Option pricing using path integrals

Option pricing using path integrals

by

Dr. Frederic D.R. Bonnet

B.Sc. (Mathematical and Computer Science with Honours), 1998.Ph.D. in Science (Theoretical and Astrophysics), 2002.

The University of Adelaide.

Thesis submitted for the degree of

Doctor of Philosophy

in

School of Electrical and Electronic Engineering

Faculty of Engineering, Computer and Mathematical Sciences

University of Adelaide, Australia

July, 2008

c© 2008

Dr. Frederic D.R. Bonnet

All Rights Reserved

Contents

Contents iii

Abstract xi

Statement of Originality xiii

Acknowledgments xv

Conventions and formsetting xvii

0.1 Typesetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

0.2 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

0.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

0.4 Mathematical Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

Publications xix

List of Figures xxi

List of Tables xxv

Chapter 1. Introduction 1

1.1 History of Financial Markets . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.1 Bachelier Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.2 History of Brownian Motion . . . . . . . . . . . . . . . . . . . . . 8

1.1.3 History of Stochastic Calculus . . . . . . . . . . . . . . . . . . . . 10

1.2 The Various Techniques Used in Finance . . . . . . . . . . . . . . . . . . . 11

1.2.1 Binomial Tree Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.2 Econometrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.3 Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.4 Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4 Statement of Original Contributions . . . . . . . . . . . . . . . . . . . . . 24

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Contents

Chapter 2. Distributions and Time series processes 27

2.1 The Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Non-Gaussian Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.1 Student t–Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.2 General Error Distribution . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.3 The Generalized Hyperbolic Levy Motion Distribution, GH(x) . 32

2.2.4 The Hyperbolic Distribution, H(x) . . . . . . . . . . . . . . . . . . 34

2.2.5 The Normal Inverse Gaussian Distribution, NIG(x) . . . . . . . . 35

2.3 Linear Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3.1 The AR(p) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3.2 The ARMA(p, q) Model . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3.3 The ARCH(p) Model . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3.4 The Linear GARCH(p, q) Model . . . . . . . . . . . . . . . . . . . 39

2.3.5 The GARCH(1, 1)− NIG(x) Model . . . . . . . . . . . . . . . . . 43

2.4 Non–Linear GARCH(p, q) Processes . . . . . . . . . . . . . . . . . . . . . 45

2.4.1 The AGARCH− I(p, q) Model . . . . . . . . . . . . . . . . . . . . 49

2.4.2 The AGARCH− II(p, q) Model . . . . . . . . . . . . . . . . . . . . 51

2.4.3 The GJR − GARCH(p, q) Model . . . . . . . . . . . . . . . . . . . 51

2.4.4 The EGARCH(p, q) Model . . . . . . . . . . . . . . . . . . . . . . 52

2.5 The Maximum Likelihood Method . . . . . . . . . . . . . . . . . . . . . . 52

2.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Chapter 3. Brownian Motion 55

3.1 Random Walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.1.1 Symmetric Random Walks . . . . . . . . . . . . . . . . . . . . . . 56

3.1.2 Scaled Symmetric Random Walks . . . . . . . . . . . . . . . . . . 58

3.2 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2.1 Filtration for Brownian Motion . . . . . . . . . . . . . . . . . . . . 62

3.2.2 Quadratic Variation . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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Contents

Chapter 4. Stochastic Calculus 69

4.1 Ito Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.1.1 Ito Integral for Simple Process . . . . . . . . . . . . . . . . . . . . . 70

4.1.2 Ito Integral for non Simple Process . . . . . . . . . . . . . . . . . . 73

4.1.3 Ito –Doeblin Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1.4 Multivariate Stochastic Calculus . . . . . . . . . . . . . . . . . . . 77

4.2 Discretization Methods (SDE Numerical Approaches) . . . . . . . . . . . 81

4.2.1 The Euler–Marayama Scheme, γ = 0.5 . . . . . . . . . . . . . . . . 84

4.2.2 The Milstein Scheme, γ = 1.0 . . . . . . . . . . . . . . . . . . . . . 85

4.2.3 The Order γ = 1.5 Strong Taylor Scheme . . . . . . . . . . . . . . 87

4.2.4 The Runge–Kutta Scheme a Strong Order γ = 1 Scheme . . . . . 92

4.3 Jump Process and Levy Processes . . . . . . . . . . . . . . . . . . . . . . . 95

4.4 Stochastic Volatility Models . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.4.1 Mean Reverting Stochastic Volatility Models . . . . . . . . . . . . 95

