Martingale Methods in Financial Modelling (Stochastic Modelling and Applied Probability)

736 Pages·2009·4.83 MB·English

by  Marek Musiela

StochasticMechanics Stochastic Modelling
RandomMedia and Applied Probability
SignalProcessing
(Formerly:
andImageSynthesis
ApplicationsofMathematics)
MathematicalEconomicsandFinance
StochasticOptimization
6
StochasticControl 3
StochasticModelsinLifeSciences
Editedby B.Rozovski˘ı
G.Grimmett
Advisory Board D.Dawson
D.Geman
I. Karatzas
F. Kelly
Y.LeJan
B.Øksendal
G.Papanicolaou
E.Pardoux

Marek Musiela Marek Rutkowski
Martingale Methods
in Financial Modelling
SecondEdition
Authors
MarekMusiela MarekRutkowski
BNPParibas TechnicalUniversityWarszawa
10HarewoodAvenue Inst.Mathematics
LondonNW16AA Pl.Politechniki1
UK 00-661Warszawa
[email protected] Poland
[email protected]
ManagingEditors
B.Rozovski˘ı G.Grimmett
DivisionofAppliedMathematics CentreforMathematicalSciences
BrownUniversity UniversityofCambridge
182GeorgeSt WilberforceRoad
Providence,RI02912 CambridgeCB30WB
USA UK
[email protected] [email protected]
Coverillustration:CoverpatterncourtesyofRickDurrett,
CornellUniversity,Ithaca,NewYork.
ISBN978-3-540-20966-9 e-ISBN978-3-540-26653-2
DOI10.1007/978-3-540-26653-2
StochasticModellingandAppliedProbabilityISSN0172-4568
LibraryofCongressControlNumber:2004114482
MathematicsSubjectClassification(2000):60Hxx,62P05,90A09
2nded.2005.Corr.3rdprinting2009
©2005,1997Springer-VerlagBerlinHeidelberg
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Preface to the Second Edition
During the seven years that elapsed between the first and second editions of the
presentbook,considerableprogresswasachievedintheareaoffinancialmodelling
and pricing of derivatives. Needless to say, it was our intention to incorporate into
thesecondeditionatleastthemostrelevantandcommonlyacceptedofthesedevel-
opments.Sinceatthesametimewehadthestrongintentionnottoexpandthebook
toanunbearablesize,wedecidedtoleaveoutfromthefirsteditionofthisbooksome
portionsofmaterialoflesserpracticalimportance.
Letusstressthatwehaveonlytakenoutfewsectionsthat,inouropinion,were
of marginal importance for the understanding of the fundamental principles of fi-
nancialmodellingofarbitragevaluationofderivatives.Inviewoftheabundanceof
new results in the area, it would be in any case unimaginable to cover all existing
approachestopricingandhedgingfinancialderivatives(nottomentionallimportant
results) in a single book, no matter how voluminous it were. Hence, several inten-
sivelystudiedareas,suchas:mean-variancehedging,utility-basedpricing,entropy-
based approach,financial models with frictions (e.g., short-selling constraints, bid-
ask spreads, transaction costs, etc.) either remain unmentioned in this text, or are
presentedverysuccinctly.Althoughtheissueofmarketincompletenessisnottotally
neglected,itisexaminedprimarilyintheframeworkofmodelsofstochastic(orun-
certain) volatility. Luckily enough, the afore-mentioned approaches and results are
coveredexhaustivelyinseveralexcellentmonographswritteninrecentyearsbyour
distinguishedcolleagues,andthusitisourpleasuretobeabletorefertheinterested
readertothesetexts.
Letuscommentbrieflyonthecontentofthesecondeditionandthedifferences
withrespecttothefirstedition.
PartIwasmodifiedtoalesserextentandthusisnotverydissimilartoPartIin
thefirstedition.However,since,aswasmentionedalready,somesectionsfromthe
firsteditionweredeliberatelytakenout,wedecidedforthesakeofbetterreadability
tomergesomechapters.Also,weincludedinPartIanewchapterentirelydevotedto
volatilityriskandrelatedmodellingissues.Asaconsequence,theissuesofhedging
ofplain-vanillaoptionsandvaluationofexoticoptionsarenolongerlimitedtothe
classicalBlack-Scholesframeworkwithconstantvolatility.Thethemeofstochastic
volatilityalsoreappearssystematicallyinthesecondpartofthebook.
vi PrefacetotheSecondEdition
PartIIhasbeensubstantiallyrevisedandthusitsnewversionconstitutesamajor
improvement of the present edition with respect to the first one. We present there
alternativeinterestratemodels,andweprovidethereaderwithananalysisofeach
ofthem,whichisverymuchmoredetailedthaninthefirstedition.Althoughwedid
noteventrytoappraisetheefficiencyofreal-lifeimplementationsofeachapproach,
wehavestressedoneachoccasionthat,whendealingwithderivativespricingmod-
els,oneshouldalwayshaveinmindaspecificpracticalperspective.Putanotherway,
weadvocatetheopinion,putforwardbymanyresearchers,thatthechoiceofmodel
shouldbetiedtoobservedrealfeaturesofaparticularsectorofthefinancialmarket
orevenaproductclass.Consequently,anecessaryfirststepinmodellingisadetailed
study of functioning of a given market we wish to model. The goal of this prelim-
inary stage is to become familiar with existing liquid primary and derivative assets
(together with their sometimes complex specifications), and to identify sources of
risksassociatedwithtradingintheseinstruments.
Itwasourhopethatbyconcentratingonthemostpertinentandwidelyaccepted
modellingapproaches,wewillbeabletoprovidethereaderwithatextfocusedon
practicalaspectsoffinancialmodelling,ratherthantheoreticalones.Weleaveit,of
course, to the reader to assess whether we have succeeded achieving this goal to a
satisfactorylevel.
MarekRutkowskiexpresseshisgratitudetoMarekMusielaandthemembersof
theFixedIncomeResearchandStrategiesTeamatBNPParibasfortheirhospitality
duringhisnumerousvisitstoLondon.
MarekRutkowskigratefullyacknowledgespartialsupportreceivedfromthePol-
ishStateCommitteeforScientificResearchundergrantPBZ-KBN-016/P03/1999.
WewouldliketoexpressourgratitudetothestaffofSpringer-Verlag.Wethank
CatrionaByrneforherencouragementandinvaluableeditorialsupervision,aswell
asSusanneDenskusforherinvaluabletechnicalassistance.
LondonandSydney MarekMusiela
September2004 MarekRutkowski
Note on the Second Printing
Thesecondprintingofthesecondeditionofthisbookexpandsandclarifiesfurther
itscontentsexposition.Severalproofspreviouslylefttothereaderarenowincluded.
The presentation of LIBOR and swap market models is expanded to include the
jointdynamicsoftheunderlyingprocessesundertherelevantprobabilitymeasures.
The appendix in completed with several frequently used theoretical results making
thebookevenmoreself-contained.Thebibliographicalreferencesarebroughtupto
dateasfaraspossible.
This printing corrects also numerous typographical errors and mistakes. We
would like the express our gratitude to Alan Bain and Imanuel Costigan who un-
coveredmanyofthem.
LondonandSydney MarekMusiela
August2006 MarekRutkowski

