Effects of Variables on Option Beta

EFFECTS OF OPTION CHARACTERISTICS AND UNDERLYING STOCK ON
OPTION BETA: AN EMPIRICAL STUDY FOR UNITED STATES MARKET

DHARMA ISWARA BAGOES OKA*

21st October, 2011

Abstract

Beta (β)1 is one of the risk management tools to capture the risk exposures of hedge-fund
investments.As most of hedge funds today trade derivative securities, the research on the
measurement of derivative beta is important. The aim of this paper is to examine the factors,
which may have impacts on option beta in the United States market.My hypothesis is
comprised into three main parts. First, I hypothesize that 5 variables (type of option, strike
price, days to maturity, firm size and book to market ratio) have linear relationship with the
option beta. Second, I hypothesize that the strength of linear relationship is varied by the type
of the industry. Third, I hypothesize that the strength of linear relationship is also varied
bythese 5 types of variables itself. To begin the process, Iuse regression method to estimate
the beta of underlying stock. Then,I estimate the option beta by multiplying the beta of
underlying stock andthe option elasticity. I then use regression method to test whether the 5
variables have linear relationship with option beta.I find that3 variables (type of option, strike
price and days to maturity) have the most significant linear relationship with option beta,
while firm size has less significant linear relationship and book to market ratio have no
significant linear relationship. Furthermore, using 2-way ANOVA, I test whether strength of
linear relationship is varied by the type of the industry and the 5 types of variables.There is
not enough evidence to infer that the strength of linear relationship between the 5 variables to
option beta is varied by the type of the industry, instead, there is enough evidence to infer that
the strength of linear relationship between the 5 variables to option betais varied by the type
of variables.

Keywords: option beta,option characteristics, linear relationship,beta of underlying stock,
option elasticity

* School of Finance, Actuarial Studies and Applied Statistics; The Australian National University. I thank to Cheng Sun,
Shengnan Zhu and He Jiang for their comments and valuable research assistance. I also thank to Wharton Research Data
Services (WRDS) for providing valuable data input for this research.
1
Beta expresses the 'sensitivity' of the portfolio relative to the 'market'. Beta is that element of return variability from a
portfolio which cannot be eliminated through diversification relative to one or several risk factors. It comprises the risk
factors common to all assets in the investment universe.

1

Table of Contents

1. Introduction ........................................................................................................................ 3

2. Aim and Research Outline ................................................................................................ 3

3. Literature Review .............................................................................................................. 3

4. Hypothesis Development ................................................................................................... 5

5. Data ..................................................................................................................................... 9

6. Methods and Models for Empirical Investigation ........................................................ 10
7. Results and Discussion of Empirical Evidence .............................................................. 12

7.1 Hypothesis 1 Test .......................................................................................................... 12

7.2 Hypothesis 2 Test .......................................................................................................... 13

7.3 Hypothesis 3 Test .......................................................................................................... 15

7.4 Hypothesis 4 Test .......................................................................................................... 16

7.5 Hypothesis 5 Test .......................................................................................................... 17

7.6 Hypothesis 6 Test .......................................................................................................... 19

7.7 Hypothesis 7 Test .......................................................................................................... 19

8. Limitations ........................................................................................................................ 21

9. Further Research ............................................................................................................. 21

10. Conclusion ........................................................................................................................ 22

11. Bibliography ..................................................................................................................... 23
12. Appendix ........................................................................................................................... 25

2

1. Introduction

Hedge funds have experienced an explosive growth in the past two decades (Chen, 2008).
Because 71% of hedge funds during 1994-2006 trade derivative securities, the use of
derivatives has becoming a great concern to investors and regulators (Chen, 2008).
Furthermore, evidence from 1998 financial crisis support the hypothesis that the effects of
derivative use are most pronounced during these periods of extreme movement (Cao, 2010).
Therefore, the risk measurement of derivatives must be carefully analysed by hedge funds.

It is already shown that hedge funds which have primary focus on equity may benefit from
the use of options on their primary asset class and this information can be used to build better
performing portfolios of funds of funds (Peltomaki, 2007). However, the research did not
show the effects of five variables (type of option, strike price, days to maturity, firm size and
book to market ratio) on the portfolio performance of hedge funds. Since the measurement of
systematic risk (beta) is important for portfolio and risk management (Jacquier, 2010), the
research to examine the effects of five variables on the option beta is essential.

2. Aim and Research Outline

The aim of this research is to provide an empirical analysis to show the effects of the five
variables (type of option, strike price, days to maturity, firm size of the underlying stock and
book to market ratio of the underlying stock) on the option beta. To do this, I will begin with
the hypothesis development to suggest the seven hypotheses based on supporting literature.
The next step is to describe the data used in the research and to explain why I choose them.
After that, I will introduce the methods for empirical investigation. Moreover, I will conduct
hypothesis tests to see whether the results are consistent with the initial hypotheses and then
analyse the results. Lastly, after discussing the limitations and further research, I will
concludemy research.

