Georgia Southern University Digital Commons@Georgia Southern Electronic Theses and Dissertations Graduate Studies, Jack N. Averitt College of Summer 2012 The Distribution of Individual Stock Returns in a Modified Black-scholes Option Pricing Model Daniel Lee Richey Follow this and additional works at: https://digitalcommons.georgiasouthern.edu/etd Part of the Finance and Financial Management Commons, and the Other Mathematics Commons Recommended Citation Richey, Daniel Lee, "The Distribution of Individual Stock Returns in a Modified Black- scholes Option Pricing Model" (2012). Electronic Theses and Dissertations. 1015. https://digitalcommons.georgiasouthern.edu/etd/1015 This thesis (open access) is brought to you for free and open access by the Graduate Studies, Jack N. Averitt College of at Digital Commons@Georgia Southern. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of Digital Commons@Georgia Southern. For more information, please contact [email protected]. THE DISTRIBUTION OF INDIVIDUAL STOCK RETURNS IN A MODIFIED BLACK-SCHOLES OPTION PRICING MODEL by DANIEL LEE RICHEY (Under the Direction of Patrica Humphrey) ABSTRACT There have been many attempts to find a model that can accurately price options. These models are built on many assumptions, including which probability distribution stock returns follow. In this paper, we test several distributions to see which best fit the log returns of 20 different companies over a period between November 1, 2006 to October 31, 2011. If a \best" distribution is found, a modified Black-Scholes model will be defined by modifying the Weiner process. We use Monte Carlo simulations to generate estimated prices under specified parameters, and compare these prices to those simulated by the model using the Weiner process. It was found the Student-t distribution did a better job at modeling the larger time intervals and the 3-parameter lognormal did a better job at modeling the smaller time intervals. We were not able to make any definite conclusion due to the cost of purchasing historical option data. Key Words: Black-Scholes, Weiner process, mathematical finance 2009 Mathematics Subject Classification: THE DISTRIBUTION OF INDIVIDUAL STOCK RETURNS IN A MODIFIED BLACK-SCHOLES OPTION PRICING MODEL by DANIEL LEE RICHEY B.S. in Physics with a minor in Economics (2008) A Thesis Submitted to the Graduate Faculty of Georgia Southern University in Partial Fulfillment of the Requirement for the Degree MASTER OF SCIENCE STATESBORO, GEORGIA 2012 ©2012 DANIEL LEE RICHEY All Rights Reserved iii THE DISTRIBUTION OF INDIVIDUAL STOCK RETURNS IN A MODIFIED BLACK-SCHOLES OPTION PRICING MODEL by DANIEL LEE RICHEY Major Professor: Patrica Humphrey Committee: Jonathan Duggins John Barkoulas Electronic Version Approved: June 2012 iv ACKNOWLEDGMENTS I would like to thank my committee members for all of their valuable comments and suggestions. Also, I would like to thank Dr. Scott Kersey, Dr. Jim Braselton, and Chasen Smith for their help with LaTex. v TABLE OF CONTENTS Page ACKNOWLEDGMENTS :::::::::::::::::::::::::::::: v LIST OF TABLES :::::::::::::::::::::::::::::::::: viii LIST OF FIGURES ::::::::::::::::::::::::::::::::: ix CHAPTER 1 Introduction ::::::::::::::::::::::::::::::: 1 2 Candidate Models for Daily Stock Returns, Derivation of the Black-Scholes Model, and Useful Theorems ::::::::::::: 8 2.1 Normal Distribution ::::::::::::::::::::::: 8 2.2 Student's t Distribution ::::::::::::::::::::: 9 2.3 Cauchy Distribution ::::::::::::::::::::::: 9 2.4 Weibull Distribution :::::::::::::::::::::: 10 2.5 3-Parameter Lognormal Distribution ::::::::::::: 11 2.6 Derivation of the Black-Scholes Model ::::::::::::: 11 3 Test and Results for Normality ::::::::::::::::::::: 17 3.1 Data and Software ::::::::::::::::::::::: 17 3.2 Decriptive Statistics ::::::::::::::::::::::: 19 3.3 Procedures :::::::::::::::::::::::::::: 19 3.4 Results of Normality :::::::::::::::::::::: 23 3.4.1 Five-year Interval ::::::::::::::::::::::: 24 3.4.2 One-Year Intervals :::::::::::::::::::::: 25 vi 3.4.3 Six-Month Intervals :::::::::::::::::::::: 27 3.4.4 Three-Month Intervals :::::::::::::::::::: 28 3.4.5 One-Month Intervals ::::::::::::::::::::: 30 3.4.6 Conclusion of Normality Examination ::::::::::: 31 4 Simulations :::::::::::::::::::::::::::::::: 32 4.1 The Models ::::::::::::::::::::::::::: 32 4.2 Simulations and Analysis :::::::::::::::::::: 34 5 Results :::::::::::::::::::::::::::::::::: 42 :::::::::::::::::::::::::::::::::::::::::::: 44 A Theorems and Definitions :::::::::::::::::::::::: 47 B Tables :::::::::::::::::::::::::::::::::: 48 C Graphs :::::::::::::::::::::::::::::::::: 66 C.1 Daily Closing Prices of Indexes Over 5-Year Interval ::::: 66 C.2 5-Year Interval :::::::::::::::::::::::::: 68 C.3 1-Year