4.4.2 Pricing With Equivalent Martingale Measure . . . . . . . . . . . . 100

4.5 Connection With Partial Differential Equations . . . . . . . . . . . . . . . 103

4.5.1 The Feynman-Kac Formula . . . . . . . . . . . . . . . . . . . . . . 105

4.5.2 The Forward Kolmogorov Equation (The Fokker–Planck Equation)107

4.5.3 The Backward Kolmogorov Equation . . . . . . . . . . . . . . . . 109

4.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Chapter 5. Option pricing and derivatives 111

5.1 Some Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.2 European Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.2.1 The Black–Scholes–Merton Model . . . . . . . . . . . . . . . . . . 115

5.3 Exotic Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.3.1 Knock In/Out Barrier Option . . . . . . . . . . . . . . . . . . . . . 125

5.3.2 Lookback Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.3.3 Asian Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.4 American Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

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Contents

Chapter 6. Path integrals in finance 133

6.1 The Path Integral in Quantum Mechanics . . . . . . . . . . . . . . . . . . 134

6.2 The Path Integral in Quantum Field Theory . . . . . . . . . . . . . . . . . 137

6.3 The Path Integral in Finance . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.3.1 The Correct Formulation of the Lagrangian Density . . . . . . . . 140

6.3.2 Standard Gaussian Path Integrals . . . . . . . . . . . . . . . . . . 144

6.3.3 Non–Standard Gaussian Path Integrals . . . . . . . . . . . . . . . 148

6.3.4 The Multifractal Random Walk Model . . . . . . . . . . . . . . . . 154

6.3.5 The Path Integral for the Multifractal Random Walk Model . . . . 157

6.4 The Instanton Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.4.1 The Instanton Method for the Non-Gaussian Model . . . . . . . . 161

6.5 Numerical Approach to the Evaluation of the Path Integral . . . . . . . . 173

6.5.1 Discretisation of the Path Integral . . . . . . . . . . . . . . . . . . 173

6.5.2 Evaluating the Path Intgeral for N = 2 . . . . . . . . . . . . . . . . 180

6.5.3 Maple Output for the Path Intgeral When N = 3 . . . . . . . . . . 181

6.5.4 Monte Carlo Methods for the Path Integral . . . . . . . . . . . . . 182

6.5.5 Pertubation Theory for the Path Integral . . . . . . . . . . . . . . 183

6.5.6 Variational Methods for the Path Integral . . . . . . . . . . . . . . 183

6.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Chapter 7. Agent models 187

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

7.2 The Minority Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

7.2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

7.2.2 The Price Function in the Minority Game . . . . . . . . . . . . . . 196

7.2.3 The Dollar Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

7.3 Financial Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

7.3.1 Positive Feedback Model With Multiplicative Noise . . . . . . . . 203

7.4 Minority Game and Dollar Game Price Function With Real Data . . . . . 206

7.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

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Contents

Chapter 8. Conclusion 215

8.1 What is the Best Method so Far? . . . . . . . . . . . . . . . . . . . . . . . 216

8.2 Open Questions for Future Work . . . . . . . . . . . . . . . . . . . . . . . 220

8.3 Summary of Original Contributions . . . . . . . . . . . . . . . . . . . . . 222

Appendix A. Mathematical concepts 225

A.1 Probability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

A.1.1 Random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

A.1.2 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

A.2 Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

A.3 Stochastic calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

A.3.1 Girsanov theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

A.4 Stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

A.4.1 Markov processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

A.5 The Chapman–Kolmogorov equation . . . . . . . . . . . . . . . . . . . . 229

A.6 Dyson series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

A.7 Integral properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

A.7.1 Normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 230

A.7.2 Gaussian identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

A.7.3 Integral identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

A.7.4 Gamma function properties . . . . . . . . . . . . . . . . . . . . . . 231

A.7.5 The Di–Gamma function definition . . . . . . . . . . . . . . . . . 231

A.8 Generating random numbers . . . . . . . . . . . . . . . . . . . . . . . . . 231

