Mathematics of Finance - An Intuitive Introduction PDF
Preview Mathematics of Finance - An Intuitive Introduction
Undergraduate Texts in Mathematics Donald G. Saari Mathematics of Finance An Intuitive Introduction Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics SeriesEditors SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA KennethRibet UniversityofCalifornia,Berkeley,CA,USA AdvisoryBoard ColinAdams,WilliamsCollege,Williamstown,MA,USA DavidA.Cox,AmherstCollege,Amherst,MA,USA L.CraigEvans,UniversityofCalifornia,Berkeley,CA,USA PamelaGorkin,BucknellUniversity,Lewisburg,PA,USA RogerE.Howe,YaleUniversity,NewHaven,CT,USA MichaelE.Orrison,HarveyMuddCollege,Claremont,CA,USA LisetteG.dePillis,HarveyMuddCollege,Claremont,CA,USA JillPipher,BrownUniversity,Providence,RI,USA FadilSantosa,UniversityofMinnesota,Minneapolis,MN,USA Undergraduate Texts in Mathematics are generally aimed at third- and fourth- year undergraduate mathematics students at North American universities. These texts strive to provide students and teachers with new perspectives and novel approaches.Thebooksincludemotivationthatguidesthereadertoanappreciation ofinterrelationsamongdifferentaspectsofthesubject.Theyfeatureexamplesthat illustratekeyconceptsaswellasexercisesthatstrengthenunderstanding. Moreinformationaboutthisseriesathttp://www.springer.com/series/666 Donald G. Saari Mathematics of Finance An Intuitive Introduction 123 DonaldG.Saari DepartmentsofEconomics andMathematics UniversityofCalifornia Irvine,CA,USA ISSN0172-6056 ISSN2197-5604 (electronic) UndergraduateTextsinMathematics ISBN978-3-030-25442-1 ISBN978-3-030-25443-8 (eBook) https://doi.org/10.1007/978-3-030-25443-8 MathematicsSubjectClassification:91Gxx,91G20,91F99 ©SpringerNatureSwitzerlandAG2019 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,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Formygoodfriend ArthurPancoe Atruefinancialwizard! Note to Instructors Thesenotesweremotivatedbyconversationswithgraduatingseniorsinmathemat- ics and economics who were headed for the finance world. Finance? From where didtheygettheirtraining?ItturnedoutatthattimeatNorthwesternUniversitythere werenorelevantundergraduatecourses.Andso,afterdiscussionswithcolleagues from the Departments of Finance and of Economics, I developed a course for the MathematicsDepartment. Based on the advantages gained by students who have taken this course, includingjobopportunitiesandanewfocusoncoursematerial,Ibelievethatsuch acourseshouldbeofferedinmostundergraduatemathprograms.Andso,thisbook is designed to help the reader grasp the fundamentals. This is both for students and for instructors who wish to teach the course but may be hesitant without a previousbackground.Courseenrollmentshavebeenahealthymixturewhereabout half are math majors and the other half are mathematically stronger students from economicsandelsewhere(e.g.,engineering,graduatestudents,etc.).Aftermymove tothesunnierclimesattheUniversityofCalifornia,Irvine,thenotesweremodified tomeettherealitiesoflargerclasssizes. Thecourseaddressesseveralneeds: 1. Capstone: Students learn a considerable amount of mathematics as undergrad- uates, but many fail to see how it is connected, what to do with it, or how it is relevant for their future. Even more, many either forgot fundamental concepts or never moved beyond technical details to grasp the absorbing power of mathematics. Asanexample,itisnotunusualtofindstudentswhodismissTaylorseriesas a side topic or as an illustration of infinite series. They most surely carried out numerousexercisesyetfailedtorecallthat,forpracticalpurposes,thispowerful toolrequiresonlyafinitenumberofterms.Then,manyfailtorememberhowto createTaylorseriesforseveralvariables.Thematerialwastaught,butforsome (notall),itwasforgotten. So,ratherthanassumingstudentsrecallmaterialfromearliercourses(includ- ing basic concepts from probability and statistics), the relevant concerns are vii viii NotetoInstructors quickly reviewed with an emphasis on their power and utility. This is not a course on these subjects, so an intuitive review, rather than a detailed, rigorous exposition,ispresented.Anicefeatureofthisfinancetopicisthatitincorporates so many mathematical concepts, which explain why the course has served as a capstoneforstudentswithonlyapassingcuriosityaboutfinance. 2. Introduction: The main purpose is to introduce students to the fundamentals of themathematicsoffinance.Mostarriveknowingnothingaboutthisarea,sothis book starts with basics and quickly moves to more complicated material. The choice of material is directed to provide a mathematical understanding of the fundamentals with an emphasis on why certain equations and concepts are of valueandwhattheyreallymean. It is standard in some courses, for instance, to present the solution for the Black–Scholes Equation without explaining from where terms come, why they shouldbeexpected,andwhattheymean.Onewaytoclosethisgapistocarefully solve the Black–Scholes Equation. But once the course’s popularity attracted classesofaround200students,timelimitationsmadethisimpossible. Fortunately,theBlack–Scholessolutioncanbeunderstoodwithoutsolvingthe partialdifferentialequation;itsufficestoappreciatehowchangesofvariables(to converttheBlack–ScholesEquationintotheheatequation)aremanifestedinthe final solution. In this manner, students understand from where and why terms ariseratherthanconfrontingaconfusingmemorylesson.Attentioncanthenbe focussedonwhatallofthismeans. 