Interval :::::::::::::::::::::::::: 70 C.4 6-Month Intervals :::::::::::::::::::::::: 75 C.5 3-Month Intervals :::::::::::::::::::::::: 85 C.6 1-Month Intervals :::::::::::::::::::::::: 90 D Matlab code ::::::::::::::::::::::::::::::: 102 vii LIST OF TABLES Table Page 4.1 Table for S0 and K ::::::::::::::::::::::::::: 37 4.2 Table for S0 and K for the S&P 500 Index :::::::::::::: 40 B.1 Company and Index Names with Ticker Symbols ::::::::::: 48 B.2 Descriptive Statistics for Logged Data on Entire Interval ::::::: 49 B.3 Descriptive Statistics for Logged Data on 1-Year Intervals ::::: 49 B.4 Descriptive Statistics for Logged Data on 6-Month Intervals :::: 50 B.5 Descriptive Statistics for Logged Data on 3-Month Intervals ::::: 50 B.6 Descriptive Statistics for Logged Data on 1-Month Intervals ::::: 52 B.7 Anderson-Darling Values (Note: * represents AD values with pvalues less than .05 and ** represents AD vales with pvalues less than .01) 55 B.8 Option Estimates for 30-day option using volatility from BAC Oct 08 with 95% Confidence Interval and Standard Deviation ::::::: 61 B.9 Option Estimates for 30-day option using volatility from BAC 5 year with 95% Confidence Interval and Standard Deviation ::::::: 62 B.10 Option Estimates for 30-day option using volatility from T Oct 08 with 95% Confidence Interval and Standard Deviation ::::::: 63 B.11 Option Estimates for 30-day option using volatility from KO Dec 07 with 95% Confidence Interval and Standard Deviation ::::::: 64 B.12 Option Estimates for 30-day option using volatility from SP Oct 08 with 95% Confidence Interval and Standard Deviation ::::::: 65 viii LIST OF FIGURES Figure Page 3.1 BAC over 5-year interval :::::::::::::::::::::::: 21 3.2 3-parameter lognormal probability plot of BAC,KO, T, S&P, Wilshire over 5-year interval :::::::::::::::::::::::::: 23 4.1 30-day option using normal distribution w/ 10 paths ::::::::: 36 C.1 DJIA over 5-year interval :::::::::::::::::::::::: 66 C.2 Nasdaq over 5-year interval ::::::::::::::::::::::: 66 C.3 S&P 500 over 5-year interval :::::::::::::::::::::: 67 C.4 Normal probability plots over 5-year interval with a 95% confidence interval ::::::::::::::::::::::::::::::::: 68 C.5 Student-t probability plots over 5-year interval :::::::::::: 68 C.6 Cauchy probability plots over 5-year interval ::::::::::::: 68 C.7 Weibull probability plots over 5-year interval with a 95% confidence interval ::::::::::::::::::::::::::::::::: 69 C.8 3-parameter lognormal probability plots over 5-year interval with a 95% confidence interval :::::::::::::::::::::::: 69 C.9 Normal probability plots over 1-year intervals with a 95% confidence interval ::::::::::::::::::::::::::::::::: 70 C.10 Student-t probability plot over 1-year intervals :::::::::::: 71 C.11 Cauchy probability plot over 1-year intervals ::::::::::::: 72 ix C.12 Weibull probability plots over 1-year intervals with a 95% confidence interval ::::::::::::::::::::::::::::::::: 73 C.13 3-Parameter Lognormal probability plots over 1-year intervals with a 95% confidence interval :::::::::::::::::::::::: 74 C.14 Normal probability plots over 6-month intervals with a 95% confidence interval ::::::::::::::::::::::::::::::::: 75 C.15 Normal probability plots over 6-month intervals with a 95% confidence interval (Cont'd) :::::::::::::::::::::::::::: 76 C.16 Student-t probability plots over 6-month intervals with a 95% confidence interval ::::::::::::::::::::::::::: 77 C.17 Student-t probability plots over 6-month intervals :::::::::: 78 C.18 Cauchy probability plots over 6-month intervals ::::::::::: 79 C.19 Cauchy probability plots over 6-month intervals (Cont'd) :::::: 80 C.20 Weibull probability plots over 6-month intervals with a 95% confidence interval ::::::::::::::::::::::::::::::::: 81 C.21 Weibull probability plots over 6-month intervals with a 95% confidence interval (Cont'd) :::::::::::::::::::::::::::: 82 C.22 3-parameter lognormal probability plots over 6-month intervals with a 95% confidence interval ::::::::::::::::::::::: 83 C.23 3-parameter lognormal probability plots over 6-month intervals with a 95% confidence interval (Cont'd) :::::::::::::::::: 84 C.24 Normal probability plots over 3-month intervals with a 95% confidence interval ::::::::::::::::::::::::::::::::: 85 C.25 Student-t probability plots over 3-month intervals :::::::::: 86 C.26 Cauchy probability plots over 3-month intervals ::::::::::: 87 C.27 Weibull probability plots over 3-month intervals with a 95% confidence interval ::::::::::::::::::::::::::::::::: 88 x C.28 3-parameter lognormal probability plots over 3-month intervals with a 95% confidence interval ::::::::::::::::::::::: 89 C.29 Normal probability plots over 1-month intervals with a 95% confidence interval :::::::::::::::::::::::::::::::::