A.8.1 Generating Uniformly Distributed Random Numbers . . . . . . . 231

A.8.2 Random Numbers with a Gaussian Probability Distribution Func-

tion, the Box–Muller random number generation . . . . . . . . . 232

Appendix B. Partial differential equations 235

B.1 Partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . 236

B.1.1 Classification partial differential equations . . . . . . . . . . . . . 236

Appendix C. Option pricing details 237

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C.1 More Different Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

C.1.1 Non-Vanilla Exercise Rights . . . . . . . . . . . . . . . . . . . . . . 238

C.1.2 ’Exotic’ Options with Standard Exercise Styles . . . . . . . . . . . 239

C.1.3 Non-vanilla path dependent “exotic” options . . . . . . . . . . . . 240

C.2 Maximum of Brownian motion with drift . . . . . . . . . . . . . . . . . . 242

Appendix D. The Fokker–Planck Equation 245

D.1 The Sturm–Liouville equation . . . . . . . . . . . . . . . . . . . . . . . . . 246

D.2 The connection between the Schroedinger equation and Fokker–Planck

equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

D.3 The wave function and the probability density in quantum mechanics . 253

Appendix E. Source code 255

E.1 The R script nasdaq.r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

E.2 The Perl script strip yahoo.pl . . . . . . . . . . . . . . . . . . . . . . . . . . 264

E.3 The discretization methods . . . . . . . . . . . . . . . . . . . . . . . . . . 270

E.3.1 The headers file for the discretization code . . . . . . . . . . . . . 270

E.3.2 The main code for the discretization code Sde main. f 90 . . . . . . 271

E.3.3 The subroutine dri f t f unc. f 90 . . . . . . . . . . . . . . . . . . . . 286

E.3.4 The subroutine di f f usion f unc. f 90 . . . . . . . . . . . . . . . . . . 287

E.3.5 The subroutine wiener. f 90 . . . . . . . . . . . . . . . . . . . . . . . 287

E.3.6 The subroutine explicit sol. f 90 . . . . . . . . . . . . . . . . . . . . 289

E.3.7 The subroutine euler. f 90 . . . . . . . . . . . . . . . . . . . . . . . . 291

E.3.8 The subroutine milstein. f 90 . . . . . . . . . . . . . . . . . . . . . . 292

E.3.9 The subroutine d f ridr 1. f 90 . . . . . . . . . . . . . . . . . . . . . . 293

E.3.10 The subroutine strg 15 taylor. f 90 . . . . . . . . . . . . . . . . . . . 294

E.3.11 The subroutine d f ridr. f 90 . . . . . . . . . . . . . . . . . . . . . . . 296

E.3.12 The subroutine gaussian. f 90 . . . . . . . . . . . . . . . . . . . . . 297

E.4 The routines used in the main for different distributions and PDF . . . . 299

E.4.1 The subroutine Gen hyper geo(x, a, y, dyda, delta), GH(x) . . . . . 300

E.4.2 The subroutine hyper geo(x, a, y, dyda, delta), H(x) . . . . . . . . . 304

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Contents

E.4.3 The subroutine hist vol, the historical volatility . . . . . . . . . . . 305

E.4.4 The subroutine for the Bessel function . . . . . . . . . . . . . . . . 307

E.4.5 The subroutine student(x, a, y, dyda, delta), f (x) . . . . . . . . . . 311

E.4.6 The subroutine gauss(x, a, y, dyda, delta), f (x) . . . . . . . . . . . 314

E.4.7 The subroutine f unc, f unc test v and f unc improper v test functions314

E.4.8 The subroutine get pd f () . . . . . . . . . . . . . . . . . . . . . . . . 315

E.4.9 The subroutine f it pd f . f 90 . . . . . . . . . . . . . . . . . . . . . . 318

E.4.10 The subroutine mrqmin. f 90 . . . . . . . . . . . . . . . . . . . . . . 320

E.4.11 The subroutine qromo md. f 90 . . . . . . . . . . . . . . . . . . . . . 322

E.4.12 The subroutine midexp md. f 90 . . . . . . . . . . . . . . . . . . . . 322

