Universitext Geon Ho Choe Stochastic Analysis for Finance with Simulations Universitext Universitext SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA VincenzoCapasso Universita` degliStudidiMilano,Milano,Italy CarlesCasacuberta UniversitatdeBarcelona,Barcelona,Spain AngusMacIntyre QueenMaryUniversityofLondon,London,UK KennethRibet UniversityofCalifornia,Berkeley,CA,USA ClaudeSabbah CNRS,E´colePolytechnique,Palaiseau,France EndreSu¨li UniversityofOxford,Oxford,UK WojborA.Woyczyn´ski CaseWesternReserveUniversity,Cleveland,OH,USA Universitext is a series of textbooksthat presents material from a wide variety of mathematicaldisciplinesatmaster’slevelandbeyond.Thebooks,oftenwellclass- testedbytheirauthor,mayhaveaninformal,personalevenexperimentalapproach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teachingcurricula,toverypolishedtexts. 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Moreinformationaboutthisseriesathttp://www.springer.com/series/223 Geon Ho Choe Stochastic Analysis for Finance with Simulations 123 GeonHoChoe DepartmentofMathematicalSciences andGraduateSchoolofFinance KoreaAdvancedInstituteofScience andTechnology Yuseong-gu,Daejeon RepublicofKorea ISSN0172-5939 ISSN2191-6675 (electronic) Universitext ISBN978-3-319-25587-3 ISBN978-3-319-25589-7 (eBook) DOI10.1007/978-3-319-25589-7 LibraryofCongressControlNumber:2016939955 MathematicsSubjectClassification:91Gxx,91G10,91G20,91G30,91G60,91G70,91G80 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Preface Thisbookisanintroductiontostochasticanalysisandquantitativefinance,includ- ingtheoreticalandcomputationalmethods,foradvancedundergraduateandgradu- atestudentsinmathematicsandbusiness,butnotexcludingpractitionersinfinance industry.Thebookisdesignedforreaderswhowanttohaveadeeperunderstanding of the delicate theory of quantitative finance by doing computer simulations in addition to theoretical study. Topics include stochastic calculus, option pricing, optimal portfolio investment, and interest rate models. Also included are simula- tions of stochastic phenomena,numericalsolutions of the Black–Scholes–Merton equation,MonteCarlomethods,andtimeseries.Basicmeasuretheoryisusedasa tool to describe probabilistic phenomena. The level of familiarity with computer programming is kept to a minimum. To make the book accessible to a wider audience,somebackgroundmathematicalfactsareincludedinthefirstpartofthe bookandalsointheappendices. Financial contracts are divided into two groups: The first group consists of primaryassetssuchassharesofstock,bonds,commodities,andforeigncurrencies. Thesecondgroupcontainsfinancialderivativessuchasoptionsandfuturesonthe underlying assets belonging to the first group. A financial derivative is a contract thatpromisespaymentincashordeliveryofanassetcontingentonthebehaviorof theunderlyingassetinthefuture.Thegoalofthisbookistopresentmathematical methods for finding how much one should pay for a financial derivative. To understandtheoptionpricingtheoryweneedideasfromvariousdisciplinesranging overpureandappliedmathematics,nottomentionfinanceitself.We trytobridge thegapbetweenmathematicsandfinancebyusingdiagrams,graphsandsimulations inadditiontorigoroustheoreticalexposition.Simulationsinthisbookarenotonly usedasthecomputationalmethodinquantitativefinance,butcanalsofacilitatean intuitiveanddeeperunderstandingoftheoreticalconcepts. SincethepublicationsbyBlack,ScholesandMertononoptionpricingin1973, thetheoryof stochasticcalculus,developedbyItoˆ,hasbecomethefoundationfor anewfieldcalledquantitativefinance.Inthisbookstochasticcalculusispresented startingfromthetheoreticalfoundation.Afterintroducingsomefundamentalideas in quantitative finance in Part I, we present mathematical prerequisites such as v vi Preface Lebesgueintegration,basicprobabilitytheory,conditionalexpectationandstochas- tic processes in Part II. After that, fundamental properties of Brownian motion, the Girsanov theorem and the reflection principle are given in Part III. In Part IV we introducethe Itoˆ integralandItoˆ’slemma, andthenpresentthe Feynman–Kac theorem.InPartVwepresentthreemethodsforpricingoptions:thebinomialtree method,theBlack–Scholes–Mertonpartialdifferentialequation,andthemartingale method. In Part VI we analyze more examples of the martingale method, and study exotic options, American options and numeraire. In Part VII the variance minimization methodfor optimal portfolioinvestmentis introducedfor a discrete