Preface to the First Edition
Theoriginofthisbookcanbetracedtocoursesonfinancialmathematicstaughtby
us at the University of New South Wales in Sydney, Warsaw University of Tech-
nology(PolitechnikaWarszawska)andInstitutNationalPolytechniquedeGrenoble.
Ourinitialaimwastowriteashorttextaroundthematerialusedintwoone-semester
graduatecoursesattendedbystudentswithdiversedisciplinarybackgrounds(mathe-
matics,physics,computerscience,engineering,economicsandcommerce).Thean-
ticipateddiversityofpotentialreadersexplainsthesomewhatunusualwayinwhich
the book is written. It starts at a very elementary mathematical level and does not
assumeanypriorknowledgeoffinancialmarkets.Later,itdevelopsintoatextwhich
requiressomefamiliaritywithconceptsofstochasticcalculus(thebasicrelevantno-
tionsandresultsarecollectedintheappendix).Overtime,whatwasmeanttobea
shorttextacquiredalifeofitsownandstartedtogrow.Thefinalversioncanbeused
asatextbookforthreeone-semestercourses–oneatundergraduatelevel,theother
twoasgraduatecourses.
The first part of the book deals with the more classical concepts and results of
arbitrage pricing theory, developed over the last thirty years and currently widely
applied in financial markets. The second part, devoted to interest rate modelling is
moresubjectiveandthuslessstandard.Aconcisesurveyofshort-terminterestrate
models is presented. However, the special emphasis is put on recently developed
modelsbuiltuponmarketinterestrates.
WearegratefultotheAustralianResearchCouncilforprovidingpartialfinancial
supportthroughoutthedevelopmentofthisbook.WewouldliketothankAlanBrace,
BenGoldys,DieterSondermann,ErikSchlögl,LutzSchlögl,AlexanderMürmann,
andAlexanderZilberman,whoofferedusefulcommentsonthefirstdraft,andBarry
Gordon,whohelpedwithediting.
Ourhopeisthatthisbookwillhelptobringthemathematicalandfinancialcom-
munities closer together, by introducing mathematicians to some important prob-
lemsarisinginthetheoryandpracticeoffinancialmarkets,andbyprovidingfinance
professionals with a set of useful mathematical tools in a comprehensive and self-
containedmanner.
Sydney MarekMusiela
March1997 MarekRutkowski