E.4.13 The subroutine trapzd. f 90 . . . . . . . . . . . . . . . . . . . . . . . 323

E.5 Some functions used in the evaluation of functions and options . . . . . 324

E.5.1 The subroutine black scholes, the Black–Scholes model and the

Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

E.5.2 The cum norm(x) function . . . . . . . . . . . . . . . . . . . . . . . 325

E.5.3 The Er f (x) function . . . . . . . . . . . . . . . . . . . . . . . . . . 325

E.5.4 The Er f c(x) function . . . . . . . . . . . . . . . . . . . . . . . . . . 326

E.5.5 The log(Γ(x)) function . . . . . . . . . . . . . . . . . . . . . . . . . 326

E.6 The minority game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

E.6.1 The headers file for the minority game code . . . . . . . . . . . . 327

E.6.2 The minority source code . . . . . . . . . . . . . . . . . . . . . . . 328

E.6.3 The routine that initialises the real data . . . . . . . . . . . . . . . 338

E.6.4 The routine real. f 90 . . . . . . . . . . . . . . . . . . . . . . . . . . 342

E.7 The code that calculates the Bubble in section 7.3 . . . . . . . . . . . . . . 345

Appendix F. Maple output for the non–Gaussian model 347

F.0.1 Maple output for the path intgeral when N = 2 . . . . . . . . . . 348

F.0.2 Maple output for the path intgeral when N = 3 . . . . . . . . . . 349

Bibliography 355

Index 367

Page ix

Contents

Biography 369

Scientific genealogy 371

Page x

Abstract

It is well established that stock market volatility has a memory of the past, moreover

it is found that volatility correlations are long ranged. As a consequence, volatility

cannot be characterized by a single correlation time in general. Recent empirical work

suggests that the volatility correlation functions of various assets actually decay as a

power law. Moreover it is well established that the distribution functions for the re-

turns do not obey a Gaussian distribution, but follow more the type of distributions

that incorporate what are commonly known as fat–tailed distributions. As a result, if

one is to model the evolution of the stock price, stock market or any financial deriva-

tive, then standard Brownian motion models are inaccurate. One must take into ac-

count the results obtained from empirical studies and work with models that include

realistic features observed on the market.

In this thesis we show that it is possible to derive the path integral for a non-Gaussian

option pricing model that can capture fat–tails. However we find that the path integral

technique can only be used on a very small set of problems, as a number of situations

of interest are shown to be intractable.

Page xi

Page xii

Statement of Originality

This work contains no material that has been accepted for the award of any other de-

gree or diploma in any university or other tertiary institution and, to the best of my

knowledge and belief, contains no material previously published or written by another

person, except where due reference has been made in the text.

I give consent to this copy of the thesis, when deposited in the University Library, being

available for loan, photocopying, and dissemination through the library digital thesis

collection, subject to the provisions of the Copyright Act 1968. Copying or publication

or use of this thesis or parts thereof for financial gain is not allowed without the authors

written permission. Due recognition shall be given the author, and the University of

Adelaide, in any scholarly use that may be made of any material in the thesis.

Signed Date

Page xiii

Page xiv

Acknowledgments

I would like to acknowledge the support that my supervisors have given me along

the course of the program. In particular I would like to thank Professor Derek Abbott

for all the financial support that he has provided me along the years that I have been

part of the Electrical and Electronic Engineering school. I would like to also take the

opportunity to thank Andrew Allison for taking the time to listen and for all of his

constructive comments. Furthermore I would like to thank John van der Hoeck for all

the support he has given me with some of the mathematical aspects.

A special thanks to my family members without whom things would not have been

possible.

I would like acknowledge the School of Electrical and Electronic Engineering and the

Australian Research Council for their financial support. I also acknowledge the SPIE

travel grant that covered some of the travel expenses.

Special thanks to Carl Chiarella (University of Technology of Sydney), Lisa Borland

(Evnine-Vaughan Associates), Jean–Philippe Bouchaud (Capital Fund Management),

Didier Sornette (Universite Nice Sophia–Antipolis) who kindly hosted my short visits

to their research labs.

