PATH-DEPENDENT OPTION PRICING: EFFICIENT METHODS …

PATH-DEPENDENT OPTION PRICING: EFFICIENT METHODS FOR LE? VY MODELS

By GUDBJORT GYLFADOTTIR

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010

c 2010 Gudbjort Gylfadottir 2

To my three favorite guys: My husband, A? rni; my dad, Gylfi; and my brother, Thro? stur 3

ACKNOWLEDGMENTS None of this would have been possible without my advisor and friend Farid. His enthusiasm and ambition inspired me and I am thankful for all the knowledge he shared with me during countless hours. Murali, of my committee, taught me a great deal of math, yet I am most appreciative for getting to know him as a friend. I would like to thank both Murali and Farid for all our inspirational conversations. Also, I would like to thank my committee members, Dr. Pardalos and Dr. Nimalendran for their support. Thanks go out to my family for their love, all the phone calls and for their lovely visits: My dad Gylfi, my brother Thro? stur, my sister-in-law Una, my nephews Thorri and Fro? di and my parents-in-law Erna and Jo? n. Also, thanks go out to my friends: Alex, Mireia, Ehsan, Kelly, Vera, Altannar, Ashwin, Shantih, Emily, May, Soheil, Behnam, Renee, Clay, Filip, Unnur, Helga Bjo? rk, Helga Bjo? rk, Svanhv?it, El?in, Anna Gyda, Ragnheidur, Lara and Jacki. Florida's nature with all its magical wonders made being here an amazing experience. And lastly, my deepest gratitude goes to A? rni, who is the most loving husband I could wish for and has been here for me all of this time. I am really grateful that we got to share this experience.

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TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

CHAPTER

1 PATH-DEPENDENT OPTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2 Asian Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Lookback Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 LE? VY PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1 Motivation for Le? vy Pricing Models . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Using Le? vy Pricing Models . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 The Fast Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 QUANTILE APPROXIMATIONS FOR ASIAN OPTIONS . . . . . . . . . . . . . 29

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Quantile Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 Distributions for Discrete Quantile Processes . . . . . . . . . . . . . . . . 32 3.4 Quantile Approximations for Fixed Strike Asian Options . . . . . . . . . . 33 3.5 Pricing in the Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . 36 3.6 Hedging Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.7 Numerical Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 PRICING OF LOOKBACK OPTIONS USING LE? VY PROCESSES . . . . . . . 43

4.1 Lookback Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 Duality and Extrema of Random Walks . . . . . . . . . . . . . . . . . . . 44 4.3 Fixed-Strike Lookback Options . . . . . . . . . . . . . . . . . . . . . . . . 48 4.4 Floating-Strike Lookback Options . . . . . . . . . . . . . . . . . . . . . . . 49 4.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

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