Effects of option characteristics and underlying stock on option beta article

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 by

these 5 types of variables itself. To begin the process, I use regression method to estimate the

beta of underlying stock. Then, I estimate the option beta by multiplying the beta of

underlying stock and the option elasticity. I then use regression method to test whether the 5

variables have linear relationship with option beta. I find that 3 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 beta is varied by the type

of variables.

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

option elasticity

1* 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.

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.

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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 Evidence12

7.1 Hypothesis 1 Test 12







8. Limitations 21

9. Further Research 21

10. Conclusion 22

11. Bibliography 23

12. Appendix 25

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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

conclude my 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

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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 with the beta of

the stock, βA to calculate the beta of the call option (βO) is βO = Ω βA 2. The elasticity of the

option (Ω) is calculated as: Ω= S

CΔ

3. Delta (Δ) is explained by Hull (2011), which is

defined as the 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

by put call parity equation. The derivation of put-call parity can be found on Nelson (1904),

Henry Deutsch (1910), Vinzene Bronzin (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-rT 4,

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 they

will 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 asset3 Where S is the price of the underlying asset, C is the value of the option and Δ is the delta4 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

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Furthermore, Rosenberg (1976) finds that firm size (market capitalization) help to predict

beta of underlying stock. In addition, Banz (1981) shows that firm size 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 thus it 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 hypotheses can be explained by the equations discussed in the literature review. Galai

and Masulis (1976) have shown that the beta of the option (βO) isβ 0=Ω β A . The elasticity of

the option (Ω) is calculated as: Ω= S

CΔ

. From put call parity equation, it is shown that value

for call option (c) is c= p+S 0−ke−rt (Hull, 2011). By manipulating the same put call parity

formula, it is also shown that the value for put option (p) is p = c−S 0−ke−rt (Hull, 2011).

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

Call option

βo=S0

p+S0−K e−rt∆ β A ………………………………………………………………………

(1)

Put option:

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βo=S0

c−S0+K e−rt ∆ β A………………………………………………………………………...

(2)

Hypothesis 1. Beta of the call option has negative 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:

βco=Sc∆co β A……………………………………………………………………………… (3)

Put option:

β po=Sp(−∆po)β A…………………………………………………………………………. (4)

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

βcoβpo

=−pc

∆co∆po

∨βco=−pc

∆co

∆po

β po………………………………………………………... (5)

Based on equation (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 applies from 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 a

lower (less sensitive) put option beta (βpo).

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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 a higher call premium (c). Thus from

equation (3), if delta (Δ) for call option is positive, then higher c (from days to maturity) will

result in a lower (less sensitive) call option beta (βco). The same result applies from equation

(2), increasing days to maturity (T) will result in a lower 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 a higher (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. This hypothesis can be

explained from the equation βO = Ω βA I 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. Because firm size and

underlying stock beta are negatively correlated (Banz, 1981), I can infer that call option beta

have a negative linear relationship with firm size, while put option beta have a 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 positive linear relationship

with book to market ratio. This hypothesis can be explained from the equation βO = Ω βA I

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. Because book to market ratio and underlying stock beta are negatively

correlated (Capaul, 1993), I can infer that call option beta have a negative linear relationship

with book to market ratio, while put option beta have a 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

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(2011), the relationship between the excess return of a security, R i to the excess return of the

index, RM is Ri (t )=α i+β i RM ( t )+ei( t) ……………………………………………....(6)

The intercept of the equation (α i) is the security’s expected excess return when the market

excess return is zero. The slope coefficient (β i) is the security beta (security’s sensitivity to

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

also be rewritten as β i=R i (t )−αi−ei(t )

RM ( t ) …………………………………………...…………

(7)

By substituting β A (beta of the asset) in equation (1) and (2) with β i (beta of the security) will

result in:

βco=S0

p+S0−K e−rt ∆

Ri ( t )−αi−ei( t)RM (t ) ………………………...

…………………………………..(8)

β po=S0

c−S0+K e−rt ∆

Ri (t )−α i−e i(t)RM (t )

……………..

……………………………………………...(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:

βco=NAVPSbook ¿

market ratio ¿p+S0−K e

−rt ∆Ri (t )−αi−ei( t)

RM (t )

……………………………………………….……(10)

β po=NAVPSbook ¿

market ratio ¿c−S0+K e

−rt ∆Ri (t )−α i−e i(t)

RM (t )

…………………………………………………….(11)

Because firm size (market capitalization) is calculated as current share price (S0) multiplied

by the number of shares outstanding (NOSO), then equation (10) and (11) can be rewritten

as:

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βco=NAVPSbook ¿

market ratio ¿

p+ firm¿¿NOSO−K e−rt∆Ri ( t )−α i−e i(t)

RM (t )¿

…………………………………...………………(12)

β po=NAVPSbook ¿

market ratio ¿

c−firm¿¿NOSO+K e−rt∆R i ( t )−αi−ei(t)

RM ( t )¿

……………………………………..…………….(13)

Thus, firm-specific effect (e i), 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 vary firm-specific effect (e i), 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.

