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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Serie: Wiley Finance Series
- Sprache: Englisch
Paul Wilmott Introduces Quantitative Finance, Second Edition is an accessible introduction to the classical side of quantitative finance specifically for university students. Adapted from the comprehensive, even epic, works Derivatives and Paul Wilmott on Quantitative Finance, Second Edition, it includes carefully selected chapters to give the student a thorough understanding of futures, options and numerical methods. Software is included to help visualize the most important ideas and to show how techniques are implemented in practice. There are comprehensive end-of-chapter exercises to test students on their understanding.
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Contents
Cover
Half Title page
Title page
Copyright page
Dedication
Preface
Chapter 1: Products and Markets: Equities, Commodities, Exchange Rates, Forwards and Futures
1.1 Introduction
1.2 Equities
1.3 Commodities
1.4 Currencies
1.5 Indices
1.6 The Time Value of Money
1.7 Fixed-Income Securities
1.8 Inflation-Proof Bonds
1.9 Forwards and Futures
1.10 More About Futures
1.11 Summary
Further Reading
Exercises
Chapter 2: Derivatives
2.1 Introduction
2.2 Options
2.3 Definition of Common Terms
2.4 Payoff Diagrams
2.5 Writing Options
2.6 Margin
2.7 Market Conventions
2.8 The Value of the Option Before Expiry
2.9 Factors Affecting Derivative Prices
2.10 Speculation and Gearing
2.11 Early Exercise
2.12 Put-Call Parity
2.13 Binaries or Digitals
2.14 Bull and Bear Spreads
2.15 Straddles and Strangles
2.16 Risk Reversal
2.17 Butterflies and Condors
2.18 Calendar Spreads
2.19 Leaps and Flex
2.20 Warrants
2.21 Convertible Bonds
2.22 Over the Counter Options
2.23 Summary
Further Reading
Exercises
Chapter 3: The Binomial Model
3.1 Introduction
3.2 Equities Can Go Down as Well as Up
3.3 The Option Value
3.4 Which Part of Our ‘Model’ didn’t We Need!
3.5 Why should this ‘Theoretical Price’ be the ‘Market Price’?
3.6 How did I Know to Sell 1/2 of the Stock for Hedging?
3.7 How does this Change if Interest Rates are Non-Zero!
3.8 Is the Stock Itself Correctly Priced!
3.9 Complete Markets
3.10 The Real and Risk-Neutral Worlds
3.11 And Now Using Symbols
3.12 An Equation for the Value of an Option
3.13 Where did the Probability p Go!
3.14 Counter-Intuitive?
3.15 The Binomial Tree
3.16 The Asset Price Distribution
3.17 Valuing Back Down the Tree
3.18 Programming the Binomial Method
3.19 The Greeks
3.20 Early Exercise
3.21 The Continuous-Time Limit
3.22 Summary
Further Reading
Welcome to My World
Appendix: Another Parameterization
Exercises
Chapter 4: The Random Behavior of Assets
4.1 Introduction
4.2 The Popular Forms of ‘Analysis’
4.3 Why We Need A Model for Randomness: Jensen’s Inequality
4.4 Similarities Between Equities, Currencies, Commodities and Indices
4.5 Examining Returns
4.6 Timescales
4.7 Estimating Volatility
4.8 The Random Walk on a Spreadsheet
4.9 The Wiener Process
4.10 The Widely Accepted Model for Equities, Currencies, Commodities and Indices
4.11 Summary
Further Reading
Exercises
Chapter 5: Elementary Stochastic Calculus
5.1 Introduction
5.2 A Motivating Example
5.3 The Markov Property
5.4 The Martingale Property
5.5 Quadratic Variation
5.6 Brownian Motion
5.7 Stochastic Integration
5.8 Stochastic Differential Equations
5.9 The Mean Square Limit
5.10 Functions of Stochastic Variables and Itô’s Lemma
5.11 Interpretation of Itô’s Lemma
5.12 Itô and Taylor
5.13 Itô in Higher Dimensions
5.14 Some Pertinent Examples
5.15 Summary
Further Reading
Exercises
Chapter 6: The Black–Scholes Model
6.1 Introduction
6.2 A Very Special Portfolio
6.3 Elimination of Risk: Delta Hedging
6.4 No Arbitrage
6.5 The Black–Scholes Equation
6.6 The Black–Scholes Assumptions
6.7 Final Conditions
6.8 Options on Dividend-Paying Equities
6.9 Currency Options
6.10 Commodity Options
6.11 Expectations and Black–Scholes
6.12 Some Other Ways of Deriving the Black-Scholes Equation
6.13 No Arbitrage in the Binomial, Black–Scholes and ‘Other’ Worlds
6.14 Forwards and Futures
6.15 Futures Contracts
6.16 Options on Futures
6.17 Summary
Further Reading
Exercises
Chapter 7: Partial Differential Equations
7.1 Introduction
7.2 Putting the Black–Scholes Equation into Historical Perspective
7.3 The Meaning of the Terms in the Black–Scholes Equation
7.4 Boundary and Initial/Final Conditions
7.5 Some Solution Methods
7.6 Similarity Reductions
7.7 Other Analytical Techniques
7.8 Numerical Solution
7.9 Summary
Further Reading
Exercises
Chapter 8: The Black–Scholes Formulae and the Greeks’