I would also like to take the opportunity to thank some of the researchers who patiently

took some of their time to listen through some of the technical discussions, and for

useful and stimulating discussions, these include Robert Elliot (University of Calgary),

Dominique Guegan (Universite Pantheon-Sorbonne), and Peter Hanggi (Universitat

Augsburg). Also acknowledged is Michel M. Dacorogna for finding the paper on the

origin of the term martingale (Mansuy 2005).

Finally I would like to thank Professor Bevan Bates for his constructive advice and

support. I would like to thank the research committee as a whole for their help and

advice.

At last I would like to thank Rose-Marie for organizing the weekly lunch meetings,

which provided nice lunch breaks.

Page xv

Page xvi

Conventions andformsetting

0.1 Typesetting

This thesis it typeset using the LATEX2e software. WinEdt build 5.4 was used as an

effective interface to LATEX (Oetiker et al. 2000). Harvard style is used for referencing

and citation in this thesis. Australian English spelling is adopted, as defined by the

Macquarie English Dictionary (Delbridge 2001).

0.2 Software

The results were obtained using mathematical packages (Mathematica, Matlab), statis-

tical packages (R).

The programming language used was Fortran 90 with parallel commands embedded

in it. Also used was C++.

The reason being that it is more realistic in our days to have a wide range of computer

skills instead of just a highly specialized one which may not be compatible elsewhere.

0.3 Data

The data was obtained from two different sources

• The yahoo finance website (Yahoo Finance 2008).

• The Tick Data website (Tick Data Global Historical Data Solutions 2008).

Page xvii

Conventions and formsetting

0.4 Mathematical Symbols

ℵ The Kurtosis.

E[ f (x)] The expected value of f (x).

Var[ f (x)] The variance of f (x).

t Time.

S(t) The asset price at time t.

O(S(t), t)) The option price of a given asset S(t) at a time t.

X(t), Y(t) The random process X(t) and Y(t) .

Ft The filtration of t.

E[X(t)] The expected value of the random process X(t).

Var[X(t)] The variance of the random process X(t).

L(θ) The loglikelihood function a distribution with parameter set θ.

L(x(t), x(t), t) The Lagrangian functional for the path integral.

A[x(t)] The action functional for the path integral.

Dx(t) Integral measure for the path integral.

Page xviii

Publications

Bonnet F. D. R., Allison, A. & Abbott, D. (2004). Review of quantum path integrals in fluctuating

markets, Proc. SPIE–Microelectronics: Design, Technology, and Packaging, Vol. 5274, Perth, Australia,

pp. 569–580.

Bonnet F. D. R., van der Hoek J., Allison, A. & Abbott, D. (2004). Path integrals in fluctuating markets,

Proc. SPIE–Noise in Complex Systems and Stochastic Dynamics II, Vol. 5471, Maspalomas Gran Canaria

Island, Spain, pp. 595–611.

Bonnet F. D. R., van der Hoek J., Allison, A. & Abbott, D. (2005). Path integrals in fluctuating mar-

kets with a non-Gaussian option pricing model, Proc. SPIE–Noise and Fluctuations in Econophysics and

Finance, Vol. 5848, Austin, TX, USA, pp. 66–85.

Bonnet F. D. R., Allison, A. & Abbott, D. (2005). Bubbles in a minority game setting with real financial

data, Proc. SPIE–Complex Systems, Vol. 6039, Brisbane, Australia, 60390C.

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Page xx

List of Figures

1.1 Louis Bachelier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Daily returns for the NASDAQ and its PDF . . . . . . . . . . . . . . . . . 12

1.3 The lattice graph for a binomial tree . . . . . . . . . . . . . . . . . . . . . 15

1.4 Daily returns for the NASDAQ and S&P500 . . . . . . . . . . . . . . . . . 17

1.5 Particle moving from A to B . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1 Distribution function of the NASDAQ and S&P500 . . . . . . . . . . . . . 28

2.2 Distribution function of the NASDAQ and S&P500 . . . . . . . . . . . . . 29

2.3 Student distribution fit to NASDAQ and S&P500 tick data . . . . . . . . 36

2.4 Time series of the NASDAQ r(t) . . . . . . . . . . . . . . . . . . . . . . . 40

2.5 Histogram of the NASDAQ r(t) . . . . . . . . . . . . . . . . . . . . . . . . 41

2.6 The qnorm − QQ plot of the NASDAQ for GARCH(1, 1) . . . . . . . . . 42

3.1 Random walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2 Sample paths for Brownian motion . . . . . . . . . . . . . . . . . . . . . . 61

