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The Black-Scholes Formula and Volatility Smile University of Louisville ThinkIR: The University of Louisville's Institutional Repository Electronic Theses and Dissertations 5-2012 The Black-Scholes formula and volatility smile. Brian Michael Butler 1969- University of Louisville Follow this and additional works at: https://ir.library.louisville.edu/etd Recommended Citation Butler, Brian Michael 1969-, "The Black-Scholes formula and volatility smile." (2012). Electronic Theses and Dissertations. Paper 188. https://doi.org/10.18297/etd/188 This Master's Thesis is brought to you for free and open access by ThinkIR: The University of Louisville's Institutional Repository. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of ThinkIR: The University of Louisville's Institutional Repository. This title appears here courtesy of the author, who has retained all other copyrights. For more information, please contact [email protected]. THE BLACK-SCHOLES FORMULA AND VOLATILITY SMILE By Brian Michael Butler B.A., Humboldt State University, 1993 A Thesis Submitted to the Faculty of the College of Arts and Sciences of the University of Louisville in Partial Fulfillment of the Requirements for the Degree of Master of Arts Department of Mathematics University of Louisville Louisville, Kentucky May 2012 THE BLACK-SCHOLES FORMULA AND VOLATILITY SMILE By Brian Michael Butler B.A, Humboldt State University, 1993 A Thesis Approved on April 23, 2012 by the following Thesis Committee: Ewa Kubicka, Thesis Director Ryan Gill ii DEDICATION This thesis is dedicated to my wife Kelly Estep whose loving support and encouragement guided me onward in my education, and to our beautiful children, Aidan and Lily, who we hope to provide with the educational opportunities we received from our parents. Also to my Great Aunt and Godmother, Hilda, who in her usual gentle and loving way encouraged me to persevere, and now it is done. III ACKNOWLEDGEMENTS The author would like to extend thanks to Professor Ewa Kubicka for suggesting this topic and providing the opportunity to explore it. Her knowledge and assistance were a great aid in the preparation of this work. IV ABSTRACT THE BLACK-SCHOLES FORMULA AND VOLATILITY SMILE Brian M. Butler April 23, 2012 This paper investigates the development and applications of the Black-Scholes formula. This well-known formula is a continuous time model used primarily to price European style options. However in recent decades, observations in financial market data have brought into question some of the basic assumptions that the model relies on. Of particular interest is the prevalence of the volatility smile in asset option prices. This is a violation of one ofthe key assumptions under this model, and as a result alternatives to and modifications of Black-Scholes have been suggested, some continuous and some discrete. This paper researches one such modification, proposed by Derman and Kani (1994), in which observed market data is used to create a discrete time implied asset price tree that correctly reflects changing volatilities, risk-neutral probabilities, and observed option prices. The results are then used to price a less conventional derivative arrangement. v TABLE OF CONTENTS PAGE ABSTRACT ...................................................................................................................... v LIST OF TABLES ............................................................................................................ vi LIST OF FIGURES .......................................................................................................... vii INTRODUCTION ............................................................................................................ 1 BACKGROUND .............................................................................................................. 4 Options ........................................................................................................................ 4 Risk ............................................................................................................................. 7 TRADING STRA TEGIES ................................................................................................ 9 Arbitrage ..................................................................................................................... 9 Market Efficiency ....................................................................................................... 10 Put-Call Parity ............................................................................................................. 11 THE PRICE PROCESS .................................................................................................... 14 The Memoryless Property and Stochastic Processes .................................................. 14 Random Walks and the Wiener Process ..................................................................... 15 The Black-Scholes Pricing Formula ........................................................................... 18 IMPLIED VOLA TIL TY IN DETAIL .............................................................................. 30 REFERENCES ................................................................................................................. 64 APPENDIX ....................................................................................................................... 66 CURRICULUM VITAE ................................................................................................... 68 VI LIST OF TABLES TABLE PAGE TABLE 4.1 ......................................................................................................................... 21 TABLE 4.2 ......................................................................................................................... 22 TABLE 5.13 ....................................................................................................................... 37 vii LIST OF FIGURES FIGURE PAGE Figure 4.1 ........................................................................................................................... 24 Figure 4.2 ........................................................................................................................... 24 Figure 4.3 ........................................................................................................................... 24 Figure 4.4 ........................................................................................................................... 25 Figure 4.5 ........................................................................................................................... 25 Figure 4.6 ........................................................................................................................... 25 Figure 4.7 ........................................................................................................................... 26 Figure 4.8 ........................................................................................................................... 26 Figure 4.9 ........................................................................................................................... 26 Figure 4.1 0 ......................................................................................................................... 27 Figure 4.11 ......................................................................................................................... 27 Figure 4.12 ......................................................................................................................... 27 Figure 4.13 ......................................................................................................................... 28 Figure 4.14 ......................................................................................................................... 28 Figure 4.15 ......................................................................................................................... 28 Figure 5.1 ........................................................................................................................... 37 Figure 5.2 ........................................................................................................................... 40 Figure 5.3 ........................................................................................................................... 41 Figure 5.4 ........................................................................................................................... 42 viii Figure 5.5 ........................................................................................................................... 44 Figure 5.6 ........................................................................................................................... 46 Figure 5.7 ........................................................................................................................... 48 Figure 5.8 ........................................................................................................................... 48 Figure 5.9 ........................................................................................................................... 54 Figure 5.1 0 ......................................................................................................................... 56 Figure 5.11 ........................................................................................................................
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