8.1 Introduction
8.2 Derivation of the Formulæ for Calls, Puts and Simple Digitals
8.3 Delta
8.4 Gamma
8.5 Theta
8.6 Speed
8.7 Vega
8.8 Rho
8.9 Implied Volatility
8.10 A Classification of Hedging Types
8.11 Summary
Further Reading
Exercises
Chapter 9: Overview of Volatility Modeling
9.1 Introduction
9.2 The Different Types of Volatility
9.3 Volatility Estimation by Statistical Means
9.4 Maximum Likelihood Estimation
9.5 Skews and Smiles
9.6 Different Approaches to Modeling Volatility
9.7 The Choices of Volatility Models
9.8 Summary
Further Reading
Appendix: How to Derive BS PDE, Minimum Fuss
Exercises
Chapter 10: How to Delta Hedge
10.1 Introduction
10.2 What if Implied and Actual Volatilities are Different?
10.3 Implied Versus Actual, Delta Hedging but Using Which Volatility?
10.4 Case I: Hedge with Actual Volatility, σ
10.5 Case 2: Hedge with Implied Volatility,
10.6 Hedging with Different Volatilities
10.7 Pros and Cons of Hedging with Each Volatility
10.8 Portfolios when Hedging with Implied Volatility
10.9 How Does Implied Volatility Behave!
10.10 Summary
Further Reading
Exercises
Chapter 11: An Introduction to Exotic and Path-Dependent Options
11.1 Introduction
11.2 Option Classification
11.3 Time Dependence
11.4 Cashflows
11.5 Path Dependence
11.6 Dimensionality
11.7 The Order of an Option
11.8 Embedded Decisions
11.9 Classification Tables
11.10 Examples of Exotic Options
11.11 Summary of Math/Coding Consequences
11.12 Summary
Further Reading
Some Formulæ for Asian Options
Some Formulæ for Lookback Options
Exercises
Chapter 12: Multi-Asset Options
12.1 Introduction
12.2 Multidimensional Lognormal Random Walks
12.3 Measuring Correlations
12.4 Options on Many Underlyings
12.5 The Pricing Formula for European Non-Path-Dependent Options on Dividend-Paying Assets
12.6 Exchanging one Asset for Another: A Similarity Solution
12.7 Two Examples
12.8 Realities of Pricing Basket Options
12.9 Realities of Hedging Basket Options
12.10 Correlation Versus Cointegration
12.11 Summary
Further Reading
Exercises
Chapter 13: Barrier Options
13.1 Introduction
13.2 Different Types of Barrier Options
13.3 Pricing Methodologies
13.4 Pricing Barriers in The Partial Differential Equation Framework
13.5 Examples
13.6 Other Features in Barrier-Style Options
13.7 Market Practice: What Volatility Should I Use?
13.8 Hedging Barrier Options
13.9 Summary
Further Reading
Exercises
Chapter 14: Fixed-Income Products and Analysis: Yield, Duration and Convexity
14.1 Introduction
14.2 Simple Fixed-Income Contracts and Features
14.3 International Bond Markets
14.4 Accrued Interest
14.5 Day-Count Conventions
14.6 Continuously and Discretely Compounded Interest
14.7 Measures of Yield
14.8 The Yield Curve
14.9 Price/Yield Relationship
14.10 Duration
14.11 Convexity
14.12 An Example
14.13 Hedging
14.14 Time-Dependent Interest Rate
14.15 Discretely Paid Coupons
14.16 Forward Rates and Bootstrapping
14.17 Interpolation
14.18 Summary
Further Reading
Exercises
Chapter 15: Swaps
15.1 Introduction
15.2 The Vanilla Interest Rate Swap
15.3 Comparative Advantage
15.4 The Swap Curve
15.5 Relationship Between Swaps and Bonds
15.6 Bootstrapping
15.7 Other Features of Swaps Contracts
15.8 Other Types of Swap
15.9 Summary
Further Reading
Exercises
Chapter 16: One-Factor Interest Rate Modeling
16.1 Introduction
16.2 Stochastic Interest Rates
16.3 The Bond Pricing Equation for the General Model
16.4 What is the Market Price of Risk?
16.5 Interpreting the Market Price of Risk, and Risk Neutrality
16.6 Named Models
16.7 Equity and Fx Forwards and Futures When Rates are Stochastic
16.8 Futures Contracts
16.9 Summary
Further Reading
Exercises
Chapter 17: Yield Curve Fitting
17.1 Introduction
17.2 Ho & Lee
17.3 The Extended Vasicek Model of Hull & White
17.4 Yield-Curve Fitting: for and Against
17.5 Other Models
17.6 Summary
Further Reading
Exercises
Chapter 18: Interest Rate Derivatives
18.1 Introduction
18.2 Callable Bonds
18.3 Bond Options
18.4 Caps and Floors
18.5 Range Notes
18.6 Swaptions, Captions and Floortions
18.7 Spread Options
18.8 Index Amortizing Rate Swaps
18.9 Contracts with Embedded Decisions
18.10 Some Examples
18.11 More Interest Rate Derivatives …
18.12 Summary
Further Reading
Exercises
Chapter 19: The Heath, Jarrow & Morton and Brace, Gatarek & Musiela Models
19.1 Introduction
19.2 The Forward Rate Equation
19.3 The Spot Rate Process
19.4 The Market Price of Risk
19.5 Real and Risk Neutral
19.6 Pricing Derivatives
19.7 Simulations
19.8 Trees
19.9 The Musiela Parameterization
19.10 Multi-Factor HJM
19.11 Spreadsheet Implementation
19.12 A Simple one-Factor Example: Ho & Lee
19.13 Principal Component Analysis
19.14 Options on Equities, Etc.
19.15 Non-Infinitesimal Short Rate
19.16 The Brace, Gatarek & Musiela Model
19.17 Simulations
19.18 Pving the Cashflows
19.19 Summary
Further Reading
Exercises
Chapter 20: Investment Lessons from Blackjack and Gambling
20.1 Introduction
20.2 The Rules of Blackjack
20.3 Beating the Dealer
20.4 The Distribution of Profit in Blackjack
20.5 The Kelly Criterion
20.6 Can You win at Roulette?
20.7 Horse Race Betting and no Arbitrage
20.8 Arbitrage
20.9 How to Bet
20.10 Summary
Further Reading
Exercises
Chapter 21: Portfolio Management
21.1 Introduction
21.2 Diversification
21.3 Modern Portfolio Theory
21.4 Where do I Want to be on the Efficient Frontier?
21.5 Markowitz in Practice
21.6 Capital Asset Pricing Model
21.7 The Multi-Index Model
21.8 Cointegration
21.9 Performance Measurement
21.10 Summary
Further Reading
Exercises
Chapter 22: Value at Risk
22.1 Introduction
22.2 Definition of Value at Risk
22.3 VaR for a Single Asset
22.4 VaR for a Portfolio
22.5 VaR for Derivatives
22.6 Simulations
22.7 Use of VaR as a Performance Measure
22.8 Introductory Extreme Value Theory
22.9 Coherence
22.10 Summary
Further Reading
Exercises
Chapter 23: Credit Risk
23.1 Introduction
23.2 The Merton Model: Equity as an Option on a Company’s Assets
23.3 Risky Bonds
23.4 Modeling the Risk of Default
