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Effects of Option Characteristics and Underlying Stock on Option Beta

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Beta (β) 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 …

Beta (β) 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.

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  • 1. EFFECTS OF OPTION CHARACTERISTICS AND UNDERLYING STOCK ON OPTION BETA: AN EMPIRICAL STUDY FOR UNITED STATES MARKET DHARMA ISWARA BAGOES OKA* 21st October, 2011 AbstractBeta (β)1 is one of the risk management tools to capture the risk exposures of hedge-fundinvestments. As most of hedge funds today trade derivative securities, the research on themeasurement 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 iscomprised into three main parts. First, I hypothesize that 5 variables (type of option, strikeprice, days to maturity, firm size and book to market ratio) have linear relationship with theoption beta. Second, I hypothesize that the strength of linear relationship is varied by the typeof the industry. Third, I hypothesize that the strength of linear relationship is also varied bythese 5 types of variables itself. To begin the process, I use regression method to estimate thebeta of underlying stock. Then, I estimate the option beta by multiplying the beta ofunderlying stock and the option elasticity. I then use regression method to test whether the 5variables 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 optionbeta, while firm size has less significant linear relationship and book to market ratio have nosignificant linear relationship. Furthermore, using 2-way ANOVA, I test whether strength oflinear relationship is varied by the type of the industry and the 5 types of variables. There isnot enough evidence to infer that the strength of linear relationship between the 5 variables tooption beta is varied by the type of the industry, instead, there is enough evidence to infer thatthe strength of linear relationship between the 5 variables to option beta is varied by the typeof 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 DataServices (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 aportfolio which cannot be eliminated through diversification relative to one or several risk factors. It comprises the riskfactors common to all assets in the investment universe. 1
  • 2. Table of Contents1. Introduction ................................................................................................................... 32. Aim and Research Outline ............................................................................................ 33. Literature Review .......................................................................................................... 34. Hypothesis Development ............................................................................................... 55. Data ................................................................................................................................ 96. Methods and Models for Empirical Investigation ...................................................... 107. 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 ..................................................................................................... 198. Limitations ................................................................................................................... 219. Further Research......................................................................................................... 2110. Conclusion ................................................................................................................... 2211. Bibliography ................................................................................................................ 2312. Appendix...................................................................................................................... 25 2
  • 3. 1. IntroductionHedge 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 ofderivatives has becoming a great concern to investors and regulators (Chen, 2008).Furthermore, evidence from 1998 financial crisis support the hypothesis that the effects ofderivative 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 fromthe use of options on their primary asset class and this information can be used to build betterperforming portfolios of funds of funds (Peltomaki, 2007). However, the research did notshow the effects of five variables (type of option, strike price, days to maturity, firm size andbook to market ratio) on the portfolio performance of hedge funds. Since the measurement ofsystematic risk (beta) is important for portfolio and risk management (Jacquier, 2010), theresearch to examine the effects of five variables on the option beta is essential.2. Aim and Research OutlineThe aim of this research is to provide an empirical analysis to show the effects of the fivevariables (type of option, strike price, days to maturity, firm size of the underlying stock andbook to market ratio of the underlying stock) on the option beta. To do this, I will begin withthe 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 conducthypothesis tests to see whether the results are consistent with the initial hypotheses and thenanalyse the results. Lastly, after discussing the limitations and further research, I willconclude my research.3. Literature ReviewIn order to examine the effects of five variables on the option beta, I need to figure out therelationship between option beta and underlying stock beta first. This relationship has beenan important research topic and many studies have been devoted to it. One such study, the 3