3. Literature Review

In order to examine the effects of five variables on the option beta, I need to figure out the
relationship between option beta and underlying stock beta first. This relationship has been an
important research topic and many studies have been devoted to it. One such study, the

3

underlying stock beta is presented in the Capital Asset Pricing Model (CAPM). The
derivation of the CAPM in a discrete framework can be found in Sharpe (1963 and 1964),
Lintner (1965a and 1965b), Mossin (1966), and Fama-Miller (1972, chs. 6 and 7); and in a
continuous time framework in Merton (1970 and 1973b). Jensen (1972) summarizes the
different approaches and provides a survey of empirical tests of the model. The option pricing
model has been derived by Black-Scholes (1973) to European-style options. In their model,
they create a perfect hedge composed one unit long/short of the underlying security and a
short/long position on the number of options, the return should be equal to the riskless return.
Additionally, Merton (1973a and 1974) has contributed to the option pricing model as well.

The option beta and underlying stock beta is connected by the option elasticity Ω. This has
been proved by Galai and Masulis (1976) by combining the option pricing model with the
CAPM yields a theoretically more complete model of corporate security pricing. One result
of their research is that the systemic risk of equity is the product of the firm’s systemic risk
and the elasticity of equity value with respect to firm’s value. Based on the argument used in
their article, the relationship between the systematic risks of the option and the underlying
stocks can be obtained. In other words, the elasticity also can be combined withthe beta of the
stock, βA to calculate the beta of the call option (βO) is βO = Ω βA2. The elasticity of the option
S 3
(Ω) is calculated as: Ω .Delta (Δ) is explained by Hull (2011), which is defined as the
C
ratio of the change in the price of the option to the change in the price of the underlying
stock.

As the option beta and underlying stock beta is connected by the option elasticity Ω, I need to
find the factors affect option elasticity and thus affect option beta. The factors are explained
byput call parity equation.The derivation of put-call parity can be found on Nelson (1904),
Henry Deutsch (1910), VinzeneBronzin (1908), Stoll (1969) and Michael Knoll (2004). From
put call parity equation, it is shown that call option premium (c) is c = p + S0 – Ke-rT4, while
put option premium (p) is p = c – S0 + Ke-rT. Therefore, I can suggest that type of option,
strike price (K) and days to maturity (T) will affect the option premium. Thus theywill affect
the option elasticity and thus affect option beta.

2
Where Ω is the elasticity of the option and βA is the beta of underlying asset
3
Where S is the price of the underlying asset, C is the value of the option and Δ is the delta
4
where S0 is the current stock price, K is the strike price of the option, r is the interest rate, T is the time to maturity and p is
the put option value with the same stock, strike price and time to maturity with the call option

4

Furthermore, Rosenberg (1976)finds that firm size (market capitalization)help to predict beta
of underlying stock. In addition, Banz(1981) shows thatfirmsize and beta of underlying stock
are negatively correlated.Furthermore, Capaul (1993) suggests that high book to market ratio
stocks typically have lower beta, not higher beta.In general, from these theories, I can infer
that firm size and book to market ratio will have effects on underlying stock beta, and thusit
will affect option beta as well.

This paper contributes to the literatures by examining the effects of type of option, strike
price, days to maturity, firm size and book to market ratio on option beta. To the best of my
knowledge, no such investigation has been carried out so far. Thus, I aim at filling a
conspicuous gap in the literatures by conducting an empirical analysis of European options
with respect to compatibility with asset-pricing theory, option pricing model and put-call
parity.

4. Hypothesis Development

In this research, I hypothesize that the five variables have linear relationship with option
beta.The hypothesescan be explained by the equations discussed in the literature review.Galai
and Masulis (1976) have shown that the beta of the option (βO) is 0 A . The elasticity of
S
the option (Ω) is calculated as: Ω . From put call parity equation, it is shown that value
C
for call option (c) is c= p S 0 ke rt (Hull, 2011). By manipulating the same put call parity
rt
formula, it is also shown that the value for put option (p) is p = c S 0 ke (Hull, 2011).

From the three equations above, I could conclude the following two equations:

Call option

………………………………………………………………………(1)

Put option:

………………………………………………………………………...(2)

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Hypothesis 1.Beta of the call option hasnegative linear relationship with beta of the put
option. This can be explained by the following reasons. The Delta (Δ) is defined as the ratio
of the change in the price of the option to the change in the price of the underlying stock
(Hull, 2011). It is the sensitivity of an option price relative to changes in the price of the
underlying stocks. The delta of a call option is positive, whereas the delta of a put is negative
(Hull, 2011).The equation (1) and (2) can be rewritten as:

Call option:

5………………………………………………………………………………(3)

Put option:

6………………………………………………………………………….(4)

Since S, c and p are always positive, and then dividing equation (3) by (4) will result in:

………………………………………………………... (5)

Based onequation (5), beta of the call option has negative linear relationship with beta of the
put option.

