Table Of ContentLecture Notes in Mathematics (cid:49)(cid:57)(cid:49)(cid:57)
Editors:
J.-M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
René A. Carmona · Ivar Ekeland
Arturo Kohatsu-Higa · Jean-Michel Lasry
Pierre-Louis Lions · Huyên Pham
Erik Taflin
Paris-Princeton Lectures
on Mathematical Finance
2004
Editorial Committee:
R.A. Carmona, E. Çinlar,
I. Ekeland, E. Jouini,
J.A. Scheinkman, N. Touzi
(cid:65)(cid:66)(cid:67)
Authors
RenéA.Carmona IvarEkeland
BendheimCenterforFinance CanadaResearchChair
DepartmentofOperationsResearch inMathematicalEconomics
andFinancialEngineering UniversityofBritishColumbia
PrincetonUniversity,Princeton DepartmentofMathematics
NJ08544,USA 1984MathematicsRoad
e-mail:[email protected] V6T1Z2Canada
e-mail:[email protected]
ArturoKohatsu-Higa Jean-MichelLasry
GraduateSchoolofEngineeringSciences CALYON
OsakaUniversity 9,QuaiduPrésidentPaulDoumer
MachikaneyamaCho1–3 92920Paris-LaDéfenseCedex,France
Osaka560-8531,Japan e-mail:[email protected]
e-mail:[email protected]
HuyênPham
Pierre-LouisLions
LaboratoiredeProbabilités
CollègedeFrance etModèlesAléatoires
11,placeMarcelinBerthelot CNRS,UMR7599
75005Paris,CX05,France UniversitéParis7andInstitutUniversitaire
e-mail:[email protected] deFrance
ParisCX05,France
ErikTaflin e-mail:[email protected]
ChairinMathematicalFinance
EISTI,EcoleInternationaledesSciences [Theaddressesofthevolumeeditors
duTraitementdel’Information appearonpageIX]
AvenueduParc
95011Cergy,France
e-mail:tafl[email protected]
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Preface
This is the third volume of the Paris-Princeton Lectures in Mathematical Finance.
The goal of this series is to publish cutting edge research in self-contained articles
preparedbywellknownleadersinthefieldorpromisingyoungresearchersinvited
bytheeditors.Particularattentionispaidtothequalityoftheexposition,andtheaim
isatarticlesthatcanserveasanintroductoryreferenceforresearchinthefield.
The series is a result of frequent exchanges between researchers in finance and
financial mathematics in Paris and Princeton. Many of us felt that the field would
benefit from timely exposés of topics in which there is important progress. René
Carmona, Erhan Cinlar, Ivar Ekeland, Elyes Jouini, José Scheinkman and Nizar
Touzi serve in the first editorial board of the Paris-Princeton Lectures in Finan-
cialMathematics.AlthoughmanyofthechaptersinvolvelecturesgiveninParisor
Princeton, we also invite other contributions. Given the current nature of the colla-
boration between the two poles, we expect to produce a volume per year. Springer
VerlagkindlyofferedtohostthisenterpriseundertheumbrellaoftheLectureNotes
inMathematicsseries,andwearethankfultoCatrionaByrneforherencouragement
andherhelpintheinitialstageoftheinitiative.
This third volume contains five chapters. In the first chapter, René Carmona
demonstrates how the HJM approach to the construction of dynamic models can
be used for different financial markets. The original proposal of Heath, Jarrow and
MortonwasframedfortheworldofTreasurybonds,butitsapplicabilitywasexten-
ded soon after its publication. However implementation of the same modeling phi-
losophyinthecaseofcreditandequitymarketshadtowait.Purelyforpedagogical
reasons, this chapter starts with a review of the original HJM approach to fixed
income markets. Then, the recent works of Sidenius, Pitterbarg and Andersen and
Schoenbucheroncreditportfoliomodelingarepresented.Finally,thelastpartofthe
chapterexplainshowCarmonaandNadtochiydevelopedtheprogramoutlinedafew
yearsagobyDermanandKaniforequitymarkets.
The second chapter, by Ivar Ekeland and Erik Taflin, also develops the HJM
framework. The emphasis here is on the optimal management of bond portfolios,
that is, on Merton’s problem of expected utility maximization. The authors intro-
duceaspecialclassofassets,therollovers,whichhaveaconstanttimetomaturity,
VI Preface
and describe the bond portfolios as a combination of rollovers rather than a com-
bination of zero-coupon bonds, as is usual in the literature. The advantage of this
approach is that a rollover does not mature, in contrast to a bond. By considering
bond portfolios as a combination of rollovers, one brings them close to stock port-
folios, which do not mature either, and one paves the way for a unified theory of
moneymarketsandequitymarkets.Inaddition,theauthorsderiveexplicitformulas
fortheoptimalportfolios,atleastinthecasewhenthedriftandvolatilityaredeter-
ministicprocesses.Theseadvantagescomeataprice:amathematicalsettingmustbe
found,whichwillaccommodatethecurvesdescribingthetermstructureofinterest
rates,andallowthemtovaryrandomly,subjecttopossiblyinfinitelymanysources
ofnoise,whileremainingsufficientlysmooth.TheMusielaparametrization(thatis,
takingtimetomaturityinsteadofdateofmaturityastherelevantvariable),although
itisnaturalinthatcontext,considerablycomplicatesmatters,foritintroducesanad-
ditional(anddiscontinuous)termintheequationsofmotion.Roughlyspeaking,that
chaptercomplementstheprecedingone:takentogether,theyhighlighttheflexibility
andgeneralityoftheHJMmodel,aswellasitsmanyunexploredconsequences.