23.5 The Poisson Process and the Instantaneous Risk of Default
23.6 Time-Dependent Intensity and the Term Structure of Default
23.7 Stochastic Risk of Default
23.8 Positive Recovery
23.9 Hedging the Default
23.10 Credit Rating
23.11 A Model For Change of Credit Rating
23.12 Copulas: Pricing Credit Derivatives with Many Underlyings
23.13 Collateralized Debt Obligations
23.14 Summary
Further Reading
Exercises
Chapter 24: RiskMetrics and Credit Metrics
24.1 Introduction
24.2 The Riskmetrics Datasets
24.3 Calculating the Parameters the Riskmetrics Way
24.4 The Creditmetrics Dataset
24.5 The Creditmetrics Methodology
24.6 A Portfolio of Risky Bonds
24.7 Creditmetrics Model Outputs
24.8 Summary
Further Reading
Chapter 25: CrashMetrics
25.1 Introduction
25.2 Why do Banks Go Broke?
25.3 Market Crashes
25.4 Crashmetrics
25.5 Crashmetrics for One Stock
25.6 Portfolio Optimization and the Platinum Hedge
25.7 The Multi-Asset/Single-Index Model
25.8 Portfolio Optimization and the Platinum Hedge in the Multi-Asset Model
25.9 The Multi-Index Model
25.10 Incorporating Time Value
25.11 Margin Calls and Margin Hedging
25.12 Counterparty Risk
25.13 Simple Extensions to Crashmetrics
25.14 The Crashmetrics Index (CMI)
25.15 Summary
Further Reading
Exercises
Chapter 26: Derivatives **** Ups
26.1 Introduction
26.2 Orange County
26.3 Proctor and Gamble
26.4 Metallgesellschaft
26.5 Gibson Greetings
26.6 Barings
26.7 Long-Term Capital Management
26.8 Summary
Further Reading
Chapter 27: Overview of Numerical Methods
27.1 Introduction
27.2 Finite-Difference Methods
27.3 Monte Carlo Methods
27.4 Numerical Integration
27.5 Summary
Further Reading
Chapter 28: Finite-Difference Methods for One-Factor Models
28.1 Introduction
28.2 Grids
28.3 Differentiation Using the Grid
28.4 Approximating θ
28.5 Approximating A
28.6 Approximating Γ
28.7 Example
28.8 Bilinear Interpolation
28.9 Final Conditions and Payoffs
28.10 Boundary Conditions
28.11 The Explicit Finite-Difference Method
28.12 The Code #1: European Option
28.13 The Code #2: American Exercise
28.14 The Code #3: 2-D Output
28.15 Upwind Differencing
28.16 Summary
Further Reading
Exercises
Chapter 29: Monte Carlo simulation
29.1 Introduction
29.2 Relationship Between Derivative Values and Simulations: Equities, Indices, Currencies, Commodities
29.3 Generating Paths
29.4 Lognormal Underlying, No Path Dependency
29.5 Advantages of Monte Carlo Simulation
29.6 Using Random Numbers
29.7 Generating Normal Variables
29.8 Real Versus Risk Neutral, Speculation Versus Hedging
29.9 Interest Rate Products
29.10 Calculating the Greeks
29.11 Higher Dimensions: Cholesky Factorization
29.12 Calculation Time
29.13 Speeding Up Convergence
29.14 Pros and Cons of Monte Carlo Simulations
29.15 American Options
29.16 Longstaff & Schwartz Regression Approach for American Options
29.17 Basis Functions
29.18 Summary
Further Reading
Exercises
Chapter 30: Numerical Integration
30.1 Introduction
30.2 Regular Grid
30.3 Basic Monte Carlo Integration
30.4 Low-Discrepancy Sequences
30.5 Advanced Techniques
30.6 Summary
Further Reading
Exercises
Appendix A: All the Math You Need… and No More (an Executive Summary)
A.1 Introduction
A.2 e
A.3 log
A.4 Differentiation and Taylor Series
A.5 Differential Equations
A.6 Mean, Standard Deviation and Distributions
A.7 Summary
Appendix B: Forecasting the Markets?, A Small Digression
B.1 Introduction
B.2 Technical Analysis
B.3 Wave Theory
B.4 Other Analytics
B.5 Market Microstructure Modeling
B.6 Crisis Prediction
B.7 Summary
Further Reading
Appendix C: A Trading Game
C.1 Introduction
C.2 AIMS
C.3 Object of the Game
C.4 Rules of the Game
C.5 Notes
C.6 How to Fill in Your Trading Sheet
Appendix D: Contents of CD Accompanying Paul Wilmott Introduces Quantitative Finance, Second Edition
Appendix E: What You Get if (When) You Upgrade to PWOQF2
Introduction
Bibliography
Index
End User License Agreement
Paul Wilmott Introduces Quantitative Finance
Second Edition
© 2007 Paul Wilmott Published by John Wiley & Sons, Ltd
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Anniversary Logo Design: Richard J. Pacifico.
Library of Congress Cataloging-in-Publication Data
Wilmott, Paul. Paul Wilmott introduces quantitative finance.—2nd ed. p. cm. ISBN 978-0-470-31958-1 1. Finance—Mathematical models. 2. Options (Finance)—Mathematical models. 3. Options (Finance)—Prices— Mathematical models. I. Title. II Title: Quantitative finance. HG173.W493 2007 332—dc22
2007015893
A catalogue record for this book is available from the British Library.
ISBN 978-0-470-31958-1 (PB)
To a rising Star
Preface
In this book I present classical quantitative finance. The book is suitable for students on advanced undergraduate finance and derivatives courses, MBA courses, and graduate courses that are mainly taught, as opposed to ones that are based on research. The text is quite self-contained, with, I hope, helpful sidebars (Time Out’) covering the more mathematical aspects of the subject for those who feel a little bit uncomfortable. Little prior knowledge is assumed, other than basic calculus, even stochastic calculus is explained here in a simple, accessible way.
By the end of the book you should know enough quantitative finance to understand most derivative contracts, to converse knowledgeably about the subject at dinner parties, to land a job on Wall Street, and to pass your exams.
The structure of the book is quite logical. Markets are introduced, followed by the necessary math and then the two are melded together. The technical complexity is never that great, nor need it be. The last three chapters are on the numerical methods you will need for pricing. In the more advanced subjects, such as credit risk, the mathematics is kept to a minimum. Also, plenty of the chapters can be read without reference to the mathematics at all. The structure, mathematical content, intuition, etc., are based on many years’ teaching at universities and on the Certificate in Quantitative Finance, and training bank personnel at all levels.