Hypothesis 2.Call option beta will be positively correlated with strike price, while put option
beta will be negatively correlated with strike price. This is because from equation (1),
increasing strike price (K) will result in a lower call premium (c). Thus from equation (3), if
delta (Δ) for call option is positive, then lower c (from increasing strike price) will result in a
higher (more sensitive)call option beta (βco). The same result appliesfrom equation (2),
increasing strike price (K) will result in a higher put premium (p). Thus from equation (4), if
delta (Δ) for put option is negative, then higher p (from increasing strike price) will result in
alower(less sensitive) put option beta (βpo).

Hypothesis 3.Call option beta will be negatively correlated with days to maturity, while put
option beta will be positively correlated with days to maturity. This is because from equation
(1), increasing days to maturity (T) will result in ahigher call premium (c). Thus from
equation (3), if delta (Δ) for call option is positive, then higher c (from days to maturity) will

5
Where is the call option beta and is the call option delta.
6
Where is the put option beta and is the put option beta.

6

result in alower (less sensitive) call option beta (βco). The same result applies from equation
(2), increasing days to maturity (T) will result in alower put premium (p). Thus from equation
(4), if delta (Δ) for put option is negative, then lower p (from increasing days to maturity) will
result in ahigher (more sensitive) put option beta (βpo).

Hypothesis 4.Call option beta will have a negative linear relationship with firm size, while
put option beta will have a positive linear relationship with firm size.Thishypothesis can be
explained from the equation βO = Ω βAI have mentioned in the literature review. Furthermore,
I suggestin hypothesis 2 thatΩ is positive for call option and negative for put option.Hence,
calloption beta will be positively related with underlying stock beta whereas put option beta
will be negatively correlated to underlying stock beta. Becausefirmsize and underlying
stockbeta are negatively correlated (Banz,1981),I can infer that call option beta havea
negative linear relationship with firm size, while put option beta havea positive linear
relationship with firm size.

Hypothesis 5.Call option beta will have a negative linear relationship with book to market
ratio of the underlying stock, while put option beta will have a positivelinear relationship
with book to market ratio.This hypothesis can be explained from the equation βO = Ω βAI
have mentioned in the literature review. Furthermore, I suggest in hypothesis 2 that Ω is
positive for call option and negative for put option.Hence, call option beta will be positively
related with underlying stock beta whereas put option beta will be negatively correlated to
underlying stock beta. Becausebook to market ratio and underlying stockbeta are negatively
correlated (Capaul,1993),I can infer that call option beta havea negative linear relationship
with book to market ratio, while put option beta havea positive linear relationship with book
to market ratio.

Hypothesis 6.The strength of linear relationship between the 5 variables (type of option,
strike price, days to maturity, firm size and book to market ratio) to option beta is varied by
the type of the industry. This hypothesis can be explained as follows. According to Bodie
(2011), the relationship between the excess return of a security, Ri to the excess return of the
index, RM is ……………………………………………....(6)

The intercept of the equation ( ) is the security’s expected excess return when the market
excess return is zero. The slope coefficient ( ) is the security beta (security’s sensitivity to

7

the index). is the zero mean, firm-specific surprise in the security return. Equation (6) can

also be rewritten as …………………………………………...…………(7)

By substituting (beta of the asset) in equation (1) and (2) with (beta of the security) will
result in:

………………………...…………………………………..(8)

……………..……………………………………………...(9)

Because book to market ratio is calculated as Net Asset Value per share (NAVPS) divided by
current share price (S0), then equation (8) and (9) can be rewritten as:

……………………………………………….……(10)

…………………………………………………….(11)

Because firm size (market capitalization) is calculated as current share price (S 0) multiplied
by the number of shares outstanding (NOSO), then equation (10) and (11) can be rewritten
as:

…………………………………...………………(12)

……………………………………..…………….(13)

Thus, firm-specific effect ( ), will vary the strength of linear relationship between the 5
variables (type of option, strike price K, days to maturity t, firm size and book to market
ratio) to option beta. Because the type of industry varyfirm-specific effect ( ), the strength of
linear relationship between the 5 variables to option beta is varied by the type of the industry
as well.

Hypothesis 7.The strength of linear relationship between each of the 5 variables to option
beta is varied by these 5 types of variables itself. This hypothesis can be explained as follows.

8

The delta of call option is N(d1) and the delta of put option is N(d1) – 1 (Hull, 2011). Because

(Hull, 2011), then equation (12) and (13) can be rewritten as:

…………………………...(14)

…….........................(15)

Because Capital Asset Pricing Model equation (Bodie, 2011)
can be rewritten as …………………………………………(16)

Substituting in equation (15) and (16) with equation (16) will result in:

…………...(17)

……(18)

Because underlying stock beta is affected by firm size (Banz,1981) andbook to market
ratio (Capaul, 1993), then equation (17) and (18) showed that the 5 variables effect (type of
option, strike price K, days to maturity t, firm size and book to market ratio)is repeated (on
the left side of the bracket and on the inside of the bracket). Thus, the strength of linear
relationship between each of the 5 variables to option beta is varied by these 5 types of
variables itself.

