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Problems and Solutions in Mathematical Finance: Stochastic Calculus, Volume I


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 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.

This book takes a dual approach using stochastic calculus to develop partial differentiation equations for pricing options and also constructs probability measures via martingale theory so that option prices can be expressed as expectations. Each chapter begins with an introduction to the fundamentals and the essential definitions and explanations needed to solve the subsequent problems.

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.

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 Tübingen and Karlsruhe respectively.

1 General Probability and Statistical Theory

11 Introduction

12 Problems and Solutions

121 Probability Spaces

122 Discrete and Continuous Random Variables

123 Properties of Expectations

2 General Statistical Theory

21 Introduction

22 Problems and Solutions

221 Parameter Estimation

222 Hypotheses Testing

223 Goodness of Fit Analysis

224 Regression Analysis

3 Wiener Process

31 Introduction

32 Problems and Solutions

321  Random Walks

322 Examples of Wiener Process

323 Markov Property

324 Martingale Property

325 First Passage Time

326 Reflection Principle

327 Quadratic Variation

4 Stochastic Differential Equations

41 Introduction

42 Problems and Solutions

421 Ito Calculus

422 One-Dimension Diffusion Process

423 Multi-Dimensional Diffusion Process

5 Change of Measure

51 Introduction

52 Problems and Solutions

521 Martingale Representation Theorem

522 Girsanov's Theorem

523 Risk Neutral Measure

6 Poisson Process

61 Introduction

62 Problems and Solutions

621 Properties of Poisson Process

622 Jump Diffusion Process

623 Change of Measure

Appendix A Mathematics Formulae

Appendix B Probability Theory Formulae

Appendix C Statistical Theory Formulae

Appendix D Differential Equations Formulae