4.1 Simple Ito process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Non–simple Ito process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3 Sample paths for the explicit solution versus the numerical schemes . . . 92

4.4 Sample paths for the explicit solution versus the numerical schemes . . . 93

5.1 Put and call graph of the payoff function for the European option . . . . 116

5.2 Knock in/out Barrier option . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.1 The graph of the probability distribution function Eq. (6.71) when q = 1.43150

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List of Figures

6.2 The graph of Z(t) and β(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.3 The evolution of the coefficient c as a function of the parameter q . . . . 152

6.4 The graph of x(t) when C1 = C2 = h(q) = 1 and γ(q) = 1 . . . . . . . . 165

6.5 The graph of P(x(t)) when C1 = C2 = h(q) = 1 and γ(q) = 1 . . . . . . 167

6.6 Same graph as in Fig. 6.5 for P(x(t)) when C1 = C2 = h(q) = 1 and

γ(q) = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.7 The graph of x(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.8 The graph of the Lagrangian functional Eq. (6.91) for a given x(t) . . . . 169

6.9 The graph of the functions γ(q) and h(q) . . . . . . . . . . . . . . . . . . 170

6.10 The graph of the coefficients C1 and C2 . . . . . . . . . . . . . . . . . . . . 171

6.11 The graph of the solution x(t) of the Euler–Lagrange eqaution . . . . . . 172

6.12 The graph of the solution x(t) of the Euler–Lagrange eqaution . . . . . . 172

6.13 The Lagrangian density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.14 The discrete path with N − 1 paths . . . . . . . . . . . . . . . . . . . . . . 174

6.15 The discrete path when N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 175




7.1 The Global efficiency σ2/N and the predictability H/N versus the criti-

cal parameter α = 2M/N . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

7.2 The global efficiency σ2/N and the predictibility H/N versus the critical

parameter α = 2M/N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

7.3 The global efficiency σ2/N versus the critical parameter α = 2M/N for

the Minority Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

7.4 The predictibility H/N versus the critical parameter α = 2M/N . . . . . 195

7.5 The price function P(t), Eq. (7.9) . . . . . . . . . . . . . . . . . . . . . . . 197

7.6 The simulated price function in the Minority Game . . . . . . . . . . . . 198

7.7 The price function in the Minority Game . . . . . . . . . . . . . . . . . . . 199

7.8 The simulated price function in the Minority Game for different sample

paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

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List of Figures

7.9 The time series for the bubble defined in Eq. (7.20) versus the time t . . . 205

7.10 Time series for the NASDAQ . . . . . . . . . . . . . . . . . . . . . . . . . 207

7.11 Time series for the S&P 500 . . . . . . . . . . . . . . . . . . . . . . . . . . 208

7.12 Time series for the NASDAQ . . . . . . . . . . . . . . . . . . . . . . . . . 209

7.13 Price function for the $–Game versus the Minority Game in the Minority

Game as a function of time t . . . . . . . . . . . . . . . . . . . . . . . . . . 210

7.14 Price function for the $–Game versus the Minority Game in the Minority

Game as a function of time t . . . . . . . . . . . . . . . . . . . . . . . . . . 211

7.15 The time series for the NASDAQ versus the $–Game price function in

the Minority Game as a function of time t . . . . . . . . . . . . . . . . . . 212

7.16 Time series for the S&P 500 versus the $–Game price function in the

Minority Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

7.17 Price function for the $–Game versus the Minority Game when the pay-

off function are set to 0 in the Minority Game as a function of time t . . . 214

A.1 The unit circle in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

C.1 Range of (M(T), W(T)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

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Page xxiv

List of Tables

2.1 Student distribution fit result . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2 The results for a mean and variance equation ARMA(0,0)+GARCH(1,1) 43






3.1 Brownian motion multiplication table . . . . . . . . . . . . . . . . . . . . 66

5.1 The results for the Black–Scholes–Merton model for a European option . 124

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Option pricing using path integrals