Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2009 Monte Carlo and Quasi-Monte Carlo Methods in Financial Derivative Pricing Ahmet Göncü Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES MONTE CARLO AND QUASI-MONTE CARLO METHODS IN FINANCIAL DERIVATIVE PRICING By AHMET GO¨NCU¨ A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Summer Semester, 2009 The members of the Committee approve the Dissertation of Ahmet G¨oncu¨ defended on June 17, 2009. Giray O¨kten Professor Directing Dissertation Fred Huffer Outside Committee Member Brian Ewald Committee Member Alec N. Kercheval Committee Member Michael Mascagni Committee Member The Graduate School has verified and approved the above named committee members. ii I dedicate this thesis to my parents; Nail G¨oncu¨, Dilek G¨oncu¨ and to my sister Sinem G¨oncu¨, for their unchanging support and confidence in me. Without their innumerable sacrifices and tolerance none of this would have been possible. Thank you. iii ACKNOWLEDGEMENTS First, I would like to acknowledge my gratitude to my advisor Dr. Giray O¨kten for his supportandunderstanding allthetime, hisguidanceandmotivationplayed avery important role in my success. I would also like to thank to my committee members; Dr. Michael Mascagni, Dr. Alec Kercheval, Dr. Brian Ewald, and Dr. Fred Huffer for their valuable advice. I am also grateful to Dr. Paul Beaumont for his valuable advice and support in my academic life. Last but not least, I am also grateful to my dear friends; Ahmet Emin Tatar, Dilek Du¨steg¨or, Yangho Park, and Emmanuel Salta for their continuous support during my study. iv TABLE OF CONTENTS List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. INTRODUCTION TO MONTE CARLO METHODS . . . . . . . . . . . . . 5 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Generating Pseudorandom Numbers . . . . . . . . . . . . . . . . . . . . 7 2.3 Statistical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Random Variate Generation . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Generating Brownian Motion Paths . . . . . . . . . . . . . . . . . . . . . 18 3. QUASI MONTE CARLO (QMC) METHODS . . . . . . . . . . . . . . . . . . 25 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Low Discrepancy Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Randomized Quasi-Monte Carlo (RQMC) . . . . . . . . . . . . . . . . . 37 4. SENSITIVITY TO NORMAL TRANSFORMATION METHODS IN QMC . 42 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Derivative Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3 Why Is Box-Muller A Good Alternative? . . . . . . . . . . . . . . . . . . 57 4.4 The Collision Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5. ERROR BOUNDS IN QMC INTEGRATION . . . . . . . . . . . . . . . . . . 63 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2 QMC Error Bounds For Pricing Financial Derivatives . . . . . . . . . . . 64 5.3 Error Bounds Under A General Probability Space . . . . . . . . . . . . . 68 5.4 Box-Muller Transformed Point Sets . . . . . . . . . . . . . . . . . . . . . 74 5.5 Stratified Box-Muller Method . . . . . . . . . . . . . . . . . . . . . . . . 83 5.6 Applications From Finance . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 v 6. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 vi LIST OF TABLES 2.1 Chi-Square Test Results for Mersenne Twister and TT800 . . . . . . . . . . 10 2.2 Kolmogorov Smirnov Test Results for Mersenne Twister and TT800 . . . . . 12 4.1 Ratchet Option: Inverse Transformation vs. Box-Muller . . . . . . . . . . . . 44 4.2 Ratchet Option RMSE: Inverse Transformation vs. Box-Muller . . . . . . . . 48 4.3 Asian Geometric Call Option: Inverse Transformation vs. Box-Muller . . . . 50 4.4 Asian Geometric Option RMSE: Inverse Transformation vs. Box-Muller . . . 51 4.5 Barrier Option: Inverse Transformation vs. Box-Muller . . . . . . . . . . . . 53 4.6 Basket Option RMSE: Inverse Transformation vs. Box-Muller . . . . . . . . 53 4.7 CMO Problem: Inverse Transformation vs. Box-Muller . . . . . . . . . . . . 55 4.8 Anderson-Darling values for the squared radii . . . . . . . . . . . . . . . . . 60 4.9 Collision Test Results: Polar rectangles. . . . . . . . . . . . . . . . . . . . . 62 4.10 Collision Test Results: Cartesian Rectangles. . . . . . . . . . . . . . . . . . . 62 5.1 Collision Test for Chi distributed radii values . . . . . . . . . . . . . . . . . . 88 5.2 Construction of a ( ,χ )-uniform point set . . . . . . . . . . . . . . . . . . 89 d M vii LIST OF FIGURES 2.1 Two dimensional plot of pseudorandom numbers . . . . . . . . . . . . . . . . 8 2.2 Example of generating Brownian motion paths . . . . . . . . . . . . . . . . . 22 2.3 Standard vs. Brownian bridge construction methods . . . . . . . . . . . . . . 24 3.1 Two dimensional projection of dimensions 30 and 40 . . . . . . . . . . . . . 29 3.2 An example of a (0,3,2)-net . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Random Shifting applied to a (0,3,2)-net . . . . . . . . . . . . . . . . . . . . 39 3.4 Random Permutation of Digits . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.1 Ratchet option with the inverse transformation method . . . . . . . . . . . . 45 4.2 Ratchet option with the Box-Muller method . . . . . . . . . . . . . . . . . . 46 4.3 Relative error plot for the ratchet option . . . . . . . . . . . . . . . . . . . . 47 4.4 Ratchet option estimates obtained from scrambled Faure sequences . . . . . 48 4.5 Relative error plot for the Asian geometric call option . . . . . . . . . . . . . 49 4.6 Asian option estimates obtained from scrambled Faure sequences . . . . . . . 50 4.7 Relative error plot for the barrier option . . . . . . . . . . . . . . . . . . . . 52 4.8 Basket option estimates obtained from scrambled Faure sequences . . . . . . 54 4.9 CMO estimates using mixed-Sobol’ sequence . . . . . . . . . . . . . . . . . . 55 4.10 Skipping the Sobol’ sequence. Barrier option. . . . . . . . . . . . . . . . . . 56 4.11 (0,3,2) net in base 2 with elementary intervals . . . . . . . . . . . . . . . . 57 − 4.12 Polar and Cartesian rectangles with transformed points . . . . . . . . . . . . 58 5.1 Asian geometric option example: numerical convergence rate . . . . . . . . . 72 viii 5.2 Partition based on radius of points: Inverse transformation vs. Box-Muller . 80 5.3 Empirical distribution of r2 under different transformation methods (4-d) . . 84 5.4 Empirical distribution of r2 under different transformation methods (20-d) . 85 5.5 Empirical distribution of r2 under different transformation methods (40-d) . 86 5.6 Integration of f( x ) = max(K e x ,0) (16-d and 40-d) . . . . . . . . . . . 90 || || || || − 5.7 Integration of cos( x ) given in Equation 5.42 (25-d and 60-d). . . . . . . . . 91 || || 5.8 Asian Geometric option pricing using 8-d Faure Sequence . . . . . . . . . . . 92 5.9 Asian Geometric option pricing using 20-d Faure Sequence . . . . . . . . . . 93 5.10 VAR for 2 stocks under normal model . . . . . . . . . . . . . . . . . . . . . . 94 5.11 VAR for 10 stocks under normal model . . . . . . . . . . . . . . . . . . . . . 95 5.12 VAR for 20 stocks under normal model . . . . . . . . . . . . . . . . . . . . . 96 ix