Universitext Rüdiger U. Seydel Tools for Computational Finance Sixth Edition Universitext Universitext Serieseditors SheldonAxler SanFranciscoStateUniversity CarlesCasacuberta UniversitatdeBarcelona AngusMacIntyre QueenMary,UniversityofLondon KennethRibet UniversityofCalifornia,Berkeley ClaudeSabbah Écolepolytechnique,CNRS,UniversitéParis-Saclay,Palaiseau EndreSüli UniversityofOxford WojborA.Woyczyn´ski CaseWesternReserveUniversity Universitext is a series of textbooksthat presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well class-tested by their author, may have an informal, personal even experimental approachtotheirsubjectmatter.Someofthemostsuccessfulandestablishedbooks intheserieshaveevolvedthroughseveraleditions,alwaysfollowingtheevolution ofteachingcurricula,toverypolishedtexts. Thus as research topics trickle down into graduate-level teaching, first textbooks writtenfornew,cutting-edgecoursesmaymaketheirwayintoUniversitext. Moreinformationaboutthisseriesathttp://www.springer.com/series/223 RuRdiger U. Seydel Tools for Computational Finance Sixth Edition 123 RuRdigerU.Seydel MathematischesInstitut UniversitaRtzuKoRln Germany ISSN0172-5939 ISSN2191-6675 (electronic) Universitext ISBN978-1-4471-7337-3 ISBN978-1-4471-7338-0 (eBook) DOI10.1007/978-1-4471-7338-0 LibraryofCongressControlNumber:2017945224 MathematicsSubjectClassification(2010):91-01,91-08,91G20,91G60,65-01 Previouseditions1-4publishedbySpringer-VerlagBerlinHeidelberg ©SpringerHeidelberg2002,2004,2006,2009 ©Springer-VerlagLondonLtd.2012,2017 Theauthor(s)has/haveassertedtheirright(s)tobeidentifiedastheauthor(s)ofthisworkinaccordance withtheCopyright,DesignsandPatentsAct1988. Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringer-VerlagLondonLtd. Theregisteredcompanyaddressis:236Gray’sInnRoad,LondonWC1X8HB,UnitedKingdom Preface to the Sixth Edition Fiveyearsafterthefiftheditioncameout,thereisaneedtoincludeadditionalresults andtoimproveexplanationsofmethodsandalgorithmsfurther.Besidesnumerous enhancementsofsmalldetails,therearenewsubjects. Newinthesixtheditionare,forexample,methodsforsmoothinginChap.1and an introduction into basic aspects of efficiency. In Chap.2, acceptance–rejection methods for generating random numbers are explained, with application to the ziggurat algorithm for calculating normal variates. Chapter 3 on Monte Carlo methods now includes a subsection on positive solutions and an outline of the antithetic variance reduction. Iterative approaches in Chap.4 have become less important,infavorofdirectmethods. To support the important role tree methods play in practice, an entire new appendix (Appendix D) is devoted to these methods. This appendix includes trinomialmethods,multidimensionaltrees,andimpliedtreesforvariablevolatility. Further,howtohandlediscretedividendsisexplained. Andthetextisenrichedbymorefigures.Tofacilitateunderstanding,manyofthe figureshavebeenrecalculatedtobecomecolored.Additionalformalizedexercises are included, and numerous hints at informal exercises are spread throughoutthe text. On the technical side, the entire book has been transferred from plainTeX to LaTeX. This has offered plenty of occasions to work the book over thoroughly. AdditionalcoloredfigurescanbefoundinthecollectionTopicsforComputational Finance(shortlyTopicsfCF)ontheInternetplatformwww.compfin.de. Köln,Germany RüdigerU.Seydel April2017 v Preface to the Fifth Edition Financial engineering and numerical computation are genuinely different disci- plines. But in finance, many computational methods are used and have become indispensable. This book explains how computational methods work in financial engineering. The main focus is on computational methods; financial engineering is the application. In this context, the numerical methods are tools, the tools for computationalfinance. Facedwiththevastandrapidlygrowingfieldoffinancialengineering,weneed to choose a subarea to avoid overloading the textbook. We choose the attractive field of option pricing, a core task of financial engineering and risk analysis. The broadfieldofoptionpricingisbothambitiousanddiverseenoughtocallforawide rangeofcomputationaltools.Confiningourselvestooptionpricingenablesamore coherenttextbookandavoidsbeingdistractedawayfromcomputationalissues.We trust that the focus on option-relatedmethodsis representativeof, or least helpful for,theentirefieldofcomputationalfinance. The book starts with an introductory Chap.1, which collects financial and stochastic background. The remaining parts of the book are devoted to compu- tational methods. Organizing computational methods, roughly speaking, leads to distinguish stochastic and deterministic approaches. By “stochastic methods,” we mean computations based on random numbers, such as Monte Carlo simulation. Chapters2and3aredevotedtosuchmethods.Incontrast,“deterministicmethods” are frequently based on solving partial differential equations. This is discussed in Chaps.4, 5, and 6. In the computer, finally, everything is deterministic. The distinction between “stochastic” and “deterministic” is mainly to motivate and derivedifferentapproaches. All of the computationalmethods must be adapted to the underlyingmodel of a financial market. Here, we meet different kinds of stochastic processes, from geometric Brownian motion to Lévy processes. Based on the chosen process, an optionmodelisselected.TheclassicalchoiceistheBlack–Scholesmodelforvanilla options with one underlying asset. This benchmark market model is “complete” in that all claims can be replicated. Established