Stochastic Finance: An Introduction in Discrete Time

609 Pages·2016·2.84 MB·English

by  Hans Föllmer

HansFöllmer,AlexanderSchied
StochasticFinance
DeGruyterGraduate
Also of Interest
IdrisAssani
ErgodicTheory:AdvancesinDynamicalSystems,2016
ISBN978-3-11-046086-5,e-ISBN(PDF)978-3-11-046151-0,
e-ISBN(EPUB)978-3-11-046091-9
YasushiIshikawa
StochasticCalculusofVariations:ForJumpProcesses,2ndEd.,
2016
ISBN978-3-11-037776-7,e-ISBN(PDF)978-3-11-037807-8,
e-ISBN(EPUB)978-3-11-039232-6
DmitriiS.Silvestrov
American-Type Options: Stochastic Approximation Methods,
Volume2,2014
ISBN978-3-11-032968-1,e-ISBN(PDF)978-3-11-032984-1,
e-ISBN(EPUB)978-3-11-038990-6
Wolf-JürgenBeyn,RaphaelKruse
NumericalAnalysisofStochasticProcesses,2017
ISBN978-3-11-044337-0,e-ISBN(PDF)978-3-11-044338-7,
e-ISBN(EPUB)978-3-11-043555-9
VidyadharS.Mandrekar
WeakConvergenceofStochasticProcesses:WithApplications
toStatisticalLimit
Theorems,2016
ISBN978-3-11-047542-5,e-ISBN(PDF)978-3-11-047631-6,
e-ISBN(EPUB)978-3-11-047545-6
www.degruyter.com
Hans Föllmer, Alexander Schied
Stochastic Finance
|
An Introduction in Discrete Time
Fourth revised and extended edition
MathematicsSubjectClassification2010
Primary:60-01,91-01,91-02;Secondary:46N10,60E15,60G40,60G42,91B08,91B16,91B30,
91B50,91B52,91B70,91G10,91G20,91G80,91G99.
Authors
Prof.Dr.HansFöllmer
Humboldt-UniversitätzuBerlin
InstitutfürMathematik
UnterdenLinden6
10117Berlin
[email protected]
Prof.Dr.AlexanderSchied
UniversitätMannheim
LehrstuhlfürWirtschaftsmathematik
A56
68159Mannheim
[email protected]
ISBN978-3-11-046344-6
e-ISBN(PDF)978-3-11-046345-3
e-ISBN(EPUB)978-3-11-046346-0
LibraryofCongressCataloging-in-PublicationData
ACIPcatalogrecordforthisbookhasbeenappliedforattheLibraryofCongress.
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2016WalterdeGruyterGmbH,Berlin/Boston
Typesetting:CompuscriptLtd,Ireland
Printingandbinding:CPIbooksGmbH,Leck
♾Printedonacid-freepaper
PrintedinGermany
www.degruyter.com
Preface to the fourth edition
In this fourth edition, we have made a number of minor improvements, added
some background material, and included more than two dozen new exercises. We
have also addressed many comments and suggestions we received from colleagues
and students during the past five years. It is a pleasure to thank Aurélien Alfonsi,
Martin Barbie, Nike Blessing, Mengyu Du, Sabina Gadzhieva, Nina Gantert, Adrian
Gorniak,RiaGrindel,NilsHansen,TimoHirscher,MariusKaiser,AlexanderKalinin,
ThomasKnispel,MouradLazgham,TobiasLutz,SimonMaurer,HeikoPreuß,Maxvon
Renesse,UweSchmock,QunWang,andIlanZlotin.SpecialthanksareduetoStefan
Gerhold.Finally,wethankourpublisherDeGruyterforencouragingustopreparea
fourtheditionandfortheirhelpfulsupport.
July 2016 HansFöllmer
AlexanderSchied
Preface to the third edition
ThisthirdeditionofourbookappearsinthedeGruytergraduatetextbookseries.We
have therefore included more than one hundred exercises. Typically, we have used
thebookasanintroductorytextfortwomajorareas,eithercombinedintoonecourse
or in two separate courses. The first area comprises static and dynamic arbitrage
theoryindiscretetime.ThecorrespondingcorematerialisprovidedinChapters1,5,
and6.Thesecondareadealswithmathematicalaspectsoffinancialriskasdeveloped
in Chapters 2, 4, and 11. Most of the exercises we have included in this edition are
therefore contained in these core chapters. The other chapters of this book can be
usedbothascomplementarymaterialfortheintroductorycoursesandasbasisfor
special-topicscourses.