ThethirdoneiswrittenbyArturoKohatsu-Higaandisbasedonashortcourse
giveninParisinNovemberandDecember2004.Itreviewsrecentresultsonmodels
forinsidertradingbasedonthetheoryofenlargementoffiltrations.Afterreviewing
the case of an insider having extra information given by the knowledge of the dis-
tribution of certain random variables, the author concentrates on the case when the
insiderbenefitsfromalmostsureadditionalinformation,leadingnaturallytotheuse
of anticipative calculus. This review can be viewed as a natural companion to the
chapterdeliveredbyFabriceBaudouininthefirstvolumeoftheseries.Thestyleof
thischapterispurposedlypedagogical,emphasizingdiscretetimeapproximationsas
awaytoillustratethedifferencesbetweenanticipativeandnon-anticipativecalculus,
andexerciseswithsolutionsasawaytoisolateproofsoftechnicalresults.
ThefourthchapteriscontributedtobyPierreLouisLionsandJeanMichelLasry.
It is concerned with the influence of hedging on the dynamics of the underlying
asset price. The problem is completely solved in the case of a large investor. The
resultsarebasedonadetailedanalysisofliquidationandindifferencepricesstudied
via the solutions of non-standards stochastic control problems. This paper is more
of a technical nature, and reads more like a research article than an introductory
review.However,theimportanceoftheissueandtheoriginalityofthemathematical
approachfullyjustifyitsinclusionintheseries.
The last chapter is concerned with applications of large deviations to problems
infinanceandinsurance.ItiscontributedbyHuyénPham.Largedeviationapprox-
imations and importance sampling methods are discussed in the context of option
pricing. Credit risk losses and portfolio benchmarking are also analyzed with the
toolsoflargedeviations.
TheEditors
Paris/Princeton
January31,2007
Contents
HJM:AUnifiedApproachtoDynamicModelsforFixedIncome,Credit
andEquityMarkets
RenéA.Carmona ................................................. 1
OptimalBondPortfolios
IvarEkelandandErikTaflin......................................... 51
ModelsforInsiderTradingwithFiniteUtility
ArturoKohatsu-Higa...............................................103
LargeInvestorTradingImpactsonVolatility
Pierre-LouisLionsandJean-MichelLasry .............................173
SomeApplicationsandMethodsofLargeDeviations
inFinanceandInsurance
HuyênPham .....................................................191
Editors
RenéA.Carmona
PaulM.Wythes’55ProfessorofEngineeringandFinance
ORFEandBendheimCenterforFinance
PrincetonUniversity
PrincetonNJ08540,USA
email: [email protected]
ErhanÇinlar
NormanJ.SollenbergerProfessorofEngineering
ORFEandBendheimCenterforFinance
PrincetonUniversity
PrincetonNJ08540,USA
email: [email protected]
IvarEkeland
CanadaResearchChairinMathematicalEconomics
DepartmentofMathematics,Annex1210
UniversityofBritishColumbia
1984MathematicsRoad
Vancouver,B.C.,CanadaV6T1Z2
email: [email protected]
ElyesJouini
CEREMADE,UFRMathématiquesdelaDécision
UniversitéParis-Dauphine
PlaceduMaréchaldeLattredeTassigny
75775ParisCedex16,France
email: [email protected]
X Editors
JoséA.Scheinkman
TheodoreWells’29ProfessorofEconomics
DepartmentofEconomicsandBendheimCenterforFinance
PrincetonUniversity
PrincetonNJ08540,USA
email: [email protected]
NizarTouzi
CentredeMathématiquesAppliquées
EcolePolytechniqueParis
F-91128PalaiseauCedex,FRANCE
email: [email protected]
HJM: A Unified Approach to Dynamic Models
for Fixed Income, Credit and Equity Markets(cid:1)
RenéA.Carmona
BendheimCenterforFinance
DepartmentofOperationsResearch&FinancialEngineering,
PrincetonUniversity,Princeton,NJ08544,USA
email: [email protected]
Summary. ThepurposeofthispaperistohighlightsomeofthekeyelementsoftheHJM
approachasoriginallyintroducedintheframeworkoffixedincomemarketmodels,toexplain
howtheverysamephilosophywasimplementedinthecaseofcreditportfolioderivativesand
toshowhowitcanbeextendedtoandusedinthecaseofequitymarketmodels.Ineachcase
weshowhowtheHJMapproachnaturallyyieldsaconsistencyconditionandano-arbitrage
condition in the spirit of the original work of Heath, Jarrow and Morton. Even though the
actual computations and the derivation of the drift condition in the case of equity models
seemstobenew,thepaperisintendedasasurveyofexistingresults,andassuch,itismostly
pedagogicalinnature.
