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.

Table of Contents:
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 It¯o 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