Problems and Solutions in Mathematical Finance, Volume 2 Equity Derivatives The Wiley Finance Series

Problems and Solutions in Mathematical Finance Volume II is an innovative reference for quantitative practitioners and students, providing guidance through a range of mathematical problems encountered in the finance industry. This volume focuses solely on equity derivatives problems, beginning with basic problems in derivatives securities before moving on to more advanced applications, including the construction of volatility surfaces to price exotic options. By providing a methodology for solving theoretical and practical problems, whilst explaining the limitations of financial models, this book helps readers to develop the skills they need to advance their careers. The text covers a wide range of derivatives pricing, such as European, American, Asian, Barrier and other exotic options. Extensive appendices provide a summary of important formulae from calculus, theory of probability, and differential equations, for the convenience of readers.

As Volume II of the four-volume Problems and Solutions in Mathematical Finance series, this book provides clear explanation of the mathematics behind equity derivatives, in order to help readers gain a deeper understanding of their mechanics and a firmer grasp of the calculations.

• Review the fundamentals of equity derivatives
• Work through problems from basic securities to advanced exotics pricing
• Examine numerical methods and detailed derivations of closed-form solutions
• Utilise formulae for probability, differential equations, and more

Mathematical finance relies on mathematical models, numerical methods, computational algorithms and simulations to make trading, hedging, and investment decisions. For the practitioners and graduate students of quantitative finance, Problems and Solutions in Mathematical Finance Volume II provides essential guidance principally towards the subject of equity derivatives.

Preface ix

About the Authors xi

1 Basic Equity Derivatives Theory 1

1.1 Introduction 1

1.2 Problems and Solutions 8

1.2.1 Forward and Futures Contracts 8

1.2.2 Options Theory 15

1.2.3 Hedging Strategies 27

2 European Options 63

2.1 Introduction 63

2.2 Problems and Solutions 74

2.2.1 Basic Properties 74

2.2.2 Black–Scholes Model 89

2.2.3 Tree-Based Methods 190

2.2.4 The Greeks 218

3 American Options 267

3.1 Introduction 267

3.2 Problems and Solutions 271

3.2.1 Basic Properties 271

3.2.2 Time-Independent Options 292

3.2.3 Time-Dependent Options 305

4 Barrier Options 351

4.1 Introduction 351

4.2 Problems and Solutions 357

4.2.1 Probabilistic Approach 357

4.2.2 Reflection Principle Approach 386

4.2.3 Further Barrier-Style Options 408

5 Asian Options 439

5.1 Introduction 439

5.2 Problems and Solutions 443

5.2.1 Discrete Sampling 443

5.2.2 Continuous Sampling 480

6 Exotic Options 531

6.1 Introduction 531

6.2 Problems and Solutions 532

6.2.1 Path-Independent Options 532

6.2.2 Path-Dependent Options 586

7 Volatility Models 647

7.1 Introduction 647

7.2 Problems and Solutions 652

7.2.1 Historical and Implied Volatility 652

7.2.2 Local Volatility 685

7.2.3 Stochastic Volatility 710

7.2.4 Volatility Derivatives 769

A Mathematics Formulae 787

B Probability Theory Formulae 797

C Differential Equations Formulae 813

Bibliography 821

Notation 825

Index 829

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 ?lafsson?(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.