time model. In Part VIII some interest rate models are introduced and used in pricingbonds.InPartIXtheNewton–Raphsonmethodoffindingimpliedvolatility, time series models for estimating volatility, Monte Carlo methods for option prices, numerical solution of the Black–Scholes–Merton equation, and numerical solutionofstochasticdifferentialequationsareintroduced.Intheappendicessome mathematical prerequisites are presented which are necessary to understand the material in the main chapters, such as point set topology,linear algebra, ordinary differentialequations, and partial differentialequations. The graphs and diagrams in thisbookwereplottedby the authorusingMATLAB and AdobeIllustrator,and thesimulationsweredoneusingMATLAB. Comments from the readers are welcome. For corrections and updates please checkauthor’shomepagehttp://shannon.kaist.ac.kr/choe/orsendanemail. ThisworkwaspartiallysupportedbyBasic ScienceResearchProgramthrough the National Research Foundation of Korea (NRF) funded by the Ministry of Science,ICTandFuturePlanning(NRF-2015R1A2A2A01006176). GeonH.Choe Acknowledgements The author wishes to thank Colin Atkinson, Suk Joon Byun, Dong-Pyo Chi, Hyeong-InChoi,DongMyungChung,DanCrisan,OleHald,YujiIto,SunYoung Jang, Intae Jeon, Duk Bin Jun, Byung-Chun Kim, Dohan Kim, Hong Jong Kim, JaewanKim,KyungSooKim,MinhyongKim,SolKim,SoonKiKim,TongSuk Kim, Ohsang Kwon, Yong Hoon Kwon, Jong Hoon Oh, Hyungju Park, Andrea Pascucci, Barnaby Sheppard, Sujin Shin, Ravi Shukla, Insuk Wee, Harry Zheng, andallhiscolleaguesintheDepartmentofMathematicalSciencesandtheGraduate SchoolofFinanceatKAIST. He wishes to thank YounghoAhn, Soeun Choi, Hyun Jin Jang, Myeong Geun Jeong,MihyunKang,Bong Jo Kim, ChihurnKim, DongHan Kim, KunheeKim, KiHwanKoo,SoonWonKwon,DongMinLee,KyungsubLee,YoungHoonNa, Jong Jun Park, Minseok Park, Byung Ki Seo and Seongjun Yoon. He also thanks many students who took various courses on quantitative finance taught over the years. The author thanks the editors Joerg Sixt, Re´mi Lodh and Catriona Byrne, and the staff at Springer, including Catherine Waite. He is also grateful to several anonymousreviewers who gave many helpful suggestions. The author thanks his mother and mother-in-law for their love and care. Finally, he wishes to thank his wifeforherloveandpatience. vii Contents PartI IntroductiontoFinancialMathematics 1 FundamentalConcepts ..................................................... 3 1.1 Risk...................................................................... 3 1.2 TimeValueofMoney................................................... 4 1.3 NoArbitragePrinciple.................................................. 4 1.4 ArbitrageFreeMarket.................................................. 5 1.5 Risk-NeutralPricingandMartingaleMeasures....................... 10 1.6 TheOnePeriodBinomialTreeModel................................. 11 1.7 ModelsinFinance ...................................................... 13 2 FinancialDerivatives........................................................ 15 2.1 ForwardContractsandFutures......................................... 15 2.2 Options.................................................................. 16 2.3 Put-CallParity .......................................................... 19 2.4 RelationsAmongOptionPricingMethods............................ 20 PartII ProbabilityTheory 3 TheLebesgueIntegral ...................................................... 25 3.1 Measures ................................................................ 25 3.2 SimpleFunctions........................................................ 28 3.3 TheLebesgueIntegral.................................................. 29 3.4 Inequalities.............................................................. 34 3.5 TheRadon–NikodymTheorem ........................................ 36 3.6 ComputerExperiments................................................. 37 4 BasicProbabilityTheory ................................................... 41 4.1 MeasureandProbability................................................ 41 4.2 CharacteristicFunctions................................................ 46 4.3 IndependentRandomVariables........................................ 50 4.4 ChangeofVariables..................................................... 56 4.5 TheLawofLargeNumbers............................................ 60 4.6 TheCentralLimitTheorem ............................................ 62 ix