Paris-Princeton lectures on mathematical finance 2004 PDF
Preview Paris-Princeton lectures on mathematical finance 2004
Lecture 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] LibraryofCongressControlNumber:(cid:50)(cid:48)(cid:48)(cid:55)(cid:57)(cid:51)(cid:48)(cid:50)(cid:50)(cid:53) MathematicsSubjectClassification((cid:50)(cid:48)(cid:48)(cid:48)(cid:41)(cid:58)91B28,91B70,60G49,49J55,60H07,90C46 ISSNprintedition:(cid:48)(cid:48)(cid:55)(cid:53)(cid:45)(cid:56)(cid:52)(cid:51)(cid:52) ISSNelectronicedition:(cid:49)(cid:54)(cid:49)(cid:55)(cid:45)(cid:57)(cid:54)(cid:57)(cid:50) ISBN(cid:57)(cid:55)(cid:56)(cid:45)(cid:51)(cid:45)(cid:53)(cid:52)(cid:48)(cid:45)(cid:55)(cid:51)(cid:51)(cid:50)(cid:54)(cid:45)(cid:51)SpringerBerlinHeidelbergNewYork DOI(cid:49)(cid:48)(cid:46)(cid:49)(cid:48)(cid:48)(cid:55)/(cid:57)(cid:55)(cid:56)(cid:45)(cid:51)(cid:45)(cid:53)(cid:52)(cid:48)(cid:45)(cid:55)(cid:51)(cid:51)(cid:50)(cid:55)(cid:45)(cid:48) Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember(cid:57), (cid:49)(cid:57)(cid:54)(cid:53),initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:176)c Springer-VerlagBerlinHeidelberg(cid:50)(cid:48)(cid:48)(cid:55) Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. TypesettingbytheauthorsandSPiusingaSpringerLATEXmacropackage Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:(cid:49)(cid:50)(cid:48)(cid:56)(cid:51)(cid:50)(cid:48)(cid:54) (cid:52)(cid:49)/SPi (cid:53)(cid:52)(cid:51)(cid:50)(cid:49)(cid:48) 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.
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