Financial Modeling: A Backward Stochastic Differential Equations Perspective PDF
Preview Financial Modeling: A Backward Stochastic Differential Equations Perspective
Springer Finance Textbooks Stéphane Crépey Financial Modeling A Backward Stochastic Diff erential Equations Perspective Springer Finance Textbooks EditorialBoard MarcoAvellaneda GiovanniBarone-Adesi MarkBroadie MarkH.A.Davis EmanuelDerman ClaudiaKlüppelberg WalterSchachermayer Springer Finance Textbooks Springer Finance is a programme of books addressing students, academics and practitionersworkingonincreasinglytechnicalapproachestotheanalysisoffinan- cialmarkets.Itaimstocoveravarietyoftopics,notonlymathematicalfinancebut foreignexchanges,termstructure,riskmanagement,portfoliotheory,equityderiva- tives,andfinancialeconomics. ThissubseriesofSpringerFinanceconsistsofgraduatetextbooks. Forfurthervolumes: http://www.springer.com/series/11355 Stéphane Crépey Financial Modeling A Backward Stochastic Differential Equations Perspective Prof.StéphaneCrépey Départementdemathématiques, LaboratoireAnalyse&Probabilités Universitéd’EvryVald’Essone Evry,France Additionalmaterialtothisbookcanbedownloadedfromhttp://extras.springer.com Password:[978-3-642-37112-7] ISBN978-3-642-37112-7 ISBN978-3-642-37113-4(eBook) DOI10.1007/978-3-642-37113-4 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013939614 MathematicsSubjectClassification: 91G20,91G60,91G80 JELClassification: G13,C63 ©Springer-VerlagBerlinHeidelberg2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To Irèneand Camille Preface Thisisabookonfinancialmodelingthatemphasizescomputationalaspects.Itgives a unified perspective on derivative pricing and hedging across asset classes and is addressedtoallthosewhoareinterestedinapplicationsofmathematicstofinance: students,quantsandacademics. The book features backward stochastic differential equations (BSDEs), which areanattractivealternativetothemorefamiliarpartialdifferentialequations(PDEs) for representing prices and Greeks of financial derivatives. First, BSDEs offer the mostunifiedsetupforpresentingthefinancialderivativespricingandhedgingtheory (asreflectedbytherelativecompactnessofthebook,givenitsratherwidescope). Second,BSDEsareatechnicallyveryflexibleandpowerfulmathematicaltoolfor elaborating the theory with all the required mathematical rigor and proofs. Third, BSDEs are also useful for the numerical solution of high-dimensional nonlinear pricingproblemssuchasthenonlinearCVAandfundingissueswhichhavebecome importantsincethegreatcrisis[30,80,81]. Structure oftheBook Part I provides a course in stochastic processes, beginning at a quite elementary levelinordertogentlyintroducethereadertothemathematicaltoolsthatareneeded subsequently.PartIIdealswiththederivationofthepricingequationsoffinancial claimsandtheirexplicitsolutionsinafewcaseswheretheseareeasilyobtained,al- thoughtypicallytheseequationshavetobesolvednumericallyasisdoneinPartIII. PartIVprovidestwocomprehensiveapplicationsofthebook’sapproachthatillus- trate the versatility of simulation/regression pricing schemes for high-dimensional pricingproblems.PartVprovidesathoroughmathematicaltreatmentoftheBSDEs andPDEsthatareoffundamentalimportanceforourapproach.Finally,PartVIis anextendedappendixwithtechnicalproofs,exercisesandcorrectedproblemsets. vii viii Preface Outline Chapters1–3provideasurveyofusefulmaterialfromstochasticanalysis.InChap.4 werecallthebasicsoffinancialtheorywhicharenecessaryforunderstandinghow therisk-neutralpricingequationofagenericcontingentclaimisderived.Thischap- tergivesa unifiedviewon thetheoryofpricingandhedgingfinancialderivatives, using BSDEs as a main tool. We then review, in Chap. 5, benchmark models on referencederivativemarkets.Chapter6isaboutMonteCarlopricingmethodsand Chaps.7and8dealwithdeterministicpricingschemes:treesinChap.7andfinite differencesinChap.8. Notethatthereisnohermeticfrontierbetweendeterministicandstochasticpric- ingschemes.Inessence,allthesenumericalschemesarebasedontheideaofprop- agating the solution, starting from a surface of the time-space domain on which it isknown(typically:thematurityofaclaim),alongsuitable(random)“characteris- tics”oftheproblem.Here“characteristics”referstoRiemann’smethodforsolving hyperbolic first-order equations (see Chap. 4 of [191]). From the point of view of controltheory,allthesenumericalschemescanbeviewedasvariantsofBellman’s dynamicprogrammingprinciple[26].MonteCarlopricingschemesmaythusbere- gardedasone-time-stepmultinomialtrees,convergingtoalimitingjumpdiffusion whenthenumberofspacediscretizationpoints(treebranches)goestoinfinity.The difference between a tree method in the usual sense and a Monte Carlo method is thataMonteCarlocomputationmeshisstochasticallygeneratedandnonrecombin- ing. Pricesofliquidfinancialinstrumentsaregivenbythemarketandaredetermined