138

CHAPTER V RESIDUAL ANALYSIS

5.1 INTRODUCTION

Residuals are the d served call option prices

and the prices predicted b s errors in our research.

Econometricians often opined that the resi eally useful in predicting

ifferences between the ob

y BS model, often called a

duals are r

the model adequacy. Gujarati, Damodar N. [61] opined that examination of the

residuals is a good visual diagnostic to detect autocorrelation or

heteroscedasticity and also for model specification errors; such as omission of

an important variable or incorrect functional form. If, in fact there are such

errors, a plot of the residuals will exhibit distinct pattern. The residuals to be of

e

d still can be improved.

In the above connotation, this study also analyzes the residuals to check

the distribution of residuals

n of residuals with the variables and parameters of the model;

Stock

constant variance. Hence this study attempts to check the model adequacy of

the BS

ra test, Bowley’s and Pearson’s

t from histograms.

randomly generated and should not have any definite pattern. Any definit

pattern may indicate that the model is not adequate an

5.2 DISTRIBUTION OF RESIDUALS

the model adequacy. The analysis is made to check

and correlatio

Price, Time to Expiration, Strike Price, Risk-Free-interest Rate and

Volatility of returns of the stock. The residuals known as white noise are

supposed to be uncorrelated, normally distributed, having a zero mean and

Model through the residual analysis. The normality of the distribution is

checked through skewness, kurtosis, Jarque-Be

skewness coefficients etc apar

139

A partial study in this angle has been made with the options data from

ournal of Management Research

ol.1, No.2, April - June 2006 [106]. (Annexure II).

As the total number of data is 95,956, there exists the same number of

as the extreme values will affect the mean in a very highly

gnificant way. Hence many econometricians use the order-based statistics like

lower

, the observations are arranged in order of

increasing magnitude. Median is the middle value of the ordered sample, and

QL or Q1 is the

valu corre pond o 23989th and 23990th number counted upward

e upper quartile Q3 or QU is

same, depth of 23,989 and

fference between Qu and QL,

amoun a table will run into pages

Figure 5.5 in page 147.

1.1.2002 to 30.7.2005 and is published in the J

V

5.2.1 FULL DATA OF RESIDUALS

residuals / errors. The normal distribution can be checked by many methods.

Using histograms will be of easy testing ways for identifying and testing the

normality of a distribution. Another popular way is using mean-based statistics

like mean, standard deviation, higher moments such as skewness and kurtosis.

But, though the mean can be proved to be an unbiased estimator, it is not a

resistant summary,

si

quartile (Q1 or QL), median (Q2), upper quartile (Q3 or QU), Lower Extreme

Value (XL), Upper Extreme Value (XU) and Inter Quartile Range (IQR) to test

the normality of a distribution which are not affected by extreme values and

hence, known as resistance summaries.

In this order-based statistics

hence, splits the list into two equal halves. In our sample of 95,956, the median is

calculated by the average of values corresponding to 47978th and 47979th of the

ordered observations, which is equal to 0.87. The lower quartile

average e s ing t

from the median; works out to -2.08. Similarly, th

calculated downward of the order relating to the

23,990; which is 3.79. Interquartile range is the di

ting to 5.87. Exhibiting all the 95,956 residuals in

and may not be appreciable, but a summary is given in

140

5.2.1.1 Histograms The residuals / errors are grouped into different categories having an

interval of ten. Then the numbers of residuals in that interval are counted and

shown below in the chart no. 5.1.

From the histogram, it may be observed that the peak frequency occurs

the frequency drops

rastically. However, it is difficult to judge whether it represents a normal

distribution.

