Paul Wilmott OnQuantitative Finance

Paul Wilmott OnQuantitative Finance

Second Edition

www.wilmott.com

Copyright 2006 Paul Wilmott

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Library of Congress Cataloging-in-Publication Data

Wilmott, Paul.Paul Wilmott on quantitative finance.—2nd ed.

p. cm.Includes bibliographical references and index.ISBN 13 978-0-470-01870-5 (cloth/cd : alk. paper)ISBN 10 0-470-01870-4 (cloth/cd : alk. paper)

1. Derivative securities—Mathematical models. 2. Options (Finance)—Mathematical models. 3. Options (Finance)—Prices—Mathematical models. I. Title.

HG6024.A3W555 2006332.64′53—dc22

2005028317

British Library Cataloguing in Publication Data

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

ISBN-13: 978-0-470-01870-5 (HB)ISBN-10: 0-470-01870-4 (HB)

Typeset in 10/12pt Times by Laserwords Private Limited, Chennai, IndiaPrinted and bound in Great Britain by Antony Rowe Ltd, Chippenham, WiltshireThis book is printed on acid-free paper responsibly manufactured from sustainable forestryin which at least two trees are planted for each one used for paper production.

In memory of Detlev Vogel

contents of volume one

Visual Basic Code xxv

Prolog to the Second Edition xxvii

PART ONE MATHEMATICAL AND FINANCIAL FOUNDATIONS; BASICTHEORY OF DERIVATIVES; RISK AND RETURN 1

1 Products and Markets 5

2 Derivatives 25

3 The Random Behavior of Assets 55

4 Elementary Stochastic Calculus 71

5 The Black–Scholes Model 91

6 Partial Differential Equations 101

7 The Black–Scholes Formulae and the ‘Greeks’ 109

8 Simple Generalizations of the Black–Scholes World 139

9 Early Exercise and American Options 151

10 Probability Density Functions and First-exit Times 169

11 Multi-asset Options 183

12 How to Delta Hedge 197

13 Fixed-income Products and Analysis: Yield, Duration and Convexity 225

14 Swaps 251

viii contents

15 The Binomial Model 261

16 How Accurate is the Normal Approximation? 295

17 Investment Lessons from Blackjack and Gambling 301

18 Portfolio Management 317

19 Value at Risk 331

20 Forecasting the Markets? 343

21 A Trading Game 359

contents ix

contents of volume twoPART TWO EXOTIC CONTRACTS AND PATH DEPENDENCY 365

22 An Introduction to Exotic and Path-dependent Derivatives 367

23 Barrier Options 385

24 Strongly Path-dependent Derivatives 417

25 Asian Options 427

26 Lookback Options 445

27 Derivatives and Stochastic Control 453

28 Miscellaneous Exotics 461

29 Equity and FX Term Sheets 481

PART THREE FIXED-INCOME MODELING AND DERIVATIVES 507

30 One-factor Interest Rate Modeling 509

31 Yield Curve Fitting 525

32 Interest Rate Derivatives 533

33 Convertible Bonds 553

34 Mortgage-backed Securities 571

35 Multi-factor Interest Rate Modeling 581

36 Empirical Behavior of the Spot Interest Rate 595

37 The Heath, Jarrow & Morton and Brace, Gatarek & Musiela Models 609

38 Fixed-income Term Sheets 627

PART FOUR CREDIT RISK 637

39 Value of the Firm and the Risk of Default 639

40 Credit Risk 649

x contents

41 Credit Derivatives 675

42 RiskMetrics and CreditMetrics 701

43 CrashMetrics 709

44 Derivatives **** Ups 731

contents xix

contents of volume threePART FIVE ADVANCED TOPICS 745

45 Financial Modeling 749

46 Defects in the Black–Scholes Model 755

47 Discrete Hedging 763

48 Transaction Costs 783

49 Overview of Volatility Modeling 813

50 Deterministic Volatility Surfaces 833

51 Stochastic Volatility 853

52 Uncertain Parameters 869

53 Empirical Analysis of Volatility 881

54 Stochastic Volatility and Mean-variance Analysis 889

55 Asymptotic Analysis of Volatility 901

56 Volatility Case Study: The Cliquet Option 915

57 Jump Diffusion 927

58 Crash Modeling 939

59 Speculating with Options 953

60 Static Hedging 969

61 The Feedback Effect of Hedging in Illiquid Markets 989

62 Utility Theory 1005

63 More About American Options and Related Matters 1013

