days. The mean error of means occur at 0.99 to 1.01 for le
EXCESS ERROR OVER MEAN OF MAES
-0.30
-0.20
-0.10 0.8
6
0.8
9
0.9
2
0.9
5
0.96
- 0.
98
0.99
- 1.
01
1.0
4
1. 1.1
0
1.1
3
1.1
6
1.2
0
0.00
0.10
0.20
0.30
0.40
0.50
0.84
-
0.87
-
0.90
-
0.93
-
1.02
-
1.05
-07
1.08
-
1.11
-
1.14
-
1.17
-
NEYNESS
EXC
ESS
ERR
OR
MO30 DAYS 31 - 60 DAYS
121
From this chart, it may be obs at the er distributed around
the options ing mean of MAE around ATM they
are not distributed evenly. ITM options
options. Options with higher life tend to e more errors than the options with
life less than 30 days. This may be due to
are more for life within 30 da
4.7.3 MEAN PERCENT ERR
The MAE will inform the magnitude of the errors but not reveal the
direction. To get more insight about the errors, mean percentage error are very
are
options namely, up to 30 days, 31 to 60
days and more than 61 days. They are exhibited in the chart no. 4.9 and the
values
MEAN PERCENTAGE ERROR OVER VARIOUS LIFE OF OPTIONS
erved th rors are
ATM. Though the errors are hav
are having lesser errors than the OTM
hav
the reason that the numbers of data
ys when compared to life between 31 to 60 days.
AGE ORS
useful and necessary. In view of the above, mean percentage errors
calculated for the three different lives of
are tabulated in table number 4.10 in the next page.
CHART 4.9
MEAN PERCENTAGE ERRORS
-40-30-20-10
01020304050
< 0.
83
0.84
86
0.87
89
0.90
92
0.93
95
0.96 0.99
0
1.02
0
1.05
07
1.08
10
1.11
13
1.14
16
1.1
19
> 1.
20
MONEYNESS
M P
E
- 0.
- 0.
- 0.
- 0.
- 0. -1
.
-1.
-1.
-1.
- 1.
- 1.
7-1.
98 1 4
<= 30 DAYS 31 TO 60 DAYS > 61 DAYS
122
TABLE 4.10
MEAN PERCENTAGE ERROR FOR VARIOUS LIFE OF OPTIONS I
MEAN PERCENTAGE ERROR
MONEYNESS < = 30 DAYS
31- 60 DAYS
> 61 DAYS
< = 0.83 34.18 31.01 17.40 0.84 - 0.86 25.23 35.19 37.21 0.87 - 0.89 30.69 22.47 -0.08 0.90 - 0.92 25.44 8.21 24.09 0.93 - 0.95 11.81 -1.76 -34.84 0.96 - 0.98 0.77 -6.11 -19.08 0.99 -1.01 -2.97 -5.14 -29.09 1.02 -1.04 -1.05 -4.63 -1.45 1.05 -1.07 0.34 -1.80 -10.27 1.08 -1.10 0.95 -2.05 -2.48 1.11 - 1.13 0.40 -1.41 1.71 1.14 - 1.16 0.97 -1.83 17.76 1.17-1.19 -1.02 -0.71 30.85 > 1.20 -3.95 -1.97 NA
From the table and chart it may be observed that the mean percentage
errors vary for different lifes of options and has a definite pattern of decreasing
over increase in moneyness. As explained earlier, the data related to options
0 including for life more than 61 days can be omitted. After
omitting that the curves are smooth and show patterns, which is displayed in
chart no. 4.10.
less than 500
123
CHART 4.10
MEAN PERCENTAGE ERROR FOR DIFFERENT LIFE OF OPTIONS FOR DATA MORE THAN 5000
MEAN PERCENTAGE ERRORS
-8-6-4-202468
101214
M P
E
0.93 - 0.95 0.96 - 0.98 0.99 -1.0 .02 -1.04 1.05 -1.07 1.08 -1.10
MONEYNESS
1 1
<= 30 DAYS 31 TO 60 DAYS
From the curves exhibit ay erved redictability and
resultant errors are not only influenced by the lifes of options but also by the
moneyness.
