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Detailed guidance on the mathematics behind equityderivatives Problems and Solutions in Mathematical Finance Volume IIis an innovative reference for quantitative practitioners andstudents, providing guidance through a range of mathematicalproblems encountered in the finance industry. This volume focusessolely on equity derivatives problems, beginning with basicproblems in derivatives securities before moving on to moreadvanced applications, including the construction of volatilitysurfaces to price exotic options. By providing a methodology forsolving theoretical and practical problems, whilst explaining thelimitations of financial models, this book helps readers to developthe skills they need to advance their careers. The text covers awide range of derivatives pricing, such as European, American,Asian, Barrier and other exotic options. Extensive appendicesprovide a summary of important formulae from calculus, theory ofprobability, and differential equations, for the convenience ofreaders. As Volume II of the four-volume Problems and Solutions inMathematical Finance series, this book provides clearexplanation of the mathematics behind equity derivatives, in orderto help readers gain a deeper understanding of their mechanics anda firmer grasp of the calculations. * Review the fundamentals of equity derivatives * Work through problems from basic securities to advanced exoticspricing * Examine numerical methods and detailed derivations ofclosed-form solutions * Utilise formulae for probability, differential equations, andmore Mathematical finance relies on mathematical models, numericalmethods, computational algorithms and simulations to make trading,hedging, and investment decisions. For the practitioners andgraduate students of quantitative finance, Problems andSolutions in Mathematical Finance Volume II provides essentialguidance principally towards the subject of equity derivatives.

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For other titles in the Wiley Finance series please see www.wiley.com/finance

Problems and Solutions in Mathematical Finance

Volume 2: Equity Derivatives

Eric Chin, Dian Nel and Sverrir Ólafsson

This edition first published 2017 © 2017 John Wiley & Sons, Ltd

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Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought.

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A catalogue record for this book is available from the British Library.

ISBN 978-1-119-96582-4 (hardback) ISBN 978-1-119-96610-4 (ebk) ISBN 978-1-119-96611-1 (ebk)          ISBN 978-1-119-19219-0 (obk)

Cover design: Cylinder Cover image: © Attitude/Shutterstock

“Blue dye is derived from the indigo plant and surpassed its parental colour”Xunzi, An Exhortation to Learning

CONTENTS

Preface

About the Authors

1 Basic Equity Derivatives Theory

1.1 Introduction

1.2 Problems and Solutions

2 European Options

2.1 Introduction

2.2 Problems and Solutions

3 American Options

3.1 Introduction

3.2 Problems and Solutions

4 Barrier Options

4.1 Introduction

4.2 Problems and Solutions

5 Asian Options

5.1 Introduction

5.2 Problems and Solutions

6 Exotic Options

6.1 Introduction

6.2 Problems and Solutions

7 Volatility Models

7.1 Introduction

7.2 Problems and Solutions

Appendix A Mathematics Formulae

Appendix B Probability Theory Formulae

Appendix C Differential Equations Formulae

Bibliography

Notation

Index

EULA

List of Tables

Chapter 1

Table 1.1

Table 1.2

Table 1.3

Table 1.4

Chapter 2

Table 2.1

Table 2.2

Table 2.3

Table 2.4

Table 2.5

Chapter 4

Table 4.1

Guide

Cover

Table of Contents

Preface

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Preface

Mathematical finance is a highly challenging and technical discipline. Its fundamentals and applications are best understood by combining a theoretically solid approach with extensive exercises in solving practical problems. That is the philosophy behind all four volumes in this series on mathematical finance. This second of four volumes in the series Problems and Solutions in Mathematical Finance is devoted to the discussion of equity derivatives. In the first volume we developed the probabilistic and stochastic methods required for the successful study of advanced mathematical finance, in particular different types of pricing models. The techniques applied in this volume assume good knowledge of the topics covered in Volume 1. As we believe that good working knowledge of mathematical finance is best acquired through the solution of practical problems, all the volumes in this series are built up in a way that allows readers to continuously test their knowledge as they work through the texts.

This second volume starts with the analysis of basic derivatives, such as forwards and futures, swaps and options. The approach is bottom up, starting with the analysis of simple contracts and then moving on to more advanced instruments. All the major classes of options are introduced and extensively studied, starting with plain European and American options. The text then moves on to cover more complex contracts such as barrier, Asian and exotic options. In each option class, different types of options are considered, including time-independent and time-dependent options, or non-path-dependent and path-dependent options.

Stochastic financial models frequently require the fixing of different parameters. Some can be extracted directly from market data, others need to be fixed by means of numerical methods or optimisation techniques. Depending on the context, this is done in different ways. In the risk-neutral world, the drift parameter for the geometric Brownian motion (Black–Scholes model) is extracted from the bond market (i.e., the returns on risk-free debt). The volatility parameter, in contrast, is generally determined from market prices, as the so-called implied volatility. However, if a stochastic process is to be fitted to known price data, other methods need to be consulted, such as maximum-likelihood estimation. This method is applied to a number of stochastic processes in the chapter on volatility models.

In all option models, volatility presents one of the most important quantities that determine the price and the risk of derivatives contracts. For this reason, considerable effort is put into their discussion in terms of concepts, such as implied, local and stochastic volatilities, as well as the important volatility surfaces.

At the end of this volume, readers will be equipped with all the major tools required for the modelling and the pricing of a whole range of different derivatives contracts. They will therefore be ready to tackle new techniques and challenges discussed in the next two volumes, including interest-rate modelling in Volume 3 and foreign exchange/commodity derivatives in Volume 4.

As in the first volume, we have the following note to the student/reader: Please try hard to solve the problems on your own before you look at the solutions!

About the Authors

Eric Chin is a quantitative analyst at an investment bank in the City of London where he is involved in providing guidance on price testing methodologies and their implementation, formulating model calibration and model appropriateness on commodity and credit products. Prior to joining the banking industry he worked as a senior researcher at British Telecom investigating radio spectrum trading and risk management within the telecommunications sector. He holds an MSc in Applied Statistics and an MSc in Mathematical Finance both from University of Oxford. He also holds a PhD in Mathematics from University of Dundee.

Dian Nel has more than 10 years of experience in the commodities sector. He currently works in the City of London where he specialises in oil and gas markets. He holds a BEng in Electrical and Electronic Engineering from Stellenbosch University and an MSc in Mathematical Finance from Christ Church, Oxford University. He is a Chartered Engineer registered with the Engineering Council UK.

Sverrir Ólafsson is Professor of Financial Mathematics at Reykjavik University; a Visiting Professor at Queen Mary University, London and a director of Riskcon Ltd, a UK based risk management consultancy. Previously he was a Chief Researcher at BT Research and held academic positions at The Mathematical Departments of Kings College, London; UMIST Manchester and The University of Southampton. Dr Ólafsson is the author of over 95 refereed academic papers and has been a key note speaker at numerous international conferences and seminars. He is on the editorial board of three international journals. He has provided an extensive consultancy on financial risk management and given numerous specialist seminars to finance specialists. In the last five years his main teaching has been MSc courses on Risk Management, Fixed Income, and Mathematical Finance. He has an MSc and PhD in mathematical physics from the Universities of Tübingen and Karlsruhe respectively.