  • 4. underlying stock beta is presented in the Capital Asset Pricing Model (CAPM). Thederivation 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 acontinuous time framework in Merton (1970 and 1973b). Jensen (1972) summarizes thedifferent approaches and provides a survey of empirical tests of the model. The option pricingmodel 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 ashort/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 hasbeen proved by Galai and Masulis (1976) by combining the option pricing model with theCAPM yields a theoretically more complete model of corporate security pricing. One resultof their research is that the systemic risk of equity is the product of the firm’s systemic riskand the elasticity of equity value with respect to firm’s value. Based on the argument used intheir article, the relationship between the systematic risks of the option and the underlyingstocks can be obtained. In other words, the elasticity also can be combined with the beta ofthe stock, βA to calculate the beta of the call option (βO) is βO = Ω βA 2. The elasticity of the Soption (Ω) is calculated as: Ω   3. Delta (Δ) is explained by Hull (2011), which is Cdefined as the ratio of the change in the price of the option to the change in the price of theunderlying stock.As the option beta and underlying stock beta is connected by the option elasticity Ω, I need tofind the factors affect option elasticity and thus affect option beta. The factors are explainedby 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 ofoption, 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 β A 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 isthe put option value with the same stock, strike price and time to maturity with the call option 4
  • 5. Furthermore, Rosenberg (1976) finds that firm size (market capitalization) help to predictbeta of underlying stock. In addition, Banz (1981) shows that firm size and beta of underlyingstock are negatively correlated. Furthermore, Capaul (1993) suggests that high book tomarket 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, strikeprice, days to maturity, firm size and book to market ratio on option beta. To the best of myknowledge, no such investigation has been carried out so far. Thus, I aim at filling aconspicuous gap in the literatures by conducting an empirical analysis of European optionswith respect to compatibility with asset-pricing theory, option pricing model and put-callparity.4. Hypothesis DevelopmentIn 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. Galaiand Masulis (1976) have shown that the beta of the option (βO) is  0  A . The elasticity of Sthe option (Ω) is calculated as: Ω   . From put call parity equation, it is shown that value Cfor call option (c) is c= p  S 0  ke rt (Hull, 2011). By manipulating the same put call parityformula, 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 ……………………………………………………………………… (1)Put option: ………………………………………………………………………... (2) 5
  • 6. Hypothesis 1. Beta of the call option has negative linear relationship with beta of the putoption. This can be explained by the following reasons. The Delta (Δ) is defined as the ratioof 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 theunderlying 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 on equation (5), beta of the call option has negative linear relationship with beta of theput option.Hypothesis 2. Call option beta will be positively correlated with strike price, while put optionbeta 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), ifdelta (Δ) for call option is positive, then lower c (from increasing strike price) will result in ahigher (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), ifdelta (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 putoption 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 fromequation (3), if delta (Δ) for call option is positive, then higher c (from days to maturity) will5 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
  • 7. 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 fromequation (4), if delta (Δ) for put option is negative, then lower p (from increasing days tomaturity) 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, whileput option beta will have a positive linear relationship with firm size. This hypothesis can beexplained 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 putoption. Hence, call option beta will be positively related with underlying stock beta whereasput option beta will be negatively correlated to underlying stock beta. Because firm size andunderlying stock beta are negatively correlated (Banz, 1981), I can infer that call option betahave a negative linear relationship with firm size, while put option beta have a positive linearrelationship with firm size.Hypothesis 5. Call option beta will have a negative linear relationship with book to marketratio of the underlying stock, while put option beta will have a positive linear relationshipwith book to market ratio. This hypothesis can be explained from the equation βO = Ω βA Ihave mentioned in the literature review. Furthermore, I suggest in hypothesis 2 that Ω ispositive for call option and negative for put option. Hence, call option beta will be positivelyrelated with underlying stock beta whereas put option beta will be negatively correlated tounderlying stock beta. Because book to market ratio and underlying stock beta are negativelycorrelated (Capaul, 1993), I can infer that call option beta have a negative linear relationshipwith book to market ratio, while put option beta have a positive linear relationship with bookto 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 bythe type of the industry. This hypothesis can be explained as follows. According to Bodie(2011), the relationship between the excess return of a security, R i to the excess return of theindex, RM is ……………………………………………....(6)The intercept of the equation ( ) is the security’s expected excess return when the marketexcess return is zero. The slope coefficient ( ) is the security beta (security’s sensitivity to 7