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

d1=ln [ StK ]+(R f+1/2σ2 )T

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

βco=NAVPSbook ¿

market ratio ¿

p+ firm¿¿NOSO−K e−rtN ( ln [ S tK ]+(R f+1/2σ2 )T

σ √T ) Ri (t )−α i−e i(t)RM (t )

¿

…………………………...(14)

β po=NAVPSbook ¿

market ratio ¿

c−firm¿¿NOSO+K e−rt(N ( ln [ S tK ]+(R f+1/2σ2 )T

σ √T )−1) Ri (t )−α i−e i(t)RM (t )

¿

…….........................(15)

Because Capital Asset Pricing Model equation E (ri )=rf+ βi [E (rM )−rf ] (Bodie, 2011) can be

rewritten as r f=E (r i)−β i [E ( rM )−rf ]…………………………………………(16)

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

βco=NAVPSbook ¿

market ratio ¿

p+ firm¿¿NOSO−K e−rtN ( ln [ S tK ]+(E (ri )−β i [E (rM )−r f ]+1/2σ2 )Tσ √T ) R i (t )−α i−e i(t)

RM ( t )¿

…………...(17)

β po=NAVPSbook ¿

market ratio ¿

c−firm¿¿NOSO+K e−rt(N ( ln [ S tK ]+(E (r i )−β i [E (rM )−r f ]+1/2σ2 )Tσ √T )−1) R i ( t )−α i−e i(t)

RM ( t )¿

……(18)

Because underlying stock beta β i is affected by firm size (Banz, 1981) and book 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

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 year period 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

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which is likely to occur during the different periods of normal and abnormal condition. All of

these data are obtained from WRDS5 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 as it is more mature compared to

others. I regard 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

elasticity is calculated using Ω= S

CΔ

. Once I have calculated the option beta, I can now test

all the hypotheses using regression method.

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).

5 Where βco is the call option beta and ∆cois the call option delta.

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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, I will 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

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]

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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]9 and [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:

Table 3: Ten companies for testing the hypotheses

Ticker Symbol Company IndustryADM Archer-Daniels-Midland Co Consumer StaplesAES AES Corp UtilitiesAMT American Tower Corp A Telecommunications ServicesBDX Becton Dickinson Health CareC Citigroup Inc. FinancialsCLF Cliffs Natural Resources Materials

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CSCO Cisco Systems Information TechnologyDHR Danaher Corp. IndustrialsSUN Sunoco Inc. EnergySWK 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-ValueADM Consumer Staples -0.97 0.00AES Utilities -1.02 0.00AMT Telecommunications Services -1.03 0.00BDX Health Care -0.93 0.00C Financials -0.96 0.00CLF Materials -1.02 0.00CSCO Information Technology -0.93 0.00DHR Industrials -0.94 0.00SUN Energy -0.98 0.00SWK 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

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

Company Industry Regression Slope Slope P-ValueADM Consumer Staples 0.03 0.11AES Utilities 1.04 0.00AMT Telecommunications Services 0.23 0.00

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BDX Health Care 0.08 0.06C Financials 0.98 0.00CLF Materials 0.06 0.03CSCO Information Technology 0.55 0.00DHR Industrials 0.09 0.02SUN Energy 0.04 0.00SWK 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


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.


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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 are 0.00, 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:


Company Industry Regression Slope Slope P-ValueADM Consumer Staples 0.01 0.00AES Utilities 0.06 0.00AMT Telecommunications Services 0.04 0.00BDX Health Care 0.04 0.00C Financials 0.13 0.00CLF Materials 0.07 0.00CSCO Information Technology 0.06 0.00DHR Industrials 0.06 0.00SUN Energy 0.02 0.00

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SWK Consumer Discretionary 0.07 0.00

From this regression, all of the listed companies show a positive regression slope.

Furthermore, the p-values of the slope are 0.00, 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



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

From this regression, nine of the listed companies show a positive regression slope, except

one with negative 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 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


17


Company Industry Regression Slope Slope P-ValueADM Consumer Staples -3.52569E-08 0.16AES Utilities -1.62284E-06 0.00AMT Telecommunications Services -6.54614E-07 0.00BDX Health Care -4.05772E-07 0.02C Financials -2.61023E-07 0.00CLF Materials 5.61455E-07 0.05CSCO Information Technology -8.43186E-08 0.00DHR Industrials -3.39261E-07 0.05SUN Energy -2.63492E-07 0.00SWK 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 positive

linear 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:


Company Industry Regression Slope Slope P-ValueADM Consumer Staples -3.04 0.49AES Utilities -5.89 0.09AMT Telecommunications Services -65.09 0.01BDX Health Care -23.15 0.71C Financials -1.06 0.19CLF Materials -21.23 0.02CSCO Information Technology -46.51 0.18DHR Industrials -79.71 0.00SUN Energy -2.85 0.50

18

SWK Consumer Discretionary 2.62 0.71

From this regression, all of the listed companies show a negative regression slope, except one

company (SWK). However, only four of the companies have 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, this result is not

consistent with the hypothesis 5.

