Preface

Objectives and Audience

This book has evolved from several mathematics courses that the authors have taught mainly within the bachelor’s and master’s programs in financial mathematics at Wilfrid Laurier University. The contents of this book are a culmination of course material that spans over a decade of the authors’ teaching experiences, as well as course and curriculum development, in financial mathematics programs at both undergraduate and master’s graduate levels. The material has been tested and refined through years of classroom teaching experience. As the title suggests, this book is a comprehensive, self-contained, and unified treatment of the main theory and application of mathematical methods behind modern day financial mathematics. In writing this book, the authors have really strived to create a single volume that can be used as a complete standard university textbook for several interrelated courses in financial mathematics at the undergraduate as well as graduate levels. As such, the authors have aimed to introduce both the financial theory and the relevant mathematical methods in a mathematically rigorous, yet student-friendly and engaging style, that includes an abundance of examples, problem exercises, and fully worked out solutions. In contrast to most published single volumes on the subject of financial mathematics, this book presents multiple problem solving approaches and hence bridges together related comprehensive techniques for pricing different types of financial derivatives. The book contains a rather complete and in-depth comprehensive coverage of both discrete-time and continuous-time financial models that form the cornerstones of financial derivative pricing theory. This book also provides a self-contained introduction to stochastic calculus and martingale theory, which are important cornerstones in quantitative finance. The material in many of the chapters is presented at a level that is mainly accessible to undergraduate students of mathematics, finance, actuarial science, economics, and other related quantitative fields. The textbook covers a breadth of material, from beginner to more advanced levels, that is required, i.e., absolutely essential, in the core curriculum courses on financial mathematics currently taught at the second, third, and senior year undergraduate levels at many universities across the globe. As well, a significant portion of the more advanced material in the textbook is meant to be used in courses at the master’s graduate level. These courses include formal derivative pricing theory, stochastic calculus, and courses in simulation (Monte Carlo) and other numerical methods. The combination of analytical and numerical methods for solving various derivative pricing problems can also be a useful reference for researchers and practitioners in quantitative finance.

The book has the following key features:

• comprehensive treatment covering a complete undergraduate program in financial mathematics as well as some master’s level courses in financial mathematics;
• student-friendly presentation with numerous fully worked out examples and exercise problems in every chapter;
• in-depth coverage of both discrete-time and continuous-time theory and methodology;
• mathematically rigorous and consistent, yet simple, style that bridges various basic and more advanced concepts and techniques;
• judicious balance of financial theory, mathematical, and computational methods.

Guide to Material

This book is divided into four main parts with each part consisting of several chapters. There are a total of eighteen chapters, and every chapter (with the exception of Chapter 9 on general probability theory and Chapter 18 on numerical applications) ends with a comprehensive and exhaustive set of exercises of varying difficulty. Part I is an introduction to pricing and management of financial securities. This part has four chapters. Chapter 1 introduces the reader to time value of money, compounding interest, and the basic concepts of fixed income markets. Chapter 2 introduces basic derivative securities and the concept of arbitrage. Chapter 3 covers standard theoretical topics of portfolio management and only requires some very basic linear algebra and optimization. Chapter 4 presents more formal definitions and gives a thorough discussion on basic options theory, including payoff replication, hedging, put-call parity relations, forwards and futures contracts, swaps, American options, and other contracts.

Part II is devoted to discrete-time financial modelling. Chapter 5, Chapter 6, Chapter 7 and Chapter 8 of Part II can be considered as a complete course on discrete-time asset pricing. Part II introduces the main financial concepts in risk-neutral pricing theory, which are further developed in Part III on continuous-time theory. Chapter 5 covers the financial and formal mathematical underpinnings of the single-period (Arrow-Debreu) economic model. Chapter 6 lays down the foundation for stochastic processes in discrete time, which is then essential for Chapter 7. The latter chapter covers the multi-period binomial market model and is considered the centerpiece of discrete-time financial derivative pricing. All the important concepts for pricing and hedging standard European, as well as path-dependent derivatives within this model, are presented. A general multi-asset, multi-period, discrete-time model is covered in Chapter 8. This chapter also presents the two fundamental theorems of asset pricing and equivalent martingale measures for discrete-time derivative pricing.

Part III is essentially the second half of the textbook and is a major part that is devoted to continuous-time modelling. In fact, on its own, Part III (i.e., Chapter 9, Chapter 10, Chapter 11, Chapter 12, Chapter 13, Chapter 14, Chapter 15 and Chapter 16) can be considered as a complete text in continuous-time financial mathematics for senior undergraduates and master’s level students. Chapter 9 is a stand-alone chapter that summarizes the main theoretical concepts in formal probability theory as it relates to measure theory. This also provides some mathematical foundation for later chapters that deal with continuous time modelling and stochastic calculus. Chapter 10 lays down the foundation for standard Brownian motion. Chapter 11 is a comprehensive coverage of stochastic (Itô) calculus that is required for a large portion of the material in the rest of the book chapters. Chapter 12 and Chapter 13 are the main chapters on continuous-time derivative pricing theory, which also include the Black-Scholes-Merton theory of European option pricing. The central concepts of dynamic hedging and replication are presented. Chapter 12 deals with derivative pricing and hedging in the Black-Scholes-Merton model with a single risky asset. Chapter 12 also covers path-dependent derivative pricing within the Black-Scholes-Merton framework. Chapter 13 extends the theory and methodology to derivative pricing and hedging with multiple underlying assets as well as the valuation of cross-currency options. The chapter combines different techniques for pricing multi-asset financial derivatives. Various option pricing formulae are then derived. Chapter 13 also presents risk-neutral asset pricing theory within a mathematically rigorous framework that incorporates equivalent martingale measures and change of numéraire methods for pricing. Chapter 14 is devoted to pricing American options on a single asset. Chapter 15 covers interest-rate modelling and derivative pricing for fixed-income products. Chapter 16 introduces some alternative asset price models, including the local volatility model and solvable state-dependent volatility (e.g., the CEV diffusion) models; stochastic volatility models; jump-diffusion and pure jump processes and variance gamma models.

Part IV concludes the book with Chapter 17 and Chapter 18. Chapter 17 is a self-contained exposition of various Monte Carlo and simulation methods that are relevant for simulating financial assets and pricing financial derivatives by simulation. Chapter 18 presents some specific algorithms for the numerical pricing of financial derivatives under various models.

The inter-relationship among the different chapters is summarized in the figure, which represents a flow chart of the material in the textbook. Each solid arrow indicates a strong connection between the material in the respective chapters, i.e., when a chapter is viewed as a prerequisite for the other. A dashed arrow indicates that a chapter is relevant but not necessarily a prerequisite for the other. Finally, the table below is a reference guide for instructors. It displays five different courses for which this book can be adopted as a required textbook at both the undergraduate and graduate levels. The relevant chapters for each course and the basic prerequisites are indicated in the table.

Acknowledgements

We would like to thank all our past and current undergraduate and graduate students for their valuable comments, feedback, and advice. We are also grateful for the feedback we have received from our colleagues in the Mathematics Department at Wilfrid Laurier University.

Giuseppe Campolieti and Roman N. Makarov

Waterloo, Ontario
January 2014