HISTOGRAM OF RESIDUALS WITH FULL DATA

around zero and as the range deviates from zero,

d

CHART 5.1

DISTRIBUTION OF RESIDUALS

0

10,000

20,000

30,000

40,000

50,000

60,000

20 70 T

20 T

70 T

20 T

70 T

20 T

70 T

20 T

70 T

200

-7

00

-2

00

30

0

800

T

130

0 T

00

TO

230

0 TO

28

00

TO

330

0 TO

38

00

TO

430

0 TO

48

00

TO

530

0 TO

58

0

RESIDUALS RANGE

FREQ

UEN

CY

-630

-580

-530

-480

-430

-380

-330

-280

-230

-180

-130 -8 -3 2 7

12 17 22 27 32 37 42 47 52 57

TO

TO

-6

-5

O

-5O

-4

O

-4O

-3

O

-3O

-2

O

-2O

-1

O

-1 T

O T

O TO

TO O O

18

141

The maximum frequencies occur around zero. The left hand side and the

right hand sides seem to be not equal. The frequency of negative residuals is

38,231 and the positive residual is higher at 57,725. But, the magnitudes of

positive residuals are lower than that of negative residuals. The chart shows

at the residuals are not normally distributed. The same is clear in the line th

chart below. Only very small numbers of residuals with large magnitude are

noticeable as the tail is large on both sides. The same can even be judged

better using the line diagram which is shown below in chart no. 5.2.

CHART 5.2

LINE CHART OF RESIDUALS WITH FULL DATA

DISTRIBUTION OF RESIDUALS

20,000

30,000

40,000

50,000

60,0 0

FREQ

UEN

CY

-10,000

10,000

0

TO

TO

TO

TO

TO

TO

-3

70

TO

-3

20

TO

-2

70

TO

-2

20

TO

-1

70

TO

-1

20

-80

TO

-70

-30

TO

-20

20 T

O

30

70 T

O

80

120

TO

130

170

TO

180

220

TO

230

270

TO

280

320

TO

330

370

TO

380

420

TO

430

470

TO

480

520

TO

530

570

TO

580

0

-620

-570

-520

-470

-420

-630

-580

-530

-480

-430

-380

-330

-280

-230

-180

-130

RESIDUALS RANGE

This can be compared with the normal curve drawn with the actual

histogram generated using SAS, shown below in chart no. 5.3.

142

CHART 5.3

From t S b e height of

the distribut double th ight of the normal distribution shown in

red colour line. The left and right tails are also larger than that of a normal

distribution. ry thin a spread e when ared to the

frequency of the corresponding als. The deviation e residuals

distribution from the normal distribution implies either heavy outliers or

inadequacy of the model.

5.2.1.2 Mean-Based Statistic

A fundamental task in many statistical analysis is to characterize the

location and variability of a data set. Mean-based statistics rely on four

measures; itself (ce read), the

coefficient of skewness (symmetry), and the coefficient of kurtosis (heavy or

thin tails). T e of mea standard deviation are well known and

need no exp kewness is a measure of symmetry, or more precisely,

the lack of symmetry. A distribution, or data set, is symmetric if it is similar to the

he above SA output, it may be clearly o served that th

ion is almost e he

They are ve nd the is larg comp

residu high of th

s

the mean ntre), the standard deviation (sp

he measur n and

lanation. S

143

left and right of the center point. To explain better sample figures of positive and

negative ske iven belo gure no.

FIGURE 5.4

POSITIVE VS NEGATIVE SKEWNESS

wness are g w in fi 5.4.

POSITIVE SKEWNESS NEGATIVE SKEWNESS These graphs illustrate the notion of skewness. Both Probability

Functions have the same expectation and variance.

The one on the left is positively skewed. The one on the right is

Density

negatively skewed.

Skewness is the third moments of the distribution which is measured by

the following formula.

Where, Yi is the residual of the ith occurrence. Ỹ is the mean of the residuals. s is

the standard deviation of the sample of residuals. N is the sample size.

Also, skewness can be measured using Bowley’s Coefficient of

Skew h is

more precisely defined in the subsequent pages.

ness and Pearson’s measure of skewness to test the distribution, whic

144

Kurtosis is a measure of whether the data are peaked or flat relative to a

normal distribution. That is, data sets with high kurtosis tend to have a distinct

peak near the mean, decline rather rapidly, and have heavy tails. Data sets with

low kurtosis tend to have a flat top near the mean rather than a sharp peak. A

uniform distribution will be the extreme case. Kurtosis is the fourth moments as

given below.

In our study both of them are ca ed and used to test the normality of

the residuals. Also, skewness, Kurtosis, Bowley’s Coefficient of Skewness and

Pearson’s measure of skewness a st the distribution.

equal; equality is the required

condition of a normal distribution.

as that of a normal distribution.

f the above, it may be concluded that the distribution of

residuals is not a normal distribution.