Even though the hypotheses can be explained theoretically, I still need to provide empirical
evidence concerning the hypotheses. Therefore, I will develop the research based on reality in
the U.S. financial markets to see whether the hypotheses are empirically correct.

5. Data

For the empirical analysis, I use data of monthly return of S&P 500 index, monthly return of
individual stocks under the list of S&P 500 index, T-bills monthly rate and option data
(option premium, option type, strike price, days to maturity, delta) corresponding to the ten

9

companies I randomly selected under S&P 500 list.To ensure my results hold for different
industries, I randomly select one company for each of ten different industries under S&P500
list.I choose these data for the 5 yearperiod from 1st January 2006 to 31st December 2010.I
think that 5 year period from 2006 to 2010 is enough to incorporate the fluctuation of boom
and bust (such as the Global Financial Crisis in 2008) so that I can address the beta change
which is likely to occur during the different periods of normal and abnormal condition. All of
these data are obtained from WRDS7 and beta of underlying stock, option elasticity and
option beta can be calculated with these data and the equation mentioned in literature review.

I conduct the empirical study based on United States market asit is more mature compared to
others. Iregard T-bills rate as the risk free rate, because it is usually assumed that there is no
chance that a government will default on an obligation denominated in its own currency. I
consider S&P 500 as the market proxy based on the fact that S&P 500 is the most popular
value-weighted index of U.S. stocks (Bodie, 2011) involved large publicly held companies
that trade on New York Stock Exchange and NASDAQ. I select monthly data as my sample
frequency since monthly data tends to be more reliable compared with daily data.

6. Methods and Models for Empirical Investigation

The method that I will use to estimate the individual stock beta (βA) is to use regression
method to regress the excess return of individual stock with the excess return of the S&P 500
portfolio index (Bodie, 2011). The excess return of individual stock can be calculated using
the return of individual stock subtracted by the corresponding T-bill rate (Bodie, 2011). The
same apply with the excess return of the S&P 500 portfolio index where I subtract the S&P
500 portfolio index return with the corresponding T-bill rate (Bodie, 2011).

As mentioned in literature review, to estimate the option beta (βO), I need to multiply the beta
of the underlying stock by the option elasticity (Ω) using equation βO = ΩβA. The option
S
elasticity is calculatedusing Ω .Once I have calculated the option beta, I can now test
C
all the hypotheses using regression method.

7
WRDS provides access to important databases in the fields of finance, accounting, banking,and economics
and so on. It is available at http://wrds.wharton.upenn.edu/

10

To test hypothesis 1, I will regress the beta of call option with the beta of put option for the
same stock. To ensure the accuracy and comparability of the test result, I will use call and put
option data with 30 days to maturity. I will repeat this method for ten companies that I
selected. I then interpret the results based on the slope of the regression and the significance
of this linear relationship (using p-value of the slope).

To test hypothesis 2, I will regress the beta of the option with the corresponding strike price.
To ensure the accuracy and comparability of the test result, I will use call option data with 30
days to maturity. I will repeat this method for ten companies that I selected. I then interpret
the results based on the slope of the regression and the significance of this linear relationship
(using p-value of the slope).

To test hypothesis 3, I will regress the beta of the option with the corresponding days to
maturity. To ensure the accuracy and comparability of the test result, I will use call and put
option data with 30, 60, 91 and 182 days to maturity. I will repeat this method for ten
companies that I selected. I then interpret the results based on the slope of the regression and
the significance of this linear relationship (using p-value of the slope).

To test hypothesis 4, I will regress the beta of the option with the corresponding firm size. To
ensure the accuracy and comparability of the test result, I will use call and put option data
with 30 days to maturity. I will repeat this method for ten companies that I selected. I then
interpret the results based on the slope of the regression and the significance of this linear
relationship (using p-value of the slope).

To test hypothesis 5, I will regress the beta of the option with the corresponding book to
market ratio of the underlying stock. To ensure the accuracy and comparability of the test
result, I will use call and put option data with 30 days to maturity. I will repeat this method
for ten companies that I selected. I then interpret the results based on the slope of the
regression and the significance of this linear relationship (using p-value of the slope).

To test hypothesis 6, Iwill see whether the strength of linear relationship (p-value) between
the 5 variables to option beta is varied by the type of the industry by using analysis of
variance method (ANOVA). I use this method because ANOVA is a procedure that tests to
determine whether differences exist between two or more population means (Keller, 2009),
that is the mean of p-values on each treatment from each of the 10 industries. Because the
samples are drawn from matched block for each treatment, the type of analysis of variance to

11

apply is two-way ANOVA (Keller, 2009). This method has an advantage that it reduces
within-treatment variation to more easily detect differences between the treatment means
(Keller, 2009). The process of performing the two-way ANOVA to test hypothesis 6 is
facilitated by table as shown in Appendix 1.

