Problems and Solutions in Mathematical Finance: Equity Derivatives, Volume 2

Problems and Solutions in Mathematical Finance

VOLUME 2

Equity Derivatives

Eric Chin, Dian Nel and Sverrir Ólafsson

The quantitative methods required for the pricing and hedging of a range of financial securities are drawn from mathematical finance, an important and rapidly growing discipline. In an increasingly complex financial world, the role of mathematical finance is indispensable. It follows that successful financial engineers or quants need to have an excellent grasp of all the major technicalities of mathematical finance to master its diverse applications in the financial industry.

Problems and Solutions in Mathematical Finance Volume 2: Equity Derivatives is the second of a four–volume set of books focusing on problems and solutions in mathematical finance.

The first volume in the series introduced the reader to all the important concepts in probability and stochastic calculus. The second volume covers a broad area of equity derivative contracts, ranging from vanilla options to various more complex options such as time dependent American, compound, barrier and volatility options. The theoretical presentation and its effective integration with a wide range of problems is clear and to the point. This approach brings the student quickly to the forefront of the modern practice of mathematical finance.

This series is unique as it provides the student with rigorous but yet intuitive explanations of some highly technical material further deepened by extensive real–world examples.

Written mainly for students, industry practitioners and those involved in teaching in this field of study, Equity Derivatives provides a valuable reference book to complement one′s further understanding of mathematical finance.

Preface 5

1 Basic Equity Derivatives Theory 7

1.1 Introduction 7

1.2 Problems and Solutions 16

1.2.1 Forward and Futures Contracts 16

1.2.2 Options Theory 24

1.2.3 Hedging Strategies 38

2 European Options 81

2.1 Introduction 81

2.2 Problems and Solutions 93

2.2.1 Basic Properties 93

2.2.2 Black–Scholes Model 108

2.2.3 Tree Based Methods 205

2.2.4 The Greeks 233

3 American Options 279

3.1 Introduction 279

3.2 Problems and Solutions 283

3.2.1 Basic Properties 283

3.2.2 Time Independent Options 304

3.2.3 Time Dependent Options 317

4 Barrier Options 361

4.1 Introduction 362

4.2 Problems and Solutions 367

4.2.1 Probabilistic Approach 367

4.2.2 Reflection Principle Approach 400

4.2.3 Further Barrier Style Options 422

5 Asian Options 455

5.1 Introduction 455

5.2 Problems and Solutions 459

5.2.1 Discrete Sampling 459

5.2.2 Continuous Sampling 496

6 Exotic Options 545

6.1 Introduction 545

6.2 Problems and Solutions 546

6.2.1 Path Independent Options 546

6.2.2 Path Dependent Options 601

7 Volatility Models 661

7.1 Introduction 661

7.2 Problems and Solutions 666

7.2.1 Historical and Implied Volatility 666

7.2.2 Local Volatility 698

7.2.3 Stochastic Volatility 724

7.2.4 Volatility Derivatives 786

A Mathematics Formulae 805

B Probability Theory Formulae 815

C Differential Equations Formulae 831

Bibliography 839

Notation 845