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Preface

Chapter 1. Introduction: A Simple Market Model
    1.1 Basic Notions and Assumptions
    1.2 No-Arbitrage Principle
    1.3 One-Step Binomial Model
    1.4 Risk and Return
    1.5 Forward Contracts
    1.6 Call and Put Options
    1.7 Managing Risk with Options 

Chapter 2. Risk-Free Assets
    2.1 Time Value of Money
            2.1.1 Simple Interest
            2.1.2 Periodic Compounding
            2.1.3 Streams of Payments
            2.1.4 Continuous Compounding
            2.1.5 How to Compare Compounding Methods
    2.2 Money Market
            2.2.1 Zero-Coupon Bonds
            2.2.2 Coupon Bonds
            2.2.3 Money Market Account

Chapter 3. Risky Assets
    3.1 Dynamics of Stock Prices
            3.1.1 Return
            3.1.2 Expected Return
    3.2 Binomial Tree Model
            3.2.1 Risk-Neutral Probability
            3.2.2 Martingale Property
    3.3 Other Models
            3.3.1 Trinomial Tree Model
            3.3.2 Continuous-Time Limit

Chapter 4. Discrete Time Market Models
    4.1 Stock and Money Market Models
            4.1.1 Investment Strategies
            4.1.2 The Principle of No Arbitrage
            4.1.3 Application to the Binomial Tree Model
            4.1.4 Fundamental Theorem of Asset Pricing
    4.2 Extended Models

Chapter 5. Portfolio Management
    5.1 Risk
    5.2 Two Securities
            5.2.1 Risk and Expected Return on a Portfolio
    5.3 Several Securities
            5.3.1 Risk and Expected Return on a Portfolio
            5.3.2 Efficient Frontier
    5.4 Capital Asset Pricing Model
            5.4.1 Capital Market Line
            5.4.2 Beta Factor
            5.4.3 Security Market Line

Chapter 6. Forward and Futures Contracts
    6.1 Forward Contracts
            6.1.1 Forward Price
            6.1.2 Value of a Forward Contract
    6.2 Futures
            6.2.1 Pricing
            6.2.2 Hedging with Futures

Chapter 7. Options: General Properties
    7.1 Definitions
    7.2 Put-Call Parity
    7.3 Bounds on Option Prices
            7.3.1 European Options
            7.3.2 European and American Calls on Non-Dividend Paying Stock
            7.3.3 American Options
    7.4 Variables Determining Option Prices
            7.4.1 European Options
            7.4.2 American Options
    7.5 Time Value of Options

Chapter 8. Option Pricing
    8.1 European Options in the Binomial Tree Model
            8.1.1 One Step
            8.1.2 Two Steps
            8.1.3 General N-Step Model
            8.1.4 Cox-Ross-Rubinstein Formula
    8.2 American Options in the Binomial Tree Model
    8.3 Black-Scholes Formula

Chapter 9. Financial Engineering
    9.1 Hedging Option Positions
            9.1.1 Delta Hedging
            9.1.2 Greek Parameters
            9.1.3 Applications
    9.2 Hedging Business Risk
            9.2.1 Value at Risk
            9.2.2 Case Study
    9.3 Speculating with Derivatives
            9.3.1 Tools
            9.3.2 Case Study

Chapter 10. Variable Interest Rates
    10.1 Maturity-Independent Yields
            10.1.1 Investment in Single Bonds
            10.1.2 Duration
            10.1.3 Portfolios of Bonds
            10.1.4 Dynamic Hedging
    10.2 General Term Structure
            10.2.1 Forward Rates
            10.2.2 Money Market Account

Chapter 11. Stochastic Interest Rates
    11.1 Binomial Tree Model
    11.2 Arbitrage Pricing of Bonds
            11.2.1 Risk-Neutral Probabilities
    11.3 Interest Rate Derivative Securities
            11.3.1 Options
            11.3.2 Swaps
            11.3.3 Caps and Floors
    11.4 Final Remarks

Solutions

Bibliography

Glossary of Symbols  

Index

True to its title, this book itself is an excellent financial investment. For the price of one volume it teaches two Nobel Prize winning theories, with plenty more included for good measure. How many undergraduate mathematics textbooks can boast such a claim?

Building on mathematical models of bond and stock prices, these two theories lead in different directions: Black-Scholes arbitrage pricing of options and other derivative securities on the one hand, and Markowitz portfolio optimisation and the Capital Asset Pricing Model on the other hand. Models based on the principle of no arbitrage can also be developed to study interest rates and their term structure. These are three major areas of mathematical finance, all having an enormous impact on the way modern financial markets operate. This textbook presents them at a level aimed at second or third year undergraduate students, not only of mathematics but also, for example, business management, finance or economics.

The contents can be covered in a one-year course of about 100 class hours. Smaller courses on selected topics can readily be designed by choosing the appropriate chapters. The text is interspersed with a multitude of worked examples and exercises, complete with solutions, providing ample material for tutorials as well as making the book ideal for self-study.

Prerequisites include elementary calculus, probability and some linear algebra. In calculus we assume experience with derivatives and partial derivatives, finding maxima or minima of differentiable functions of one or more variables, Lagrange multipliers, the Taylor formula and integrals. Topics in probability include random variables and probability distributions, in particular the binomial and normal distributions, expectation, variance and covariance, conditional probability and independence. Familiarity with the Central Limit Theorem would be a bonus. In linear algebra the reader should be able to solve systems of linear equations, add, multiply, transpose and invert matrices, and compute determinants. In particular, as a reference in probability theory we recommend our book: M. Capinski and T. Zastawniak, Probability Through Problems, Springer-Verlag, New York, 2001.

In many numerical examples and exercises it may be helpful to use a computer with a spreadsheet application, though this is not absolutely essential.

We are indebted to Nigel Cutland for prompting us to steer clear of an inaccuracy frequently encountered in other texts, of which more will be said in Remark 4.1. It is also a great pleasure to thank our students for their feedback on preliminary versions of various chapters.

Marek Capinski and Tomasz Zastawniak
January 2003