[Appendix 1 here]

From data arranged in Appendix 1, the hypotheses to be tested are as follows:

H0: Ten means ([T]1, [T]2, [T]3,[T]4, [T]5, [T]6,[T]7, [T]8, [T]9and [T]10) do not differ

H1: At least two means differ

Before I will run the F-Test of the two-way ANOVA, I must check whether the required
conditions are met, that is the random variable must be normally distributed and the
population variances must be equal (Keller, 2009). After I run the two-way ANOVA using
Minitab, I will conclude whether the strength of linear relationship between the 5 variables to
option beta is varied by the type of the industry from interpreting the F-Statistics.

To test hypothesis 7, I will see whether the strength of linear relationship between each of the
5 variables to option beta is varied by these 5 types of variables by using the same method as
testing hypothesis 6. The process of performing the two-way ANOVA to test hypothesis 7 is
facilitated by table as shown in Appendix 2.

[Appendix 2 here]

7. Results and Discussion of Empirical Evidence

6.1 Hypothesis 1 Test

The first test is to determine whether beta of the call option has negative linear relationship
with beta of the put option. There are ten different types of industries under S&P500 list. I
randomly select one stock from each of ten industries as the test objects. Table below shows
the stocks I have randomly selected from S&P500 list:

12

Table 3: Ten companies for testing the hypotheses

Ticker Symbol Company Industry
ADM Archer-Daniels-Midland Co Consumer Staples
AES AES Corp Utilities
AMT American Tower Corp A Telecommunications Services
BDX Becton Dickinson Health Care
C Citigroup Inc. Financials
CLF Cliffs Natural Resources Materials
CSCO Cisco Systems Information Technology
DHR Danaher Corp. Industrials
SUN Sunoco Inc. Energy
SWK Stanley Black & Decker Consumer Discretionary

By regressing beta of call option with the beta of put option for each of ten companies
mentioned in table 3, the result is as follows:

Table 4: Regression Result to Test Hypothesis 1

Company Industry Regression Slope Slope P-Value
ADM Consumer Staples -0.97 0.00
AES Utilities -1.02 0.00
AMT Telecommunications Services -1.03 0.00
BDX Health Care -0.93 0.00
C Financials -0.96 0.00
CLF Materials -1.02 0.00
CSCO Information Technology -0.93 0.00
DHR Industrials -0.94 0.00
SUN Energy -0.98 0.00
SWK Consumer Discretionary -0.96 0.00

From this regression, all of the listed companies show a regression slope very close to -1.
Furthermore, the p-value of the slope is very small(in excess of 20 decimals), which shows an
overwhelming evidence that linear relationship exists. Thus, from this regression slope and
the p-value, I suggest that there is a tendency of a perfect negative linear relationship between
call option beta and put option beta.This evidence strongly supports hypothesis 1.


The second test is to determine whether call option beta will be positively correlated with
strike price, while put option beta will be negatively correlated with strike price. By

13

regressing beta of the call option with the corresponding strike price for each company

Table 5: Regression Result to Test Hypothesis 2 for Call Option

ADM Consumer Staples 0.03 0.11
AES Utilities 1.04 0.00
AMT Telecommunications Services 0.23 0.00
BDX Health Care 0.08 0.06
C Financials 0.98 0.00
CLF Materials 0.06 0.03
CSCO Information Technology 0.55 0.00
DHR Industrials 0.09 0.02
SUN Energy 0.04 0.00
SWK Consumer Discretionary 0.43 0.00

From this regression, all of the listed companies show a positive regression slope. However,
one company (ADM) slope p-value is not statistically significant (p-value of 0.11), while the
slope p-values of the remaining companies are found to be statistically significant. Thus,
from this regression slope and the p-value, I suggest that there is a tendency of a positive
linear relationship between call option beta and strike price. This evidence supports my
hypothesis 2.

By regressing beta of the put option with the corresponding strike price for each company

Table 6: Regression Result to Test Hypothesis 2 for Put Option


14

From this regression, all of the listed companies show a negative regression slope. However,
one company (ADM) slope p-value is not statistically significant (p-value of 0.17), while the
slope p-values of the remaining companies are found to be statistically significant. Thus,
from this regression slope and the p-value, I suggest that there is a tendency of a negative
linear relationship between put option beta and strike price. This evidence supports my
hypothesis 2.


The third test is to determine whether call option beta will be negatively correlated with days
to maturity, while put option beta will be positively correlated with days to maturity.By
regressing beta of the call option with the corresponding days to maturity for each company



From this regression, all of the listed companies show a negative regression slope.
Furthermore, the p-values of the slope are0.00, which shows overwhelming evidence that
linear relationship exists. Thus, from this regression slope and the p-value, I suggest that
there is a tendency of a negative linear relationship between call option beta and days to
maturity. This evidence supports my hypothesis 3.

By regressing beta of the put option with the corresponding days to maturity for each
company mentioned in table 3, the result is as follows:

15



From this regression, all of the listed companies show a positive regression slope.
Furthermore, the p-values of the slope are0.00, which shows overwhelming evidence that
linear relationship exists. Thus, from this regression slope and the p-value, I suggest that
there is a tendency of a positive linear relationship between put option beta and days to
maturity. This evidence supports my hypothesis 3.