The accompanying CD contains spreadsheets and Visual Basic programs implementing many of the techniques described in the text. The CD icon will be seen throughout the book, indicating material to be found on the CD, naturally. There is also a full list of its contents at the end of the book.
You can also find an Instructors Manual at www.wiley.com/go/pwiqf2 containing answers to the end-of-chapter questions in this book. The questions are, in general, of a mathematical nature but suited to a wide range of financial courses.
This book is a shortened version of Paul Wilmott on Quantitative Finance, second edition. It’s also more affordable than the ‘full’ version. However, I hope that you’ll eventually upgrade, perhaps when you go on to more advanced, research-based studies, or take that job on The Street.
PWOQF is, I am told, a standard text within the banking industry, but in Paul Wilmott Introduces Quantitative Finance I have specifically the university student in mind.
The differences between the university and the full versions are outlined at the end of the book. And to help you make the leap, we’ve included a form for you to upgrade, giving you a nice discount. Roughly speaking, the full version includes a great deal of non-classical, more modern approaches to quantitative finance, including several non-probabilistic models. There are more mathematical techniques for valuing exotic options and more markets are covered. The numerical methods are described in more detail.
If you have any problems understanding anything in the book, find errors, or just want a chat, email me at [email protected]. I’ll do my very best to respond as quickly as possible. Or visit www.wilmott.com to discuss quantitative finance, and other subjects, with other people in this business.
I would like to thank the following people. My partners in various projects: Paul and Jonathan Shaw and Gil Christie at 7city, unequaled in their dedication to training and their imagination for new ideas. Also Riaz Ahmad, Seb Lleo and Siyi Zhou who have helped make the Certificate in Quantitative Finance so successful, and for taking some of the pressure off me. Everyone involved in the magazine, especially Aaron Brown, Alan Lewis, Bill Ziemba, Caitlin Cornish, Dan Tudball, Ed Lound, Ed Thorp, Elie Ayache, Espen Gaarder Haug, Graham Russel, Henriette Präst, Jenny McCall, Kent Osband, Liam Larkin, Mike Staunton, Paula Soutinho and Rudi Bogni. I am particularly fortunate and grateful that John Wiley & Sons have been so supportive in what must sometimes seem to them rather wacky schemes. I am grateful to James Fahy for his work on my websites, and apologies for always failing to provide a coherent brief. Thanks also to David Epstein for help with the exercises, again; to Ron Henley, the best hedge fund partner a quant could wish for: “It’s just a jump to the left. And then a step to the right”; to John Morris of Fulcrum, interesting times; to all my lawyers for keeping the bad people away, Jared Stamell, Richard Schager, John Crow, Harry Issler, David Price and Kathryn van Gelder; and, of course, to Nassim Nicholas Taleb for entertaining chats.
Thanks to John, Grace, Sel and Stephen, for instilling in me their values. Values which have invariably served me well. And to Oscar and Zachary who kept me sane throughout many a series of unfortunate events!
Finally, thanks to my number one fan, Andrea Estrella, from her number one fan, me.
ABOUT THE AUTHOR
Paul Wilmott’s professional career spans almost every aspect of mathematics and finance, in both academia and in the real world. He has lectured at all levels, and founded a magazine, the leading website for the quant community, and a quant certificate program. He has managed money as a partner in a very successful hedge fund. He lives in London, is married, and has two sons. Although he enjoys quantitative finance his ideal job would be designing Kinder Egg toys.
You will see this icon whenever a method is implemented on the CD.
More info about the particular meaning of an icon is contained in its ‘speech box’.
CHAPTER 1
products and markets: equities, commodities, exchange rates, forwards and futures
The aim of this Chapter…
… is to describe some of the basic financial market products and conventions, to slowly introduce some mathematics, to hint at how stocks might be modeled using mathematics, and to explain the important financial concept of ‘no free lunch.’ By the end of the chapter you will be eager to get to grips with more complex products and to start doing some proper modeling.
In this Chapter…
1.1 INTRODUCTION
This first chapter is a very gentle introduction to the subject of finance, and is mainly just a collection of definitions and specifications concerning the financial markets in general. There is little technical material here, and the one technical issue, the ‘time value of money,’ is extremely simple. I will give the first example of ‘no arbitrage.’ This is important, being one part of the foundation of derivatives theory. Whether you read this chapter thoroughly or just skim it will depend on your background.
1.2 EQUITIES
The most basic of financial instruments is the equity, stock or share. This is the ownership of a small piece of a company. If you have a bright idea for a new product or service then you could raise capital to realize this idea by selling off future profits in the form of a stake in your new company. The investors may be friends, your Aunt Joan, a bank, or a venture capitalist. The investor in the company gives you some cash, and in return you give him a contract stating how much of the company he owns. The shareholders who own the company between them then have some say in the running of the business, and technically the directors of the company are meant to act in the best interests of the shareholders. Once your business is up and running, you could raise further capital for expansion by issuing new shares.
This is how small businesses begin. Once the small business has become a large business, your Aunt Joan may not have enough money hidden under the mattress to invest in the next expansion. At this point shares in the company may be sold to a wider audience or even the general public. The investors in the business may have no link with the founders. The final point in the growth of the company is with the quotation of shares on a regulated stock exchange so that shares can be bought and sold freely, and capital can be raised efficiently and at the lowest cost.
Figures 1.1 and 1.2 show screens from Bloomberg giving details of Microsoft stock, including price, high and low, names of key personnel, weighting in various indices, etc. There is much, much more info available on Bloomberg for this and all other stocks. We’ll be seeing many Bloomberg screens throughout this book.
Figure 1.1 Details of Microsoft stock.
Source: Bloomberg L.P.
Figure 1.2 Details of Microsoft stock continued.
Source: Bloomberg L.P.
In Figure 1.3 I show an excerpt from The Wall Street Journal Europe of 14th April 2005. This shows a small selection of the many stocks traded on the New York Stock Exchange. The listed information includes highs and lows for the day as well as the change since the previous day’s close.
Figure 1.3The Wall Street Journal Europe of 14th April 2005.
The behavior of the quoted prices of stocks is far from being predictable. In Figure 1.4 I show the Dow Jones Industrial Average over the period January 1950 to March 2004. In Figure 1.5 is a time series of the Glaxo–Wellcome share price, as produced by Bloomberg.