  • 8. the index). is the zero mean, firm-specific surprise in the security return. Equation (6) canalso be rewritten as …………………………………………...…………(7)By substituting (beta of the asset) in equation (1) and (2) with (beta of the security) willresult in: ………………………...…………………………………..(8) ……………..……………………………………………...(9)Because book to market ratio is calculated as Net Asset Value per share (NAVPS) divided bycurrent 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) multipliedby the number of shares outstanding (NOSO), then equation (10) and (11) can be rewrittenas: …………………………………...………………(12) ……………………………………..…………….(13)Thus, firm-specific effect ( ), will vary the strength of linear relationship between the 5variables (type of option, strike price K, days to maturity t, firm size and book to marketratio) to option beta. Because the type of industry vary firm-specific effect ( ), the strengthof linear relationship between the 5 variables to option beta is varied by the type of theindustry as well.Hypothesis 7. The strength of linear relationship between each of the 5 variables to optionbeta is varied by these 5 types of variables itself. This hypothesis can be explained as follows. 8
  • 9. 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) and book to marketratio (Capaul, 1993), then equation (17) and (18) showed that the 5 variables effect (type ofoption, strike price K, days to maturity t, firm size and book to market ratio) is repeated (onthe left side of the bracket and on the inside of the bracket). Thus, the strength of linearrelationship between each of the 5 variables to option beta is varied by these 5 types ofvariables itself.Even though the hypotheses can be explained theoretically, I still need to provide empiricalevidence concerning the hypotheses. Therefore, I will develop the research based on reality inthe U.S. financial markets to see whether the hypotheses are empirically correct.5. DataFor the empirical analysis, I use data of monthly return of S&P 500 index, monthly return ofindividual 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
  • 10. companies I randomly selected under S&P 500 list. To ensure my results hold for differentindustries, I randomly select one company for each of ten different industries under S&P500list. I choose these data for the 5 year period from 1st January 2006 to 31st December 2010. Ithink that 5 year period from 2006 to 2010 is enough to incorporate the fluctuation of boomand bust (such as the Global Financial Crisis in 2008) so that I can address the beta changewhich is likely to occur during the different periods of normal and abnormal condition. All ofthese data are obtained from WRDS7 and beta of underlying stock, option elasticity andoption 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 toothers. I regard T-bills rate as the risk free rate, because it is usually assumed that there is nochance that a government will default on an obligation denominated in its own currency. Iconsider S&P 500 as the market proxy based on the fact that S&P 500 is the most popularvalue-weighted index of U.S. stocks (Bodie, 2011) involved large publicly held companiesthat trade on New York Stock Exchange and NASDAQ. I select monthly data as my samplefrequency since monthly data tends to be more reliable compared with daily data.6. Methods and Models for Empirical InvestigationThe method that I will use to estimate the individual stock beta (βA) is to use regressionmethod to regress the excess return of individual stock with the excess return of the S&P 500portfolio index (Bodie, 2011). The excess return of individual stock can be calculated usingthe return of individual stock subtracted by the corresponding T-bill rate (Bodie, 2011). Thesame apply with the excess return of the S&P 500 portfolio index where I subtract the S&P500 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 betaof the underlying stock by the option elasticity (Ω) using equation βO = ΩβA. The option Selasticity is calculated using Ω   . Once I have calculated the option beta, I can now test Call the hypotheses using regression method.7 WRDS provides access to important databases in the fields of finance, accounting, banking, and economicsand so on. It is available at http://wrds.wharton.upenn.edu/ 10
