UNTAG Universitas 17 Agustus 1945 Samarinda

4.1214.1314.1414.1514.1614.1714.1814.2.8 Amortization14.2.9 Call provisionInternational bond markets14.3.1 United States of America14.3.2 United Kingdom14.3.3 JapanAccrued interestDay-count conventionsContinuously and discretely compounded interestMeasures of yield14.7.1 Current yield14.7.2 The yield to maturity (YTM) or internal rate of return (IRR)The yield curvePrice/yield relationshipDurationConvexityAn exampleHedgingTime-dependent interest rateDiscretely paid couponsForward rates and bootstrapping14.16.1 Discrete data14.16.2 On a 34434615 Swaps15.1 Introduction15.2 The vanilla interest rate swap15.3 Comparative advantage15.4 The swap curve15.5 Relationship between swaps and bonds15.6 Bootstrapping15.7 Other features of swaps contracts15.8 Other types of swap15.8.1 Basis rate swap15.8.2 Equity swaps15.8.3 Currency swaps15.9 Summary34935035035135335435535635735735735735816 One-factor Interest Rate Modeling16.1 Introduction16.2 Stochastic interest rates16.3 The bond pricing equation for the general model16.4 What is the market price of risk?16.5 Interpreting the market price of risk, and risk neutrality16.6 Named models359360361362365366366

contents16.716.816.916.6.1 Vasicek16.6.2 Cox, Ingersoll & Ross16.6.3 Ho & Lee16.6.4 Hull & WhiteEquity and FX forwards and futures when rates are stochastic16.7.1 Forward contractsFutures contracts16.8.1 The convexity adjustmentSummary36636736836936936937037137217 Yield Curve Fitting17.1 Introduction17.2 Ho & Lee17.3 The extended Vasicek model of Hull & White17.4 Yield-curve fitting: For and against17.4.1 For17.4.2 Against17.5 Other models17.6 Summary37337437437537637637638038018 Interest Rate Derivatives18.1 Introduction18.2 Callable bonds18.3 Bond options18.3.1 Market practice18.4 Caps and floors18.4.1 Cap/floor parity18.4.2 The relationship between a caplet and a bond option18.4.3 Market practice18.4.4 Collars18.4.5 Step-up swaps, caps and floors18.5 Range notes18.6 Swaptions, captions and floortions18.6.1 Market practice18.7 Spread options18.8 Index amortizing rate swaps18.8.1 Other features in the index amortizing rate swap18.9 Contracts with embedded decisions18.10 Some examples18.11 More interest rate derivatives. . .18.12 9439439639739840040119 The Heath, Jarrow & Morton and Brace, Gatarek & Musiela Models19.1 Introduction19.2 The forward rate equation19.3 The spot rate process19.3.1 The non-Markov nature of HJM403404404404406xv

19.1319.1419.1519.1619.1719.1819.19The market price of riskReal and risk neutral19.5.1 The relationship between the risk-neutral forward rate driftand volatilityPricing derivativesSimulationsTreesThe Musiela parameterizationMulti-factor HJMSpreadsheet implementationA simple one-factor example: Ho & LeePrincipal Component Analysis19.13.1 The power methodOptions on equities, etc.Non-infinitesimal short rateThe Brace, Gatarek & Musiela modelSimulationsPVing the 41541641641741941942020 Investment Lessons from Blackjack and Gambling20.1 Introduction20.2 The rules of blackjack20.3 Beating the dealer20.3.1 Summary of winning at blackjack20.4 The distribution of profit in blackjack20.5 The Kelly criterion20.6 Can you win at roulette?20.7 Horse race betting and no arbitrage20.7.1 Setting the odds in a sporting game20.7.2 The mathematics20.8 Arbitrage20.8.1 How best to profit from the opportunity?20.9 How to bet20.10 3821 Portfolio Management21.1 Introduction21.2 Diversification21.2.1 Uncorrelated assets21.3 Modern portfolio theory21.3.1 Including a risk-free investment21.4 Where do I want to be on the efficient frontier?21.5 Markowitz in practice21.6 Capital Asset Pricing Model21.6.1 The single-index model21.6.2 Choosing the optimal portfolio441442442443445447447450451451453

contents21.721.821.921.10The multi-index modelCointegrationPerformance measurementSummary45445445545622 Value at Risk22.1 Introduction22.2 Definition of Value at Risk22.3 VaR for a single asset22.4 VaR for a portfolio22.5 VaR for derivatives22.5.1 The delta approximation22.5.2 The delta/gamma approximation22.5.3 Use of valuation models22.5.4 Fixed-income portfolios22.6 Simulations22.6.1 Monte Carlo22.6.2 Bootstrapping22.7 Use of VaR as a performance measure22.8 Introductory Extreme Value Theory22.8.1 Some EVT results22.9 Coherence22.10 6946947047023 Credit Risk23.1 Introduction23.2 The Merton model: equity as an option on a company’s assets23.3 Risky bonds23.4 Modeling the risk of default23.5 The Poisson process and the instantaneous risk of default23.5.1 A note on hedging23.6 Time-dependent intensity and the term structure of default23.7 Stochastic risk of default23.8 Positive recovery23.9 Hedging the default23.10 Credit rating23.11 A model for change of credit rating23.12 Copulas: pricing credit derivatives with many underlyings23.12.1 The copula function23.12.2 The mathematical definition23.12.3 Examples of copulas23.13 Collateralized debt obligations23.14 8948949049049224 RiskMetrics and CreditMetrics24.1 Introduction24.2 The RiskMetrics datasets495496496xvii