The statistics o

lculat

re used to te

The above mean-based statistics are calculated and tested for normality

in the residuals distribution. The observations are:

i. Mean = -0.071 Median = 0.87 Mode = 0.53

ii. The mean, median and mode are not

iii. The skewness is -2.74 and not zero

iv. The kurtosis is 193.76 against three for normal distribution.

In view o

btained by SAS output is given below in figure 5.5.

145

FIGURE 5.5

TRIBUTION OF RESIDUALS ecember 14, 2007

The UNIVARIATE Procedure Variable: RESIDUALS

Moments

N 9595 m Weights 95956 Mean .07101 Sum Observations -6813.98 Std Deviati .9151082 riance 6.120884 Skewness Kurtosis Uncorrected 7455213. rrected SS 454729.5 Coeff Variation Std Error Mean

ic Statistical Measures

Sign M 9759 Pr >= |M| <.0001 3.9409E8 Pr >= |S| <.0001

Quantiles (Definition 5)

38.08

95% 14.05 90% 8.39 75% Q3 3.79 50% Median 0.87

-17.48 -

0% Min -625.01

DIS D

6 Su

-0on 16

-2.7390091 Va 28

193.77488SS 2

-23820.2363 Co 27

0.05460579

Bas

Location Variability

Mean -0.07101 Std Deviation 16.91511 Median 0.87000 Variance 286.12088 Mode 0.53000 Range 1227

Interquartile Range 5.87000

Tests for Location: Mu0=0

Test -Statistic- -----p Value------

Student's t t -1.30044 Pr > |t| 0.1935

Signed Rank S

Quantile Estimate

100% Max 602.1099%

25% Q1 -2.08 10% -9.13 5% 1%

58.33

146

5.2.1.2.1 Pearson ness Me e:

Another famo easure of no distribution is Pearson’s measure of

skewness, which s below

Pearson’s measure of skewness9 = 3 (Mean – Median) / standard deviation

O

earson’s measure of skewness = (Mean – Mode) / standard deviation

e – Bera (JB) Test of Normality:

It is the practice of econometricians to use mean-based statistics to test

e normality of exploratory data. The third moment of the distribution skewness

nd the fourth moment of the distribution kurtosis are used to test the normality.

be calculated using the formula and comparing it

e test with 2 degree of freedom.

-------------------------------------------------------------------------------------- Mosteller, Fredrick and Tukey, John W. (1977), Data Analysis and Regression: A

cond Course in Statistics, Reading, MA; Addison-Wesley.

’s Skew asur

us m rmal

is defined a :

r

P

The coefficient of skewness, based on median is -0.1668, and based on

mode is -0.0355. It indicates a negative skewness but the magnitude of skewness

is not significant as it is below one.

5.2.1.3 Jarqu

th

a

The JB statistic may

with Chi-squar

S2 (K – 3)2

J B9 = n ------ + ----------

6 24

where S is the skewness coefficient and K is that of the kurtosis and n is the

number of observations.

------------------9

Se

147

The result of 145,610,863 is far higher than the critical value of the Chi-

e distribution

not equal to zero and hence the test for normality is rejected.

U 3

Lower extreme value (XL) = -625.01

the range

is the difference between lower extreme value (XL) and higher extreme value

(XU). IQR is the resistance summ not affected by a few extreme

valu

The same is depicted in the Box-plot as

square statistic with two degree of freedom of 5.99 at the five percent level of

significance. Thus the distribution of residuals is proved that it is not following

normal distribution.

More tests like Student's t test, Sign, and Signed Rank are used to test

the location of the mean of the distribution and the values of -1.30, 9759 and

394,090,000 are well above the critical values and the mean of th

is

5.2.1.4 Order-Based Statistics

The order-based statistics are only five; median (Q2), lower quartile (QL

or Q1), upper quartile (QU or Q3), lower extreme value (XL) and higher extreme

value (XU). These famously known as five-number summary are:

Median (Q2) = 0.87 Lower quartile (QL or Q1) = -2.08

Upper quartile (Q or Q ) = 3.79

Higher extreme value (XU) = 602.10

The dispersion of the residuals can be measured by Inter Quartile Range

(IQR) and the range. IQR is the difference between QU and QL where

ary which is

es but the range is not resistance summary.

in chart no. 5.6

148

CHART 5.6

LOT OF RESIDUABOX-P LS

.2.1.4.1 Bowley’s Coefficient of Skewness

wness, and the Box-plot indicates that the residuals got more

outliers. That is why the mean-based statistics differ significantly with the order-

based statistics.