Figure 1.4 A time series of the Dow Jones Industrial Average from January 1950 to March 2004.
Figure 1.5 Glaxo–Wellcome share price (volume below).
Source: Bloomberg L.P.
If we could predict the behavior of stock prices in the future then we could become very rich. Although many people have claimed to be able to predict prices with varying degrees of accuracy, no one has yet made a completely convincing case. In this book I am going to take the point of view that prices have a large element of randomness. This does not mean that we cannot model stock prices, but it does mean that the modeling must be done in a probabilistic sense. No doubt the reality of the situation lies somewhere between complete predictability and perfect randomness, not least because there have been many cases of market manipulation where large trades have moved stock prices in a direction that was favorable to the person doing the moving. Having said that, I will digress slightly in Appendix B where I describe some of the popular methods for supposedly predicting future stock prices.
Figure 1.6 A simulation of an asset price path?
Figure 1.7 Simple spreadsheet to simulate the coin-tossing experiment.
See the simulation on the CD
Time Out…
More about coin tossing
Notice how in the above experiment I’ve chosen to multiply each ‘asset price’ by a factor, either 1.01 or 0.99. Why didn’t I simply add a fixed amount, 1 or −1, say? This is a very important point in the modeling of asset prices; as the asset price gets larger so do the changes from one day to the next. It seems reasonable to model the asset price changes as being proportional to the current level of the asset, they are still random but the magnitude of the randomness depends on the level of the asset. This will be made more precise in later chapters, where we’ll see how it is important to model the return on the asset, its percentage change, rather than its absolute value. And, of course, in this simple model the ‘asset price’ cannot go negative.
If we use the multiplicative rule we get an approximation to what is called a lognormal random walk, also geometric random walk. If we use the additive rule we get an approximation to a Normal or arithmetic random walk.
As an experiment, using Excel try to simulate both the arithmetic and geometric random walks, and also play around with the probability of a rise in asset price; it doesn’t have to be one half. What happens if you have an arithmetic random walk with a probability of rising being less than one half?
1.2.1 Dividends
The owner of the stock theoretically owns a piece of the company. This ownership can only be turned into cash if he owns so many of the stock that he can take over the company and keep all the profits for himself. This is unrealistic for most of us. To the average investor the value in holding the stock comes from the dividends and any growth in the stock’s value. Dividends are lump sum payments, paid out every quarter or every six months, to the holder of the stock.
The amount of the dividend varies from time to time depending on the profitability of the company. As a general rule companies like to try to keep the level of dividends about the same each time. The amount of the dividend is decided by the board of directors of the company and is usually set a month or so before the dividend is actually paid.
When the stock is bought it either comes with its entitlement to the next dividend (cum) or not (ex). There is a date at around the time of the dividend payment when the stock goes from cum to ex. The original holder of the stock gets the dividend but the person who buys it obviously does not. All things being equal a stock that is cum dividend is better than one that is ex dividend. Thus at the time that the dividend is paid and the stock goes ex dividend there will be a drop in the value of the stock. The size of this drop in stock value offsets the disadvantage of not getting the dividend.
This jump in stock price is in practice more complex than I have just made out. Often capital gains due to the rise in a stock price are taxed differently from a dividend, which is often treated as income. Some people can make a lot of risk-free money by exploiting tax ‘inconsistencies.’
1.2.2 Stock splits
Stock prices in the US are usually of the order of magnitude of $100. In the UK they are typically around £1. There is no real reason for the popularity of the number of digits, after all, if I buy a stock I want to know what percentage growth I will get, the absolute level of the stock is irrelevant to me, it just determines whether I have to buy tens or thousands of the stock to invest a given amount. Nevertheless there is some psychological element to the stock size. Every now and then a company will announce a stock split. For example, the company with a stock price of $90 announces a three-for-one stock split. This simply means that instead of holding one stock valued at $90, I hold three valued at $30 each.1
Figure 1.8 Stock split info for Microsoft.
Source: Bloomberg L.P.
1.3 COMMODITIES
Commodities are usually raw products such as precious metals, oil, food products, etc. The prices of these products are unpredictable but often show seasonal effects. Scarcity of the product results in higher prices. Commodities are usually traded by people who have no need of the raw material. For example they may just be speculating on the direction of gold without wanting to stockpile it or make jewelry. Most trading is done on the futures market, making deals to buy or sell the commodity at some time in the future. The deal is then closed out before the commodity is due to be delivered. Futures contracts are discussed below.
Figure 1.9 shows a time series of the price of pulp, used in paper manufacture.
Figure 1.9 Pulp price.
Source: Bloomberg L.P.
1.4 CURRENCIES
Another financial quantity we shall discuss is the exchange rate, the rate at which one currency can be exchanged for another. This is the world of foreign exchange, or Forex or FX for short. Some currencies are pegged to one another, and others are allowed to float freely. Whatever the exchange rates from one currency to another, there must be consistency throughout. If it is possible to exchange dollars for pounds and then the pounds for yen, this implies a relationship between the dollar/pound, pound/yen and dollar/yen exchange rates. If this relationship moves out of line it is possible to make arbitrage profits by exploiting the mispricing.
Figure 1.10 is an excerpt from The Wall Street Journal Europe of 22nd August 2006. At the bottom of this excerpt is a matrix of exchange rates. A similar matrix is shown in Figure 1.11 from Bloomberg.
Figure 1.10The Wall Street Journal Europe of 22nd August 2006, currency exchange rates.
Figure 1.11 Key cross currency rates.
Source: Bloomberg L.P.
Although the fluctuation in exchange rates is unpredictable, there is a link between exchange rates and the interest rates in the two countries. If the interest rate on dollars is raised while the interest rate on pounds sterling stays fixed we would expect to see sterling depreciating against the dollar for a while. Central banks can use interest rates as a tool for manipulating exchange rates, but only to a degree.
At the start of 1999 Euroland currencies were fixed at the rates shown in Figure 1.12.
Figure 1.12 Euro fixing rates.
Source: Bloomberg L.P.