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



Company Industry Regression Slope Slope P-ValueADM Consumer Staples 2.85 0.52AES Utilities 6.08 0.12AMT Telecommunications Services 66.77 0.01BDX Health Care 26.88 0.68C Financials 2.01 0.03CLF Materials 20.59 0.01CSCO Information Technology 48.71 0.19DHR Industrials 84.76 0.00SUN Energy 2.87 0.53SWK Consumer Discretionary 2.62 0.71

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

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.


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:

19

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.


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:

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

20

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 type4.51105E-

27 1

Days to Maturity (Call Option)1.90628E-

11 2Days to Maturity (Put Option) 8.3459E-11 3Strike Price (Call Option) 0.02296202 4

Strike Price (Put Option)0.02647624

4 5

Firm Size (Call Option)0.04650519

9 6

Firm Size (Put Option)0.04731148

3 7

Book to Market Ratio (Put Option)0.28110888

9 8

Book to Market Ratio (Call Option)0.29060612

1 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 βP=∑i−1

n+1

w iβ i (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

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

21

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.

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

22

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 few puzzles 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

- 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.

23

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.

Capaul C., Rowley I., and Sharpe W. F., International Value and Growth Stock Returns,

Financial Analysts Journal (January/February1993), Page 27-36.

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.

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.

24

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.

Appendix 1 : Process of two-way ANOVA to test whether the strength of linear relationship

between the 5 variables to option beta is varied by the type of the industry

Block Treatment (Type of Industry) Block

25

MeanC

onsu

mer

Sta

ple

s

Uti

liti

es

Tel

ecom

mu

nic

atio

ns

Ser

vice

s

Hea

lth

Car

e

Fin

anci

als

Mat

eria

ls

Info

rmat

ion

Tec

hn

olog

y

Ind

ust

rial

s

En

ergy

Con

sum

er D

iscr

etio

nar

y

Regression SlopeP-Value of Option type

effect p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

[B]1

Regression SlopeP-Value of Strike Price

effect (Call Option) p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

[B]2

Regression SlopeP-Value of Strike Price

effect (Put Option) p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

[B]3

Regression SlopeP-Value of Days to

Maturity effect (Call Option)

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

…

Regression SlopeP-Value of Days to Maturity effect (Put

Option)

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

…

Regression SlopeP-Value of Firm Size effect (Call Option) p-

valu

e

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

…

Regression SlopeP-Value of Firm Size effect (Put Option) p-

valu

e

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

…

Regression SlopeP-Value of Book to Market Ratio effect

(Call Option)

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

…

Regression SlopeP-Value of Book to

Market Ratio effect (Put Option)

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

[B]9

Treatment Mean [T]1 [T]2 [T]3 … … … … … … [T]10

Appendix 2: Process of two-way ANOVA to test whether the strength of linear relationship between the 5 variables to option beta is varied by the type of variables

Block Treatment (Type of Variable) Block

26

MeanReg

ress

ion

Slo

pe

P-V

alu

e of

Op

tion

ty

pe

effe

ct

Reg

ress

ion

Slo

pe

P-V

alu

e of

Str

ike

Pri

ce e

ffec

t (C

all O

pti

on)

Reg

ress

ion

Slo

pe

P-V

alu

e of

Str

ike

Pri

ce e

ffec

t (P

ut

Op

tion

)

Reg

ress

ion

Slo

pe

P-V

alu

e of

Day

s to

M

atu

rity

eff

ect

(Cal

l Op

tion

)

Reg

ress

ion

Slo

pe

P-V

alu

e of

Day

s to

M

atu

rity

eff

ect

(Pu

t O

pti

on)

Reg

ress

ion

Slo

pe

P-V

alu

e of

Fir

m S

ize

effe

ct (

Cal

l Op

tion

)

Reg

ress

ion

Slo

pe

P-V

alu

e of

Fir

m S

ize

effe

ct (

Pu

t O

pti

on)

Reg

ress

ion

Slo

pe

P-V

alu

e of

Boo

k t

o M

ark

et R

atio

eff

ect

(Cal

l Op

tion

)

Reg

ress

ion

Slo

pe

P-V

alu

e of

Boo

k t

o M

ark

et R

atio

eff

ect

(Pu

t O

pti

on)

Consumer Staples

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

[B]1

Utilities

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

[B]2

Telecom’s Services

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

[B]3

Health Care

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

…

Financials

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

…

Materials

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

…

Information Technology

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

…

Industrials

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

…

Energy

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

…

Consumer Discretionary

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

p-va

lue

[B]10

Treatment Mean

[T]1 [T]2 [T]3 … … … … … [T]9

27

Economy & Finance

Effects of option characteristics and underlying stock on option beta article