The above chart gives a clear picture that there are many outliers in the

residuals and relying the mean-based statistics will not be correct.

5

Bowley’s Coefficient of Skewness measure using order-based statistics is

defined as

Bowley’s Coefficient of Skewness S kb = (Q1 + Q3 - 2 Q2) / IQR

Bowley’s Coefficient of Skewness varies from -1 to +1. The negative values

imply the negative skew and the positive values indicate the positive skew that

is the right hand side. For the sample, the value of residuals is -0.005, that is,

the skewness is slightly negative but the magnitude is insignificant.

The difference in the Bowley’s Coefficient of Skewness and Pearson’s

measure of ske

149

5.2.2 RESIDUALS WITHOUT OUTLIERS

Now, let us remove the outliers from the residuals and study further.

Outlier10

is the data with value less than QL - 1.5IQR or more than QU +1.5IQR,

where QL is the lower quartile value and QU is the upper quartile value.

data. In the negative side there are 5,574 (5.81%) and at the positive side 8,096

5.2.2.1 Histograms

The residuals after elimination of outliers are categorized into 24

categories with interval of one. The corresponding frequencies are counted,

summarized and tabulated in table no. 5.1. From this table, it may be observed

here are 30,314 errors that

are negative and 52,153 positive errors in l of

outliers.

----------------------------- -------------------- -------------------------- 10

Mukherjee, Chandan, Howard White, and Marc Wuyts, nometrics and Data Analysis for Developing Countries’, Routledge, Lo

Accordingly, values less than -10.88 or greater than 12.59 are outliers in the

(8.44%) and totaling 13,670 (14.25%) residuals identified as outliers. 85.75% of

the options are having residuals confined to -10.88 and 12.59.

that most of the residuals are near to zero and distributed around zero. More

than 65% of the residuals are within ± 5 % errors. T

the options after the remova

----------------- ------------

(1998), ‘Econdon

150

TABLE 5.1 RESIDUALS AND THEIR FREQUENCY AFTER

REMOVAL OF OUTLIERS

Range of Residuals Frequency Percentage Cumulative

Percentage

-11 To -10 701 0.73 0.73 -10 To -9 933 0.97 1.70 -9 To -8 1,066 1.11 2.81 -8 To -7 1,175 1.22 4.04 -7 To -6 1,428 1.49 5.53 -6 To -5 1,789 1.86 7.39 -5 To -4 2,284 2.38 9.77 -4 To -3 2,866 2.99 12.76 -3 To -2 4,006 4.17 16.93 -2 To -1 5,631 5.87 22.80 -1 To 0 8,255 8.60 31.40 0 To 1 11,096 11.56 42.97 1 To 2 9,783 10.20 53.16 2 To 3 7,961 8.30 61.46 3 To 4 6,057 6.31 67.77 4 To 5 4,638 4.83 72.61 5 To 6 3,504 3.65 76.26 6 To 7 2,549 2.66 78.91 7 To 8 1,914 1.99 80.91 8 To 9 1,477 1.54 82.45 9 To 10 1 1.19 83.64 ,14610 To 11 901 0.94 84.58 11 To 12 723 0.75 85.33 12 To 13 404 85.75 0.42

The histogram, line diagram and Box ting the outliers are

exhibited in chart no. 5.6 to 5.8 respectively. Also, the new box-plot chart is

given in chart no. 5.9.

-plot after elimina

151

FIGURE 5.7

This chart is to be compared with the chart numbers 5.1 to 5.3. The

distribution of residuals after the elimination of outliers looks better than the full

siduals. Still, even though it appears in the shape of a normal distribution, the

f a

normal distribution. Tails are also broader than the normal distribution. But, it is

seen that the distribution of the r proved and closer to a normal

distribu

re

sides are not equal and the height of the distribution also more than that o

esiduals has im

tion.

