Description
Book SynopsisMathematical 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.
Table of ContentsPreface 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
