Problems and Solutions in Mathematical Finance: Stochastic Calculus

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.

1. General Probability and Statistical Theory 1.1 Introduction 1.2 Problems and Solutions 1.2.1 Probability Spaces 1.2.2 Discrete and Continuous Random Variables 1.2.3 Properties of Expectations 2. General Statistical Theory 2.1 Introduction 2.2 Problems and Solutions 2.2.1 Parameter Estimation 2.2.2 Hypotheses Testing 2.2.3 Goodness of Fit Analysis 2.2.4 Regression Analysis 3. Wiener Process 3.1 Introduction 3.2 Problems and Solutions 3.2.1  Random Walks 3.2.2 Examples of Wiener Process 3.2.3 Markov Property 3.2.4 Martingale Property 3.2.5 First Passage Time 3.2.6 Reflection Principle 3.2.7 Quadratic Variation 4. Stochastic Differential Equations 4.1 Introduction 4.2 Problems and Solutions 4.2.1 Ito Calculus 4.2.2 One–Dimension Diffusion Process 4.2.3 Multi–Dimensional Diffusion Process 5. Change of Measure 5.1 Introduction 5.2 Problems and Solutions 5.2.1 Martingale Representation Theorem 5.2.2 Girsanov′s Theorem 5.2.3 Risk Neutral Measure 6. Poisson Process 6.1 Introduction 6.2 Problems and Solutions 6.2.1 Properties of Poisson Process 6.2.2 Jump Diffusion Process 6.2.3 Change of Measure Appendix A Mathematics Formulae Appendix B Probability Theory Formulae Appendix C Statistical Theory Formulae Appendix D Differential Equations Formulae

Dr. Eric Chin (London, UK) is a quantitative analyst at Standard Chartered Bank where he is involved in providing guidance on price testing methodologies and their implementation, formulating model calibration and model appropriateness across all asset classes. Dian Nel (London, UK) is a quantitative analyst currently working for Norwegian Energy and has many years experience in energy markets where his main interests include exotic options, portfolio optimisation and hedging in incomplete markets. Dr. Sverrir Olafsson (Reykjavik, Iceland) is a professor in the School of Business at the University of Reykjavik, Iceland and a visiting professor in the Department of Electrical Engineering and Computer Science at Queen Mary University of London. He is also the director of Riskcon Ltd a UK based consultancy on risk management.