64 Advanced Dividend Modeling 1035

65 Serial Autocorrelation in Returns 1045

66 Asset Allocation in Continuous Time 1051

xx contents

67 Asset Allocation Under Threat of a Crash 1061

68 Interest-rate Modeling Without Probabilities 1077

69 Pricing and Optimal Hedging of Derivatives, the Non-probabilisticModel Cont’d 1099

70 Extensions to the Non-probabilistic Interest-rate Model 1117

71 Modeling Inflation 1129

72 Energy Derivatives 1141

73 Real Options 1151

74 Life Settlements and Viaticals 1161

75 Bonus Time 1175

PART SIX NUMERICAL METHODS AND PROGRAMS 1189

76 Overview of Numerical Methods 1191

77 Finite-difference Methods for One-factor Models 1199

78 Further Finite-difference Methods for One-factor Models 1227

79 Finite-difference Methods for Two-factor Models 1253

80 Monte Carlo Simulation 1263

81 Numerical Integration 1285

82 Finite-difference Programs 1295

83 Monte Carlo Programs 1311

Appendix A All the Math You Need. . . and No More (An Executive Summary) 1317

Bibliography 1329

Index 1351

visual basic code

Implied volatility, Newton–Raphson 130Cumulative distribution for Normal variable 131The binomial method, European option 286The binomial method, American option 290Double knock-out barrier option, finite difference 490Instalment knock-out barrier option, finite difference 493Range Note, finite difference 497Lookback, finite difference 501Index Amortizing Rate Swap, finite difference 634Cliquet option, uncertain volatility, finite difference 923Optimization subroutine 983Setting up final condition, finite difference 1212Finite difference time loop, first example 1213European option, finite difference, three dimensions 1215American option, finite difference, three dimensions 1219European or American option, finite difference, two dimensions 1221Upwind differencing, interest rate 1225LU decomposition 1234Matrix solution 1235Successive over relaxation 1238Successive over relaxation, early exercise 1246Jump condition for discrete dividends 1248Jump condition for path-dependent quantities 1249Two-factor explicit finite difference 1257Convertible bond constraint 1257Box–Muller 1269Cholesky factorization 1276Numerical integration, Monte Carlo 1287Halton number generation 1290Kolmogorov equation, explicit finite difference 1295Convertible bond time stepping fragment, explicit finite difference 1297American option, implicit finite difference 1298Parisian option, explicit finite difference 1299Passport option, explicit finite difference 1300

xxii visual basic code

Chooser Passport option, explicit finite difference 1301Stochastic volatility, explicit finite difference 1303Uncertain volatility, gamma rule 1304Crash model, finite difference code fragment 1305Epstein–Wilmott model, finite difference 1305Risky bond, explicit finite difference 1307Basket option, Monte Carlo 1311Basket option, quasi Monte Carlo 1313American option, Monte Carlo 1314

prolog to the second edition

This book is a greatly updated and expanded version of the first edition. The content continuesto reflect my own interests and prejudices, based on my skills, such as they are. In the periodbetween the first and second editions, the financial markets have expanded, the tools availableto the modeler have expanded, and my girth has expanded. On a personal basis I have spent asmuch time being a practitioner in a hedge fund as being an independent researcher. Much of thenew material therefore represents both my desire as a scientist to build the best, most accuratemodels, and my need as a practitioner to have models that are fast and robust and simple tounderstand. As I said, this book is a very personal account of my areas of expertise. Since thesubject of quant finance has been galloping apace of late, I advise that you supplement thisbook with the specialized books that I recommend throughout, and in particular those in thequant library at the end.