Inrecentyears,therehasbeenanincreasingawareness,bothamongpractitioners
andinacademia,oftheproblemofmodeluncertaintyinfinanceandeconomics,often
calledKnightianuncertainty;see,e.g.,[270].Inthisthirdeditionwehaveputmore
emphasisonthisissue.Thetheoryofriskmeasurescanbeseenasacasestudyhow
to deal with model uncertainty in mathematical terms. We have therefore updated
Chapter 4 on static risk measures and added the new Chapter 11 on dynamic risk
measures.Moreover,inSection2.5wehaveextendedthecharacterizationofrobust
preferencesintermsofriskmeasuresfromthecoherenttotheconvexcase.Wehave
alsoincludedthenewSections3.5and8.3onrobustvariantsoftheclassicalproblems
ofoptimalportfoliochoiceandefficienthedging.
It is a pleasure to express our thanks to all students and colleagues whose
comments have helped us to prepare this third edition, in particular to Aurélien
Alfonsi, Günter Baigger, Francesca Biagini, Julia Brettschneider, Patrick Cheridito,
SamuelDrapeau,MarenEckhoff,Karl-TheodorEisele,DamirFilipovic,ZichengHong,
Kostas Kardaras, Thomas Knispel, Gesine Koch, Heinz König, Volker Krätschmer,
ChristophKühn,MichaelKupper,MouradLazgham,SvenLickfeld,MareikeMassow,
IrinaPenner,ErnstPresman,MichaelScheutzow,MelvinSim,AllaSlynko,Stephan
Sturm,GregorSvindland,LongTeng,FlorianWerner,WiebkeWittmüß,andLeiWu.
SpecialthanksareduetoYuliyaMishuraandGeorgiyShevchenko,ourtranslatorsfor
theRussianedition.
November2010 HansFöllmer
AlexanderSchied
Preface to the second edition
Since the publication of the first edition we have used it as the basis for several
courses. These include courses for a whole semester on Mathematical Finance in
Berlin and also short courses on special topics such as risk measures given at the
InstitutHenriPoincaréinParis,attheDepartmentofOperationsResearchatCornell
University,attheAcademiaSinicainTaipei,andatthe8thSymposiumonProbability
and Stochastic Processes in Puebla. In the process we have made a large number
ofminorcorrections,wehavediscoveredmanyopportunitiesforsimplificationand
clarification,andwehavealsolearnedmoreaboutseveraltopics.Asaresult,major
parts of this book have been improved or even entirely rewritten. Among them are
thoseonrobustrepresentationsofriskmeasures,arbitrage-freepricingofcontingent
claims,exoticderivativesintheCRRmodel,convergencetoBlack–Scholesprices,and
stabilityunderpastingwithitsconnectionstodynamicallyconsistentcoherentrisk
measures.Inaddition,thissecondeditioncontainsseveralnewsections,includinga
systematicdiscussionoflaw-invariantriskmeasures,ofconcavedistortions,andof
therelationsbetweenriskmeasuresandChoquetintegration.
Itisapleasuretoexpressourthankstoallstudentsandcolleagueswhosecom-
mentshavehelpedustopreparethissecondedition,inparticulartoDirkBecherer,
HansBühler,Rose-AnneDana,UlrichHorst,MesropJanunts,ChristophKühn,Maren
Liese, Harald Luschgy, Holger Pint, Philip Protter, Lothar Rogge, Stephan Sturm,
StefanWeber,WiebkeWittmüß,andChing-TangWu.SpecialthanksareduetoPeter
BankandtoYuliyaMishuraandGeorgiyShevchenko,ourtranslatorsfortheRussian
edition.Finally,wethankIreneZimmermannandManfredKarbeofdeGruyterVerlag
forurgingustowriteasecondeditionandfortheirefficientsupport.
Berlin,September2004 HansFöllmer
AlexanderSchied
Preface to the first edition
This book is an introduction to probabilistic methods in Finance. It is intended for
graduatestudentsinmathematics,anditmayalsobeusefulformathematiciansin
academiaandinthefinancialindustry.Ourfocusisonstochasticmodelsindiscrete
time.Thislimitationhastwoimmediatebenefits.First,theprobabilisticmachinery
is simpler, and we can discuss right away some of the key problems in the theory
ofpricingandhedgingoffinancialderivatives.Second,theparadigmofacomplete
financialmarket,whereallderivativesadmitaperfecthedge,becomestheexception
ratherthantherule.Thus,thediscrete-timesettingprovidesashortcuttosomeofthe
morerecentliteratureonincompletefinancialmarketmodels.