152

CHART 5.8

DISTRIBUTION OF RESIDUALS AFTER REMOVAL OF OUTLIERS

DISTRIBUTION OF RESIDUALS

0

2,000

4,000

6,000

8,000

10,000

12,000

-11

To

-10

-10

To

-9

-9

To

-8 -8

To

-7

-7

To

-6 -6

To

-5

-5

To

-4 -4

To

-3

-3

To

-2 -2

To

-1

-1

To

0 0

To

1

1

To

2 2

To

3

3

To

4 4

To

5

5

To

6 6

To

7

7

To

8 8

To

9

9

To 1

010

To

11

11

To 1

212

To

13

RANGE OF RESIDUALS

FREQ

UEN

CY

CHART 5.9

UPPER QUARTILE

LOWER QUARTILE MEDIAN

153

Now, the residuals in box-plot are more clearly and evenly distributed on

both sides. The red color box represents middle fifty percentages of the total

residuals; the black line inside the box indicates the median. Still some outliers

are observed in orange colors after the extreme limits, shown in black lines

outside the box.

5.2.2.2 Mean-Based Statistics

fter elimination of the outliers, the statistics are calculated and exhibited

in table

TABLE 5.2 COMPARISION OF MEAN - BASED STATISTICS

A

no. 5.2.

Statistics Full Data Without outliers

Number 95,956 82,287

Percentage 100 85.75

Mean -0.071 1

Median 0.87 0.99

Mode 0.53 0.47

Standard Deviation

16.915 4.25

Skewness -2.74 -0.11

Kurtosis 193.77 0.42

Pearson’s Skewness

-0.1668 0.007

After elimination of outliers, the mean and median is closer and almost

equal but not equal to zero as that of a normal distribution. Mode is also closer

to the mean and median but at the left hand side of them. Skewness is almost

come near to zero but still negative. Kurtosis is one which reduced drastically

om 193 to 0.42. The distribution which is leptokurtic with full data is now fr

154

become platykurtic after elimination of outliers. Definitely, the residuals are

loser to normal distribution but not exactly a normal distribution.

.2.2.3 Order-Based Statistics

The lower and upper quartiles and median of the residuals distribution

ithout outliers are calculated and given along with the statistics of the full data

re given below in table number 5.3

TABLE 5.3 COMPARISION OF ORDER-BASED STATISTICS

c

5

w

a

Statistics Full Data Without outliers

Q1 -2.08 -1.19

Median 0.87 0.99

Q3 3.79 3.43

Bowley’s Skewness

-0.005 0.056

Extreme Low -625.01 -10.88

Extreme High 12.59 602.10

is not

equal to a normal distribution due to the asons that median has increased and

ley’s coefficient of skewness has increased (but near

to zero only).

Truly there is an improvement in the distribution of residuals but it

re

not equal to zero, and Bow

155

5.3 CORRELATION OF RESIDUALS WITH VARIABLES AND

PARAMETERS

is that the determinants of the model

sho als. This study will pin point

abo hich has to be improved on its form

or ant

correla y

hel n with

the its methodology of

cal ame is precisely and

xtensively covered.

(So/X). Likewise,

in this study also, moneyness is used to measure the correlation.

One more important requirement

uld not have any correlation with the residu

ut the particular variables / parameters w

structure. The variables and parameters which are not having signific

tion with residuals can be inferred as well defined in the model and ma

p in predicting the price correctly. If a variable is having high correlatio

residuals, then, that variable has to be studied for

culation, form or structural relationship. The s

e

5.3.1 CORRELATION OF RESIDUALS WITH MONEYNESS The most sensitive factor of option pricing is the stock price and almost

equal to it is strike price. Stock price of different companies are not comparable

with each other in the whole sample as the other parameters and variables like

volatility of stock return and the strike price differ for different companies. To

overcome this difficulty, the researchers abroad are combining the stock price

and strike price to obtain a relative measure called moneyness

The moneyness and categorization of the same are defined in chapters II

and IV early, and the coefficient of correlation of the residuals with moneyness

is calculated and is at -0.004. It may be noted that the call option price is highly

sensitivity to stock price, strike price and moneyness as explained in the

chapter III; the ratio of percentage of change in call option price to percentage

change in stock price is 11.64 and that of strike price is -11.95. Also, there

exists bias of predictability over moneyness as explained in chapter IV. In spite

156

of these, the coefficient of correlation is very negligible and insignificant. The

meaning is that the functional form and the structure of the model with respect

to stock price and strike price are good and sufficient.