xviiicontents24.324.424.524.624.724.8Calculating the parameters the RiskMetrics way24.3.1 Estimating volatility24.3.2 CorrelationThe CreditMetrics dataset24.4.1 Yield curves24.4.2 Spreads24.4.3 Transition matrices24.4.4 CorrelationsThe CreditMetrics methodologyA portfolio of risky bondsCreditMetrics model 25 CrashMetrics25.1 Introduction25.2 Why do banks go broke?25.3 Market crashes25.4 CrashMetrics25.5 CrashMetrics for one stock25.6 Portfolio optimization and the Platinum hedge25.6.1 Other ‘cost’ functions25.7 The multi-asset/single-index model25.7.1 Assuming Taylor series for the moment25.8 Portfolio optimization and the Platinum hedge in the multi-assetmodel25.8.1 The marginal effect of an asset25.9 The multi-index model25.10 Incorporating time value25.11 Margin calls and margin hedging25.11.1 What is margin?25.11.2 Modeling margin25.11.3 The single-index model25.12 Counterparty risk25.13 Simple extensions to CrashMetrics25.14 The CrashMetrics Index (CMI)25.15 Summary50550650650650750851051151151726 Derivatives **** Ups26.1 Introduction26.2 Orange County26.3 Proctor and Gamble26.4 Metallgesellschaft26.4.1 Basis risk26.5 Gibson Greetings26.6 Barings26.7 Long-Term Capital Management26.8 22522522524524524525526

contents27 Overview of Numerical Methods27.1 Introduction27.2 Finite-difference methods27.2.1 Efficiency27.2.2 Program of study27.3 Monte Carlo methods27.3.1 Efficiency27.3.2 Program of study27.4 Numerical integration27.4.1 Efficiency27.4.2 Program of study27.5 Summary54154254254354354454554554654654654728 Finite-difference Methods for One-factor Models28.1 Introduction28.2 Grids28.3 Differentiation using the grid28.4 Approximating θ28.5 Approximating 28.6 Approximating 28.7 Example28.8 Bilinear interpolation28.9 Final conditions and payoffs28.10 Boundary conditions28.11 The explicit finite-difference method28.11.1 The Black–Scholes equation28.11.2 Convergence of the explicit method28.12 The Code #1: European option28.13 The Code #2: American exercise28.14 The Code #3: 2-D output28.15 Upwind differencing28.16 6757157357557829 Monte Carlo Simulation29.1 Introduction29.2 Relationship between derivative values and simulations: equities,indices, currencies, commodities29.3 Generating paths29.4 Lognormal underlying, no path dependency29.5 Advantages of Monte Carlo simulation29.6 Using random numbers29.7 Generating Normal variables29.7.1 Box–Muller29.8 Real versus risk neutral, speculation versus hedging29.9 Interest rate products29.10 Calculating the greeks29.11 Higher dimensions: Cholesky 94xix

xxcontents29.12 Calculation time29.13 Speeding up convergence29.13.1 Antithetic variables29.13.2 Control variate technique29.14 Pros and cons of Monte Carlo simulations29.15 American options29.16 Longstaff & Schwartz regression approach for American options29.17 Basis functions29.18 Summary30 Numerical Integration30.1 Introduction30.2 Regular grid30.3 Basic Monte Carlo integration30.4 Low-discrepancy sequences30.5 Advanced techniques30.6 13614AAll the Math You Need. . . and No More (An Executive n and Taylor seriesA.5Differential equationsA.6Mean, standard deviation and recasting the Markets? A Small DigressionB.1IntroductionB.2Technical analysisB.2.1PlottingB.2.2Support and resistanceB.2.3TrendlinesB.2.4Moving averagesB.2.5Relative strengthB.2.6OscillatorsB.2.7Bollinger bandsB.2.8Miscellaneous patternsB.2.9Japanese candlesticksB.2.10 Point and figure chartsB.3Wave theoryB.3.1Elliott waves and Fibonacci numbersB.3.2Gann chartsB.4Other analyticsB.5Market microstructure modelingB.5.1Effect of demand on 638638640640

contentsB.6B.7B.5.2Combining market microstructure and option theoryB.5.3ImitationCrisis predictionSummary641641641641CA Trading GameC.1IntroductionC.2AimsC.3Object of the gameC.4Rules of the gameC.5NotesC.6How to fill in your trading sheetC.6.1During a trading roundC.6.2At the end of the game643643643643643644645645645DContents of CD accompanying Paul Wilmott Introduces QuantitativeFinance, second edition649What you get if (when) you upgrade to PWOQF2653EBibliography659Index683xxi