So far the options were categorized into ee different types in the same
style of NSE. Now, in another angle, the options were categorized into eight,
each having te as d b he table no.4.11. During the
categorization it was observed that there exist small number of (18) outliers,
which distort the error pattern in a sign
using SAS and al me hes were id and taken off
from the sample. The calculated Mean percentage errors are tabulated in the
table no. 4.11.
ed, it m be obs that the p
thr
n days life detaile elow in t
ificant manner. Box-plots were drawn
by graphic thod t e outliers entified
124
TABLE 4.11
MEAN PERCENTAGE ERROR OVER VARIOUS LIVES OF OPTIONS II
LIFE IN DAYS
No. of data
Mean Absolute
Error
Mean Percentage
Error Percentage
of data Cumulative Percentage
of Data
0 - 10 20,160 0.16 3.35 21.01 21.01
11 - 20 23,635 0.19 3.52 24.64 45.65
21 - 30 31,062 0.20 -1.72 32.38 78.03
31 - 40 15,283 0.23 -1.70 15.93 93.96
41 - 50 4,480 0.26 -4.14 4.67 98.63
51 - 60 1,126 0.25 -8.43 1.17 99.80
61 - 70 133 0.22 -11.25 0.14 99.94
71 - 80 59 0.31 -17.34 0.06 100.00
The mean percentage errors vary from 3.35 to -17.34. If the mean
percentage errors for the data which are more than 5,000 at least are
considered reliable, then the errors are within 3.35 to -1.70 only. The table
enlightens us that the most of the traded options are having life within forty days
e sample, 93.96% of options got Mean percentage errors within
only. Out of th
3.35 to -1.70. If one includes the life within 41- 50 days, then the mean
percentage error slightly increases to - 4.14 but covers 98.63% of options of our
sample. To understand better, the same are furnished in the chart no. 4.11.
125
CHART 4.11
MEAN PERCENTAGE ERROR OVER VARIOUS LIFE OF OPTIONS
MEAN PERCENTAGE ERROR
-
5.00
-20.00
15.00
LIFE IN DAYS
-10.00
-5.00
0.000 - 10 11- 20 21 - 30 31 - 40 41 - 50 51 - 60 61 - 70 71 - 80
EM
P
Though the mean percentage errors were very low, the mean
percentage errors were still show a patte m distribution.
The errors are positive for the op p to 20 day. The options having
more t
model over-prices the options within life of 20 days and under-prices the options
having life more than 20 days. The errors are stribut omly around
m stra p
be inferred that the biases exist with lifes of th s also
rn and not having rando
tions with life u
han 20 days have negative mean percentage errors. In other words, BS
not di
attern can be
e option
ed rand
observed. Ththe mean, but ore or less a ight line
the
us it can
.
126
4.8 BIASES OVE ATILIT F S RETU
the s e and strike price, a lained ter III,
luencing par r of the model is the volatility of stoc rns. M
over, i
4.8.1 MEAN ABSOLUTE ERRORS
The sample is observed for the actual volatility for all the twenty-eight
companies. The volatility varies from 0.16 to 0.95, with some meager data with
specific v ly
Dr.Reddys and Infosys. In view of the ab mple is categorized from
0.10 in steps of 0.10 till the volatility of 1.00 and volatilities 1.21 and 4. The
ratios. It can be observed that the figures related to less than 5000 are marked
R VOL IES O TOCK RNS
Next to tock pric s exp in chap the
more inf amete k retu ore
t is the only factor that is estimated in the model as other variables and
parameters are observable and taken directly. Hence, the study about volatility
is important and interesting. In the following sub chapters the analysis and the
findings are enumerated in detail.
olatility of 1.21 to 1.23 and 4.08 related to only two companies name
ove, the sa
categorization and the respective MAE are given in the table number 4.12. The
errors are given in rupees and rounded off to one rupee. MAE are expressed as
with shades and the figures may not represent the true value.
127
TABLE 4.12
L F S V S RETURNS
MEAN ABSO UTE ERRORS OR VARIOU OLATILITIE OF STOCK
Volatility No. of Data
Total Observed
Price Rs.