3. Developing mathematical intuition: Students who have taken this course have directlyenteredthefinanceworldormovedtograduateprogramstolearnmore. This means that they must develop intuition for what is being presented, the limitationsofvariousconclusions,1 andwhattopicsareopenforresearch. Limitations are emphasized throughout the book starting with the intro- duction. Students catch on: they begin to appreciate the importance of those hypothesesthat,inthepast,theymighthaveignored.Betterstudentsrecognize whereaddedresearchisrequired. To help students develop an instinctive understanding of the material, the approachofthisbookdiffersfromatraditionalcourse:newmaterialisintroduced intermsofwhattheycanreadilygrasp.Thatis,topicsarelaunchedwithstories or closely related themes.2 The definition and significance of arbitrage, for instance, are introduced in the first chapter with a simple gambling example. The limitations of the “Efficient Market Hypothesis” are compared with the constraintsofaquadraticTaylorseriesrepresentationfory =cos(x). 1This is critical: After the 2008 crash, some government experts attended an NRC committee meeting to explore what they missed. A couple were surprised to discover that key equations fromthisareaarenotalwaysapplicable.Whenonewonderedhowtodiscoversuchinformation,I volunteeredmybetterstudents. 2Thisappealtogeneralconceptsavoidsthecommonproblemwhereyoumustknowsomething aboutfinancebeforeyoucanstudy(orteachacourse)inthisarea. NotetoInstructors ix Teaching always involves compromises between time and depth of coverage, suchasinquarterlengthcourses.Myadviceistopacethecoursetoensurethatthe messagesofChapters6,7,and8arecovered.Notdoingsowouldbeakintoreading anAgathaChristiemysterynovelonlytodiscoverthatthelastchaptersaremissing. Problems at the end of each chapter are roughly in the order of the chapter’s presentation,whichmakesiteasierforassigninghomework.Comingupwithother problems,bymimickingexamplesinthetext,iseasy.Mystudentsareresponsible foralltheproblemsincoveredchapters,andmyspotquizzestypicallycomefrom changingnumbersinassignedproblems. Ideally,thecourseshouldemphasizehowthepowerofmathematicssignificantly assists developing a sense, an intuition, about the market. When the material becomesmathematicallymoretechnical,thereisadangerthatstudentswillfocuson mathematicaldetailsattheexpenseofdevelopingintuitionaboutfinancialoptions. Tocounterthis,“intuitionbreaks”areinsertedinvariouslocations.Manyserveas reasonablehomeworkproblems. Subjecttotimeconstraints,otherfeaturescanbeaddressed.Tosuggestopportu- nitieswhenteachingacourseondynamics,itispossibletoappealtoNewton’slaw orblocksslidingdowninclinedboards.Butfortheuninitiated,muchofeconomics is a mysterious world; instinct, experience, and intuition should be provided. OK. How? Central to the material is the “buy low, sell high” phrase. A way to experience this cliché (time permitting) is with an experiment where a portion of the class, thesuppliers,producewidgets:infact,theybuythemfromanimaginary“Sue.”A differentportionoftheclassarethebuyersorconsumers.Thevalueeachattaches towidgetsisdeterminedasfollows:eachconsumercanpurchasewidgetsfromthe suppliersandsellthemtome. Reflecting different levels of manufacturing expertise, Sue offers different studentsdifferentprices(asspecifiedonslipsofpaperthatthestudentsdrawfroma bowl);theyhavenoideawhatpricestheothersuppliershave.Similarly,eachbuyer drawsaslipfromabowlwhichstateswhatIwillpayforawidget;differentbuyers havedifferentpricestoreflectthedifferentvaluesconsumersplaceonwidgets. The market opens as soon as some student offers to buy, or sell, a widget at a price they specify: if someone agrees, a sale is made, and the two are out of the market.Eachpersonmarksontheslipthesalepricetodeterminepersonalprofits. (The instructor’s role is to explain the process, ensure that the auction starts, and maintainorder—whateverhappenswiththebiddinghappens.)Everythingcontinues untilnomoresalescanbemade.Forinstance,Sue(i.e.,theslipsofpaperdrawnby thestudentswhoaresuppliers)mayoffereachofsixstudentsapricefrom2,4,5,7, 9,and10;similarly,myoffersmaycomefrom3,5,6,6,8,and9.(Efficientgroup sizesrangefrom25to30each;theyneednotbethesamesize.)Norealmoneyis involved,butstudents,evenobservers,quicklybecomecaptivated. Thisgameiscarriedoutseveraltimes(thereisalearningprocess;inlaterrounds, studentsbecomemorestrategicandsophisticated)whereeachroundusesdifferent prices.(So,ifastudentwitharawdealendsupinasubsequentround,sheorhemay drawabetterchoice.)Themessageiswhenthesupplyanddemandcurvesfromeach x NotetoInstructors experiment are plotted on the board, the intersection is close to the price obtained throughthegames:notonlydoesthisexerciseprovidevalidityforthesupplyversus demandstory(forparticularsettings),butthebiddingprocessdemonstrateshowthe “wisdomofthecrowd”influencesthesearchforanequilibriumprice. Thiscourseisenjoyabletoteach!OfhelpforreadersaremyYouTubelectures foundunder“Math176,MathematicsofFinance,”whichcovermostofthematerial in this book. As for required background, students who have finished the calculus sequence(severalvariables)andanintroductorycourseinprobabilityandstatistics havebeensuccessful. Finally, my thanks to Dan Jessie for corrections and suggested changes in the notesafterhetaughtthiscourseseveraltimes.SantiagoGuisasolamodifiedportions ofthesenotestoteachgiftedhighschoolstudentsinasummercamp.Mythanksto AnneliDuffinandKatriSiebergfortheirassistanceduringthedevelopmentofthis material.Thankstothefivereviewersfortheirusefulcomments.And,inparticular, mythankstothemanystudentsfortheirfeedback!
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