  • 11. To test hypothesis 1, I will regress the beta of call option with the beta of put option for thesame stock. To ensure the accuracy and comparability of the test result, I will use call and putoption data with 30 days to maturity. I will repeat this method for ten companies that Iselected. I then interpret the results based on the slope of the regression and the significanceof 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 30days to maturity. I will repeat this method for ten companies that I selected. I then interpretthe 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 tomaturity. To ensure the accuracy and comparability of the test result, I will use call and putoption data with 30, 60, 91 and 182 days to maturity. I will repeat this method for tencompanies that I selected. I then interpret the results based on the slope of the regression andthe 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. Toensure the accuracy and comparability of the test result, I will use call and put option datawith 30 days to maturity. I will repeat this method for ten companies that I selected. I theninterpret the results based on the slope of the regression and the significance of this linearrelationship (using p-value of the slope).To test hypothesis 5, I will regress the beta of the option with the corresponding book tomarket ratio of the underlying stock. To ensure the accuracy and comparability of the testresult, I will use call and put option data with 30 days to maturity. I will repeat this methodfor ten companies that I selected. I then interpret the results based on the slope of theregression 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) betweenthe 5 variables to option beta is varied by the type of the industry by using analysis ofvariance method (ANOVA). I use this method because ANOVA is a procedure that tests todetermine 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 thesamples are drawn from matched block for each treatment, the type of analysis of variance to 11
  • 12. apply is two-way ANOVA (Keller, 2009). This method has an advantage that it reduceswithin-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 isfacilitated 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]9 and [T]10) do not differH1: At least two means differBefore I will run the F-Test of the two-way ANOVA, I must check whether the requiredconditions are met, that is the random variable must be normally distributed and thepopulation variances must be equal (Keller, 2009). After I run the two-way ANOVA usingMinitab, I will conclude whether the strength of linear relationship between the 5 variables tooption 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 the5 variables to option beta is varied by these 5 types of variables by using the same method astesting hypothesis 6. The process of performing the two-way ANOVA to test hypothesis 7 isfacilitated by table as shown in Appendix 2. [Appendix 2 here]7. Results and Discussion of Empirical Evidence6.1 Hypothesis 1 TestThe first test is to determine whether beta of the call option has negative linear relationshipwith beta of the put option. There are ten different types of industries under S&P500 list. Irandomly select one stock from each of ten industries as the test objects. Table below showsthe stocks I have randomly selected from S&P500 list: 12
  • 13. Table 3: Ten companies for testing the hypothesesTicker 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 MaterialsCSCO Cisco Systems Information TechnologyDHR Danaher Corp. IndustrialsSUN Sunoco Inc. EnergySWK Stanley Black & Decker Consumer DiscretionaryBy regressing beta of call option with the beta of put option for each of ten companiesmentioned in table 3, the result is as follows:Table 4: Regression Result to Test Hypothesis 1Company 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 0.00From 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 showsan overwhelming evidence that linear relationship exists. Thus, from this regression slope andthe p-value, I suggest that there is a tendency of a perfect negative linear relationship betweencall option beta and put option beta. This evidence strongly supports hypothesis 1.6.2 Hypothesis 2 TestThe second test is to determine whether call option beta will be positively correlated withstrike price, while put option beta will be negatively correlated with strike price. By 13
  • 14. regressing beta of the call option with the corresponding strike price for each companymentioned in table 3, the result is as follows:Table 5: Regression Result to Test Hypothesis 2 for Call OptionCompany Industry Regression Slope Slope P-ValueADM Consumer Staples 0.03 0.11AES Utilities 1.04 0.00AMT Telecommunications Services 0.23 0.00BDX 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 0.00From 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 theslope 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 positivelinear relationship between call option beta and strike price. This evidence supports myhypothesis 2.By regressing beta of the put option with the corresponding strike price for each companymentioned in table 3, the result is as follows:Table 6: Regression Result to Test Hypothesis 2 for Put OptionCompany Industry Regression Slope Slope P-ValueADM Consumer Staples -0.02 0.17AES Utilities -1.02 0.00AMT Telecommunications Services -0.23 0.00BDX Health Care -0.10 0.02C Financials -1.00 0.00CLF Materials -0.05 0.04CSCO Information Technology -0.52 0.00DHR Industrials -0.09 0.04SUN Energy -0.03 0.00SWK Consumer Discretionary -0.44 0.00 14