PrefaceIn this book I present classical quantitative finance. The book is suitable for students onadvanced undergraduate finance and derivatives courses, MBA courses, and graduatecourses that are mainly taught, as opposed to ones that are based on research. Thetext is quite self-contained, with, I hope, helpful sidebars (‘Time Out’) covering the moremathematical aspects of the subject for those who feel a little bit uncomfortable. Little priorknowledge is assumed, other than basic calculus, even stochastic calculus is explainedhere in a simple, accessible way.By the end of the book you should know enough quantitative finance to understandmost 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 thenecessary math and then the two are melded together. The technical complexity is neverthat great, nor need it be. The last three chapters are on the numerical methods you willneed for pricing. In the more advanced subjects, such as credit risk, the mathematicsis kept to a minimum. Also, plenty of the chapters can be read without reference tothe mathematics at all. The structure, mathematical content, intuition, etc., are based onmany years’ teaching at universities and on the Certificate in Quantitative Finance, andtraining bank personnel at all levels.The accompanying CD contains spreadsheets and Visual Basic programs implementingmany of the techniques described in the text. The CD icon will be seen throughout thebook, indicating material to be found on the CD, naturally. There is also a full list of itscontents at the end of the book.You can also find an Instructors Manual at www.wiley.com/go/pwiqf2 containinganswers to the end-of-chapter questions in this book. The questions are, in general, of amathematical nature but suited to a wide range of financial courses.This book is a shortened version of Paul Wilmott on Quantitative Finance, secondedition. It’s also more affordable than the ‘full’ version. However, I hope that you’lleventually 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 WilmottIntroduces Quantitative Finance I have specifically the university student in mind.The differences between the university and the full versions are outlined at the end ofthe book. And to help you make the leap, we’ve included a form for you to upgrade,

xxivprefacegiving you a nice discount. Roughly speaking, the full version includes a great deal ofnon-classical, more modern approaches to quantitative finance, including several nonprobabilistic models. There are more mathematical techniques for valuing exotic optionsand 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 wanta chat, email me at paul@wilmott.com. I’ll do my very best to respond as quickly aspossible. 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 andJonathan Shaw and Gil Christie at 7city, unequaled in their dedication to training andtheir imagination for new ideas. Also Riaz Ahmad, Seb Lleo and Siyi Zhou who havehelped make the Certificate in Quantitative Finance so successful, and for taking someof the pressure off me. Everyone involved in the magazine, especially Aaron Brown, AlanLewis, Bill Ziemba, Caitlin Cornish, Dan Tudball, Ed Lound, Ed Thorp, Elie Ayache, EspenGaarder Haug, Graham Russel, Henriette Präst, Jenny McCall, Kent Osband, Liam Larkin,Mike Staunton, Paula Soutinho and Rudi Bogni. I am particularly fortunate and gratefulthat John Wiley & Sons have been so supportive in what must sometimes seem to themrather wacky schemes. I am grateful to James Fahy for his work on my websites, andapologies for always failing to provide a coherent brief. Thanks also to David Epsteinfor help with the exercises, again; to Ron Henley, the best hedge fund partner a quantcould wish for: ‘‘It’s just a jump to the left. And then a step to the right’’; to John Morrisof Fulcrum, interesting times; to all my lawyers for keeping the bad people away, JaredStamell, 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 whichhave invariably served me well. And to Oscar and Zachary who kept me sane throughoutmany a series of unfortunate events!Finally, thanks to my number one fan, Andrea Estrella, from her number one fan, me.ABOUT THE AUTHORPaul Wilmott’s professional career spans almost every aspect of mathematics andfinance, in both academia and in the real world. He has lectured at all levels, and foundeda 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 wouldbe 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 1products and markets:equities, commodities,exchange rates,forwards and futuresThe aim of this Chapter. . . . . is to describe some of the basic financial market products and conventions, toslowly introduce some mathematics, to hint at how stocks might be modeled usingmathematics, and to explain the important financial concept of ‘no free lunch.’ By theend of the chapter you will be eager to get to grips with more complex productsand to start doing some proper modeling.In this Chapter. . . an introduction to equities, commodities, currencies and indices the time value of money fixed and floating interest rates futures and forwards no-arbitrage, one of the main building blocks of finance theory

2Paul Wilmott introduces quantitative finance1.1INTRODUCTIONThis first chapter is a very gentle introduction to the subject of finance, and is mainlyjust a collection of definitions and specifications concerning the financial markets ingeneral. There is little technical material here, and the one technical issue, the ‘timevalue of money,’ is extremely simple. I will give the first example of ‘no arbitrage.’ This isimportant, being one part of the foundation of derivatives theory. Whether you read thischapter thoroughly or just skim it will depend on your background.1.2EQUITIESThe most basic of financial instruments is the equity, stock or share. This is the ownershipof a small piece of a company. If you have a bright idea for a new product or servicethen 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 a venture capitalist. The investor in the company gives you some cash, and in returnyou give him a contract stating how much of the company he owns. The shareholderswho own the company between them then have some say in the running of the business,a

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 British Library Cataloguing in .