Actuarial Notes

Problems and Solutions in Mathematical Finance: Stochastic Calculus

By Eric Chin, Dian Nel and Sverrir Ólafsson · Wiley · 2014

Study guide: Exam INV-201

Volume 1 of a four-volume Wiley Finance set of problems and solutions in mathematical finance. It introduces the basic stochastic calculus concepts required for the study of mathematical finance through a large number of worked examples, which build the foundation for the more practically oriented problems of the later volumes. It is written mainly for students, industry practitioners and those involved in teaching in the field.

On the syllabus

  • Exam INV-201 — page 52 (Definitions 2.1 and 2.2; Theorems 2.3 and 2.4); pages 57-58 (Question 4); pages 72-73 (Question 3); pages 97-98 (Theorem 3.3); pages 113-114 (Itô Isometry); pages 128-130 (Questions 6 and 7); pages 132-137 (Questions 10-12); pages 146-147 (Question 19); pages 186-188 (Definition 4.1(b), Definition 4.1(f) and Theorem 4.2); pages 221-227 (Questions 1-5); pages 233-242 (Questions 12-17); outcomes 2a, 2b, 2d and 2f for each. The introductory study note lists errata for the book.

Preface

Prologue

About the Authors

1 General Probability 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 Wiener Process

  • 2.1 Introduction
  • 2.2 Problems and Solutions
    • 2.2.1 Basic Properties
    • 2.2.2 Markov Property
    • 2.2.3 Martingale Property
    • 2.2.4 First Passage Time
    • 2.2.5 Reflection Principle
    • 2.2.6 Quadratic Variation

3 Stochastic Differential Equations

  • 3.1 Introduction
  • 3.2 Problems and Solutions
    • 3.2.1 Ito Calculus
    • 3.2.2 One-Dimensional Diffusion Process
    • 3.2.3 Multi-Dimensional Diffusion Process

4 Change of Measure

  • 4.1 Introduction
  • 4.2 Problems and Solutions
    • 4.2.1 Martingale Representation Theorem
    • 4.2.2 Girsanov's Theorem
    • 4.2.3 Risk-Neutral Measure

5 Poisson Process

  • 5.1 Introduction
  • 5.2 Problems and Solutions
    • 5.2.1 Properties of Poisson Process
    • 5.2.2 Jump Diffusion Process
    • 5.2.3 Girsanov's Theorem for Jump Processes
    • 5.2.4 Risk-Neutral Measure for Jump Processes

Appendix A Mathematics Formulae

Appendix B Probability Theory Formulae

Appendix C Differential Equations Formulae

Bibliography

Notation

Index

Sources