SPRINGER BRIEFS IN MATHEMATICS Peter Carr · Qiji Jim Zhu Convex Duality and Financial Mathematics 123 SpringerBriefs in Mathematics SeriesEditors NicolaBellomo MicheleBenzi PalleJorgensen TatsienLi RoderickMelnik OtmarScherzer BenjaminSteinberg LotharReichel YuriTschinkel GeorgeYin PingZhang SpringerBriefsinMathematicsshowcasesexpositionsinallareasofmathematics andappliedmathematics.Manuscriptspresentingnewresultsorasinglenewresult inaclassicalfield,newfield,oranemergingtopic,applications,orbridgesbetween newresultsandalreadypublishedworks,areencouraged.Theseriesisintendedfor mathematiciansandappliedmathematicians. Moreinformationaboutthisseriesathttp://www.springer.com/series/10030 Peter Carr • Qiji Jim Zhu Convex Duality and Financial Mathematics 123 PeterCarr QijiJimZhu DepartmentofFinanceandRiskEngineering DepartmentofMathematics TandonSchoolofEngineering WesternMichiganUniversity NewYorkUniversity Kalamazoo,MI,USA NewYork,NY,USA ISSN2191-8198 ISSN2191-8201 (electronic) SpringerBriefsinMathematics ISBN978-3-319-92491-5 ISBN978-3-319-92492-2 (eBook) https://doi.org/10.1007/978-3-319-92492-2 LibraryofCongressControlNumber:2018946786 Mathematics Subject Classification: 26B25, 49N15, 52A41, 60J60, 90C25, 91B16, 91B25, 91B26, 91B30,91G10,91G20 ©TheAuthor(s),underexclusivelicencetoSpringerNatureSwitzerlandAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland ToCarolandOlivia ToLilly andCharles. AndinmemoryofJonathanBorwein (1951–2016)withrespect. Preface Convex duality plays an essential role in many important financial problems. For example, it arises both in the minimization of convex risk measures and in the maximization of concave utility functions. Together with generalized convex duality, they also appear when an optimization is not immediately apparent, for instance in implementing dynamic hedging of contingent claims. Recognizing the role of convex duality in financial problems is crucial for several reasons. First, considering the primal and dual problem together gives the financial modeler the option to tackle the more accessible problem first. Usually, knowledge of the solutionofonehelpsinsolvingtheother.Moreover,thesolutiontothedualproblem can usually be given a financial interpretation. As a result, the dual problem often illuminatesanalternativeperspective,whichisnoteasilyachievedbyexaminingthe primal problem in isolation. When flipping from the primal to the dual, a surprise insighttypicallyawaits,irrespectiveofpastexperience.Finally,asanaddedbenefit, the primal and the dual can often be paired together to provide better numerical solutionsthanwheneithersideisconsideredinisolation. Thegoalofthisbookistoprovideaconciseintroductiontothisgrowingresearch field.Ourtargetaudienceisgraduatestudentsandresearchersinrelatedareas.We begin in Chapter 1 with a quick introduction of convex duality and related tools. WeemphasizetherelationshipbetweenconvexdualityandtheLagrangemultiplier rule for constrained optimization problems. We then give a quick overview of the intrinsicdualityrelationshipinseveraldiversefinancialproblems. In Chapter 2, we consider the simplest possible financial market model. In particular,weconsideraone-periodeconomywithafinitenumberofpossiblestates. Usingthissimplefinancialmarketmodel,weshowcaseconvexdualityinanumber of important financial problems. We begin with the Markowitz portfolio theory, which involves a particularly simple convex programming problem: optimizing a quadratic function with linear constraints. Duality plays two important roles in Markowitzportfoliotheory.First,whiletheprimalproblemmayinvolvehundreds oreventhousandsofvariablesrepresentingtheriskyassetspotentiallyincludedin vii viii Preface theportfolio,thedualproblemhasonlytwovariablesrelatedtothetwoconstraints on the initial endowment and the expected return. In fact, the key observation of Markowitzisthatonecanevaluatetheperformanceofaportfoliointhedualspace usingthevariance-expectedreturnpair.Second,thedualityrelationshipbetweenthe primalMarkowitzportfolioproblemanditsdualhelpsustounderstandthattheset ofoptimalportfoliosisanaffineset,whichleadstotheimportanttwo-fundtheorem. Thecoremethodologyofoptimizingaquadraticfunctionwithlinearconstraintswas