I would like to re-thank those people I mentioned in the prolog to the first edition: Arefin Huq,Asli Oztukel, Bafkam Bim, Buddy Holly, Chris McCoy, Colin Atkinson, Daniel Bruno, DaveThomson, David Bakstein, David Epstein, David Herring, David Wilson, Edna Hepburn-Ruston,Einar Holstad, Eli Lilly, Elisabeth Keck, Elsa Cortina, Eric Cartman, Fouad Khennach, GlenMatlock, Henrik Rassmussen, Hyungsok Ahn, Ingrid Blauer, Jean Laidlaw, Jeff Dewynne, JohnLydon, John Ockendon, Karen Mason, Keesup Choe, Malcolm McLaren, Mauricio Bouabci,Patricia Sadro, Paul Cook, Peter Jackel, Philip Hua, Philipp Schonbucher, Phoebus Theologites,Quentin Crisp, Rich Haber, Richard Arkell, Richard Sherry, Sam Ehrlichman, Sandra Maler,Sara Statman, Simon Gould, Simon Ritchie, Stephen Jefferies, Steve Jones, Truman Capote,Varqa Khadem, and Veronika Guggenbichler.

I would also like to thank the following people. My partners in various projects: Paul andJonathan Shaw at 7city, unequaled in their dedication to training and their imagination for newprojects. Also Riaz Ahmad and Seb Lleo who have helped make the Certificate in QuantitativeFinance so successful, and for taking some of the pressure off me; Everyone involved in themagazine, especially Aaron Brown, Alan Lewis, Bill Ziemba, Caitlin Cornish, Dan Tudball, EdLound, Ed Thorp, Elie Ayache, Espen Gaarder Haug, Graham Russel, Henriette Prast, JennyMcCall, Kent Osband, Liam Larkin, Mike Staunton, Paula Soutinho and Rudi Bogni. I amparticularly fortunate and grateful that John Wiley & Sons have been so supportive in whatmust sometimes seem to them rather wacky schemes; Thanks to Ron Henley, the best hedgefund partner a quant could wish for, ‘It’s just a jump to the left. And then a step to theright.’ And to John Morris of Fulcrum, interesting times; and to Nassim Nicholas Taleb forinteresting chats.

xxiv prolog to the second edition

Thanks to, John, Grace, Sel and Stephen, for instilling in me their values: values which haveinvariably served me well. And to Oscar and Zachary who kept me sane throughout many aseries 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, inboth academia and in the real world. He has lectured at all levels, founded a magazine, theleading website for the quant community, and a quant certificate program. He has managedmoney as a partner in a very successful hedge fund. He lives in London, is married, and hastwo sons. His only remaining dream is to get some sleep.

prolog to the second edition xxv

More info about the particular meaning of an icon is contained in its ‘speech box.’

You will see this icon whenever a method is implemented on the CD.

PART ONE

mathematical and financialfoundations; basic theoryof derivatives; risk and

return

The first part of the book contains the fundamentals of derivatives theory and practice. We lookat both equity and fixed income instruments. I introduce the important concepts of hedging andno arbitrage, on which most sophisticated finance theory is based. We draw some insight fromideas first seen in gambling, and we develop those into an analysis of risk and return.

The assumptions, key concepts and results in Part One make up what is loosely known as the‘Black–Scholes world,’ named for Fischer Black and Myron Scholes who, together with RobertMerton, first conceived them. Their original work was published in 1973, after some resistance(the famous equation was first written down in 1969). In October 1997 Myron Scholes andRobert Merton were awarded the Nobel Prize for Economics for their work, Fischer Blackhaving died in August 1995. The New York Times of Wednesday, 15th October 1997 wrote:‘Two North American scholars won the Nobel Memorial Prize in Economic Science yesterdayfor work that enables investors to price accurately their bets on the future, a breakthroughthat has helped power the explosive growth in financial markets since the 1970’s and plays aprofound role in the economics of everyday life.’1

Part One is self contained, requiring little knowledge of finance or any more than elementarycalculus.

Chapter 1: Products and Markets An overview of the workings of the financial markets andtheir products. A chapter such as this is obligatory. However, my readers will fall into one oftwo groups. Either they will know everything in this chapter and much, much more besides.Or they will know little, in which case what I write will not be enough.

