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.

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