alsousedinthecapitalassetpricingmodel,whichleadstothewidelyusedSharpe ratio.Dualityalsoplaysacrucialroleinthisproblem. Next, we consider portfolio optimization from the perspective of maximizing expected utility. There has been a very long history of using utility functions in economics.Infinancialproblems,utilityfunctionsareincreasingconcavefunctions of wealth. The concavity of the utility function captures the risk aversion of an investor. Arrow and Pratt introduced widely used measures of the level of risk aversion. It turns out that there is a precise way of using generalized convexity to characterizePratt–Arrowriskaversion.Thisapplicationillustratestherelevanceof generalizedconvexityindealingwithfinancialproblems.Itisevenmoreinteresting to consider the dual of the expected utility maximization problem. It turns out that in the absence of arbitrage, solutions to the dual problem are in essence the equivalent martingale measures (also called risk-neutral probabilities), which are widely used in pricing financial derivatives. Considering the expected utility maximization problem along with its dual leads us to rediscover the fundamental theorem of asset pricing. An added benefit of this alternative approach is that martingalemeasurescanberelatedtotheriskaversionofagentsinthemarket. ThelastapplicationthatwecoverinChapter2concernsthedualrepresentation of coherent risk measures. Coherent risk measures are motivated by the common regulatory practice of assigning each position in a risky asset with the appropriate amountofcashreserves.Hence,theyarewidelyusedtoanalyzerisks.Mathemat- ically, a coherent risk measure is characterized by a sublinear function: a convex function with positive homogeneity. It is well known that the dual of a sublinear function is an indicator function. Thus, using dual representation, a coherent risk measureisjustthesupportfunctionofaclosedconvexset.Financially,wecanview the generating set of a coherent riskmeasure as the probabilities assigned torisky scenariosinastresstest.Dualityalsogeneratesnumericalmethodsforcalculating someimportantcoherentriskmeasuressuchastheconditionalvalueatrisk. Weexpandourdiscussiontoamoregeneralmultiperiodfinancialmarketmodel in Chapter 3. This more general setting allows us to model dynamic trading. The added complexity in dealing with a multiperiod model mainly involves capturing the increase in information using an information structure. After laying out the multiperiodfinancialmarketmodel,weshowthatthefundamentaltheoremofasset pricingalsoarisesinamultiperiodfinancialmarketmodel.Afterthatwealsodis- cusstwonewtopics:super(sub)hedgingandconicfinance.Ingeneral,theabsence ofarbitrageleadstomultiple(usuallyinfinitelymany)pricingmartingalemeasures Preface ix inanincompletemarket.Thus,thenoarbitrageprincipleusuallydeterminesaprice rangeforacontingentclaimwithupperandlowerbounds,whicharegivenbythe supremum and the infimum of the expectation of the payoff under the martingale measures, respectively. If a market price falls outside of these bounds, then an arbitrageopportunityoccurs.Itturnsoutthatthedualsolutiontotheoptimization problem of finding the upper or lower no arbitrage bounds provides a trading strategythatonecanusetotakeadvantageofsuchanarbitrageopportunity.Conic finance is used to describe financial markets for which the absolute value of the price depends on whether one is buying or selling. In other words, conic finance describesrealisticfinancialmarketswithastrictlypositivebid-askspread.Insuch amodel,thecashflowsthatcanbeachievedfromimplementingacceptabletrading strategiesformaconvexcone.Thisobservationprovidestherationaleforthename conic finance. Despite the added complication of dealing with a conic constraint, weshowthatmostofthedualityrelationshipsthatareobservedunderzerobid–ask spreadstillprevailwhenthespreadispositive. We then move to continuous-time financial models in Chapter 4. The most noteworthydualityrelationshipdevelopedinthischapteristheobservationthatthe classicalBlack-Scholesformulaforpricingacontingentclaimwithaconvexpayoff is, in fact, a Fenchel-Legendre transform. We show that the function describing cash borrowings while delta