1 We’ll be hearing more about these two in Chapter 44 on ‘Derivatives **** Ups.’

2 Part One mathematical and financial foundations

Chapter 2: Derivatives An introduction to options, options markets, market conventions.Definitions of the common terms, simple no arbitrage, put-call parity and elementary tradingstrategies.

Chapter 3: The Random Behavior of Assets An examination of data for various financialquantities, leading to a model for the random behavior of prices. Almost all of sophisticatedfinance theory assumes that prices are random, the question is how to model that randomness.

Chapter 4: Elementary Stochastic Calculus We’ll need a little bit of theory for manipulatingour random variables. I keep the requirements down to the bare minimum. The key concept isIto’s lemma which I will try to introduce in as accessible a manner as possible.

Chapter 5: The Black–Scholes Model I present the classical model for the fair value of optionson stocks, currencies and commodities. This is the chapter in which I describe delta hedgingand no arbitrage and show how they lead to a unique price for an option. This is the foundationfor most quantitative finance theory and I will be building on this foundation for much, but byno means all, of the book.

Chapter 6: Partial Differential Equations Partial differential equations play an important rolein most physical applied mathematics. They also play a role in finance. Most of my readerstrained in the physical sciences, engineering and applied mathematics will be comfortable withthe idea that a partial differential equation is almost the same as ‘the answer,’ the two beingseparated by at most some computer code. If you are not sure of this connection I hope thatyou will persevere with the book. This requires some faith on your part; you may have to readthe book through twice: I have necessarily had to relegate the numerics, the real ‘answer,’ tothe last few chapters.

Chapter 7: The Black–Scholes Formulae and the ‘Greeks’ From the Black–Scholes partialdifferential equation we can find formulae for the prices of some options. Derivatives of optionprices with respect to variables or parameters are important for hedging. I will explain someof the most important such derivatives and how they are used.

Chapter 8: Simple Generalizations of the Black–Scholes World Some of the assumptionsof the Black–Scholes world can be dropped or stretched with ease. I will describe several ofthese. Later chapters are devoted to more extensive generalizations.

Chapter 9: Early Exercise and American Options Early exercise is of particular importancefinancially. It is also of great mathematical interest. I will explain both of these aspects.

Chapter 10: Probability Density Functions and First-exit Times The random nature of finan-cial quantities means that we cannot say with certainty what the future holds in store. For thatreason we need to be able to describe that future in a probabilistic sense.

Chapter 11: Multi-asset Options Another conceptually simple generalization of the basicBlack–Scholes world is to options on more than one underlying asset. Theoretically simple,this extension has its own particular problems in practice.

Chapter 12: How to Delta Hedge Not everyone believes in no arbitrage, the absence of freelunches. In this chapter we see how to profit if you have a better forecast for future volatilitythan the market.

mathematical and financial foundations Part One 3

Chapter 13: Fixed-income Products and Analysis: Yield, Duration and Convexity Thischapter is an introduction to the simpler techniques and analyses commonly used in the market.In particular I explain the concepts of yield, duration and convexity. In this and the next chapterI assume that interest rates are known, deterministic quantities.

Chapter 14: Swaps Interest-rate swaps are very common and very liquid. I explain the basicsand relate the pricing of swaps to the pricing of fixed-rate bonds.

Chapter 15: The Binomial Model One of the reasons that option theory has been so successfulis that the ideas can be explained and implemented very easily with no complicated mathematics.The binomial model is the simplest way to explain the basic ideas behind option theory usingonly basic arithmetic. That’s a good thing, right? Yes, but only if you bear in mind that themodel is for demonstration purposes only, it is not the real thing. As a model of the financialworld it is too simplistic, as a concept for pricing it lacks the elegance that makes other methodspreferable, and as a numerical scheme it is prehistoric. Use once and then throw away, that’smy recommendation.

Chapter 16: How Accurate is the Normal Approximation? One of the major assumptions offinance theory is that returns are Normally distributed. In this chapter we take a look at whywe make this assumption, and how good it really is.

Chapter 17: Investment Lessons from Blackjack and Gambling We draw insights and inspi-ration from the not-unrelated world of gambling to help us in the treatment of risk, return, andmoney/risk management.