  • 15. 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 theslope 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 negativelinear relationship between put option beta and strike price. This evidence supports myhypothesis 2.6.3 Hypothesis 3 TestThe third test is to determine whether call option beta will be negatively correlated with daysto maturity, while put option beta will be positively correlated with days to maturity. Byregressing beta of the call option with the corresponding days to maturity for each companymentioned in table 3, the result is as follows:Table 7: Regression Result to Test Hypothesis 3 for Call OptionCompany 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 0.00SWK Consumer Discretionary -0.07 0.00From 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 thatlinear relationship exists. Thus, from this regression slope and the p-value, I suggest thatthere is a tendency of a negative linear relationship between call option beta and days tomaturity. This evidence supports my hypothesis 3.By regressing beta of the put option with the corresponding days to maturity for eachcompany mentioned in table 3, the result is as follows: 15
  • 16. Table 8: Regression Result to Test Hypothesis 3 for Put OptionCompany 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 0.00SWK Consumer Discretionary 0.07 0.00From 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 thatlinear relationship exists. Thus, from this regression slope and the p-value, I suggest thatthere is a tendency of a positive linear relationship between put option beta and days tomaturity. This evidence supports my hypothesis 3.6.4 Hypothesis 4 TestThe fourth test is to determine whether call option beta will have a negative linearrelationship with firm size, while put option beta will have a positive linear relationship withfirm size. By regressing beta of the call option with the corresponding firm size for eachcompany mentioned in table 3, the result is as follows:Table 9: Regression Result to Test Hypothesis 4 for Call OptionCompany 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 16
  • 17. From this regression, nine of the listed companies show a positive regression slope, exceptone with negative regression slope. Furthermore, the slope p-value of the eight companies isstatistically significant, while two other have no statistical significance. Thus, from thisregression slope and the p-value, I suggest that there is a tendency of a positive linearrelationship between call option beta and firm size. Therefore, from these findings, I rejecthypothesis 4.By regressing beta of the put option with the corresponding firm size for each companymentioned in table 3, the result is as follows:Table 10: Regression Result to Test Hypothesis 4 for Put OptionCompany 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 0.19From this regression, nine of the listed companies show a negative regression slope, exceptone with positive regression slope. Furthermore, the slope p-value of the eight companies isstatistically significant, while two other have no statistical significance. Thus, from thisregression slope and the p-value, I suggest that there is a tendency of a negative linearrelationship between put option beta and firm size. Therefore, from these findings, I rejecthypothesis 4.6.5 Hypothesis 5 TestThe fifth test is to determine whether call option beta will have a negative linear relationshipwith 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 thecorresponding book to market ratio for each company mentioned in table 3, the result is asfollows: 17
  • 18. Table 11: Regression Result to Test Hypothesis 5 for Call OptionCompany 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 0.50SWK Consumer Discretionary 2.62 0.71From this regression, all of the listed companies show a negative regression slope, except onecompany (SWK). However, only four of the companies have slope p-value that is statisticallysignificant. Thus, from this regression slope and the p-value, I suggest book to market ratiohave no significant linear relationship with call option beta. Therefore, this result is notconsistent with the hypothesis 5.By regressing beta of the put option with the corresponding book to market ratio for eachcompany mentioned in table 3, the result is as follows:Table 12: Regression Result to Test Hypothesis 5 for Put OptionCompany 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 0.71From 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
  • 19. regression slope and the p-value, I suggest book to market ratio have no significant linearrelationship with put option beta. Therefore, this result is not consistent with the hypothesis 5.6.6 Hypothesis 6 TestAfter arranging all p-values which appear from table 4 to table 12 in accordance with theformat shown in table 1, I initially check whether the required conditions are met. Since thepreliminary requirements of variance equality is not satisfied, I can address this issue byusing the logarithmic transformation of these p-values before proceeding to run the two-wayANOVA. 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 betweenthe 5 variables to option beta is varied by the type of the industryThe F-statistic and p-value to determine whether differences exist between 10 industries is0.49 and 0.877, respectively. Against a standard of 10% significance level, the p-value of0.877 suggests that there is no sufficient evidence to infer that at least two of the industriesdiffer. Thus, this finding is not consistent with hypothesis 6.6.7 Hypothesis 7 TestAfter arranging all p-values which appear from table 4 to table 12 in accordance with theformat shown in table 2, I initially check whether the required conditions are met. Since thepreliminary requirements of variance equality is not satisfied, I can address this issue byusing the logarithmic transformation of these p-values before proceeding to run the two-wayANOVA. By using the transformed values of the p-values as the inputs for Minitab, the two-way ANOVA result is as follows: 19