Total Absolute
Errors Rs.
Mean Absolute
Error
0.10-0.20 956 23,848 5,934 0.25
0.20-0.30 2 10,39, 1,80,263 0.1732,93 443
0.30-0.40 9 12,39,0 2,17,059 0.1830,18 08
0.40-0.50 6,35,6 1,20,958 0.1919,805 19
0.50-0.60 1,83,600 46,908 0.266,708
0.60-0.70 2,406 1,37,810 43,281 0.31
0.70-0.80 1,607 1,14,065 35,060 0.31
0.80-0.90 809 15,401 6,792 0.44
0.90-1.00 315 5,424 4,471 0.82
1.21-1.23 199 9,361 12,457 1.33
> 4 30 5,096 9,886 1.94 Rounded off to rupee
The MAE vary from 0.17 to 1.94. As usual, after omitting the options less
than 5,000, the errors systematically increases from 0.17 to 0.26 as the volatility
increases from 0.20 to 0.60. A pattern is observed as there exists a direct,
positive relation between the errors and the volatility of stock returns.
The errors seem to be low at low volatilities 0.20 - 0.30, 0.30-0.40, and
0.40-0.50. For the volatility is higher to 0.50, the errors increases at a faster rate
to 0.26. The pattern is much easily observed from the chart no. 4.12
128
CHART 4.12
MEAN ABSOLUTE ERROR FOR VARIOUS VOLATILITIES
MEAN A0.35
BSOLUTE ERROR FOR VOLATILITIES
0.30
0.00
0.05
0.10
0.15
0.20
0.25
0.20-0.30 0.30-0.40 0.40-0.50 0.50-0.60 0.60-0.70 0.70-0.80
VOLATILITY
M A
E
The errors are increasing with volatility and the model has prediction bias
over the volatility of stock returns. Some portion of these high errors may be
due the fact that the numbers of data are relatively low.
It is analyzed in another angle, for each category of moneyness, the
volatility is changed and the MAE are calculated, summarized and tabulated.
The MAE are then drawn in a chart and displayed in chart no. 4.13. The
influence of moneyness is seen as ITM options are having flat curves and
curves of OTM options tend to be steeper.
129
CHART 4.13
MEAN ABSOLUTE ERROR OVER VARIOUS VOLATILITY AND MONEYNESS
Mean Absolute Errors
0.58
0.080.30-0.40 0.40-0.50 0.50-0.60 0.60-0
0.18
.70
0.28
0.38
0.48
VOLATILITY
M A
E
0.90-0.92 0.93-0.95 0.96-0.98 0.99-1.01 1.02-1.04 1.05-1.07 1.08-1.10 1.11-1.13
130
4.8.2 MEAN PERCENTAGE ERRORS
As followed for other variables and parameters, mean percentage errors
re also calculated for different volatility, summarized and tabulated in table no.
.13. The volatility ranging from 0.20 to 0.50 contains 86.42 % of the data. It
cludes volatility of 0.50 - 0.60, 93.41% data are covered. The other
insignificant data were shown in shade. The plots of the mean percentage
errors are shown in chart no. 4.14
TABL 4.13
MEAN PERCENTAGE ERROR FOR VARIOUS VOLATILITY
a
4
in
E
Volatility No. of Data
Observed Option Price
Percentage Errors
0.10-0.20 956 23,848 28.68
0.20-0.30 32,932 039,443 13.24 1,
0.30-0.40 30,189 239,008 4.62 1,
0.40-0.50 19,805 635,619 -7.09
0.50-0.60 6,708 183,600 -20.93
0.60-0.70 2,406 137,810 -44.44
0.70-0.80 1,607 114,065 -32.77
0.80-0.90 809 15,401 -59.26
0.90-1.00 315 5,424 -114.51
1.21-1.23 199 9,361 -157.64
> 4 30 5,096 -206.79
131
CHART 4.14
DETAILS OF MEAN P RIOUS VOLATILITY ERCENTAGE ERROR FOR VA
MEAN PERCENTAGE ERROR
-25
-20
-15
-10
-5
0
5
0.20-0.30 0.30-0.40 0.40-0.50 0.50-0.60VOLATILITY
M P
E
10
15
entage error is positive and high at 13.24 for the volatility
40 it drops to 4.62 and is negative
for the volatility range of 0.40 – 0.50 but magnitude-wise it increases to around
The mean perc
of 0.20 - 0.30. For next category of 0.30 – 0.