5.3.2 CORRELATION OF RESIDUALS WITH VOLATILITY OF STOCK

RETURNS The coefficients of correlation of residuals with volatility of stock returns

for various moneyness are calculated and tabulated in Table no. 5.4

TABLE 5.4

COEFFICIENT OF CORRELATION BETWEEN THE RESIDUALS AND VOLATILITY FOR VARIOUS CATEGORIES OF MONEYNESS

MONEYNESS COEFFICIENT OF CORRELATION

0.84 - 0.86 -0.4970

0.87 - 0.89 -0.5501

0.90 - 0.92 -0.4737

0.93 - 0.95 -0.4553

0.96 - 0.98 -0.4204

0.99 - 1.01 -0.4083

1.02 - 1.04 -0.3466

1.05 - 1.07 -0.4931

1.08 - 1.10 -0.6430

1.11 - 1.13 -0.3304

1.14 - 1.16 -0.0445

1.17 - 1.19 -0.0004

> 1.20 0.0336

157

For different moneyness, the coefficient of correlation changes, but they

are high and negatively correlated. It varies from -0.35 to 0.64 if options less

than 5000 are eliminated. However, most of them are around -0.4.

The inference is that the residuals are negatively correlated with the

parameter volatility. It is neither perfectly correlated nor perfectly uncorrelated. It

has a g

ole data is taken without categorizing into

moneyness, then the coefficient of correlation is -0.40731. If we eliminate the

229 hi

LIFE OF OPTION

The coefficients of correlation of residuals with life of the options for

5.5.

ood amount of correlation in the negative direction.

In another angle, if the wh

ghly volatility options (0.23% of total data of 95,956), then the coefficient

of correlation comes down to -0.2668. These outliers belong to only two

companies namely Dr.Reddys Laboratories and Infosys Technologies Ltd. The

options are related to the specific period of 14-7-2006 to 20-11-2006 in the case

of Infosys Technologies Ltd. and 02-01-2002 to 25-02-2002 for the Dr.Reddys

Laboratories.

It may be recalled that the sensitivity of call option price to volatility is

only 0.86 compared to 11.64 of stock price and -11.95 of strike price. But the

coefficient of correlation is significantly high at -0.2668. It is clearly a concern for

the model adequacy and predictability.

5.3.3 CORRELATION OF RESIDUALS WITH

various categories of moneyness is found and tabulated below in table no.

158

TABLE 5.5 COEFFICIENT OF CORRELATION OF RESIDUALS WITH LIFE OF OPTION FOR VARIOUS CATEGORIES OF MONEYNESS

MONEYNESS COEFFICIENT

OF CORRELATION

0.84 - 0.86 0.1107

0.87 - 0.89 -0.0087

0.90 - 0.92 -0.0307

0.93 - 0.95 -0.0837

0.96 - 0.98 -0.0746

0.99 - 1.01 -0.0511

1.02 - 1.04 -0.0725

1.05 - 1.07 -0.0632

1.08 - 1.10 -0.0457

1.11 - 1.13 -0.0191

1.14 - 1.16 0.0201

1.17 - 1.19 -0.0239

> 1.20 -0.0266

The coefficients of correlation are very low at the second digit of the

decimal, but like others, it is in negative sign mostly. In our study, the coefficient

of correlation of the residuals with Time to expiration (life) is ranging from

-0.0457 to -0.0837 with a mean of -0.065. When coefficient of correlation is

calculated without categorizing, coefficient of correlation is only -0.054. By

which, it may be concluded that the correlation between the residuals and the

time to

incorporated in the model.

expiration is insignificant.

Even though the sensitivity is 0.61 near to that of volatility figures of 0.86,

the correlation is low and insignificant. Life of the option may be adequately

159

5.3.4

he parameter risk - free interest rate is the least sensitivity one in the

model

S WITH RISK-FREE- INTEREST RATES FOR VARIOUS CAT GORIES OF MONEYNESS

CORRELATION OF RESIDUALS WITH RISK - FREE INTEREST RATE

T

and the coefficient of correlation also found insignificantly low as detailed

below.