by Black, Merton, Scholes, and others, this model is the main application of methods explained in Chaps.2–6. vii viii PrefacetotheFifthEdition Chapter 7 goes beyond and addresses more general models. Allowing for jump processes, transaction costs, multiasset underlyings, or more complicated payoffs leads to incomplete markets. Computational methods for incomplete markets are brieflydiscussedinChap.7. Thisbookhasbeenpublishedinseveraleditions.ThefirstGermanedition(2000) was mainly absorbed by the Black–Scholes equation.Later editions (first English edition2002)werecarefullyopenedtomoregeneralmodelsandawiderselection of methods. The book has grown with the developmentof the field. Faced with a largevarietyofpossiblecomputationaltools,thisbookattemptstobalancetheneed for a sufficient number of powerfulalgorithms with the limitations of a textbook. The balance has been gradually shifting over the years and editions. Numerous investigationsin our research group have influencedthe choice of covered topics. Wehaveimplementedandtestedmanydozensofalgorithmsandgainedinsightand experience.Asignificantpartofthisknowledgehasenteredthebook. Readership Thisbookiswrittenfromtheperspectiveofanappliedmathematician.Thelevelof mathematicsistailoredtoadvancedundergraduatescienceandengineeringmajors. Apart from this basic knowledge, the book is self-contained and can be used for a course on the subject. The intended readership is interdisciplinary and includes professionalsinfinancialengineering,mathematicians,andscientistsofmanyother fields. An expository style may attract a readership ranging from students to practi- tioners.Methodsareintroducedastoolsforimmediateapplication.Formulatedand summarizedasalgorithms,astraightforwardimplementationincomputerprograms shouldbepossible.Inthisway,thereadermaylearnbycomputationalexperiment. Learning by calculating will be a possible way to explore several aspects of the financial world. In some parts, this book provides an algorithmic introduction to computationalfinance.Tokeepthetextreadableforawideaudience,someaspects of proofs and derivations are exported to exercises at which hints are frequently given. New intheFifth Edition The revisionsto this fifth edition are much more extensivethan those of previous editions. Compared to the fourth edition, the page count has increased by about 100pages.Themain additionisChap.7,whichis devotedto incompletemarkets. It beginswith an introductionto nonlinearBlack–Scholes-type partial differential equations,astheyarisefromconsideringtransactioncostsorrangesforastochastic volatility. Numerical approaches require instruments that converge to viscosity PrefacetotheFifthEdition ix solutions.Thesesolutionsareintroducedinanappendix.Theroleofmonotonicity of numerical schemes is outlined. Lévy processes, with a focus on Merton’s jumpdiffusionandanumericalapproachtotheresultingpartialintegrodifferential equation, are then addressed. The chapter ends with an exposition on how the Fourier transform can be applied to option pricing. To complete the introduction of more general models and methods, the Dupire equation is outlined in a new appendix. Inadditionto thenewChap.7,severallargerextensionsandnewsectionshave beenwrittenforthisedition.ThecalculationofGreeksisdescribedinmoredetail, includingthe methodof adjointsfor a sensitivity analysis(new Sect.3.7).Penalty methodsareintroducedandappliedtoatwo-factormodelinthenewSect.6.7.More materialispresentedinthefieldofanalyticalmethods;inparticular,Kim’sintegral representationanditscomputationhavebeenaddedtoChap.4.Tentativeguidelines on how to comparedifferentalgorithmsand judgeefficiencyare givenin the new Sect.4.9. The chapter on finite elements has been extended with a discussion of two-assetoptions. Apartfromadditionalmateriallistedabove,theentirebookhasbeenthoroughly revised. The clarity of the expository parts has been improved; all sections have beentestedintheclassroom.Numerousamendments,furtherfigures,exercises,and manyreferenceshave beenadded. For example,the principalcomponentanalysis and its applications are included, and the role of different boundary conditions is outlinedinmoredetail. HowtoUse ThisTextbook Exercisesarestatedattheendofeachchapter.Theyrangefromeasyroutinetasks to laborious projects. In addition to these explicitly formulated exercises, plenty of “hidden” exercises are spread throughout the book, with comments such as “the reader may check.” Of course, the reader is encouragedto fill in those small intermediatestepsthatareexcludedfromthetext. This book explains the basic ideas of several approaches, presenting more material than is accomplishable in one semester. The following guidelines have provedsuccessfulinteaching: FirstCourse: Chapter1withoutSect.1.6.2 Chapter2 Chapter3withoutSect.3.7 Chapter4,withoneanalyticmethodoutofSect.4.8 andwithoutSect.4.9 Chapter6,orpartsofit x PrefacetotheFifthEdition SecondCourse: Theremainingparts,inparticular Chapter5andChap.7 Dependingonthedetailofexplanation,thefirstcoursecouldbeforundergraduate students.Thesecondcoursemayattractgraduatestudents. Extensionsinthe Internet ThereisanaccompanyingInternetpage: www.compfin.de This is intended to serve the needs of the computational finance community and providescomplementarymaterial to this book. In particular, the collection Topics for Computational Finance, which is under construction, presents several of our findingsorfiguresthatwouldgobeyondthelimitedscopeofatextbook.Initsfinal state,TopicsisanticipatedasacompanionsourcetotheTools.