hedging a short position in a contingent claim is just theFenchelconjugateofthecontingentclaimpricingfunction.Theflipsideisthat the contingent claim pricing function can itself be viewed as a Fenchel conjugate of the function describing thesecash borrowings. This provides anew perspective on the convex function linking the price of the contingent claim to the underlying spot price. With the availability of many tradable contingent claims such as those embeddedinETFs,theabilitytodynamicallyhedgeacontingentclaimwithother contingent claims is increasingly becoming a financial reality. Interestingly, when using contingent claims as hedging instruments, one discovers a similar duality relationshipbetweenthecontingentclaimpricingfunctionandthecashborrowings function in terms of generalized convexity. Many useful applications are also discussed in this chapter. We examine the convexity and generalized convexity of the Bachelier and Black-Scholes option pricing formulae with respect to volatility aswell.Generalizationsofthesepropertiesmightbeusefulindealingwithfinancial productsrelatedtovolatilityandbeapotentiallyfruitfulfutureresearchdirection. The material in this book grew out of slides used to teach a joint doctoral seminaratNewYorkUniversity’sCourantInstituteinthefallof2015.Partofthe materialshasalsobeenusedpreviouslyforgraduatetopiccoursesonoptimization and modeling at Western Michigan University. We thank our colleagues at both NYUandWMUforprovidinguswithsupportiveresearchenvironments.Professor Robert Kohn helped to arrange us becoming neighbors, which facilitated our collaboration in no small part. Conversations with Professors Marco Avellaneda, JonathanGoodman,andFang-HuaLinhavebeenmosthelpful.Wearealsoindebted totheparticipantsofthesecoursesformanystimulatingdiscussions.Inparticular, x Preface we thank Monty Essid, Tom Li, Matthew Foreman, Sanjay Karanth, Jay Treiman, MehdiVazifadan,andGuolinYuwhosedetailedcommentsonvariouspartsofour lecturenoteshavebeenincorporatedintothetext. NewYork,NY,USA PeterCarr Kalamazoo,MI,USA QijiJimZhu April,2017 Contents 1 ConvexDuality............................................................... 1 1.1 ConvexSetsandFunctions............................................. 1 1.1.1 Definitions ...................................................... 1 1.1.2 ConvexProgramming .......................................... 3 1.2 SubdifferentialandLagrangeMultiplier............................... 4 1.2.1 Definition........................................................ 5 1.2.2 NonemptinessofSubdifferential............................... 5 1.2.3 Calculus......................................................... 6 1.2.4 RoleinConvexProgramming.................................. 9 1.3 FenchelConjugate ...................................................... 12 1.3.1 TheFenchelConjugate......................................... 12 1.3.2 TheFenchel–YoungInequality................................. 12 1.3.3 GraphicIllustrationandGeneralizations ...................... 14 1.4 ConvexDualityTheory................................................. 15 1.4.1 RockafellarDuality............................................. 16 1.4.2 FenchelDuality................................................. 18 1.4.3 LagrangeDuality ............................................... 21 1.4.4 GeneralizedFenchel–YoungInequality ....................... 23 1.5 GeneralizedConvexity,ConjugacyandDuality....................... 28 2 FinancialModelsinOnePeriodEconomy................................ 35 2.1 Portfolio ................................................................. 35 2.1.1 MarkowitzPortfolio ............................................ 36 2.1.2 CapitalAssetPricingModel ................................... 40 2.1.3 SharpeRatio .................................................... 45 2.2 UtilityFunctions ........................................................ 46 2.2.1 UtilityFunctions................................................ 46 2.2.2 MeasuringRiskAversion....................................... 48 2.2.3 GrowthOptimalPortfolioTheory ............................. 50 2.2.4 EfficiencyIndex................................................. 53 xi