7. As volatility further increases, the mean percentage error raised to -20.93.
This high error may be due to the fact that the number of data is low at 6708
only for a period of 5 years and 10 months. A definite pattern of errors is
evidenced from the above chart. Thus, pricing biases do exist for volatility of
stock returns also.
132
4.9 R
ive. The model has biases over all the other
categorizes the option sample into
different RFR, analyses them and explains the findings in detail.
RISK – FREE INTEREST RATES I
ISK - FREE INTEREST RATE
Risk - free interest rate is the last parameter under study for which the
call option price is least sensit
variables and parameters. This chapter
4.9.1 MEAN ABSOLUTE ERRORS
The actual risk - free interest rates are ranging from 4.08% to 11.81% as
per the money market conditions prevailing during the period of study. They are
converted into continuously compounded interest rates and used in the BS
model. These continuously compounded interest rates are categorized from 4%
to 11% in steps of one percentage. The MAE are measured and tabulated in
table No. 4.14.
TABLE 4.14
MEAN ABSOLUTE ERRORS AND
Actual Risk-Free-
Interest Rate
R %
Continuously Compounded Interest rate
r %
No. of Data
Total Observed
Price
Rs.
Total Absolute
Errors
Rs.
Mean Absolute
Error
4.08 - 5.13 4 - 5 30,112 1,085,765 234,634 0.22
5.14 - 6.18 5 - 6 32,256 1,112,029 191,459 0.17
6.19 - 7.25 6 - 7 19,166 690,916 155,832 0.23
7.26 - 8.32 7 - 8 10,437 399,729 79,361 0.20
8.33 - 9.41 8 - 9 2,472 75,338 13,714 0.18
9.42 - 10.52 9 - 10 1,205 36,739 6,371 0.17
10.53 - 11.63 10 - 11 245 6,434 1,235 0.19
> 11.64 > 11 63 1,668 305 0.18
133
As per the data of the table, most of the options are traded during the
period in which the risk - free interest rates are within 4% to 8%. This covers
95.85%
how no major pattern and seem to be flat at
around
CHART 4.15
MEAN ABSOLUTE ERROR AND RISK - FREE - INTEREST RATE WITH DATA MORE THAN 5000
of the sample. The errors are varying from 0.17 to 0.23 for various risk-
free interest rates. If the low volumes related data are eliminated from the study,
as shown in shades, the figures s
0.22. The same is plotted in chart no. 4.15.
MEAN ABSOLUTE ERROR
0.150.160.170.180.19
20
4 - 5 5 - 6 6 - 7 7 - 8
RFR
M A
0.0.210.220.23
E
From the chart no.4.15, it may be observed that the curve is almost flat
except for two points. Hence, it may be concluded that the errors are having no
certain pattern with risk-free-intere reason is very simple. Risk-free-
interest rate is ption price in
the BS model.
4.9.2 MEAN PERCENTAGE ERRORS
Mean percentage errors are measured against the various risk - free
interest rates and tabulated in the table no.4.15.
st rate. The
the least sensitive factor of determinants of call o
134
TABLE 4.15
MEAN PERCENTAGE ERROR AND RISK - FREE - INTEREST RATE II
Actual Risk - Free - Interest Rate
R
%
Continuously Compounded Interest rate
r
%
No. of Data
Total
Observed Price
Rs.
Mean
Percentage Error
Rs.