  • 20. Figure 2 : Two-way ANOVA result to test whether the strength of linear relationship betweenthe 5 variables to option beta is varied by the type of the variablesThe F-statistic and p-value to determine whether differences exist between the variables is52.43 and 0.000, respectively. Against a standard of 5% significance level, the p-value of0.000 suggests that there overwhelming evidence to infer that at least two of the variablesdiffer. Furthermore, I am interested to see which under which variables the linear relationshipbetween the 5 variables to option beta have the strongest linear relationship, and rank theminto order. By observing the mean of the p-values on each variables, the rank of linearrelationship strength are as follows:Table 13: Rank list of linear relationship strength of variablesVariables p-value Rank of linear relationship strengthOption type 4.51105E-27 1Days to Maturity (Call Option) 1.90628E-11 2Days to Maturity (Put Option) 8.3459E-11 3Strike Price (Call Option) 0.02296202 4Strike Price (Put Option) 0.026476244 5Firm Size (Call Option) 0.046505199 6Firm Size (Put Option) 0.047311483 7Book to Market Ratio (Put Option) 0.281108889 8Book to Market Ratio (Call Option) 0.290606121 9Based from table 13, I can see that the variables which have strongest linear relationship tooption beta is option type, and the lowest is book to market ratio. In addition to portfolio betaequation ∑ (Bodie, 2011), this information can be used to help investors toform strategy to achieve desired beta of portfolios of option. That is, based from table 13, it is 20
  • 21. 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
  • 22. 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 predictbeta of underlying stock, it would be valuable to examine how these variables will affectunderlying stock and whether they have impacts on option beta.Thirdly, since I have examined the effects of five variables on option beta, it would bemeaningful to figure out how these effects could be used in reality. For example, this papercould be able to help investors with different risk aversion to form investment portfolioswhich 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 industry10. ConclusionFrom 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
  • 23. - 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. BibliographyBlack, F. and Scholes M., 1973, The Pricing of Option and Corporate Liabilities, Journal ofPolitical Economy, Volume 81, Page 637-654.Bodie Z., Kane A. and Marcus A.J., 2011, Investments and Portfolio Management, NewYork: McGraw Hill.Cao C., Eric G. and Frank H., 2010, Derivatives Do Affect Mutual Fund Returns: Evidencefrom 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, NewYork).Jacquier E., Titman S. and Yalcin A., 2010, Predicting Systematic Risk: Implications fromGrowth Options.Linter J., 1965a, The Valuation of Risk Assets and the Selection of Risky Investments inStock Portfolios and Capital Budgets, Review of Economics and Statistics, pp13-37. 23
  • 24. Linter J., 1965b, Security Prices, Risk and Maximal Gains from Diversification, Journal ofFinance, 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 itsApplication to the Pricing of the Capital Structure of the Firm.Merton R.C., 1973a, Theory of Rational Option Pricing. Bell journal of Economics andManagement 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, 2ndEdition.. 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 CengageLearning.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 1and 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 Conditionsof Risk. Journal of Finance, 429-442. 24
  • 25. 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) 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 Block Mean25
  • 26. Block Staples Energy Utilities Services Materials Financials Consumer Consumer Industrials Telecom’s Health Care Information Technology Discretionary Treatment Mean 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 p-value p-value p-value p-value p-value p-value p-value p-value p-value p-value [T]2 Price effect (Call Option) Regression Slope P-Value of Strike p-value p-value p-value p-value p-value p-value p-value p-value p-value p-value [T]3 Price effect (Put Option) Regression Slope P-Value of Days to … p-value p-value p-value p-value p-value p-value p-value p-value p-value p-value Maturity effect (Call Option) Regression Slope P-Value of Days to … p-value p-value p-value p-value p-value p-value p-value p-value p-value p-value Maturity effect (Put Option) Regression Slope P-Value of Firm Size … p-value p-value p-value p-value p-value p-value p-value p-value p-value p-value effect (Call Option) Regression Slope P-Value of Firm Size Treatment (Type of Variable) … p-value p-value p-value p-value p-value p-value p-value p-value p-value p-value effect (Put Option) between the 5 variables to option beta is varied by the type of variables Regression Slope P-Value of Book to … p-value p-value p-value p-value p-value p-value p-value p-value p-value p-value Market Ratio effect (Call Option) Regression Slope P-Value of Book to p-value p-value p-value p-value p-value p-value p-value p-value p-value p-value [T]9 Market Ratio effect (Put Option) … … … … … … [B]3 [B]2 [B]1 [B]10 Block Mean Appendix 2: Process of two-way ANOVA to test whether the strength of linear relationship26
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