1.5 INDICES
For measuring how the stock market/economy is doing as a whole, there have been developed the stock market indices. A typical index is made up from the weighted sum of a selection or basket of representative stocks. The selection may be designed to represent the whole market, such as the Standard & Poor’s 500 (S&P500) in the US or the Financial Times Stock Exchange index (FTSE100) in the UK, or a very special part of a market. In Figure 1.4 we saw the DJIA, representing major US stocks. In Figure 1.13 is shown JP Morgan’s Emerging Market Bond Index.
Figure 1.13 JP Morgan’s EMBI+.
The EMBI+ is an index of emerging market debt instruments, including external-currency-denominated Brady bonds, Eurobonds and US dollar local markets instruments. The main components of the index are the three major Latin American countries, Argentina, Brazil and Mexico. Bulgaria, Morocco, Nigeria, the Philippines, Poland, Russia and South Africa are also represented.
Figure 1.14 shows a time series of the MAE All Bond Index which includes Peso and US dollar denominated bonds sold by the Argentine Government.
Figure 1.14 A time series of the MAE All Bond Index.
Source: Bloomberg L.P.
1.6 THE TIME VALUE OF MONEY
The simplest concept in finance is that of the time value of money; $1 today is worth more than $1 in a year’s time. This is because of all the things we can do with $1 over the next year. At the very least, we can put it under the mattress and take it out in one year. But instead of putting it under the mattress we could invest it in a gold mine, or a new company. If those are too risky, then lend the money to someone who is willing to take the risks and will give you back the dollar with a little bit extra, the interest. That is what banks do, they borrow your money and invest it in various risky ways, but by spreading their risk over many investments they reduce their overall risk. And by borrowing money from many people they can invest in ways that the average individual cannot. The banks compete for your money by offering high interest rates. Free markets and the ability to quickly and cheaply change banks ensure that interest rates are fairly consistent from one bank to another.
Time Out…
Symbols
It had to happen sooner or later, and the first chapter is as good as anywhere. Our first mathematical symbol is nigh.
Please don’t be put off by the use of symbols if you feel more comfortable with numbers and concrete examples. I know that math is the one academic subject that can terrify adults, just because of poor teaching in schools. If you fall into this category, just go with the flow, concentrate on the words, the examples and the Time Outs, and before you know it…
I am going to denote interest rates by r. Although rates vary with time I am going to assume for the moment that they are constant. We can talk about several types of interest. First of all there is simple and compound interest. Simple interest is when the interest you receive is based only on the amount you initially invest, whereas compound interest is when you also get interest on your interest. Compound interest is the only case of relevance. And compound interest comes in two forms, discretely compounded and continuously compounded. Let me illustrate how they each work.
Suppose I invest $1 in a bank at a discrete interest rate of r paid once per annum. At the end of one year my bank account will contain
If the interest rate is 10% I will have one dollar and ten cents. After two years I will have
or one dollar and twenty-one cents. After n years I will have (1 + r)n. That is an example of discrete compounding.
Now suppose I receive m interest payments at a rate of r/m per annum. After one year I will have
(1.1)
Now I am going to imagine that these interest payments come at increasingly frequent intervals, but at an increasingly smaller interest rate: I am going to take the limit m → ∞. This will lead to a rate of interest that is paid continuously. Expression (1.1) becomes2
That is how much money I will have in the bank after one year if the interest is continuously compounded. And similarly, after a time t I will have an amount
(1.2)
in the bank. Almost everything in this book assumes that interest is compounded continuously.
Time Out…
The math so far
Let’s see m getting larger and larger in an example. I produced the next figure in Excel.
These functions are all plotted on a spreadsheet
As m gets larger and larger, so the curve seems to get smoother and smoother, eventually becoming the exponential function. We’ll be seeing this function a lot. In Excel the exponential function ex (also written exp(x)) is EXP().
What mathematics have we seen so far? To get to (1.2) all we needed to know about are the two functions, the exponential functione (or exp) and the logarithm log, and Taylor series. Believe it or not, you can appreciate almost all finance theory by knowing these three things together with ‘expectations.’ I’m going to build up to the basic Black–Scholes and derivatives theory assuming that you know all four of these. Don’t worry if you don’t know about these things, in Appendix AI review these requisites.
En passant, what would the above figures look like if interest were simple rather than compound? Which would you prefer to receive?
Another way of deriving the result (1.2) is via a differential equation. Suppose I have an amount M(t) in the bank at time t, how much does this increase in value from one day to the next? If I look at my bank account at time t and then again a short while later, time t + dt, the amount will have increased by
where the right-hand side comes from a Taylor series expansion. But I also know that the interest I receive must be proportional to the amount I have, M, the interest rate, r, and the time step, dt. Thus
Dividing by dt gives the ordinary differential equation
the solution of which is
Time Out…
Differential equations
This is our first differential equation; hang on in there, it’ll become second nature soon. Whenever you see d something over d something else you know you’re looking at a slope, or gradient, also known as rate of change or sensitivity. So here we’ve got the rate of change of money with time, i.e. rate of growth of money in the bank. You don’t need to know how I solved this differential equation really. In Appendix A I explain all about slope, sensitivities and differential equations.
This first differential equation is an example of an ordinary differential equation, there is only one independent variablet. M is the dependent variable, its value depends on t. We’ll also be seeing partial differential equations where there is more than one independent variable. And we’ll also see quite a few stochastic differential equations. These are equations with a random term in them, used for modeling the randomness in the financial world. For the next few chapters there will be no more mention of differential equations. Whew.
This equation relates the value of the money I have now to the value in the future. Conversely, if I know I will get one dollar at time T in the future, its value at an earlier time t is simply
The present value is clearly less than the future value.
Interest rates are a very important factor determining the present value of future cashflows. For the moment I will only talk about one interest rate, and that will be constant. In later chapters I will generalize.
1.7 FIXED-INCOME SECURITIES
In lending money to a bank you may get to choose for how long you tie your money up and what kind of interest rate you receive. If you decide on a fixed-term deposit the bank will offer to lock in a fixed rate of interest for the period of the deposit, a month, six months, a year, say. The rate of interest will not necessarily be the same for each period, and generally the longer the time that the money is tied up the higher the rate of interest, although this is not always the case. Often, if you want to have immediate access to your money then you will be exposed to interest rates that will change from time to time, since interest rates are not constant.