The fourth test is to determine whether call option beta will have a negative linear
relationship with firm size, while put option beta will have a positive linear relationship with
firm size.By regressing beta of the call option with the corresponding firm size for each


ADM Consumer Staples 3.89156E-08 0.11
AES Utilities 1.6553E-06 0.00
AMT Telecommunications Services 1.87992E-07 0.00
BDX Health Care 3.28754E-07 0.06
C Financials 2.52109E-07 0.00
CLF Materials -5.90386E-07 0.05
CSCO Information Technology 8.99038E-08 0.00
DHR Industrials 3.42312E-07 0.04
SUN Energy 2.66451E-07 0.00
SWK Consumer Discretionary 4.97836E-07 0.22

16

From this regression, nine of the listed companies show a positive regression slope, except
one with negative regression slope. Furthermore, theslope p-value of the eight companies is
statistically significant, while two other have no statistical significance. Thus, from this
regression slope and the p-value, I suggest that there is a tendency of a positive linear
relationship between call option beta and firm size. Therefore, from these findings, I reject
hypothesis 4.

By regressing beta of the put option with the corresponding firm size for each company


ADM Consumer Staples -3.52569E-08 0.16
AES Utilities -1.62284E-06 0.00
AMT Telecommunications Services -6.54614E-07 0.00
BDX Health Care -4.05772E-07 0.02
C Financials -2.61023E-07 0.00
CLF Materials 5.61455E-07 0.05
CSCO Information Technology -8.43186E-08 0.00
DHR Industrials -3.39261E-07 0.05
SUN Energy -2.63492E-07 0.00
SWK Consumer Discretionary -5.2944E-07 0.19

From this regression, nine of the listed companies show a negative regression slope, except
one with positive regression slope. Furthermore, the slope p-value of the eight companies is
statistically significant, while two other have no statistical significance. Thus, from this
regression slope and the p-value, I suggest that there is a tendency of a negative linear
relationship between put option beta and firm size. Therefore, from these findings, I reject
hypothesis 4.


The fifth test is to determine whether call option beta will have a negative linear relationship
with book to market ratio of the underlying stock, while put option beta will have a
positivelinear relationship with book to market ratio.By regressing beta of the call option with
the corresponding book to market ratio for each company mentioned in table 3, the result is
as follows:

17



From this regression, all of the listed companies show a negative regression slope, except one
company (SWK). However, only four of the companieshave slope p-value that is statistically
significant. Thus, from this regression slope and the p-value, I suggest book to market ratio
have no significant linear relationship with call option beta. Therefore, thisresult is not
consistent with the hypothesis 5.

By regressing beta of the put option with the corresponding book to market ratio for each



From this regression, all of the listed companies show a positive regression slope. However,
only four of the companies have slope p-value that is statistically significant. Thus, from this

18

regression slope and the p-value, I suggest book to market ratio have no significant linear
relationship with put option beta. Therefore, this result is not consistent with the hypothesis 5.

6.6Hypothesis 6 Test

After arranging all p-values which appear from table 4 to table 12 in accordance with the
format shown in table 1, I initially check whether the required conditions are met. Since the
preliminary requirements of variance equality is not satisfied, I can address this issue by
using the logarithmic transformation of these p-values before proceeding to run the two-way
ANOVA. By using the transformed values of the p-values as the inputs for Minitab, the two-
way ANOVA result is as follows:

Figure 1 : Two-way ANOVA result to test whether the strength of linear relationship between
the 5 variables to option beta is varied by the type of the industry

The F-statistic and p-value to determine whether differences exist between 10 industries is
0.49 and 0.877, respectively. Against a standard of 10% significance level, the p-value of
0.877 suggests that there is no sufficient evidence to infer that at least two of the industries
differ. Thus, this finding is not consistent with hypothesis 6.

6.7Hypothesis 7 Test

After arranging all p-values which appear from table 4 to table 12 in accordance with the
format shown in table 2, I initially check whether the required conditions are met. Since the
preliminary requirements of variance equality is not satisfied, I can address this issue by
using the logarithmic transformation of these p-values before proceeding to run the two-way
ANOVA. By using the transformed values of the p-values as the inputs for Minitab, the two-
way ANOVA result is as follows:

19

Figure 2 : Two-way ANOVA result to test whether the strength of linear relationship between
the 5 variables to option beta is varied by the type of the variables

The F-statistic and p-value to determine whether differences exist between the variables is
52.43 and 0.000, respectively. Against a standard of 5% significance level, the p-value of
0.000 suggests that there overwhelming evidence to infer that at least two of the variables
differ. Furthermore, I am interested to see which under which variables the linear relationship
between the 5 variables to option beta have the strongest linear relationship, and rank them
into order. By observing the mean of the p-values on each variables, the rank of linear
relationship strength are as follows:

