Monte Carlo and Quasi-Monte Carlo Methods in Financial Derivative Pricing

116 Pages·2015·1.18 MB·English

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
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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.
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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.
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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.
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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
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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
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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
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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
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5.7 Integration of cos( x ) given in Equation 5.42 (25-d and 60-d). . . . . . . . . 91
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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
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