These two types of interest payments, fixed and floating, are seen in many financial instruments. Coupon-bearing bonds pay out a known amount every six months or year, etc. This is the coupon and would often be a fixed rate of interest. At the end of your fixed term you get a final coupon and the return of the principal, the amount on which the interest was calculated. Interest rate swaps are an exchange of a fixed rate of interest for a floating rate of interest. Governments and companies issue bonds as a form of borrowing. The less creditworthy the issuer, the higher the interest that they will have to pay out. Bonds are actively traded, with prices that continually fluctuate.
1.8 INFLATION-PROOF BONDS
A very recent addition to the list of bonds issued by the US Government is the index-linked bond. These have been around in the UK since 1981, and have provided a very successful way of ensuring that income is not eroded by inflation.
In the UK inflation is measured by the Retail Price Index or RPI. This index is a measure of year-on-year inflation, using a ‘basket’ of goods and services including mortgage interest payments. The index is published monthly. The coupons and principal of the index-linked bonds are related to the level of the RPI. Roughly speaking, the amounts of the coupon and principal are scaled with the increase in the RPI over the period from the issue of the bond to the time of the payment. There is one slight complication in that the actual RPI level used in these calculations is set back eight months. Thus the base measurement is eight months before issue and the scaling of any coupon is with respect to the increase in the RPI from this base measurement to the level of the RPI eight months before the coupon is paid. One of the reasons for this complexity is that the initial estimate of the RPI is usually corrected at a later date.
Figure 1.15 shows the UK gilts prices published in The Financial Times of 22nd August 2006. The index-linked bonds are on the right.
Figure 1.15 UK gilts prices from The Financial Times of 22nd August 2006.
In the US the inflation index is the Consumer Price Index (CPI). A time series of this index is shown in Figure 1.16.
Figure 1.16 The CPI index.
Source: Bloomberg LP.
I will not pursue the modeling of inflation or index-linked bonds in this book. I would just like to say that the dynamics of the relationship between inflation and short-term interest rates is particularly interesting. Clearly the level of interest rates will affect the rate of inflation directly through mortgage repayments, but also interest rates are often used by central banks as a tool for keeping inflation down.
1.9 FORWARDS AND FUTURES
A forward contract is an agreement where one party promises to buy an asset from another party at some specified time in the future and at some specified price. No money changes hands until the delivery date or maturity of the contract. The terms of the contract make it an obligation to buy the asset at the delivery date, there is no choice in the matter. The asset could be a stock, a commodity or a currency.
The amount that is paid for the asset at the delivery date is called the delivery price. This price is set at the time that the forward contract is entered into, at an amount that gives the forward contract a value of zero initially. As we approach maturity the value of this particular forward contract that we hold will change in value, from initially zero to, at maturity, the difference between the underlying asset and the delivery price.
In the newspapers we will also see quoted the forward price for different maturities. These prices are the delivery prices for forward contracts of the quoted maturities, should we enter into such a contract now.
Try to distinguish between the value of a particular contract during its life and the specification of the delivery price at initiation of the contract. It’s all very subtle. You might think that the forward price is the market’s view on the asset value at maturity; this is not quite true as we’ll see shortly. In theory, the market’s expectation about the value of the asset at maturity of the contract is irrelevant.
A futures contract is very similar to a forward contract. Futures contracts are usually traded through an exchange, which standardizes the terms of the contracts. The profit or loss from the futures position is calculated every day and the change in this value is paid from one party to the other. Thus with futures contracts there is a gradual payment of funds from initiation until maturity.
Because you settle the change in value on a daily basis, the value of a futures contract at any time during its life is zero. The futures price varies from day to day, but must at maturity be the same as the asset that you are buying.
I’ll show later that provided interest rates are known in advance, forward prices and futures prices of the same maturity must be identical.
Forwards and futures have two main uses, in speculation and in hedging. If you believe that the market will rise you can benefit from this by entering into a forward or futures contract. If your market view is right then a lot of money will change hands (at maturity or every day) in your favor. That is speculation and is very risky. Hedging is the opposite, it is avoidance of risk. For example, if you are expecting to get paid in yen in six months’ time, but you live in America and your expenses are all in dollars, then you could enter into a futures contract to lock in a guaranteed exchange rate for the amount of your yen income. Once this exchange rate is locked in you are no longer exposed to fluctuations in the dollar/yen exchange rate. But then you won’t benefit if the yen appreciates.
1.9.1 A first example of no arbitrage
Although I won’t be discussing futures and forwards very much they do provide us with our first example of the no-arbitrage principle. I am going to introduce some more mathematical notation now, it will be fairly consistent throughout the book. Consider a forward contract that obliges us to hand over an amount $F at time T to receive the underlying asset. Today’s date is t and the price of the asset is currently $S(t), this is the spot price, the amount for which we could get immediate delivery of the asset. When we get to maturity we will hand over the amount $F and receive the asset, then worth $S(T). How much profit we make cannot be known until we know the value $S(T), and we cannot know this until time T. From now on I am going to drop the ‘$’ sign from in front of monetary amounts.
We know all of F, S(t), t and T, but is there any relationship between them? You might think not, since the forward contract entitles us to receive an amount S(T) − F at expiry and this is unknown. However, by entering into a special portfolio of trades now we can eliminate all randomness in the future. This is done as follows.
Enter into the forward contract. This costs us nothing up front but exposes us to the uncertainty in the value of the asset at maturity. Simultaneously sell the asset. It is called going short when you sell something you don’t own. This is possible in many markets, but with some timing restrictions. We now have an amount S(t) in cash due to the sale of the asset, a forward contract, and a short asset position. But our net position is zero. Put the cash in the bank, to receive interest.
When we get to maturity we hand over the amount F and receive the asset, this cancels our short asset position regardless of the value of S(T). At maturity we are left with a guaranteed −F in cash as well as the bank account. The word ‘guaranteed’ is important because it emphasizes that it is independent of the value of the asset. The bank account contains the initial investment of an amount S(t) with added interest, this has a value at maturity of
Our net position at maturity is therefore
Since we began with a portfolio worth zero and we end up with a predictable amount, that predictable amount should also be zero. We can conclude that
(1.3)
This is the relationship between the spot price and the forward price. It is a linear relationship, the forward price is proportional to the spot price.
The cashflows in this special hedged portfolio are shown in Table 1.1.
Table 1.1 Cashflows in a hedged portfolio of asset and forward.