4.08 - 5.13 4 - 5 30,112 1,085,765 -1.45
5.14 - 6.18 5 - 6 32,256 1,112,029 5.26
6.19 - 7.25 6 - 7 19,166 690,916 -5.71
7.26 - 8.32 7 - 8 10,437 399,729 1.89
8.33 - 9.41 8 - 9 2,472 75,338 7.00
- 10.52 9.42 9 - 10 1,205 36,739 -7.94
10.53 - 11.63 10 - 11 245 6,434 -8.66
> 11.64 > 11 63 1,668 -0.87
CHART 4.16
MEAN PERCENTAGE ERROR AND RISK - FREE - INTEREST RATE
MEAN PERCENTAGE ERROR
2
4
6
-8
-6
-4
-2
04 - 5 5 - 6 6 - 7 7 - 8
RFR
M P
E
135
From the above table and chart, it may be inferred that the mean
uted against the risk – free interest rate and
no definite pattern is available. Hence, there is no bias on the model. This is
coincid
individual absolute errors is anal of the options it is less than or
equal to Rs.1.27, the next 25% options got it in between Rs.1.27 to Rs.3.12.
The n of the options have
it less than Rs. 15.62 and greater than Rs. 7.03. When compared to the
observed call option price as high as Rs.1200 and above, these errors may not
be high. The Mean percentage errors vary from -3.51 to 7.10 depending upon
the moneyness. That is the BS model predicts the call option price with a
minimum accuracy of 92.90% and a maximum of 96.49%; mean 95.4%.
Howe side than
the low 84.42%;
the n 9%. The
predic od. The
research paper of Bakshi, Gurdip, Cao, Charles, and Chen, Zhiwu [10],
“Empirical Performance of Alternative Option pricing Models” where, 38,749 call
options (S&P 500) for a period of June 1988 to May 1991, the Mean percentage
pricing
ictability varies from 34.22% to 97.51%. Fortune, Peter
[53], in his findings inferred that the errors vary from 10 to 100%. It may be
percentage error is randomly distrib
ing with the findings of MAE above.
4.10 CONCLUSION
The predictability of the model is checked by mainly measuring MAE and
mean percentage errors. The median-based percentiles also calculated. As far
as the MAE are concerned, it varied from 0.12 to 0.43 for the moneyness of
1.10 to 0.93. The average is 0.24. That is according to the MAE predictability of
the model vary from 57% to 88%; mean being 76%. When the distribution of
yzed, for 25%
ext 25% have it in between Rs. 3.12 to Rs. 7.03; 15%
ver, the predictability of most of the options is biased on higher
er side. Middle 50% of options are having the predictability of
ext 30% of options are having the predictability of 76.2
tability of the model in Indian Option Market is extremely go
errors vary from 2.49% to -65.78% for the moneyness range of 1.06 to
0.94. That is, the pred
136
concluded that the model’s predictability in Indian Option Market is comparably
better than the predictability of the model in overseas option markets as per the
above research papers published abroad.
ther as
oneyness and all the others are individually tested. The MAE and mean
percen
or of the model.
Merville [94], Bakshi, Gurdip [10], and Rubinston [118] also
confirmed the existence of the biases. But the biases are not in the same
direction in all the above cases. In Indian market, the model systematically
underpriced deep OTM options and the deep ITM options. The near OTM
options and ATM options are overpriced. This is against the prediction of Black,
Fisher
reverse of the
biases reported by Fisher Black [23].
e used to rectify the errors
also. For example, if the model under-prices the options of moneyness 1.02 -
1.04 by 4.50%; the investor can adjust
error. Thu
correct pr rd of caution is that the systematic
biases may change over different periods of observation like, recession period
All the variables and parameters of the model are studied towards
prediction biases, if any. The stock price and strike price are used toge
m
tage errors are showing definite biases over change in moneyness,
volatility and life of the options. The model does not exhibit any bias over risk -
free - interest rate, which is the least sensitive fact
Our findings are in line with the biases that are explained in the papers of
Geske, Robert and Roll, Richard in their paper [55], “The Black –Scholes call
option pricing model is subject to systematic empirical biases. These biases
have been documented with respect to the option’s exercise price, its time to
expiration, and the underlying common stock’s volatility…”. Black, Fisher [23],
MacBeth and
[23] that the model systematically underpriced deep OTM options and
overprices the deep ITM options. MacBeth and Merville [94] confirmed by the
strike biases for the period 1975-1976, but found that it was the
But, these biases are systematic and can b
the price accordingly and correct the
s as far as the biases are systematic it can be used to predict the
ice by adjustments. However, a wo