Problems and Solutions in Mathematical Finance, Volume 1 Stochastic Calculus

An increasingly popular field of study at universities and an essential skill for investment bank employees, mathematical finance has changed dramatically in recent years, but its roots remain in stochastic calculus. Problems and Solutions in Mathematical Finance: Volume 1 provides a comprehensive explanation of stochastic calculus and probability theory focusing on their relationship with mathematical finance. Quantitative analysts Dr. Eric Chin and Dian Nel and Professor Sverrir Olafsson portray stochastic calculus' role in generating partial differentiation equations for pricing options and constructing probability measures in conjunction with martingale theory. Mathematical and computational finance rely on computational intelligence, numerical methods, and computer simulations to make trading, hedging, and investment decisions, to determine the risk of those decisions, and to define price derivatives.

• Includes chapter-by-chapter introduction of the fundamental tenets, essential definitions, and detailed explanations needed to solve financial problems
• Offers advice from experts in price testing methodologies, model calibration, energy markets, hedging in incomplete markets, and risk management

Problems and Solutions in Mathematical Finance: Volume 1 functions as either an independent information text or a study supplement for students and practitioners eager to recover the basics of mathematical finance.

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.

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