Problems and Solutions in Mathematical Finance, Volume 1 Stochastic Calculus The Wiley Finance Series

Mathematical finance requires the use of advanced mathematical techniques drawn from the theory of probability, stochastic processes and stochastic differential equations. These areas are generally introduced and developed at an abstract level, making it problematic when applying these techniques to practical issues in finance.

Problems and Solutions in Mathematical Finance Volume I: Stochastic Calculus is the first of a four-volume set of books focusing on problems and solutions in mathematical finance.

This volume introduces the reader to the basic stochastic calculus concepts required for the study of this important subject, providing a large number of worked examples which enable the reader to build the necessary foundation for more practical orientated problems in the later volumes. Through this application and by working through the numerous examples, the reader will properly understand and appreciate the fundamentals that underpin mathematical finance.

Written mainly for students, industry practitioners and those involved in teaching in this field of study, Stochastic Calculus provides a valuable reference book to complement one?s further understanding of mathematical finance.

Preface ix

Prologue xi

About the Authors xv

1 General Probability Theory 1

1.1 Introduction 1

1.2 Problems and Solutions 4

1.2.1 Probability Spaces 4

1.2.2 Discrete and Continuous Random Variables 11

1.2.3 Properties of Expectations 41

2 Wiener Process 51

2.1 Introduction 51

2.2 Problems and Solutions 55

2.2.1 Basic Properties 55

2.2.2 Markov Property 68

2.2.3 Martingale Property 71

2.2.4 First Passage Time 76

2.2.5 Reflection Principle 84

2.2.6 Quadratic Variation 89

3 Stochastic Differential Equations 95

3.1 Introduction 95

3.2 Problems and Solutions 102

3.2.1 Itō Calculus 102

3.2.2 One-Dimensional Diffusion Process 123

3.2.3 Multi-Dimensional Diffusion Process 155

4 Change of Measure 185

4.1 Introduction 185

4.2 Problems and Solutions 192

4.2.1 Martingale Representation Theorem 192

4.2.2 Girsanov’s Theorem 194

4.2.3 Risk-Neutral Measure 221

5 Poisson Process 243

5.1 Introduction 243

5.2 Problems and Solutions 251

5.2.1 Properties of Poisson Process 251

5.2.2 Jump Diffusion Process 281

5.2.3 Girsanov’s Theorem for Jump Processes 298

5.2.4 Risk-Neutral Measure for Jump Processes 322

Appendix A Mathematics Formulae 331

Appendix B Probability Theory Formulae 341

Appendix C Differential Equations Formulae 357

Bibliography 365

Notation 369

Index 373

Eric Chin is a quantitative analyst at an investment bank in the City of London where he is involved in providing guidance on price testing methodologies and their implementation, formulating model calibration and model appropriateness on commodity and credit products. Prior to joining the banking industry he worked as a senior researcher at British Telecom investigating radio spectrum trading and risk management within the telecommunications sector. Eric Chin holds an MSc in Applied Statistics and an MSc in Mathematical Finance both from University of Oxford. He also holds a PhD in Mathematics from University of Dundee.

Dian Nel has more than 10 years of experience in the commodities sector. He currently works in the City of London where he specialises in oil and gas markets. He holds a BEng in Electrical and Electronic Engineering from StellenboschUniversity and an MSc in Mathematical Finance from ChristChurch, OxfordUniversity. He is a Chartered Engineer registered with the Engineering Council UK.

Sverrir Olafsson is Professor of Financial Mathematics at Reykjavik University; a Visiting Professor at QueenMaryUniversity, London and a director of Riskcon Ltd, a UK based risk management consultancy. Previously he was a Chief Researcher at BT Research and held academic positions at The Mathematical Departments of Kings College, London; UMIST Manchester and The University of Southampton. Dr Olafsson is the author of over 95 refereed academic papers and has been a key note speaker at numerous international conferences and seminars. He is on the editorial board of three international journals. He has provided an extensive consultancy on financial risk management and given numerous specialist seminars to finance specialists. In the last five years his main teaching has been MSc courses on Risk Management, Fixed Income, and Mathematical Finance.
Dr Olafsson has an MSc and PhD in mathematical physics from the Universities of Tübin