Keywords: Implied volatilty surface, Local Volatility surface, Market models,
Arbitrage-freetermstructuredynamics,Health–Jarrow–Mortontheory
MathematicsSubjectClassification(2000)91B24
JELClassification(2000)G13
1 Introduction
Themotivationforthispapercanbefoundinthedesiretounderstandrecentattempts
to implement the HJM philosophy in the valuation of options on credit portfolios.
Several proposals appeared almost simultaneously in the literature on credit port-
folio valuation. They were written independently by N. Bennani [3], J. Sidenius,
V. Piterbarg and L. Andersen [26] and P. Shönbucher [41], the latter being most
influentialinthepreparationofthepresentsurvey.Afterasharpincreaseinvolume
andliquidityduetothecomingofageofthesingletranchesyntheticCDOs,markets
for these credit portfolios came to a stand still due to the lack of dynamic mod-
els needed to price forward starting contracts, options on options, .... So the need
(cid:1)ThisresearchwaspartiallysupportedbyNSFDMS-0456195.
2 R.A.Carmona
fordynamicmodelspromptedtheseauthorstobuildanalogiesbetweentheoriginal
HJMapproachtointerestratederivativesandderivativesoncreditportfoliolosses.
Thecommon startingpointofthesethreepapers isthelithanyofwelldocumented
shortcomings of the market standard for the valuation of Collaterized Debt Oblig-
ations (CDOs). The standard Gaussian copula model is intrinsically a one period
staticmodelwhichcannotbeusedtopriceforwardstartingcontracts.Thevaluation
byexpectationoftheseforwardstartingcontractsrequiretheanalysisofatermstruc-
tureofforwardlossprobabilities.TheHJMmodelingofthedynamicsoftheforward
instantaneous interest rates, suggests how to choose dynamic models for these for-
ward loss probabilities. The three papers mentioned above try to take advantage of
thisanalogywithvariousdegreesofgeneralityandsuccess.
The goal of this paper is to review the salient features of the HJM modeling
philosophyastheycanbeappliedtothreedifferentmarkets:thefixedincomemar-
ketsoriginallyconsideredbyHeath,JarrowandMorton,thecreditmarketsandthe
equitymarkets.Ineachofthethreecasesconsideredinthispaper,thefinancialmar-
ketmodelisbasedonasetoffinancialsecuritieswhichareassumedtobeliquidly
traded.Abasicassumptionisthatthepriceofeachsuchsecurityisobservable,and
anyquantityofthesecuritycanbesoldorboughtatthisobservedprice.Theseprices
areusedtoencapsulatewhatthemarketistellingthemodeler,andthethrustofthe
HJM modeling approach is to postulate dynamical equations for the prices of all
theseliquidinstrumentsandtocheckthatthemultitudeofalltheseequationsdonot
introduceinconsistenciesandarbitrageopportunitiesinthemarketmodel.
TheclassicalHJMapproachisreviewedinSection3.Ourinformalpresentation
doesnotdojusticetothedepthoftheoriginalcontribution[14]ofHeath,Jarrowand
Morton.Itismeantasalightintroductiontothemodelingphilosophy,ourmaingoal
beingtointroducenotationwhichareusedthroughoutthepaper,andtoemphasize
thecrucialstepswhichwillrecurinthediscussionoftheothermarketmodels.Sec-
tion5isdevotedtothediscussionoftherecentworks[26]ofSidenius,Pitterbargand
Andersenand[41]Schoenbucherontheconstructionofdynamicmodelsforcredit
portfoliosinthespiritoftheHJMapproach.Thesetwopapersareattherootofour
renewedinterestintheHJMmodelingphilosophy.Itiswhilereadingthemthatwe
realized the impact they could have on the classical equity models. The latter are
usually calibrated to market prices by constructing an implied volatility surface, or
equivalentlyalocalvolatilitysurfaceasadvocatedbyDupireandDermanandKani
in a series of influential works [19][16]. As we explain in Section 6, the construc-
tion of these surfaces is only the first step in the construction of a dynamic model.
AdynamicversionoflocalvolatilitymodelingwastoutedbyDermanandKaniina
paper[17]mostlyknownforitsdiscussionofimpliedtreemodels.Motivatedbythe
factthatthetechnicalpartsof[17]dealingwithcontinuousmodelsareratherinfor-
malandlackingmathematicalproofs,CarmonaandNadtochiydevelopedin[7]the
programoutlinedin[17].Onthetopofprovidingarigorousmathematicalderivation
of the so-called drift condition, they also provide calibration and Monte Carlo im-
plementationrecipes,andtheyanalyzetheclassicalMarkovianspotmodelsaswell
as stochastic volatility models in a generalized HJM framework. We present their
resultsinthelastsectionofthispaper.
