TheEconomicsofContinuous-TimeFinance TheEconomicsofContinuous-TimeFinance BernardDumasandElisaLuciano TheMITPress Cambridge,Massachusetts London,England ©2017MassachusettsInstituteofTechnology Allrightsreserved.Nopartofthisbookmaybereproducedinanyformbyanyelectronicormechanicalmeans (includingphotocopying,recording,orinformationstorageandretrieval)withoutpermissioninwritingfromthe publisher. This book was set in Times Roman by diacriTech, Chennai. LibraryofCongressCataloging-in-PublicationData Names:Dumas,Bernard,author.|Luciano,Elisa,author. Title:Theeconomicsofcontinuous-timefinance/BernardDumasandElisaLuciano. Description:Cambridge,MA:MITPress,[2017]|Includesbibliographical referencesandindex. Identifiers:LCCN2016051903|ISBN9780262036542(hardcover:alk.paper) Subjects:LCSH:Finance–Mathematicalmodels.|Finance–Econometricmodels. Classification:LCCHG106.D862017|DDC332.01/519233–dc23LCrecordavailableat https://lccn.loc.gov/2016051903 10 9 8 7 6 5 4 3 2 1 Tothosewhogavemeasecondchance. Tomykids,whosmileandfight. —Elisa TothememoryofProfessorJackHirshleiferwhosetextbookInvestment,Interestand CapitalilluminatedmyPh.D.years. —Bernard Contents 1 Introduction 1 1.1 Motivation 1 1.2 Outline 3 1.3 HowtoUseThisBook 4 1.4 Apologies 5 1.5 Acknowledgments 5 I DISCRETE-TIMEECONOMIES 7 2 PricingofRedundantSecurities 9 2.1 Single-PeriodEconomies 10 2.1.1 TheMostElementaryProbleminFinance 10 2.1.2 Uncertainty 13 2.1.3 SecuritiesPayoffsandPrices,andInvestors’BudgetSet 14 2.1.4 AbsenceofArbitrageandRisk-NeutralPricing 16 2.1.5 CompleteversusIncompleteMarkets 22 2.1.6 CompleteMarketsandState-PriceUniqueness 23 2.1.7 BenchmarkExample 24 2.1.8 ValuationofRedundantSecurities 25 2.2 MultiperiodEconomies 27 2.2.1 InformationArrivaloverTimeandStochasticProcesses 28 2.2.2 Self-FinancingConstraintandRedundantSecurities 33 2.2.3 Arbitrage,NoArbitrage,andRisk-NeutralPricing 35 2.2.4 ValuationofRedundantSecurities 40 2.2.5 StaticallyversusDynamicallyCompleteMarkets 40 2.2.6 BenchmarkExample 42 2.3 Conclusion 50 viii Contents 3 InvestorOptimalityandPricingintheCaseofHomogeneousInvestors 53 3.1 One-PeriodEconomies 54 3.1.1 InvestorOptimalityandSecurityPricingunderCertainty 54 3.1.2 InvestorOptimalityandSecurityPricingunderUncertainty 57 3.1.3 Arrow-DebreuSecurities 58 3.1.4 ComplexorReal-WorldSecurities 63 3.1.5 RelationwiththeNo-ArbitrageApproach 66 3.1.6 TheDualProblem 68 3.2 ABenchmarkExample 69 3.2.1 IsoelasticUtility 69 3.2.2 SecuritiesPricing 70 3.2.3 FromSecurityPricestoStatePrices,Risk-Neutral Probabilities,andStochasticDiscountFactors 71 3.3 MultiperiodModel 72 3.3.1 OptimizationMethods 74 3.3.2 RecursiveApproach 74 3.3.3 GlobalApproach 77 3.3.4 SecuritiesPricing 81 3.4 BenchmarkExample(continued) 85 3.5 Conclusion 87 4 EquilibriumandPricingofBasicSecurities 91 4.1 One-PeriodEconomies 91 4.2 CompetitiveEquilibrium 94 4.2.1 EqualizationofStatePrices 99 4.2.2 RiskSharing 101 4.2.3 SecurityPricingbytheRepresentativeInvestorandtheCAPM 103 4.2.4 TheBenchmarkExample(continued) 106 4.3 IncompleteMarket 107 4.4 Multiple-PeriodEconomies 110 4.4.1 RadnerEquilibrium 110 4.4.2 StatePricesandRepresentativeInvestor:FromRadnerto Arrow-DebreuEquilibria 112 4.4.3 SecuritiesPricing 113 4.4.4 RiskSharing 115 4.4.5 ASideCommentonTime-AdditiveUtilityFunctions 116 4.5 Conclusion 117 Contents ix II PRICINGINCONTINUOUSTIME 119 5 BrownianMotionandItôProcesses 121 5.1 MartingalesandMarkovProcesses 121 5.2 ContinuityforStochasticProcessesandDiffusions 123 5.3 BrownianMotion 125 5.3.1 IntuitiveConstruction 125 5.3.2 AFinancialMotivation 130 5.3.3 Definition 131 5.4 ItôProcesses 133 5.5 BenchmarkExample(continued) 135 5.5.1 TheBlack-ScholesModel 135 5.5.2 ConstructionfromDiscrete-Time 138 5.6 Itô’sLemma 140 5.6.1 Interpretation 142 5.6.2 Examples 144 5.7 DynkinOperator 145 5.8 Conclusion 146 6 Black-ScholesandRedundantSecurities 149 6.1 Replicating-PortfolioArgument 151 6.1.1 BuildingtheBlack-ScholesPDE 151 6.1.2 SolvingtheBlack-ScholesPDE 154 6.2 Martingale-PricingArgument 155 6.3 Hedging-PortfolioArgument 157 6.3.1 ComparingtheArguments:Intuition 159 6.4 Extensions:Dividends 160 6.4.1 DividendPaidontheUnderlying 160 6.4.2 DividendPaidontheOption 162 6.5 Extensions:APartiallyGeneralizedBlack-ScholesModel 163 6.5.1 Replicating-PortfolioArgument 163 6.5.2 Martingale-PricingArgument 165 6.5.3 HedgingArgument 165 6.6 ImpliedProbabilities 165 6.7 ThePriceofRiskofaDerivative 166 6.8 BenchmarkExample(continued) 168 6.9 Conclusion 172