Holding
Worth today (
t
)
Worth at maturity (
T
)
Forward
0
S
(
T
) −
F
−Stock
−
S
(
t
)
−
S
(
T
)
Cash
S
(
t
)
S
(
t
)
e
r(
T-t
)
Total
0
S
(
t
)
e
(
T−t
)
−
F
Time Out…
No arbitrage again
Example: The spot asset price S is 28.75, the one-year forward price F is 30.20 and the one-year interest rate is 4.92%. Are these numbers consistent with no arbitrage?
This is effectively zero to the number of decimal places quoted.
If we know any three out of S, F, r and T − t we can find the fourth, assuming there are no arbitrage possibilities. Note that the forward price in no way depends on what the asset price is expected to do, whether it is expected to increase or decrease in value.
In Figure 1.17 is a path taken by the spot asset price and its forward price. As long as interest rates are constant, these two are related by (1.3).
Figure 1.17 A time series of a spot asset price and its forward price.
If this relationship is violated then there will be an arbitrage opportunity. To see what is meant by this, imagine that F is less than S(t)er(T-t). To exploit this and make a riskless arbitrage profit, enter into the deals as explained above. At maturity you will have S(t)er(T-t) in the bank, a short asset and a long forward. The asset position cancels when you hand over the amount F, leaving you with a profit of S(t)er(T-t) − F. If F is greater than that given by (1.3) then you enter into the opposite positions, going short the forward. Again you make a riskless profit. The standard economic argument then says that investors will act quickly to exploit the opportunity, and in the process prices will adjust to eliminate it.
1.10 MORE ABOUT FUTURES
Futures are usually traded through an exchange. This means that they are very liquid instruments and have lots of rules and regulations surrounding them. Here are a few observations on the nature of futures contracts.
Available assets A futures contract will specify the asset which is being invested in. This is particularly interesting when the asset is a natural commodity because of non-uniformity in the type and quality of the asset to be delivered. Most commodities come in a variety of grades. Oil, sugar, orange juice, wheat, etc. futures contracts lay down rules for precisely what grade of oil, sugar, etc. may be delivered. This idea even applies in some financial futures contracts. For example, bond futures may allow a range of bonds to be delivered. Since the holder of the short position gets to choose which bond to deliver he naturally chooses the cheapest.
The contract also specifies how many of each asset must be delivered. The quantity will depend on the market.
Delivery and settlement The futures contract will specify when the asset is to be delivered. There may be some leeway in the precise delivery date. Most futures contracts are closed out before delivery, with the trader taking the opposite position before maturity. But if the position is not closed then delivery of the asset is made. When the asset is another financial contract settlement is usually made in cash.
Margin I said above that changes in the value of futures contracts are settled each day. This is called marking to market. To reduce the likelihood of one party defaulting, being unable or unwilling to pay up, the exchanges insist on traders depositing a sum of money to cover changes in the value of their positions. This money is deposited in a margin account. As the position is marked to market daily, money is deposited or withdrawn from this margin account.
Margin comes in two forms, the initial margin and the maintenance margin. The initial margin is the amount deposited at the initiation of the contract. The total amount held as margin must stay above a prescribed maintenance margin. If it ever falls below this level then more money (or equivalent in bonds, stocks, etc.) must be deposited. The levels of these margins vary from market to market.
Margin has been much neglected in the academic literature. But a poor understanding of the subject has led to a number of famous financial disasters, most notably Metallgesellschaft and Long Term Capital Management. We’ll discuss the details of these cases in Chapter 26, and we’ll also be seeing how to model margin and how to margin hedge.
1.10.1 Commodity futures
Futures on commodities don’t necessarily obey the no-arbitrage law that led to the asset/future price relationship explained above. This is because of the messy topic of storage. Sometimes we can only reliably find an upper bound for the futures price. Will the futures price be higher or lower than the theoretical no-storage-cost amount? Higher. The holder of the futures contract must compensate the holder of the commodity for his storage costs. This can be expressed in percentage terms by an adjustment s to the risk-free rate of interest.
But things are not quite so simple. Most people actually holding the commodity are benefiting from it in some way. If it is something consumable, such as oil, then the holder can benefit from it immediately in whatever production process they are engaged in. They are naturally reluctant to part with it on the basis of some dodgy theoretical financial calculation. This brings the futures price back down. The benefit from holding the commodity is commonly measured in terms of the convenience yieldc:
Observe how the storage cost and the convenience yield act in opposite directions on the price. Whenever
the market is said to be in backwardation. Whenever
the market is in contango.
1.10.2 FX futures
There are no problems associated with storage when the asset is a currency. We need to modify the no-arbitrage result to allow for interest received on the foreign currency rf. The result is
The confirmation of this is an easy exercise.
1.10.3 Index futures
Futures contracts on stock indices are settled in cash. Again, there are no storage problems, but now we have dividends to contend with. Dividends play a role similar to that of a foreign interest rate on FX futures. So
Here q is the dividend yield. This is clearly an approximation. Each stock in an index receives a dividend at discrete intervals, but can these all be approximated by one continuous dividend yield?
1.11 SUMMARY
The above descriptions of financial markets are enough for this introductory chapter. Perhaps the most important point to take away with you is the idea of no arbitrage. In the example here, relating spot prices to futures prices, we saw how we could set up a very simple portfolio which completely eliminated any dependence on the future value of the stock. When we come to value derivatives, in the way we just valued a forward, we will see that the same principle can be applied albeit in a far more sophisticated way.
FURTHER READING
For general financial news visit
www.bloomberg.com
and
www.reuters.com
. CNN has online financial news at
www.cnnfn.com
. There are also online editions of
The Wall Street Journal
,
www.wsj.com
,
The Financial Times
,
www.ft.com
and
Futures and Options World
,
www.fow.com
.
For more information about futures see the Chicago Board of Trade website
www.cbot.com
.
Many, many financial links can be found at Wahoo!,
www.io.com/~gibbonsb/wahoo.html
.
See Bloch (1995) for an empirical analysis of inflation data and a theoretical discussion of pricing index-linked bonds.
In the main, we’ll be assuming that markets are random. For insight about alternative hypotheses see Schwager (1990, 1992).
See Brooks (1967) for how the raising of capital for a business might work in practice.
Cox,
et al.
(1981) discuss the relationship between forward and future prices.
EXERCISES
1 In the UK this would be called a two-for-one split.
2 The symbol ~, called ‘tilde,’ is like ‘approximately equal to,’ but with a slightly more technical, in a math sense, meaning. The symbol → means ‘tends to.’