Table 13: Rank list of linear relationship strength of variables

Variables p-value Rank of linear relationship strength
Option type 4.51105E-27 1
Days to Maturity (Call Option) 1.90628E-11 2
Days to Maturity (Put Option) 8.3459E-11 3
Strike Price (Call Option) 0.02296202 4
Strike Price (Put Option) 0.026476244 5
Firm Size (Call Option) 0.046505199 6
Firm Size (Put Option) 0.047311483 7
Book to Market Ratio (Put Option) 0.281108889 8
Book to Market Ratio (Call Option) 0.290606121 9

Based from table 13, I can see that the variables which have strongest linear relationship to
option beta is option type, and the lowest is book to market ratio. In addition to portfolio beta
equation (Bodie, 2011), this information can be used to help investors to
form strategy to achieve desired beta of portfolios of option. That is, based from table 13, it is

20

suggested that the investor’s top priority of strategy to manipulate the beta of option
portfolios is by varying the option type (switching from put/call option to call/put option).
This is reasonable since option type have the highest rank of linear relationship strength to
option beta. In other words, changing option type will have more predictability of changing
beta compared to any other type of strategy. If this strategy is not feasible, for example, if
there is no put counterpart of the call option (or vice versa), the investor can choose the next
strategy according to the order of the rank, and so on. Therefore, the result in table 13 can
provide a useful guidance to help investors form strategy to vary the 5 variables in option
portfolio to achieve the target beta.

8. Limitations

Beta is one of the most popular tools to manage risk, while it has some drawbacks to capture
the risk exposures of portfolio investments. The drawbacks are:

Beta looks backward and history is not always an accurate predictor of the future.
Investors can find beta is a good tool to measure risk in short-term decision-making, where
price volatility is important. However, as a single predictor of risk for a long-term investor,
beta has too many flaws.

Beta suggests a stock’s price volatility relative to the whole market, but that volatility can
be upward as well as downward movement. That means the movement could be favorable.
In a sustained advancing market, a stock that is outperforming the whole market would
have a beta greater than 1.

Beta expresses the 'sensitivity' of the portfolio relative to the 'market'which cannot be
eliminated through diversification. It is so called 'systematic risk'. However, it cannot
measure the total risk of the stock, which is expressed by the volatility of the stock.

9. Further research

This paper provides a platform for further work in this area, which should focus on a number
of issues. Firstly, this paper just focus on the options in American market, a detailed research
would be needed to explore the effects of option characteristics in other markets as well.

21

Secondly, as suggested by Rosenberg (1976) that variance of earnings, variance of cash flow,
growth in earnings per share, dividend yield and debt-to-asset ratio are also helpful to predict
beta of underlying stock, it would be valuable to examine how these variables will affect
underlying stock and whether they have impacts on option beta.

Thirdly, since I have examined the effects of five variables on option beta, it would be
meaningful to figure out how these effects could be used in reality. For example, this paper
could be able to help investors with different risk aversion to form investment portfolios
which meet their risk preferences and return requirements.

Lastly, this research indicates fewpuzzles to look at into further research:

- There is a tendency of a positive linear relationship between call option beta and firm size,
while there is a tendency of negative linear relationship between put option beta and firm
size
- Book to market ratio have no significant linear relationship with call option beta, nor with
put option beta
- There is not enough evidence to infer that the strength of linear relationship between the 5
variables to option beta is varied by the type of the industry

10. Conclusion

From this research, I can conclude that:

- There is a tendency of a perfect negative linear relationship between call option beta and
put option beta
- There is a tendency of a positive linear relationship between call option beta and strike
price, while there is a tendency of negative linear relationship between put option beta and
strike price
- There is a tendency of a negative linear relationship between call option beta and days to
maturity, while there is a tendency of positive linear relationship between put option beta
and days to maturity
- There is a tendency of a positive linear relationship between call option beta and firm size,
while there is a tendency of negative linear relationship between put option beta and firm
size

22

- Book to market ratio have no significant linear relationship with call option beta, nor with
put option beta
- There is not enough evidence to infer that the strength of linear relationship between the 5
variables to option beta is varied by the type of the industry, while there is enough
evidence to infer that the strength of linear relationship between the 5 variables to option
beta is varied by the type of variables.

11. Bibliography

Black, F. and Scholes M., 1973,The Pricing of Option and Corporate Liabilities, Journal of
Political Economy, Volume 81, Page 637-654.

Bodie Z., Kane A. and Marcus A.J., 2011, Investments and Portfolio Management, New
York: McGraw Hill.

Cao C., Eric G. and Frank H., 2010, Derivatives Do Affect Mutual Fund Returns: Evidence
from the Financial Crisis of 1998.

Chen Y., 2008, Derivatives Use and Risk Taking: Evidence from the Hedge Fund Industry,
Doctoral Dissertation paper, Virginia Tech.

CapaulC., Rowley I., and Sharpe W. F., International Value and Growth Stock Returns,
Financial Analysts Journal (January/February1993), Page27-36.

Fama, E. and Miller M., 1972,The theory of finance (Holt, Rinehart and Winston, New
York).