Chapter 18: Portfolio Management If you are willing to accept some risk how should youinvest? I explain the classical ideas of Modern Portfolio Theory and the Capital Asset PricingModel.

Chapter 19: Value at Risk How risky is your portfolio? How much might you conceivablylose if there is an adverse market move? These are the topics of this chapter.

Chapter 20: Forecasting the Markets? Although almost all sophisticated finance theoryassumes that assets move randomly, many traders rely on technical indicators to predict thefuture direction of assets. These indicators may be simple geometrical constructs of the assetprice path or quite complex algorithms. The hypothesis is that information about short-termfuture asset price movements are contained within the past history of prices. All traders usetechnical indicators at some time. In this chapter I describe some of the more common tech-niques.

Chapter 21: A Trading Game Many readers of this book will never have traded anything moresophisticated than baseball cards. To get them into the swing of the subject from a practicalpoint of view I include some suggestions on how to organize your own trading game based onthe buying and selling of derivatives. I had a lot of help with this chapter from David Epsteinwho has been running such games for several years.

CHAPTER 1

products and markets

In this Chapter. . .

• the time value of money• an introduction to equities, commodities, currencies and indices• fixed and floating interest rates• futures and forwards• no-arbitrage, one of the main building blocks of finance theory

1.1 INTRODUCTION

This first chapter is a very gentle introduction to the subject of finance, and is mainly just acollection of definitions and specifications concerning the financial markets in general. There islittle technical material here, and the one technical issue, the ‘time value of money,’ is extremelysimple. I will give the first example of ‘no arbitrage.’ This is important, being one part of thefoundation of derivatives theory. Whether you read this chapter thoroughly or just skim it willdepend on your background; mathematicians new to finance may want to spend more time onit than practitioners, say.

1.2 THE TIME VALUE OF MONEY

The simplest concept in finance is that of the time value ofmoney; $1 today is worth more than $1 in a year’s time. Thisis because of all the things we can do with $1 over the nextyear. At the very least, we can put it under the mattress and takeit out in one year. But instead of putting it under the mattresswe could invest it in a gold mine, or a new company. If thoseare too risky, then lend the money to someone who is willingto take the risks and will give you back the dollar with a little bit extra, the interest. That iswhat banks do, they borrow your money and invest it in various risky ways, but by spreadingtheir risk over many investments they reduce their overall risk. And by borrowing money frommany people they can invest in ways that the average individual cannot. The banks competefor your money by offering high interest rates. Free markets and the ability to change banksquickly and cheaply ensure that interest rates are fairly consistent from one bank to another.

6 Part One mathematical and financial foundations

I am going to denote interest rates by r . Although rates vary with time I am going to assumefor the moment that they are constant. We can talk about several types of interest. First of allthere is simple and compound interest. Simple interest is when the interest you receive isbased only on the amount you invest initially, whereas compound interest is when you alsoget interest on your interest. Compound interest is the main case of relevance. And compoundinterest comes in two forms, discretely compounded and continuously compounded. Let meillustrate how they each work.

Suppose I invest $1 in a bank at a discrete interest rate of r paid once per annum. At theend of one year my bank account will contain

$1 × (1 + r).

If the interest rate is 10% I will have one dollar and ten cents. After two years I will have

$1 × (1 + r) × (1 + r) = (1 + r)2,

or one dollar and twenty-one cents. After n years I will have (1 + r)n dollars. That is anexample of discrete compounding.

Now suppose I receive m interest payments at a rate of r/m per annum. After one year Iwill have

(1 + r

m

)m

. (1.1)

(I have dropped the $ sign, taking it as read from now on.)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 leadto a rate of interest that is paid continuously. Expression (1.1) becomes

(1 + r

m

)m

= em log(1+ rm ) ∼ er .