bysupply-and-demand.Liquidmarketpricesarethusactuallyusedbymodelsinthe “reverse-engineering” mode that consists in calibrating a model to market prices. This calibration process is the topic of Chap. 9. Once calibrated to the market, a model can be used for Greeking and/or for pricing more exotic claims (Greeking meanscomputingrisksensitivitiesinordertoset-uparelatedhedge). Analogiesanddifferencesbetweensimulationanddeterministicpricingschemes aremostclearlyvisibleinthecontextofpricingbysimulationclaimswithearlyex- ercise features (American and/or cancelable claims). Early exercisable claims can be priced by hybrid “nonlinear Monte Carlo” pricing schemes in which dynamic programmingequations,similartothoseusedindeterministicschemes,areimple- mentedonstochasticallygeneratedmeshes.Suchhybridschemesarethetopicsof Chaps.10and11,indiffusionandinpurejumpsetups,respectively.Again,thisis presentlybecomingquitetopicalforthepurposeofCVAcomputations. Chapters 12–14 develop, within a rigorous mathematical framework, the con- nection between backward stochastic differential equations and partial differential equations. This is done in a jump-diffusion setting with regime switching, which coversallthemodelsconsideredinthebook. FinallyChap.15gathersthemostdemandingproofsofPartV,Chap.16isde- votedtoexercisesforPartIandChap.17providessolvedproblemsetsforPartsII andIII. Preface ix Fig.1 Gettingstartedwiththebook:roadmapof“afirstandpartialreading”fordifferentaudi- ences.Green:Students.Blue:Quants.Red:Academics Roadmap Giventhedualnatureoftheproposedaudience(scholarsandquants),wehavepro- videdmorebackgroundmaterialonstochasticprocesses,pricingequationsandnu- mericalmethodsthanisneededforourmainpurposes.Yetwehavenotavoidedthe sometimesdifficultmathematicaltechniquethatisneededfordeepunderstanding. So, for the convenience of readers, we signal sections that contain advanced ma- terial with an asterisk (*) or even a double asterisk (**) for the still more difficult portions. Our ambition is, of course, that any reader should ultimately benefit from all partsofthebook.Weexpectthatanaveragereaderwillneedtwoorthreeattempts atreadingatdifferentlevelsforachievingthisobjective.Toprovideadditionalguid- ance,weproposethefollowingroadmapofwhata“firstandpartial”readingofthe bookcouldbeforthree“stylized”readers(seeFig.1forapictorialrepresentation): a student (in “green” on the figure), a quant (“blue” audience) and an academic (“red”;the“blueandred”boxinthechartmayrepresentavaluablefirstreadingfor bothquantsandacademics): • for a graduate student (“green”), we recommend a first reading of the book at a classical quantitative and numerical finance textbook level, as follows in this order: – startbyChaps.1–3(exceptforthestarredsectionsofChap.3),alongwiththe accompanying(generallyclassical)exercisesofChap.16,1 1Solutionsoftheexercisesareavailableforcourseinstructors. x Preface – thenjumptoChaps.5to9,dothecorrectedproblemsofChap.17andrunthe accompanyingMatlabscripts(http://extras.springer.com); • foraquant(“blue”): – startwiththestarredsectionsofChap.3,followedbyChap.4, – thenjumptoChaps.10and11; • foranacademicsoraPhDstudent(“red”): – startwiththestarredsectionsofChap.3,followedbyChap.4, – thenjumptoChaps.12to14,alongwiththerelatedproofsinChap.15. TheRoleofBSDEs Althoughthisbookisn’texclusivelydedicatedtoBSDEs,itfeaturestheminvarious contextsasacommonthreadforguidingreadersthroughtheoreticalandcomputa- tional aspects of financial modeling. For readers who are especially interested in BSDEs,werecommend: • Section3.5foramathematicalintroductionofBSDEsataheuristiclevel, • Chapter4fortheirgeneralconnectionwithhedging, • Section6.10andChap.10fornumericalaspectsand • PartVandChap.15fortherelatedmathematics. InSect.11.5wealsogiveaprimerofCVAcomputationsusingsimulation/regression techniquesthataremotivatedbyBSDEnumericalschemes,eventhoughnoBSDEs appear explicitly. More on this will be found in [30], for which the present book shouldbeausefulcompanion. BibliographicGuidelines Toconcludethispreface,hereareafewgeneralreferences: • on random processes and stochastic analysis, often with connections to finance (Chaps.1–3):[149,159,167,174,180,205,228]; • onmartingalemodelinginfinance(Chap.4):[93,114,159,191,208,245]; • onmarketmodels(Chap.5):[43,44,58,131,146,208,230,241]; • onMonteCarlomethods(Chap.6):[133,176,226]; • ondeterministicpricingschemes(Chaps.7and8):[1,12,27,104,172,174,207, 248,254,256]; • onmodelcalibration(Chap.9):[71,116,213]; • onsimulation/regressionpricingschemes(Chaps.10and11):[133,136]; • onBSDEsandPDEs,especiallyinconnectionwithfinance(Chaps.12–14):[96, 113,114,122,164,197,224].
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