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
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CHAPTER IV PREDICTABILITY OF THE BS MODEL IN INDIAN
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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
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.
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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
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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