This is a simple application of Taylor series when r/m is small. And that is how much moneyI will have in the bank after one year if the interest is continuously compounded. Similarly,after a time t I will have an amount

ert (1.2)

in the bank. Almost everything in this book assumes that interest is compounded continuously.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 thenext? 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

M(t + dt) − M(t) ≈ dM

dtdt + · · · ,

where the right-hand side comes from a Taylor series expansion of M(t + dt). But I also knowthat the interest I receive must be proportional to the amount I have, M , the interest rate, r ,and the time step, dt . Thus

dM

dtdt = rM(t) dt.

products and markets Chapter 1 7

Dividing by dt gives the ordinary differential equation

dM

dt= rM(t)

the solution of which is

M(t) = M(0) ert .

If the initial amount at t = 0 was $1 then I get (1.2) again.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

e−r(T −t).

I can relate cashflows in the future to their present value by multiplying by this factor. As anexample, suppose that r is 5% i.e. r = 0.05, then the present value of $1,000,000 to be receivedin two years is

$1,000,000 × e−0.05×2 = $904,837.

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 laterchapters I will generalize.

Important Aside

What mathematics have we seen so far? To get to (1.2) allwe needed to know about are the two functions e (or exp)and log, and Taylor series. Believe it or not, you can appre-ciate almost all finance theory by knowing these three thingstogether with ‘expectations.’ I’m going to build up to the basicBlack–Scholes and derivatives theory assuming that you knowall four of these. Don’t worry if you don’t know about thesethings yet, take a look at Appendix A where I review these requisites and show how to interpretfinance theory and practice in terms of the most elementary mathematics.

Just because you can understand derivatives theory in terms of basic math doesn’t mean thatyou should. I hope that there’s enough in the book to please the Ph.D.s1 as well.

1.3 EQUITIES

The most basic of financial instruments is the equity, stock or share. This is the owner-ship of a small piece of a company. If you have a bright idea for a new product or service

1 And Nobel laureates.

8 Part One mathematical and financial foundations

then you could raise capital to realize this idea by selling off future profits in the form ofa stake in your new company. The investors may be friends, your Aunt Joan, a bank, or aventure capitalist. The investor in the company gives you some cash, and in return you givehim a contract stating how much of the company he owns. The shareholders who own thecompany between them then have some say in the running of the business, and technicallythe directors of the company are meant to act in the best interests of the shareholders. Onceyour business is up and running, you could raise further capital for expansion by issuing newshares.

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 nextexpansion. At this point shares in the company may be sold to a wider audience or even thegeneral public. The investors in the business may have no link with the founders. The final pointin the growth of the company is with the quotation of shares on a regulated stock exchangeso that shares can be bought and sold freely, and capital can be raised efficiently and at thelowest cost.

Figures 1.1 and 1.2 show screens from Bloomberg giving details of Microsoft stock, includingprice, high and low, names of key personnel, weighting in various indices (see below) etc. Thereis much, much more info available on Bloomberg for this and all other stocks. We’ll be seeingmany Bloomberg screens throughout this book.

Figure 1.1 Details of Microsoft stock. Source: Bloomberg L.P.

products and markets Chapter 1 9

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. Thelisted information includes highs and lows for the day as well as the change since the previousday’s close.

The behavior of the quoted prices of stocks is far from being predictable. In Figure 1.4 I showthe Dow Jones Industrial Average over the period January 1950 to March 2004. In Figure 1.5is a time series of the Glaxo–Wellcome share price, as produced by Bloomberg.

If we could predict the behavior of stock prices in the future then we could become veryrich. Although many people have claimed to be able to predict prices with varying degrees ofaccuracy, no one has yet made a completely convincing case. In this book I am going to takethe point of view that prices have a large element of randomness. This does not mean that wecannot model stock prices, but it does mean that the modeling must be done in a probabilisticsense. No doubt the reality of the situation lies somewhere between complete predictabilityand perfect randomness, not least because there have been many cases of market manipulationwhere large trades have moved stock prices in a direction that was favorable to the persondoing the moving.

To whet your appetite for the mathematical modeling later, I want to show you a simple wayto simulate a random walk that looks something like a stock price. One of the simplest randomprocesses is the tossing of a coin. I am going to use ideas related to coin tossing as a modelfor the behavior of a stock price. As a simple experiment start with the number 100 which you

10 Part One mathematical and financial foundations

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