TABLE 5.6

COEFFICIENT OF CORRELATION OF RESIDUALE

MONEYNESS COEFFICIENT

OF CORRELATION

0.84 – 0.86 -0.113

0.87 – 0.89 -0.004

0.90 – 0.92 -0.019

0.93 – 0.95 -0.005

0.96 – 0.98 -0.012

0.99 – 1.01 -0.017

1.02 – 1.04 -0.016

1.05 – 1.07 -0.031

1.08 – 1.10 -0.044

1.11 – 1.13 -0.042

1.14 – 1.16 0.009

1.17 – 1.19 0.006

> 1.20 0.034

nt of co ion vary -0.0 -0 ith a m

-0. wit and is 7 for the le da itho egoriza

The coefficie rrelat from 05 to .044 w ean

02 h categorization -0.01 who ta w ut cat tion;

160

a completely insignificant leve e mode esig de ly well

res ct t free - interest r

to the the f econom s, h rre and pa

of residuals with one of the pa ters of ode ate eakne

the d el can mproved . M rec it indicates

incorrect functional form or structure of the model.

The above inferences can be better o ed fr e c o. 5.10

ART 5.1

CORRELATION OF RESIDUALS W HE V AB ND AMETE

l. Th l is d ned a quate with

pe o risk - ate.

According ory o etric igh co lation ttern

rame the m l indic s a w ss in

mo el and the mod be i upon ore p isely,

bserv om th hart n

CH 0 ITH T ARI LES A

PAR RS

CORRELATION OF A

-0.70

-0.20

-0

0

0

0

84 -

0.

87 -

0.

90 -

0.92

93 -

0.95

96 -

0.

99 -

01

02 -

05 -

07

08 -

11 -

13

14 -

17 -

19

REL

ATI

ON

EF

FIC

IE

RESIDU LS

.10 86 89

.00

.10

.20

98

1.

1.

1. 1.

1.

1.

1.

> 1.

04 10 16 20NT

-0.60

-0.50

-0.40

-0.30 0. 0. 0. 0. 0. 0. 1. 1. 1. 1. 1. 1.

MONEYNESS

CO

RC

O

VOLATILITY RISK FREE RATE LIFE OF OPTION

161

5.4 CONCLUSION The above analysis confirms that the distribution of residuals is not

exactly a normal distribution. But almost near to it, if the outliers are eliminated

om the sample. It may be inferred that most of the variables and parameters

t stock price, strike

rice, time to expiration (life) and risk - free interest rate are having no pattern

reveals that it is having a significant correlation

with the residuals in the negative directi . The chart no. 5.10 points a pattern

of the distribution of coefficients of correlation of volatility with residuals. Thus, it

may be concluded that volatility of stock returns is the only parameters of the

model which is not satisfactorily / adequately incorporated in the model. The

predictability of call option price by the BS model can be still improved if the

volatility is adequately incorporated. In

structure of the volatility in the model may improve the predictability.

calculation of the volatility of stock return and the actual market volatilit

turn.

The volatility is calculated from t torica ck pr nd the

re s. e up to wh stoc e are taken fo

calculating the volatility is not defined by under the m ne ma

ta 60 120 days or 180 d he lo he pe ccordin

to the theory of statistics, will minimize t ror. B gicall the dat

further away from the date of volatility calculations, they may not correctly

fr

are correct and the form and structure also near adequate. But, they are not

exactly adequate.

The analysis of coefficients of correlation of the residuals with various

variables and the parameters of the model indicates tha

p

and insignificantly low. In may be inferred that the above four out of five factors

of call option price are well defined in the model. However, study of the volatility

of stock returns of the model

on

other words, improving the form and

The reason for the same can be understood if we analyze the method of

y of the

stock re

he his l sto ices a ir

turn The duration of the tim ich the k pric to be r

the fo s of odel. O y

ke days, 90 days, ays. T nger t riod, a g

he er ut, lo y, as a

162

re se stock r . Bot oppo oints bu

logically correct. Practically, the stock pri n va re tha averag

his rica to good / bad n and conom nditions

Practically, there are some jumps

Thus, the improper / inadequate form of volati f stock ns in th

model can be explainable.

T its form struc s far ck price

strike price, life of the option and RFR are concerned. Though latility i

lo lly correct in the model, the estimatio ds pre n and vement.

pre nt the current trend of the eturns h are sing p t

ces ca ry mo n the e

to l volatility due ews the e ic co .

in the returns of the stock.

lity o retur e

he model is well defined in and ture a as sto ,

the vo s

gica n nee cisio impro