Asatextbookformathematicians,itisanintroductionatanintermediatelevel,
withspecialemphasisonmartingalemethods.Sinceitdoesnotusethecontinuous-
timemethodsofItôcalculus,itneedslesspreparationthanmoreadvancedtextssuch
as[101],[102],[111],[177],[263].Ontheotherhand,itistechnicallymoredemanding
thantextbookssuchas[225]:Weworkongeneralprobabilityspaces,andsothetext
capturestheinterplaybetweenprobabilitytheoryandfunctionalanalysiswhichhas
beencrucialforsomeoftherecentadvancesinmathematicalfinance.
ThebookisbasedonournotesforfirstcoursesinMathematicalFinancewhich
bothofusareteachinginBerlinatHumboldtUniversityandatTechnicalUniversity.
These courses are designed for students in mathematics with some background in
probability.Sometimes,theyaregiveninparalleltoasystematiccourseonstochastic
processes.Atothertimes,martingalemethodsindiscretetimearedevelopedinthe
course,astheyareinthisbook.Usuallythecourseisfollowedbyasecondcourseon
MathematicalFinanceincontinuoustime.Thereitturnsouttobeusefulthatstudents
arealreadyfamiliarwithsomeofthekeyideasofMathematicalFinance.
ThecoreofthisbookisthedynamicarbitragetheoryinthefirstchaptersofPartII.
Whenteachingacourse,wefounditusefultoexplainsomeofthemainargumentsin
themoretransparentone-periodmodelbeforeusingtheminthedynamicalsetting.
So one approach would be to start immediately in the multi-period framework of
Chapter 5, and to go back to selected sections of Part I as the need arises. As an
alternative,onecouldfirstfocusontheone-periodmodel,andthenmoveontoPartII.
WeincludeinChapter2abriefintroductiontothemathematicaltheoryofexpec-
tedutility,eventhoughthisisaclassicaltopic,andthereisnoshortageofexcellent
expositions;see,forinstance,[193]whichhappenstobeourfavorite.Wehavethree
reasonsforincludingthischapter.Ourfocusinthisbookisonincompleteness,and
incompletenessinvolves,inoneformoranother,preferencesinthefaceofriskand
uncertainty. We feel that mathematicians working in this area should be aware, at
leasttosomeextent,ofthelonglineofthoughtwhichleadsfromDanielBernoullivia
vonNeumann–MorgensternandSavagetosomemorerecentdevelopmentswhichare
motivatedbyshortcomingsoftheclassicalparadigm.Thisisourfirstreason.Second,
Prefacetothefirstedition | ix
theanalysisofriskmeasureshasemergedasamajortopicinmathematicalfinance,
andthisiscloselyrelatedtoarobustversionoftheSavagetheory.Third,butnotleast,
ourexperienceisthatthispartofthecoursewasfoundparticularlyenjoyable,both
bythestudentsandbyourselves.
We acknowledge our debt and express our thanks to all colleagues who have
contributed, directly or indirectly, through their publications and through informal
discussions, to our understanding of the topics discussed in this book. Ideas and
methodsdevelopedbyFreddyDelbaen,DarrellDuffie,NicoleElKaroui,DavidHeath,
YuriKabanov,IoannisKaratzas,DimitriKramkov,DavidKreps,StanleyPliska,Chris
Rogers,SteveRoss,WalterSchachermayer,MartinSchweizer,DieterSondermannand
ChristopheStrickerplayakeyroleinourexposition.Weareobligedtomanyothers;
forinstancethetextbooks[76],[101],[102],[160],and[200]wereagreathelpwhenwe
startedtoteachcoursesonthesubject.
Wearegratefultoallthosewhoreadpartsofthemanuscriptandmadeuseful
suggestions,inparticulartoDirkBecherer,UlrichHorst,SteffenKrüger,IrinaPenner,
andtoAlexanderGiesewhodesignedsomeofthefigures.Specialthanksaredueto
PeterBankforalargenumberofconstructivecomments.Wealsoexpressourthanksto
ErhanÇinlar,AdamMonahan,andPhilipProtterforimprovingsomeofthelanguage,
andtotheDepartmentofOperationsResearchandFinancialEngineeringatPrinceton
Universityforitshospitalityduringtheweekswhenwefinishedthemanuscript.
Berlin,June2002 HansFöllmer
AlexanderSchied