Jacquier E., Titman S. and Yalcin A., 2010, Predicting Systematic Risk: Implications from
Growth Options.

Linter J., 1965a, The Valuation of Risk Assets and the Selection of Risky Investments in
Stock Portfolios and Capital Budgets, Review of Economics and Statistics, pp13-37.

23

Linter J., 1965b, Security Prices, Risk and Maximal Gains from Diversification, Journal of
Finance, 587-616.

Mossin J., 1966,Equibrium in a Capital Asset Market,Econometrica, 768-783.

Merton R.C., 1970,A Dynamic General Equilibrium Model of the Asset Market and its
Application to the Pricing of the Capital Structure of the Firm.

Merton R.C., 1973a, Theory of Rational Option Pricing. Bell journal of Economics and
Management Science 4, 141-183.

Nelson, Samuel Armstrong, 1904.The A.B.C. of Options and Arbitrage. New York.

Henry Deutsch, 1910. Arbitrage in Bullion, Coins, Bills, Stocks, Shares and Options, 2nd
Edition.. London: Engham Wilson.

Hull J.C., 2011.Options, Futures and Other Derivatives. New Jersey: Pearson Prentice Hall.

Galai D. and Masulis R.W., 1976, The Option Pricing Model and the Risk Factor of Stock,
Journal of Financial Economics 3, 53-81.

Keller G., 2009, Statistics for Management and Economics. Mason: South-Western Cengage
Learning.

Peltomaki J., 2007, The Use of Options and Hedge Fund Performance, University of Vaasa.

Rosenberg B. and Guy J., 1976, Prediction of Beta from Investment Fundamentals, Parts 1
and 2, Financial Analysts Journal, May-June and July-August 1976.

Sharpe W.F., 1963, A Simple Model for Portfolio Analysis, Management Science, 377-392.

Sharpe W.F., 1964, Capital Asset Prices: A Theory of Market Equilibrium Under Conditions
of Risk. Journal of Finance, 429-442.

24

effect
Block

Option)
Option)

Option)
(Call Option)
Regression Slope
Regression Slope
Regression Slope
Regression Slope
Regression Slope
Regression Slope
Regression Slope
Regression Slope
Regression Slope

effect (Put Option)
effect (Put Option)

P-Value of Days to
P-Value of Days to

P-Value of Book to
P-Value of Book to
Market Ratio effect
effect (Call Option)

Maturity effect (Put
Maturity effect (Call

P-Value of Firm Size
P-Value of Firm Size
P-Value of Strike Price
P-Value of Strike Price
P-Value of Option type

Market Ratio effect (Put

Treatment Mean
p-value p-value p-value p-value p-value p-value p-value p-value p-value Consumer Staples

[T]1
p-value p-value p-value p-value p-value p-value p-value p-value p-value Utilities

p-value p-value p-value p-value p-value p-value p-value p-value p-value Telecommunications Services

[T]2 [T]3
…
p-value p-value p-value p-value p-value p-value p-value p-value p-value Health Care

…
p-value p-value p-value p-value p-value p-value p-value p-value p-value Financials

…
p-value p-value p-value p-value p-value p-value p-value p-value p-value Materials

…
p-value p-value p-value p-value p-value p-value p-value p-value p-value Information Technology
Treatment (Type of Industry)

…
p-value p-value p-value p-value p-value p-value p-value p-value p-value Industrials
between the 5 variables to option beta is varied by the type of the industry

…
p-value p-value p-value p-value p-value p-value p-value p-value p-value Energy

p-value p-value p-value p-value p-value p-value p-value p-value p-value Consumer Discretionary

[T]10
…
…
…
…
…
Appendix 1 : Process of two-way ANOVA to test whether the strength of linear relationship

[B]3
[B]2
[B]1

[B]9
Mean
Block

25

Block

Staples

Energy
Utilities

Services

Materials
Financials

Consumer
Consumer

Industrials
Telecom’s

Information
Health Care

Technology

Discretionary

Mean
Treatment
Regression Slope P-Value of Option
p-value p-value p-value p-value p-value p-value p-value p-value p-value p-value

[T]1
type effect
Regression Slope P-Value of Strike

[T]2
Price effect (Call Option)
Regression Slope P-Value of Strike

[T]3
Price effect (Put Option)
Regression Slope P-Value of Days to

…
Maturity effect (Call Option)
Regression Slope P-Value of Days to

…
Maturity effect (Put Option)

Regression Slope P-Value of Firm Size

…

Regression Slope P-Value of Firm Size
Treatment (Type of Variable)

…
effect (Put Option)
between the 5 variables to option beta is varied by the type of variables

Regression Slope P-Value of Book to

…
Market Ratio effect (Call Option)
Regression Slope P-Value of Book to

[T]9
Market Ratio effect (Put Option)

…
…
…
…
…
…
[B]3
[B]2
[B]1

[B]10
Mean
Block
Appendix 2: Process of two-way ANOVA to test whether the strength of linear relationship

26

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